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![4 Introduction
prefilter or precoder postfilter or equalizer
F(jω)
s(t)
q(t)
s(t)
G(jω)
+
y(t)
LTI channel
H(jω)
x(t)
Figure 1.3. The analog communication system represented in terms of frequency
responses.
1.3 Digitial communication systems
In the preceding section, the signal s(t) was regarded as a continuous time signal
with continuous (unquantized) amplitude. In a digital communication system,
the messages are quantized amplitudes, transmitted in discrete time. Figure 1.4
shows the schematic of a digital communication system. Here we have a discrete-
time message or signal s(n) which we wish to transmit over a continuous-time
channel. The amplitudes of s(n) are “digitized,” that is they come from a finite
set of symbols. This collection of symbols is called a constellation. We shall
come to details of digitization later.2
Since s(n) is a discrete-time signal and the
channel is continuous-time, the signal is first converted into a continuous-time
signal x(t) as indicated in the figure.
The conversion from s(n) to x(t) can be described schematically in two steps.
The building block indicated as D/C is a discrete-to-continuous-time converter,
and it converts s(n) to a signal sc(t) given by
sc(t) =
∞
n=−∞
s(n)δc(t − nT). (1.6)
Here δc(t) is the impulse or Dirac delta function [Oppenheim and Willsky, 1997].
Thus, the sample s(n) is converted into an impulse positioned at time nT. The
sample spacing T determines the speed with which the message samples are
conveyed. Since we have 1/T symbols per second, the symbol rate is given by
fs =
1
T
Hz. (1.7)
The prefilter F(jω) at the transmitter performs a convolution to produce the
output
x(t) =
∞
n=−∞
s(n)f(t − nT), (1.8)
2Examples of constellations include PAM and QAM systems to be described in Sec. 2.2.](https://image.slidesharecdn.com/65604-250620202834-0ce82231/75/Signal-Processing-and-Optimization-for-Transceiver-Systems-1st-Edition-P-P-Vaidyanathan-25-2048.jpg)
![1.4 MIMO channels 7
precoder equalizer
s(n) s(n)
+
channel
F (z)
d G (z)
d
H (z)
d
q (n)
d
Figure 1.6. An all-discrete equivalent of the digital communication system.
Here Hd(z) is the transfer function of an equivalent discrete-time channel. It
is the z-transform of an equivalent digital channel impulse response hd(n), that
is
Hd(z) =
∞
n=−∞
hd(n)z−n
. (1.13)
Similarly, Fd(z) and Gd(z) are the transfer functions of the discrete-time pre-
coder and equalizer. The subscript d (for “discrete”), which is just for clarity,
is usually dropped. In practice Hd(z) is causal and can be approximated by a
finite impulse response, or FIR, system so that
Hd(z) =
L
n=0
hd(n)z−n
. (1.14)
The problem of optimizing the precoder Fd(z) and equalizer Gd(z) for fixed
channel Hd(z) and fixed noise statistics will be addressed in later chapters.
1.4 MIMO channels
The transceivers described so far have one input signal s(n) and a corresponding
output
s(n). These are called single-input single-output, or SISO, transceivers.
An important communication system that comes up frequently in this book is
the multi-input multi-output, or MIMO, channel. Figure 1.7 shows a MIMO
channel assumed to be linear and time-invariant with a transfer function matrix
H(z), usually an FIR system:
H(z) =
L
n=0
h(n)z−n
. (1.15)
The sequence h(n), called the MIMO impulse response, is a sequence of matrices.
If the channel has P inputs and J outputs then H(z) has size J × P, and so
does each of the matrices h(n). The MIMO communication channel is used to
transmit a vector signal s(n) with M components:
s(n) = [ s0(n) s1(n) . . . sM−1(n) ]
T
. (1.16)](https://image.slidesharecdn.com/65604-250620202834-0ce82231/75/Signal-Processing-and-Optimization-for-Transceiver-Systems-1st-Edition-P-P-Vaidyanathan-28-2048.jpg)
![1.5 Scope and outline 9
include Proakis [1995], Oppenheim and Willsky [1997], Oppenheim and Schafer
[1999], Lathi [1998], Haykin [2001], Mitra [2001], and Antoniou [2006], among
other excellent texts. Advanced material related to the topics in this book can
also be found in Ding and Li [2001], and Giannakis, et al. [2001]. Books on the
very important related areas of wireless and multiuser communications include
Rappaport [1996], Verdu [1998], Goldsmith [2005], Haykin and Moher [2005],
and Tse and Viswanath [2005].
The book is divided into four parts. Part 1 contains introductory material
on digital communication systems and signal processing aspects. In part 2 we
discuss the optimization of transceivers, with emphasis on MIMO channels. Part
3 provides mathematical background material for optimization of transceivers.
This part can be used as a reference, and will be very useful for readers wishing
to pursue more detailed literature on optimization. Part 4 contains eight appen-
dices on commonly used material such as matrix theory, Wiener filtering, and
so forth.
The history of digital communication theory is fascinating. It is a humbling
experience to look back and reflect on the tremendous insights and accomplish-
ments of the communications and signal processing society in the last six decades.
Needless to say, much of the recent research is built upon six to seven decades of
this solid foundation. A detour into history will be provided in Chap. 9, where
we present a historical perspective of transceiver design, equalization, and opti-
mization, all of which originated in the early 1960s, and have continued to this
day to be research topics. All references to literature will be given in the specific
chapters as appropriate. An extensive reference list is given at the end of the
book. In what follows we briefly describe the four parts of the book.
Part 1: Communication fundamentals
Part 1 consists of Chapters 1 to 8. In Chap. 2 we review basic topics in digital
communication systems, such as signal constellations, carrier modulation, and
so forth. Formulas for probabilities of error in symbol detection are derived.
Matched filtering, which is used in some receiver systems, is discussed in some
detail. In Chap. 3 we describe digital communication systems using the language
of multirate filter banks. Such a representation is very useful for transceivers with
or without redundancy, and has many applications as we shall see throughout
the book.
In Chap. 4 we describe digital communication systems using discrete-time
language. This chapter also introduces symbol spaced equalizers (SSE) and
fractionally spaced equalizers (FSE). The minimum mean square error (MMSE)
equalizer is also introduced in this chapter. Chapter 5 discusses a number of fun-
damental techniques that are commonly used in digital communications. First
a detailed discussion of the matched filter is provided. Then we discuss opti-
mal sequence estimators, such as the maximum likelihood (ML) detector and
the Viterbi alogrithm. Nonlinear methods, such as the decision feedback equal-
izer and nonlinear precoders, are introduced. Chapter 6 is a brief discussion of
channel capacity with emphasis on MIMO channels.
Chapter 7 introduces redundant precoders, including zero-padded and cyclic-](https://image.slidesharecdn.com/65604-250620202834-0ce82231/75/Signal-Processing-and-Optimization-for-Transceiver-Systems-1st-Edition-P-P-Vaidyanathan-30-2048.jpg)
![2
Review of basic ideas
from digital communication
2.1 Introduction
In this chapter we briefly review introductory material from the theory and
practice of digital communication. The reader familiar with such introductory
material can use this chapter primarily as a reference for later chapters. For
a more detailed treatement one should consult standard communication texts
such as Proakis [1995], Lathi [1998], or Haykin [2001].
2.1.1 Chapter overview
The schematic representation of digital communication systems described earlier
in Sec. 1.3 is reproduced in Fig. 2.1. As mentioned in that section, the input
symbol stream has sample values s(n) chosen from a finite set of values called the
symbol alphabet, constellation, or code words – we shall consistently use the term
constellation. In Sec. 2.2 we shall describe two commonly used constellations
called the PAM (pulse amplitude modulation) and the QAM (quadrature PAM)
constellations.
Given the symbol stream s(n) the transmitter generates the baseband wave-
form
x(t) =
∞
n=−∞
s(n)f(t − nT),
12](https://image.slidesharecdn.com/65604-250620202834-0ce82231/75/Signal-Processing-and-Optimization-for-Transceiver-Systems-1st-Edition-P-P-Vaidyanathan-33-2048.jpg)
![14 Review of digital communications
symbol sest(n) can sometimes be different from s(n), resulting in an error. Under
appropriate assumptions on the statistics of the reconstruction error it is possible
to compute the probability of error in symbol identification. This is done in Sec.
2.3. In practice the baseband signal x(t) is used to modulate a high-frequency
carrier signal, and the modulated signal is transmitted. At the receiver this
signal is demodulated to extract the (noisy version of) the baseband signal. The
details of this modulation and demodulation are different for PAM and QAM
constellations, and will be presented in Sec. 2.4.
Returning to the postfilter G(jω), we mentioned in Chap. 1 that it is called
the equalizer, and compensates for the channel distortions. In practice, this
equalization is often performed with a digital filter after the sampling process.
The analog filter G(jω) then plays a different role called matched filtering. The
purpose of this filter is to maximize the signal-to-noise ratio at the sample lo-
cations, and its optimal choice depends on the channel H(jω) as well as on the
power spectrum of the noise q(t). Matched filtering is reviewed in Sec. 2.5, and
some practical details are discussed in Sec. 2.6.
2.2 Signal constellations
Two of the popular constellations widely used today are the PAM (pulse am-
plitude modulation) and the QAM (quadrature amplitude modulation) constel-
lations. The PAM constellation has real-valued numbers for s(n), whereas the
QAM constellation has complex numbers. A b-bit PAM or QAM constella-
tion has M = 2b
allowed numbers, called constellation symbols or codewords
(sometimes written as code words), and is also called an M-PAM or M-QAM
constellation.
Figure 2.3 shows a PAM constellation with M codewords. Note that M is
even (a power of 2), and adjacent codewords are separated by a fixed amount
2A. No codeword has value zero. The positive number A can be chosen to adjust
the average energy per codeword as we shall see later. Figure 2.4 shows a QAM
constellation with M = 16 codewords. This is a 4-bit constellation. Once again,
no codeword has value zero. Note that any 4-bit QAM code word has the form
z = x + jy,
where x and y are 2-bit PAM words. The components x and y are sometimes
called the in-phase and quadrature components. A b-bit QAM constellation
has real and imaginary parts coming from (b/2)-bit PAM constellations. The
QAM constellations we use here are called square QAM constellations. More
generally they can be rectangular constellations with b1 bits for x and b2 bits
for y. There are more general types of QAM constellations including circular
ones [Proakis, 1995], but we shall not elaborate on those here. Unless mentioned
otherwise, we always imply square constellations when we use the term QAM.
So the number of bits b is an even integer. As in PAM, any word in the square
QAM constellation has nearest neighbors at distance 2A.](https://image.slidesharecdn.com/65604-250620202834-0ce82231/75/Signal-Processing-and-Optimization-for-Transceiver-Systems-1st-Edition-P-P-Vaidyanathan-35-2048.jpg)
![24 Review of digital communications
of the transmitted symbol stream s(n), of the form
s(n) = s(n) + e(n), (2.20)
where e(n) is the reconstruction error. Recall here that s(n), and hence
s(n),
are complex numbers. It is usually reasonable to assume that the error e(n) is
complex and has the form
e(n) = er(n) + jei(n),
where er(n) and ei(n) are independent zero-mean Gaussian random variables
with identical variance 0.5σ2
e . In this case the total variance of the complex error
e(n) is
0.5σ2
e + 0.5σ2
e = σ2
e .
The joint pdf of the variables [er(n), ei(n)] is given by
fE(er, ei) =
e−e2
r/σ2
e
πσ2
e
×
e−e2
i /σ2
e
πσ2
e
=
e−(e2
r+e2
i )/σ2
e
πσ2
e
. (2.21)
Figure 2.13 demonstrates this for σ2
e = 0.01. This is a special case of a so-called
circularly symmetric complex Gaussian random variable.4
Next, the complex
symbol s(n) is of the form
sr(n) + jsi(n),
where sr(n) and si(n) are PAM symbols. Since
|s(n)|2
= s2
r(n) + s2
i (n),
it follows that the average energy of the constellation is the sum of the average
energies of the real and imaginary parts. Thus, for a b-bit QAM constellation
with average energy Eave, the real and imaginary parts are (b/2)-bit PAM con-
stellations with average energy Eave/2, and each of these PAM constellations
sees an error source with variance σ2
e /2. For the real-part PAM the probability
of error can be obtained from Eq. (2.17) by replacing b, Eave, and σ2
e with half
their values:
Pe,re = 2(1 − 2−b/2
)Q
3Eave
(2b − 1)σ2
e
. (2.22)
Since the factor of one-half cancels out in the ratio Eave/σ2
e , this is nothing but
the error probability for a (b/2)-bit PAM constellation with energy Eave and
noise variance σ2
e . Similarly for the imaginary part
Pe,im = 2(1 − 2−b/2
)Q
3Eave
(2b − 1)σ2
e
. (2.23)
The QAM symbol is detected correctly if the real part and imaginary part are
both detected correctly. The probability for this is
(1 − Pe,re)2
.
4A detailed discussion of circularly symmetric complex random variables can be found in
Sec. 6.6.](https://image.slidesharecdn.com/65604-250620202834-0ce82231/75/Signal-Processing-and-Optimization-for-Transceiver-Systems-1st-Edition-P-P-Vaidyanathan-45-2048.jpg)
![26 Review of digital communications
Example 2.2: Two-bit QAM or QPSK
Consider the case of 2-bit QAM, also known as a QPSK (quadrature phase-
shift keying) constellation. This is shown in Fig. 2.14. Setting b = 2 in Eq.
(2.26), the average error probability is given by
Pe,QP SK = 2Q
Eave
σ2
e
. (2.28)
For both the PAM and QAM systems, note that the error probability depends on
the ratio Eave/σ2
e , rather than the individual values of the energy Eave and the
error variance σ2
e . This ratio is called the signal-to-error ratio or signal-to-noise
ratio SNR at the input of the detector:
SNR =
Eave
σ2
e
(2.29)
Figure 2.15 shows plots of the symbol error probability Pe,P AM as a function of
this SNR for PAM systems, for various values of the number of bits b. Figure 2.16
shows similar plots for QAM systems. To compare PAM and QAM systems, it
is useful to introduce the bit error rates (BER), which are related to the symbol
error rates. Before doing this we have to introduce a binary coding system called
the Gray code.
2.3.3 Gray codes
In digital communication systems we are often required to transmit binary
streams. These streams can be converted to PAM or QAM symbols by ap-
propriately grouping the bits. This is called the symbol modulation process. For
example, if the binary stream is divided into blocks of size 3:
. . . 010 001 111 011 100 . . .
then each 3-bit block can be turned into a 3-bit PAM symbol. More generally, a
b-bit block can be translated into an M-word constellation with M = 2b
. There
are many ways to define the mapping from the binary representation to the
constellation words. Figure 2.17 shows an example of such a representation for
a 3-bit PAM constellation. Thus the preceding binary sequence is converted to
. . . − A, −5A, 3A, −3A, 7A, . . .
In this example the binary words are assigned such that adjacent symbols in
the constellation (Fig. 2.17) differ only in one of the bit locations. Such a
representation is called a Gray code [Proakis, 1995].](https://image.slidesharecdn.com/65604-250620202834-0ce82231/75/Signal-Processing-and-Optimization-for-Transceiver-Systems-1st-Edition-P-P-Vaidyanathan-47-2048.jpg)
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Signal Processing and Optimization for Transceiver Systems 1st Edition P. P. Vaidyanathan
- 1. Signal Processing andOptimization for Transceiver Systems 1st Edition P. P. Vaidyanathan pdf download https://ebookgate.com/product/signal-processing-and-optimization- for-transceiver-systems-1st-edition-p-p-vaidyanathan/ Get Instant Ebook Downloads – Browse at https://ebookgate.com
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- 4. SIGNAL PROCESSING ANDOPTIMIZATION FOR TRANSCEIVER SYSTEMS Presenting the first complete treatment of MIMO transceiver optimization, this self- contained book provides all the mathematical information needed to understand transceiver optimization in a single volume. It begins with a review of digital com- munication fundamentals, and then moves on to a detailed study of joint transceiver optimization, starting from simple single-input single-output channels all the way to minimum bit error rate transceivers for MIMO channels. Crucial background mate- rial is covered, such as Schur-convex functions, matrix calculus, and constrained optimization, together with eight appendices providing further background mate- rial on topics such as matrix theory, random processes, and sampling theory. A final ninth appendix provides a grand summary of all the optimization results. With 360 illustrations, over 70 worked examples, and numerous summary tables provided to aid understanding of key concepts, this book is ideal for graduate students, practitioners, and researchers in the fields of communications and signal processing. p. p. vaidyanathan is a Professor of Electrical Engineering at the California Institute of Technology, where he has been a faculty member since 1983. He is an IEEE Fellow and has co-authored over 400 technical papers and two previous books in the area of signal processing. He has received numerous awards, including four awards for journal papers, the Award for Excellence in Teaching at the California Institute of Technology three times, and the Technical Achievement Award of the IEEE Signal Processing Society. see-may phoong is a Professor in the Graduate Institute of Communication Engineering and the Department of Electrical Engineering at the National Taiwan University. He is a recipient of the Charles H. Wilts Prize for outstanding indepen- dent doctoral research at the California Institute of Technology and the Chinese Institute of Electrical Engineering’s Outstanding Youth Electrical Engineer Award. yuan-pei lin is a Professor in Electrical Engineering at the National Chiao Tung University, Taiwan. She is a recipient of the Ta-You Wu Memorial Award, the Chinese Institute of Electrical Engineering’s Outstanding Youth Electrical Engineer Award, and of the Chinese Automatic Control Society’s Young Engineer in Auto- matic Control Award.
- 6. SIGNAL PROCESSING AND OPTIMIZATIONFOR TRANSCEIVER SYSTEMS P. P. VAIDYANATHAN California Institute of Technology SEE-MAY PHOONG National Taiwan University YUAN-PEI LIN National Chiao Tung University, Taiwan
- 7. cambridge university press Cambridge,New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521760799 C Cambridge University Press 2010 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2010 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN 978-0-521-76079-9 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 8. To Usha, Vikram,Sagar, and my parents — P. P. Vaidyanathan To our families — See-May Phoong and Yuan-Pei Lin
- 10. Contents at aglance Part 1: Communication fundamentals 1. Introduction 1 2. Review of basic ideas from digital communication 12 3. Digital communication systems and filter banks 70 4. Discrete-time representations 113 5. Classical transceiver techniques 167 6. Channel capacity 216 7. Channel equalization with transmitter redundancy 244 8. The lazy precoder with a zero-forcing equalizer 295 Part 2: Transceiver optimization 9. History and outline 317 10. Single-input single-output transceiver optimization 332 11. Optimal transceivers for diagonal channels 370 12. MMSE transceivers with zero-forcing equalizers 397 13. MMSE transceivers without zero forcing 430 14. Bit allocation and power minimization 452 15. Transceivers with orthonormal precoders 477 16. Minimization of error probability in transceivers 494 17. Optimization of cyclic-prefix transceivers 528 18. Optimization of zero-padded systems 577 19. Transceivers with decision feedback equalizers 592 Part 3: Mathematical background 20. Matrix differentiation 660 21. Convexity, Schur convexity and majorization theory 694 22. Optimization with equality and inequality constraints 730 Part 4: Appendices A. Inner products, norms, and inequalities 750 B. Matrices: a brief overview 753 C. Singular value decomposition 766 D. Properties of pseudocirculant matrices 771 E. Random processes 779 F. Wiener filtering 792 G. Review of concepts from sampling theory 802 H. Euclid’s algorithm 808 I. Transceiver optimization: summary and tables 812 Glossary 825 Acronyms 826 References 827 Index 845
- 12. Contents Part 1: Communicationfundamentals 1 Introduction 1 1.1 Introduction 1 1.2 Communication systems 1 1.3 Digital communication systems 4 1.4 MIMO channels 7 1.5 Scope and outline 8 1.6 Commonly used notations 11 2 Review of basic ideas from digital communication 12 2.1 Introduction 12 2.2 Signal constellations 14 2.3 Error probability 17 2.4 Carrier-frequency modulation 30 2.5 Matched filtering 38 2.6 Practical considerations in matched filtering 54 2.7 Concluding remarks 58 Appendix 60 Problems 66 3 Digital communication systems and filter banks 70 3.1 Introduction 70 3.2 Multirate building blocks 70 3.3 Decimation filters 81 3.4 Interpolation filters 82 3.5 Blocking and unblocking 86 3.6 Parsing a scalar signal into a vector signal 88 3.7 Decimation and interpolation in polyphase form 89 3.8 The transmultiplexer system 94
- 13. x Contents 3.9 Analysisof the transmultiplexer system 99 3.10 Concluding remarks 105 Problems 106 4 Discrete-time representations 113 4.1 Introduction 113 4.2 Conversion between continuous and discrete time 114 4.3 Discrete-time representations of channels 116 4.4 The raised-cosine function 123 4.5 MIMO systems and multiuser systems 127 4.6 Digital equalization 128 4.7 Oversampling the received signal 130 4.8 Fractionally spaced equalizers 132 4.9 Noble identities and digital design of filters 149 4.10 MMSE equalization 152 4.11 Concluding remarks 161 Problems 162 5 Classical transceiver techniques 167 5.1 Introduction 167 5.2 Matched filtering and reconstructibility 167 5.3 Sampled-noise whitening receiver filter 178 5.4 Vector space interpretation of matched filtering 181 5.5 Optimal estimates of symbols and sequences 184 5.6 The Viterbi algorithm for channel equalization 190 5.7 Decision feedback equalizers 201 5.8 Precoders for pre-equalization of a channel 204 5.9 Controlled ISI and partial-response signals 208 5.10 Concluding remarks 212 Appendix 213 Problems 214 6 Channel capacity 216 6.1 Introduction 216 6.2 Ideal lowpass channel 216 6.3 SNR gap for PAM signals 218 6.4 Capacity of frequency-dependent channel 219 6.5 Splitting the channel into subbands 220 6.6 Circularly symmetric complex random vectors 224 6.7 Capacity for MIMO and complex channels 234 6.8 Concluding remarks 241 Problems 242
- 14. Contents xi 7 Channelequalization with transmitter redundancy 244 7.1 Introduction 244 7.2 Zero padding 244 7.3 Introduction of the cyclic prefix 253 7.4 The circulant matrix representation 261 7.5 Variations of the cyclic-prefix system 264 7.6 The discrete multitone system 268 7.7 Concluding remarks 273 Problems 277 8 The lazy precoder with a zero-forcing equalizer 295 8.1 Introduction 295 8.2 Noise amplification and Frobenius norm 298 8.3 Frobenius norm of left inverse as A grows taller 300 8.4 Application in equalization 300 8.5 Autocorrelation property 302 8.6 Effect of increasing the block size 307 8.7 Concluding remarks 308 Appendix 312 Problems 315 Part 2: Transceiver optimization 9 History and outline 317 9.1 Introduction 317 9.2 A brief history of transceiver optimization 318 9.3 Outline for Part 2 328 10 Single-input single-output transceiver optimization 332 10.1 Introduction 332 10.2 Optimization of the SISO communication system 333 10.3 The all-discrete SISO channel 341 10.4 General forms of optimal filters 347 10.5 Excess bandwidth and oversampling 356 10.6 Optimal pulse shape in single-pulse case 360 10.7 Concluding remarks 367 Problems 368
- 15. xii Contents 11 Optimaltransceivers for diagonal channels 370 11.1 Introduction 370 11.2 Minimizing MSE under the ZF constraint 372 11.3 Minimizing MSE without ZF constraint 376 11.4 Maximizing channel capacity 380 11.5 Minimizing the symbol error rate 382 11.6 Examples of optimal diagonal transceivers 388 11.7 Concluding remarks 395 Problems 396 12 MMSE transceivers with zero-forcing equalizers 397 12.1 Introduction 397 12.2 Assumptions on noise and signal statistics 398 12.3 Problem formulation 401 12.4 Solution to the ZF-MMSE optimization problem 407 12.5 Optimizing the noise-to-signal ratio 417 12.6 Concluding remarks 419 Appendices 420 Problems 428 13 MMSE transceivers without zero forcing 430 13.1 Introduction 430 13.2 Formulation of the problem 431 13.3 MMSE equalizer for fixed precoder 432 13.4 Formulating the optimal precoder problem 434 13.5 Solution to the optimal precoder problem 437 13.6 Structure of the MMSE transceiver 441 13.7 Concluding remarks 446 Appendix 447 Problems 449 14 Bit allocation and power minimization 452 14.1 Introduction 452 14.2 Error probabilities, bit rates, and power 453 14.3 Minimizing transmitter power with bit allocation 455 14.4 Optimizing the precoder and equalizer 457 14.5 Optimal transceiver in terms of channel SVD 460 14.6 Further properties of optimal solutions 464 14.7 Coding gain due to bit allocation 471 14.8 Concluding remarks 473 Appendix 474 Problems 475
- 16. Contents xiii 15 Transceiverswith orthonormal precoders 477 15.1 Introduction 477 15.2 Orthonormal precoders restricted to be square 478 15.3 Rectangular orthonormal precoder matrices 486 15.4 Concluding remarks 492 Problems 493 16 Minimization of error probability in transceivers 494 16.1 Introduction 494 16.2 Minimizing error probability in ZF transceivers 494 16.3 Bias in the reconstruction error 500 16.4 Minimizing error probability without ZF 505 16.5 Bias-removed MMSE versus ZF-MMSE 508 16.6 Concluding remarks 511 Appendices 513 Problems 526 17 Optimization of cyclic-prefix transceivers 528 17.1 Introduction 528 17.2 Optimal cyclic-prefix systems: preliminaries 528 17.3 Cyclic-prefix systems optimized for MSE: details 533 17.4 CP systems with minimum error probability 539 17.5 DMT systems optimized for power 544 17.6 The cyclic-prefix system with unitary precoder 547 17.7 Cyclic-prefix optimization examples 552 17.8 Increasing the block size in cyclic-prefix systems 561 17.9 Power minimization using bit allocation 564 17.10 Concluding remarks 572 Appendix 573 Problems 576 18 Optimization of zero-padded systems 577 18.1 Introduction 577 18.2 Zero-padded optimal transceivers 577 18.3 Effect of increasing M in zero-padded systems 585 18.4 Concluding remarks 590 Problems 591 19 Transceivers with decision feedback equalizers 592 19.1 Introduction 592 19.2 Fundamentals of decision feedback equalizers 592
- 17. xiv Contents 19.3 OptimalDFE system with zero forcing 597 19.4 Optimal DFE system without zero forcing 608 19.5 Minimizing error probability in DFE transceivers 619 19.6 Examples of optimal transceivers with DFE 622 19.7 DFE optimization and mutual information 637 19.8 Other algorithms related to decision feedback 640 19.9 Concluding remarks 645 Appendices 646 Problems 657 Part 3: Mathematical background 20 Matrix differentiation 660 20.1 Introduction 660 20.2 Real matrices and functions 660 20.3 Complex gradient operators 668 20.4 Complex matrices and derivatives 673 20.5 Optimization examples 678 20.6 Being careful with interpretations ... 683 20.7 Summary and conclusions 685 Problems 691 21 Convexity, Schur convexity and majorization theory 694 21.1 Introduction 694 21.2 Review of convex functions 694 21.3 Schur-convex functions 706 21.4 Examples of Schur-convex functions 709 21.5 Relation to matrix theory 717 21.6 Multiplicative majorization 724 21.7 Summary and conclusions 725 22 Optimization with equality and inequality constraints 730 22.1 Introduction 730 22.2 Setting up the problem 730 22.3 Maximizing channel capacity 735 22.4 MMSE transceiver 739 22.5 KKT conditions are only necessary conditions 747 22.6 Concluding remarks 749
- 18. Contents xv Part 4:Appendices Appendix A Inner products, norms, and inequalities 750 A.1 Inner products and norms 750 A.2 Cauchy-Schwartz inequality 751 A.3 The AM-GM inequality 752 Appendix B Matrices: a brief overview 753 B.1 Introduction 753 B.2 Determinant and trace 754 B.3 Rank 756 B.4 Eigenvalues and eigenvectors 757 B.5 Matrices with special properties 761 B.6 Positive definite matrices 763 B.7 Rayleigh-Ritz principle 765 Appendix C Singular value decomposition 766 C.1 Introduction 766 C.2 Left inverse computed from SVD 767 C.3 Frobenius norm and SVD 768 C.4 Frobenius norm of the left inverse 769 Appendix D Properties of pseudocirculant matrices 771 D.1 Introduction 771 D.2 Circulant matrices 771 D.3 Diagonalization of pseudocirculants 773 D.4 Further properties of pseudocirculants 775 Appendix E Random processes 779 E.1 Introduction 779 E.2 Wide sense stationary processes 779 E.3 Cyclo WSS processes 784 E.4 Linear combinations of random variables 787
- 19. xvi Contents Appendix FWiener filtering 792 F.1 Introduction 792 F.2 Theory of statistically optimal filtering 792 F.3 Wiener filter for zero-mean uncorrelated noise 798 F.4 Concluding remarks 801 Appendix G Review of concepts from sampling theory 802 G.1 Introduction 802 G.2 Noble identities for C/D and D/C converters 803 G.3 The generalized alias-free(T) band 804 G.4 Alias-free(T) signals with identical samples 806 Appendix H Euclid’s algorithm 808 Appendix I Transceiver optimization: summary and tables 812 Glossary 825 Acronyms 826 References 827 Index 845
- 20. xvii Preface Digital communication systemshave been studied for many decades, and they have become an integral part of the technological world we live in. Many excel- lent books in recent years have told the story of this communication revolution, and have explained in considerable depth the theory and applications. Since the late 1990s particularly, there have been a number of significant contributions to digital communications from the signal processing community. This book presents a number of these recent developments, with emphasis on the use of filter bank precoders and equalizers. Optimization of these systems will be one of the main themes in this book. Both multiple-input multiple-output (MIMO) systems and single-input single-output (SISO) systems will be considered. The book is divided into four parts. Part 1 contains introductory material on digital communication systems and signal processing aspects. In Part 2 we discuss the optimization of transceivers, with emphasis on MIMO channels. Part 3 provides mathematical background material for optimization of transceivers. This part can be used as a reference, and will be useful for readers wishing to pursue more detailed literature on optimization. Part 4 contains eight appen- dices on commonly used material such as matrix theory, Wiener filtering, and so forth. Thus, while it is assumed that the reader has some exposure to digital communications and signal processing at the introductory level, there is plenty of review material at the introductory level (Part 1) and at the advanced level (Parts 3 and 4). The material in Parts 2 and 3 will be useful for students wishing to pursue advanced work in the field, which is still a very active area for research. A detailed outline of the book can be found in Sec. 1.5 of Chap. 1. Some of the material herein has been tested in the classroom, and a con- siderable part has benefited students at an advanced level. While many of the results in Part 2 can be regarded as results which appeared since the late 1990s, the mathematical foundation for this material is much older. Starting with the days of Shannon and Nyquist, there have been many giants in the field who contributed to this strong foundation since the 1940s. However, because of tech- nological advances and the directions in which applications evolved, such as wireless communication and DSL technology, some of the theoretical problems have been revisited and some new problems solved by researchers in recent years. This freshness and novelty in the midst of old grandeur can clearly been seen from the combination of topics covered in Parts 1, 2, and 3 of the book. We have endeavored to come up with a text that will be useful in the class- room, and which will serve as a research reference for advanced students. The writing style is in the form of an easy-to-read text book with detailed theory, plenty of examples, discussions, and homework problems. It is self-contained for students with an introductory background in signal processing and communica- tions.
- 21. xviii Acknowledgements The wonderful environmentprovided by the California Institute of Technology, and the generous support from the Office of Naval Research and the National Science Foundation, have been crucial in developing the material covered in this book. As mentioned in the introductory and historical review sections of this book, many great minds have been involved in making the fields of communi- cation and signal processing what they are today. Without their fundamental contributions this book would have been impossible. Many graduate students have participated in extensive discussions relating to the material in this book. It is my pleasure to thank them here, and especially acknowledge the extensive discussions I have had with Chun-Yang Chen and Ching-Chih Weng regarding the material in Part 2 of the book. For a project of this magnitude, long hours of hard work and concentration are absolutely essential. I have to thank Usha for creating the peaceful atmo- sphere which is crucial for the success of such a project. She has shown infinite patience during the long evenings and weekends of my absorption in this book. Her total unconditional love and sincere support, and the enthusiasm and love from Vikram and Sagar, are much appreciated! P. P. Vaidyanathan California Institute of Technology, Pasadena, CA It is our pleasure to acknowledge the support of National Taiwan University and National Chiao-Tung University. We would also like to thank the National Science Council, Taiwan, for continued support of our research. See-May Phoong National Taiwan University, Taipei, Taiwan and Yuan-Pei Lin National Chiao Tung University, Hsinchu, Taiwan
- 22. 1 Part 1 Communicationfundamentals 1 Introduction 1.1 Introduction Digital communication systems have been studied for many decades, and they have become an integral part of the technological world we live in. Many excel- lent books in recent years have told the story of this communication revolution, and have explained in considerable depth the theory and applications. Since the late 1990s particularly, there have been a number of significant contributions to digital communications from the signal processing community. This book presents a number of these recent developments, with emphasis on the use of filter bank precoders and equalizers. Optimization of these systems will be one of the main themes in this book. Both multiple-input multiple-output (MIMO) systems and single-input single-output (SISO) systems will be considered. It is assumed that the reader has had some exposure to digital communications and signal processing at the introductory level. Many text books cover this prerequisite, and some are mentioned at the beginning of Sec. 1.5. Before we describe the contents of the book we first give an introductory description of analog and digital communication systems in the next few sections. The scope and outline of the book will be described in Sec. 1.5. 1.2 Communications systems Figure 1.1(a) shows the schematic of a simple analog communication system. Here we have a message signal s(t) which is transmitted over a channel to produce the signal y(t) at the receiver end. In many practical systems the channel can be modeled as a linear time invariant, or LTI, system followed by an additive noise source q(t). This is shown in Fig. 1.1(b), where the channel impulse response is indicated as h(t).
- 23. 2 Introduction q(t) + y(t) LTIchannel h(t) s(t) noise y(t) s(t) channel postfilter or equalizer s(t) q(t) s(t) g(t) + y(t) LTI channel h(t) (a) (b) (c) Figure 1.1. An analog communication system. (a) Channel with input s(t) and output y(t). (b) The channel modeled as a linear time invariant system followed by an additive noise source. (c) The channel followed by a postfilter or equalizer at the receiver. The received signal y(t) in Fig. 1.1(b) can be expressed in the form y(t) = ∞ −∞ h(τ)s(t − τ)dτ + q(t). (1.1) The first term above represents a convolution integral. In practice the channel is causal so that h(t) is zero for t 0. In this case the lower limit of the integral can be taken as 0 rather than −∞. Next consider Fig. 1.1(c), where the received signal y(t) is processed using an LTI system called the equalizer or the postfilter. The purpose of an equalizer is to compensate for the distortion caused by the convolution with the channel h(t), and to reduce the effect of the channel noise. The equalizer should be designed by taking into account the knowledge of h(t) and whatever knowledge might be available about the statistics of the noise q(t). The reconstructed signal s(t) then serves as an approximation of s(t). The reconstruction error is given by e(t) = s(t) − s(t). (1.2) Figure 1.2 shows a further enhancement at the transmitter end. The message signal s(t) is first passed through an LTI system called the prefilter or precoder. This system has impulse response f(t). The prefilter “shapes” the message s(t) appropriately.
- 24. 1.2 Communications systems3 prefilter or precoder at the transmitter postfilter or equalizer at the receiver f(t) s(t) q(t) s(t) g(t) + y(t) LTI channel h(t) x(t) Figure 1.2. An analog communication system with a linear precoder at the transmit- ter and a linear equalizer at the receiver. Given our knowledge of the channel and the noise statistics, it is possible to choose the prefilter (jointly with the equalizer) such that s(t) approximates s(t) as best as possible. For this we have to specify some error criterion such as, for example, the mean square error. Also, appropriate constraints on the trans- mitted power have to be specified. The precoder at the transmitter and the equalizer at the receiver are together referred to as a transceiver. Thus, we often talk of optimal design of a transceiver {f(t), g(t)} for a given channel.1 In later chapters we will consider many recent variations of this classical problem, espe- cially in the context of digital communications. The multi-input multi-output (MIMO) version of this problem is especially important as we shall see. It is of- ten convenient to use a frequency domain representation for the communication system. Thus, let H(jω) denote the Fourier transform of h(t), that is, H(jω) = ∞ −∞ h(t)e−jωt dt. (1.3) This is called the frequency response of the channel. Similarly, let F(jω) and G(jω) represent the frequency responses of the precoder f(t) and equalizer g(t). Figure 1.3 shows a redrawing of Fig. 1.2 in terms of these Fourier transform quantities. In terms of this notation the effective channel from s(t) to s(t) is Heff (jω) = G(jω)H(jω)F(jω). (1.4) With (s ∗ f)(t) denoting the convolution of s(t) with f(t), the effective channel impulse response can be written in the form heff (t) = (g ∗ h ∗ f)(t). (1.5) There are many variations of the above channel model. In some situations, like mobile communications, the channel is modeled as a slowly time varying system rather than an LTI system. In some scenarios the channel impulse response h(t) is regarded as a random variable drawn from a known distribution. The noise source q(t) is often modeled as a Gaussian random process with known power spectrum. We shall come to the details later. 1Sometimes the entire system in the figure is loosely referred to as the transceiver.
- 25. 4 Introduction prefilter orprecoder postfilter or equalizer F(jω) s(t) q(t) s(t) G(jω) + y(t) LTI channel H(jω) x(t) Figure 1.3. The analog communication system represented in terms of frequency responses. 1.3 Digitial communication systems In the preceding section, the signal s(t) was regarded as a continuous time signal with continuous (unquantized) amplitude. In a digital communication system, the messages are quantized amplitudes, transmitted in discrete time. Figure 1.4 shows the schematic of a digital communication system. Here we have a discrete- time message or signal s(n) which we wish to transmit over a continuous-time channel. The amplitudes of s(n) are “digitized,” that is they come from a finite set of symbols. This collection of symbols is called a constellation. We shall come to details of digitization later.2 Since s(n) is a discrete-time signal and the channel is continuous-time, the signal is first converted into a continuous-time signal x(t) as indicated in the figure. The conversion from s(n) to x(t) can be described schematically in two steps. The building block indicated as D/C is a discrete-to-continuous-time converter, and it converts s(n) to a signal sc(t) given by sc(t) = ∞ n=−∞ s(n)δc(t − nT). (1.6) Here δc(t) is the impulse or Dirac delta function [Oppenheim and Willsky, 1997]. Thus, the sample s(n) is converted into an impulse positioned at time nT. The sample spacing T determines the speed with which the message samples are conveyed. Since we have 1/T symbols per second, the symbol rate is given by fs = 1 T Hz. (1.7) The prefilter F(jω) at the transmitter performs a convolution to produce the output x(t) = ∞ n=−∞ s(n)f(t − nT), (1.8) 2Examples of constellations include PAM and QAM systems to be described in Sec. 2.2.
- 26. 1.3 Digitial communicationsystems 5 prefilter postfilter F(jω) s(n) q(t) s(n) G(jω) + y(t) D/C C/D T T s (t) c 0 −T T t −1 0 1 n x(t) t t 0 −1 1 n s (t) c channel H(jω) noise detector s (n) est Figure 1.4. A digital communication system. where f(t) is the impulse response of F(jω). Typically f(t) is a smooth, finite duration function, as demonstrated in Fig. 1.5(a). In practice, f(t) is causal, that is, it is zero for negative time. In the figure it is shown to be noncausal for generality. Note that x(t) is a weighted sum of uniformly shifted versions of the impulse response f(t). The weight on the nth shifted version f(t − nT) is the nth message sample s(n). This construction of x(t) from s(n) is demonstrated in Fig. 1.5(b). The signal x(t) is then transmitted over the continuous-time channel H(jω), which also adds noise q(t): y(t) = ∞ −∞ h(τ)x(t − τ)dτ + q(t). (1.9) This is the signal that will be observed at the receiver. The goal at the receiver is to reconstruct the original discrete signal s(n) from this noisy and distorted continuous-time signal y(t). First, the postfilter G(jω) at the receiver processes y(t) to produce sc(t), which is then sampled at the rate fs = 1/T to obtain a reconstructed version of s(n): s(n) = sc(nT). (1.10) The box labeled C/D is the continuous-to-discrete-time converter, and performs the sampling operation (1.10). The reconstruction error is given by e(n) = s(n) − s(n). (1.11) Given the knowledge of the channel H(jω) and the noise statistics, it is possible to design the filters F(jω) and G(jω) to minimize an appropriate measure of reconstruction error. A simple example of an “appropriate measure” is the mean square error (i.e., the average value of |e(n)|2 ).
- 27. 6 Introduction 0 t f(t) 0 T2T t f(t) s(0) f(t−T) s(1) s(0) s(1) (a) (b) Figure 1.5. (a) Impulse response f(t) of the prefilter F(jω), and (b) the signal x(t) generated from the samples s(n) by interpolation with the function f(t). Note that, while s(n) belongs to the signal constellation, the quantity s(n) does not. In practice there is a device called the detector at the receiver (Fig. 1.4), which obtains an estimated constellation symbol sest(n) from the quantity s(n). The probability of symbol error is defined to be the probability that sest(n) differs from s(n). The minimization of this probability is another important optimization problem. Such optimizations and several generalizations will be discussed in appropriate sections of the book. Bandlimiting. In practice the bandwidth allowed for a transceiver is limited. This bandlimiting is enforced by using lowpass filters at the transmitter and receiver. These filters can be incorporated as parts of F(jω) and G(jω). The transmitted signal x(t) therefore occupies a fairly narrow bandwidth of the form −σ ω σ, (1.12) and is called the baseband signal. The bandwidth can be a few kHz to MHz, depending on application. The signal x(t) is actually used to modulate a high- frequency carrier, and the modulated signal is transmitted either wirelessly using antennas or on wirelines. A discussion of carrier modulation is included in Sec. 2.4. The channel model discussed above is called the baseband model, as it does not show the carrier explicitly. Similarly the continuous-time system described in Sec. 1.2 also represents a baseband model. 1.3.1 Discrete-time equivalent We will see in later sections that the problem of designing the digital communi- cation system of Fig. 1.4 can be reformulated entirely in terms of discrete-time transfer functions as in Fig. 1.6.
- 28. 1.4 MIMO channels7 precoder equalizer s(n) s(n) + channel F (z) d G (z) d H (z) d q (n) d Figure 1.6. An all-discrete equivalent of the digital communication system. Here Hd(z) is the transfer function of an equivalent discrete-time channel. It is the z-transform of an equivalent digital channel impulse response hd(n), that is Hd(z) = ∞ n=−∞ hd(n)z−n . (1.13) Similarly, Fd(z) and Gd(z) are the transfer functions of the discrete-time pre- coder and equalizer. The subscript d (for “discrete”), which is just for clarity, is usually dropped. In practice Hd(z) is causal and can be approximated by a finite impulse response, or FIR, system so that Hd(z) = L n=0 hd(n)z−n . (1.14) The problem of optimizing the precoder Fd(z) and equalizer Gd(z) for fixed channel Hd(z) and fixed noise statistics will be addressed in later chapters. 1.4 MIMO channels The transceivers described so far have one input signal s(n) and a corresponding output s(n). These are called single-input single-output, or SISO, transceivers. An important communication system that comes up frequently in this book is the multi-input multi-output, or MIMO, channel. Figure 1.7 shows a MIMO channel assumed to be linear and time-invariant with a transfer function matrix H(z), usually an FIR system: H(z) = L n=0 h(n)z−n . (1.15) The sequence h(n), called the MIMO impulse response, is a sequence of matrices. If the channel has P inputs and J outputs then H(z) has size J × P, and so does each of the matrices h(n). The MIMO communication channel is used to transmit a vector signal s(n) with M components: s(n) = [ s0(n) s1(n) . . . sM−1(n) ] T . (1.16)
- 29. 8 Introduction precoder equalizer channel s(n) q(n) s(n) MP J M F(z) H(z) G(z) x(n) y(n) Figure 1.7. A digital communication system. The precoder F(z) transforms this sequence s(n) into another sequence x(n). We will see that the choice of F(z) plays an important role in the performance of the communication system. The channel produces the inevitable distortion represented by the transfer function H(z) and the noise vector q(n). Thus the signal obtained at the receiver is y(n) = L k=0 h(k)x(n − k) + q(n). (1.17) The equalizer G(z) seeks to reconstruct s(n) from this distorted version: s(n) = k g(k)y(n − k). (1.18) The joint design of the transceiver {F(z), G(z)} is an important problem in modern digital communications. The MIMO transceiver shown in the figure can be used to transmit messages sk(n), 0 ≤ k ≤ M − 1, (1.19) from M separate users. It can also be used to transmit information from one user by representing the message s(n) from the user in the form of a vector s(n); such systems are called block-based transceivers for SISO channels. They have many advantages as we shall see. MIMO channels also arise from the use of multiple antennas for single users; a detailed discussion of how MIMO channels arise will be given in Sec. 4.5. A special case of the MIMO system arises when the channel is memoryless, that is, the transfer function H(z) is just a constant H. This corresponds to the situation where L = 0 in Eq. (1.15). Optimization of transceivers for memoryless MIMO channels will be the focus of some of the chapters in this book. 1.5 Scope and outline The reader is assumed to have some familiarity with introductory topics in com- munications and signal processing. References for such background material
- 30. 1.5 Scope andoutline 9 include Proakis [1995], Oppenheim and Willsky [1997], Oppenheim and Schafer [1999], Lathi [1998], Haykin [2001], Mitra [2001], and Antoniou [2006], among other excellent texts. Advanced material related to the topics in this book can also be found in Ding and Li [2001], and Giannakis, et al. [2001]. Books on the very important related areas of wireless and multiuser communications include Rappaport [1996], Verdu [1998], Goldsmith [2005], Haykin and Moher [2005], and Tse and Viswanath [2005]. The book is divided into four parts. Part 1 contains introductory material on digital communication systems and signal processing aspects. In part 2 we discuss the optimization of transceivers, with emphasis on MIMO channels. Part 3 provides mathematical background material for optimization of transceivers. This part can be used as a reference, and will be very useful for readers wishing to pursue more detailed literature on optimization. Part 4 contains eight appen- dices on commonly used material such as matrix theory, Wiener filtering, and so forth. The history of digital communication theory is fascinating. It is a humbling experience to look back and reflect on the tremendous insights and accomplish- ments of the communications and signal processing society in the last six decades. Needless to say, much of the recent research is built upon six to seven decades of this solid foundation. A detour into history will be provided in Chap. 9, where we present a historical perspective of transceiver design, equalization, and opti- mization, all of which originated in the early 1960s, and have continued to this day to be research topics. All references to literature will be given in the specific chapters as appropriate. An extensive reference list is given at the end of the book. In what follows we briefly describe the four parts of the book. Part 1: Communication fundamentals Part 1 consists of Chapters 1 to 8. In Chap. 2 we review basic topics in digital communication systems, such as signal constellations, carrier modulation, and so forth. Formulas for probabilities of error in symbol detection are derived. Matched filtering, which is used in some receiver systems, is discussed in some detail. In Chap. 3 we describe digital communication systems using the language of multirate filter banks. Such a representation is very useful for transceivers with or without redundancy, and has many applications as we shall see throughout the book. In Chap. 4 we describe digital communication systems using discrete-time language. This chapter also introduces symbol spaced equalizers (SSE) and fractionally spaced equalizers (FSE). The minimum mean square error (MMSE) equalizer is also introduced in this chapter. Chapter 5 discusses a number of fun- damental techniques that are commonly used in digital communications. First a detailed discussion of the matched filter is provided. Then we discuss opti- mal sequence estimators, such as the maximum likelihood (ML) detector and the Viterbi alogrithm. Nonlinear methods, such as the decision feedback equal- izer and nonlinear precoders, are introduced. Chapter 6 is a brief discussion of channel capacity with emphasis on MIMO channels. Chapter 7 introduces redundant precoders, including zero-padded and cyclic-
- 31. 10 Introduction prefixed precoders.The redundant precoder is an integral part of many of the transceiver designs today. For example, cyclic prefix systems are employed in orthogonal frequency division multiplexing (OFDM) systems and discrete mul- titone (DMT) systems, used in digital subscriber loop (DSL) technology. The introduction of redundancy allows us to compensate or equalize the effects of a linear channel very efficiently – for example, an FIR channel can be equalized without the use of IIR equalizers. In Chap. 8 we discuss zero-padded systems in greater detail and introduce zero-forcing FIR equalizers, which can perfectly equalize FIR channels by exploiting the redundancy in the transmitted symbol stream. A number of properties of such equalizers are studied. Part 2: Transceiver optimization Part 2 consists of Chapters 9 to 19. Chapter 9 gives a brief historical introduction to transceiver optimization, and provides a detailed outline for Chapters 10 to 19. Briefly, Chap. 10 discusses the optimization of transceivers for scalar channels, and Chap. 11 discusses the optimization of transceivers for MIMO diagonal channels. Chapters 12 and 13 discuss the minimization of mean square error in trans- ceivers (MMSE transceivers) for general (nondiagonal) channels with and with- out the so-called zero-forcing constraint. Chapter 14 discusses the minimization of transmitted power for fixed performance criteria (such as error probability). This chapter also shows how one can perform bit allocation among the symbol streams optimally. Chapter 15 discusses transceiver optimization for the special case where the precoder at the transmitter is constrained to be orthogonal. In Chap. 16 we consider the minimization of symbol error rates or bit error rates (BER), which are more directly related to practical performance than mean square errors. There is a close connection between MMSE transceivers and minimum-BER transceivers as we shall see in that chapter. The results of transceiver optimiza- tion are applied in Chaps. 17 and 18 to the case of cyclic-prefix systems and zero-padded systems, respectively. These are single-input single-output (SISO) channels turned into multi-input multi-output (MIMO) channels by introducing redundancy as described in Chap. 7. Chapter 19 discusses the decision feedback equalizer for MIMO channels. The joint optimization of transceiver matrices with decision feedback is discussed in detail. Part 3: Mathematical background Part 3 consists of Chapters 20 to 22. Some of the mathematical background needed for the optimization chapters is given in these chapters. This includes matrix calculus, Schur convex functions, and nonlinear optimization tools. Ma- trix calculus is a less commonly reviewed topic, so Chap. 20 offers a detailed review. Schur convex functions have played a major role in transceiver opti- mization in recent years, and the review in Chap. 21 will be useful to readers wishing to pursue the literature in depth. Chapter 22 is a review of constrained optimization theory, which is useful in some of the chapters on transceiver opti-
- 32. 1.6 Notations 11 mization. Part4: Appendices There are eight appendices at the end of the book. They contain short discus- sions on useful topics from inequalities, matrix theory, singular value decompo- sitions, random processes, Wiener filtering, sampling theory, and so forth. In addition, there are appendices at the ends of some individual chapters, which contain useful material relevant to those chapters. Book appendices are num- bered as Appendix A, Appendix B, and so forth. Chapter appendices are num- bered as Appendix 2.A (App. A at the end of Chap. 2), and so forth. Appendix I at the end of the book gives a summary of the main optimization results in Part 2 of the book, with each major result summarized in one page. 1.6 Commonly used notations Bold-faced letters, such as A and v, indicate matrices and vectors. Superscript T, ∗, and †, as in AT , A∗ , and A† denote, respectively, the transpose, conjugate, and transpose-conjugate of a matrix. The determinant of a square matrix A is denoted as det (A), and the trace as Tr (A), with brackets omitted when redundant. Given two Hermitian matrices A and B, the notation A ≥ B means that A − B is positive semidefinite, and A B means that A − B is positive definite (Appendix B). For a continuous-time function h(t) the Laplace transform is denoted as H(s) and the Fourier transform as H(jω). The frequency variable f = ω/2π is also sometimes used. For a discrete-time function g(n) the z- transform is denoted as G(z) and the Fourier transform as G(ejω ). The tilde notation on a function of z is defined as follows: H(z) = H† (1/z∗ ). Thus, H(z) = n h(n)z−n ⇒ H(z) = n h† (n)zn , so that the tilde notation effectively replaces all coefficients with the transpose conjugates, and replaces z with 1/z. For example, H(z) = h(0) + h(1)z−1 ⇒ H(z) = h∗ (0) + h∗ (1)z, and H(z) = a0 + a1z−1 1 + b1z−1 ⇒ H(z) = a∗ 0 + a∗ 1z 1 + b∗ 1z Note that H(ejω ) = H† (ejω ). That is, the tilde notation reduces to transpose conjugation on the unit circle.
- 33. 2 Review of basicideas from digital communication 2.1 Introduction In this chapter we briefly review introductory material from the theory and practice of digital communication. The reader familiar with such introductory material can use this chapter primarily as a reference for later chapters. For a more detailed treatement one should consult standard communication texts such as Proakis [1995], Lathi [1998], or Haykin [2001]. 2.1.1 Chapter overview The schematic representation of digital communication systems described earlier in Sec. 1.3 is reproduced in Fig. 2.1. As mentioned in that section, the input symbol stream has sample values s(n) chosen from a finite set of values called the symbol alphabet, constellation, or code words – we shall consistently use the term constellation. In Sec. 2.2 we shall describe two commonly used constellations called the PAM (pulse amplitude modulation) and the QAM (quadrature PAM) constellations. Given the symbol stream s(n) the transmitter generates the baseband wave- form x(t) = ∞ n=−∞ s(n)f(t − nT), 12
- 34. 2.1 Introduction 13 prefilterpostfilter F(jω) s(n) q(t) s(n) G(jω) + y(t) D/C C/D T T s (t) c 0 −T T t −1 0 1 n x(t) t t 0 −1 1 n s (t) c channel H(jω) noise detector s (n) est Figure 2.1. A digital communication system. 0 t f(t) 0 T 2T t f(t) s(0) f(t−T) s(1) s(0) s(1) (a) (b) Figure 2.2. (a) Impulse response f(t) of the prefilter F(jω), and (b) the signal x(t) generated from the samples s(n) by interpolation with the function f(t). where f(t) is a prefilter waveform. We also say that f(t) is the transmitted pulse (and sometimes we use the notation p(t)). This waveform is usually time-limited, and most of its energy is confined to a narrow band of frequencies such as |ω| σ called the baseband (sometimes written as base band). So f(t) can be considered to be nearly bandlimited. A typical f(t) is shown in Fig. 2.2(a), and a pictorial description of the waveform x(t) is reproduced in Fig. 2.2(b) (from Sec. 1.3). The signal x(t) is filtered by the channel H(jω), and the receiver receives a noisy version of this signal. The received signal is further filtered by G(jω) and then sampled at the symbol rate 1/T. From the sampled version s(n) the receiver has to identify or detect the symbol s(n) which was actually transmitted. Since the channel introduces distortions (due to H(jω) and noise q(t)), the detected
- 35. 14 Review ofdigital communications symbol sest(n) can sometimes be different from s(n), resulting in an error. Under appropriate assumptions on the statistics of the reconstruction error it is possible to compute the probability of error in symbol identification. This is done in Sec. 2.3. In practice the baseband signal x(t) is used to modulate a high-frequency carrier signal, and the modulated signal is transmitted. At the receiver this signal is demodulated to extract the (noisy version of) the baseband signal. The details of this modulation and demodulation are different for PAM and QAM constellations, and will be presented in Sec. 2.4. Returning to the postfilter G(jω), we mentioned in Chap. 1 that it is called the equalizer, and compensates for the channel distortions. In practice, this equalization is often performed with a digital filter after the sampling process. The analog filter G(jω) then plays a different role called matched filtering. The purpose of this filter is to maximize the signal-to-noise ratio at the sample lo- cations, and its optimal choice depends on the channel H(jω) as well as on the power spectrum of the noise q(t). Matched filtering is reviewed in Sec. 2.5, and some practical details are discussed in Sec. 2.6. 2.2 Signal constellations Two of the popular constellations widely used today are the PAM (pulse am- plitude modulation) and the QAM (quadrature amplitude modulation) constel- lations. The PAM constellation has real-valued numbers for s(n), whereas the QAM constellation has complex numbers. A b-bit PAM or QAM constella- tion has M = 2b allowed numbers, called constellation symbols or codewords (sometimes written as code words), and is also called an M-PAM or M-QAM constellation. Figure 2.3 shows a PAM constellation with M codewords. Note that M is even (a power of 2), and adjacent codewords are separated by a fixed amount 2A. No codeword has value zero. The positive number A can be chosen to adjust the average energy per codeword as we shall see later. Figure 2.4 shows a QAM constellation with M = 16 codewords. This is a 4-bit constellation. Once again, no codeword has value zero. Note that any 4-bit QAM code word has the form z = x + jy, where x and y are 2-bit PAM words. The components x and y are sometimes called the in-phase and quadrature components. A b-bit QAM constellation has real and imaginary parts coming from (b/2)-bit PAM constellations. The QAM constellations we use here are called square QAM constellations. More generally they can be rectangular constellations with b1 bits for x and b2 bits for y. There are more general types of QAM constellations including circular ones [Proakis, 1995], but we shall not elaborate on those here. Unless mentioned otherwise, we always imply square constellations when we use the term QAM. So the number of bits b is an even integer. As in PAM, any word in the square QAM constellation has nearest neighbors at distance 2A.
- 36. 2.2 Signal constellations15 A −A 0 −3A −(M−1)A 3A (M−1)A 2A Figure 2.3. A PAM constellation with M codewords. 0 2A 2A codewords Re Im Figure 2.4. A 4-bit QAM constellation with 24 = 16 codewords. 2.2.1 Average energy in a PAM constellation It is useful to have an expression for the average energy in the symbols. In a PAM constellation, the words are real numbers of the form (2n + 1)A, where n is an integer. The energy of a symbol is simply (2n+1)2 A2 . Note that for every positive codeword there is a corresponding negative codeword with the same magnitude. So the average energy of a b-bit PAM constellation (with M = 2b words) is Eave,P AM = 2A2 M 12 + 32 + 52 + . . . + (M − 1)2 . (2.1) If the M codewords are equally likely, then this represents the average energy
- 37. 16 Review ofdigital communications per sample in the sequence s(n). Using the fact (see below) that 12 + 32 + 52 + . . . + (M − 1)2 = M(M2 − 1) 6 , (2.2) we therefore have Eave,P AM = (M2 − 1)A2 3 = (22b − 1)A2 3 , (2.3) so that A = 3Eave,P AM 22b − 1 . Since b = log2 M, the quantity Eb,P AM = Eave,P AM log2 M = (M2 − 1)A2 3 log2 M , (2.4) is called the energy per bit. Derivation of Eq. (2.2). It is well known that n k=1 k2 = n(n+1)(2n+1)/6. Now, for even M we can write 1 + 32 + . . . + (M − 1)2 = M k=1 k2 − (22 + 42 + . . . + M2 ) = M k=1 k2 − 4 1 + 22 + . . . + ( M 2 )2 = M(M + 1)(2M + 1) 6 − 1 6 × 4M 2 M 2 + 1 (M + 1) = M(M + 1) 6 2M + 1 − (M + 2) = M(M2 − 1) 6 , which proves Eq. (2.2). 2.2.2 Average energy in a QAM constellation Now consider a b-bit QAM constellation. There are M = 2b codewords of the form z = x + jy, where x and y belong to (b/2)-bit constellations with √ M words each. Thus the average energy in the real part is as in Eq. (2.3) with M replaced by √ M = 2b/2 : Eave,x = (M − 1)A2 3 .
- 38. 2.3 Error probability17 The same is true for the imaginary part y. So the average energy of a b-bit QAM constellation (with M = 2b words) is Eave,QAM = 2(M − 1)A2 3 = 2(2b − 1)A2 3 . (2.5) The energy per bit is therefore Eb,QAM = Eave,QAM log2 M = 2(M − 1)A2 3 log2 M . (2.6) 2.3 Error probability In a digital communication system the receiver constructs an approximation s(n) of the transmitted symbol stream s(n). The reconstructed version s(n) differs from s(n) because of errors introduced by the channel H(jω) and the noise q(t). We can write s(n) = s(n) + e(n), (2.7) where e(n) is the reconstruction error, which can often be modeled as random noise. Thus, even though s(n) is a codeword belonging to a constellation such as a PAM constellation, the reconstructed number s(n) is not. In practice s(n) is processed by a decision making device, called the detector, which takes the signal s(n) and estimates the transmitted codeword s(n) (Fig. 2.5). There is a nonzero probability that this estimated codeword sest(n) is different from the original codeword s(n). This error probability depends on the statistics of the error term e(n) in Eq. (2.7). Figure 2.6 shows how the received symbols s(n) get spread out into a “cloud” owing to the noise e(n) in a QAM constellation. The received signal can be anywhere in the shaded areas. If the shaded area for a symbol overlaps with the corresponding shaded area of an adjacent symbol, there is nonzero probability of symbol error. In this section we derive mathematical expressions for error probabilities. 2.3.1 Error probability for PAM signals First consider the PAM constellation shown in Fig. 2.7 for 3 bits. We have indicated small vertical lines called decision boundaries. These are placed exactly midway between every pair of symbols. If s(n) falls within a pair of decision boundaries, then the unique codeword within those boundaries is assumed to be transmitted because it is the closest codeword. This is the estimated symbol sest(n) corresponding to the transmitted symbol s(n). To demonstrate, Fig. 2.8(a) shows the threshold detector characteristics for 1-bit PAM. This figure says that the 1-bit PAM detector estimates the symbol according to the rule sest(n) = A if s(n) ≥ 0 −A if s(n) 0. (2.8)
- 39. 18 Review ofdigital communications detector s(n) estimated symbol in the constellation s (n) est Figure 2.5. The detector, or decision device, at the receiver takes the reconstructed symbol s(n) and maps it into a symbol sest(n) in the constellation, which is regarded as an estimate of the transmitted symbol s(n). Re Im Figure 2.6. Noise cloud associated with a 4-bit QAM constellation. Figure 2.8(b) shows the threshold detector characteristics for an arbitrary num- ber of bits. As a specific example, assume that the transmitted symbol at time n was s(n) = 3A, as highlighted in Fig. 2.7. If the error term e(n) (which is a random variable) is such that |e(n)| A (2.9) then s(n) is within the shaded box shown in Fig. 2.7, and the symbol estimation is correct, that is, sest(n) = s(n). If e(n) has magnitude larger than A then the symbol is estimated to be A (for e(n) −A) or 5A (for e(n) A), and there is an error. The probability of error in the decision can be calculated if we know the probability density function (pdf) of the additive error term e(n) in Eq. (2.7). Figure 2.7 shows an example of this pdf, denoted as fE(e).
- 40. 2.3 Error probability19 A −A −3A −7A 3A 7A 2A 5A −5A decision boundaries f (e) E low noise, small variance noise pdf high noise, large variance symbols Figure 2.7. Explanation of how the detector works, and how decision errors occur in a PAM constellation. See text. (b) (a) A −A s sest A sest 3A 5A 2A 4A s Figure 2.8. Threshold detector characteristics (a) for 1-bit PAM and (b) for a general PAM. The probability of incorrectly deciding that 5A was transmitted instead of 3A is the probability that e(n) A, that is, P(3A detected as 5A) = ∞ A fE(e)de. (2.10) The probability of incorrectly deciding that A was transmitted instead of 3A is the probability that e(n) −A, that is, P(3A detected as A) = −A −∞ fE(e)de. (2.11)
- 41. 20 Review ofdigital communications The probability of an incorrect decision is therefore the sum of the above two integrals.3 For a boundary symbol such as 7A or −7A, the contribution to error comes only from one integral. For example, since there are no symbols past 7A, the probability of error is just the integral in (2.11), and similarly for the transmitted symbol −7A, the probability of error is the integral in (2.10). Only for the M − 2 interior symbols is the error probability the sum of the integrals in (2.10) and (2.11). 2.3.1.A Case of Gaussian noise In many communication systems it is reasonable to assume that the error e(n) is a Gaussian random variable with zero mean, that is fE(e) = 1 2πσ2 e e−(e2 /2σ2 e ) , (2.12) where σ2 e is the variance of e(n). Since e(n) is usually also assumed to be white, we say that it is additive white Gaussian noise, or AWGN. Figure 2.9 demon- strates this pdf for two values of the variance σ2 e . Since fE(e) is symmetric, the integrals (2.10) and (2.11) are identical in this case. In fact this integral can be expressed elegantly in terms of the so-called Q-function. This function is defined as the integral Q(v) = ∞ v fN (u)du, (2.13) where fN (u) is the normal density e−u2 /2 / √ 2π, that is, Gaussian with zero mean and unit variance. Thus Q(v) = 1 √ 2π ∞ v e−u2 /2 du. (2.14) The Q-function is related to the complementary error function erfc(x) by Q(x) = 0.5 erfc(x/ √ 2), (2.15) or equivalently erfc(x) = 2Q( √ 2x). A plot is shown in Fig. 2.10. Notice how it decreases monotonically. In Sec. 21.2.3 we will see that Q(x) is a convex function. Using the Q-function we can express the integral (2.10) as ∞ A fE(e)de = 1 2πσ2 e ∞ A e−(e2 /2σ2 e ) de = 1 √ 2π ∞ A/σe e−(u2 /2) du = Q(A/σe), where we have used the change of variables u = e/σe. 3Note that if the error pdf fE(e) is confined to the range −A e A then the symbol error probability is zero. That is, the receiver can perfectly identify the transmitted symbol even though there is channel noise.
- 42. 2.3 Error probability21 -0.5 -0.25 0 0.25 0.5 0 3 6 9 12 e f E (e) var. = 0.007 var. = 0.001 Figure 2.9. Examples of the Gaussian density function for zero mean and two values of the variance σ2 e . For large variance, the plot is more spread out. For small variance the plot is taller. The area under each curve is unity. 0 1 2 3 4 0.1 0.2 0.3 0.4 0.5 x Q(x) Figure 2.10. A plot of Q(x) = ∞ x e−u2 /2 du/ √ 2π, for x ≥ 0. Thus the probability of error for any interior symbol is 2Q(A/σe) whereas the probability of error for each boundary symbol is just Q(A/σe). Assuming all codewords are equally likely, the average error probability is therefore given by Pe,P AM = 2(M − 2)Q(A/σe) + Q(A/σe) + Q(A/σe) M = 2Q(A/σe)(M − 1) M . Substituting M = 2b , where b is the number of bits, we therefore obtain Pe,P AM = 2(1 − 2−b )Q(A/σe). (2.16)
- 43. 22 Review ofdigital communications Using the expression for the average energy (2.3) in a QAM constellation, we can rewrite this as Pe,P AM = 2(1 − 2−b )Q 3Eave (22b − 1)σ2 e , (2.17) where the subscript PAM on Eave has been deleted for simplicity. The error probabilities are also called the symbol error rates (SER) because they tell us what fraction of symbols are expected to be in error, given a long symbol stream. Summary. Let the detector input have the form s(n) = s(n) + e(n), where s(n) is a b-bit PAM symbol and e(n) is zero-mean Gaussian with variance σ2 e , and let the average energy of the PAM constellation be Eave. Then the average error probability in detecting s(n) is given by Eq. (2.17). The error probability can also be written as (2.16), where A is the amplitude of the smallest codeword (Fig. 2.3). In these expressions, Q(.) is the integral defined in Eq. (2.14). Example 2.1: One-bit PAM Consider the case of 1-bit PAM, also known as a binary antipodal or PSK (phase-shift keying) or BPSK (binary phase-shift keying) constellation. This is shown in Fig. 2.11. Setting b = 1 in Eq. (2.17), the average error probability is given by Pe,P SK = Q(A/σe). (2.18) We can also use (2.17) to get the equivalent expression Pe,P SK = Q Eave σ2 e . (2.19) Figure 2.12 shows a typical error pdf fE(e). Also shown is the pdf of the received signal when the symbol −A is transmitted for two different noise variances. The shaded area, which represents the probability of error (prob- ability that a −A is judged as an A), is smaller when the noise variance is smaller.
- 44. 2.3 Error probability23 A −A 0 Figure 2.11. The 1-bit PAM constellation. 0 (b) (c) 0 pdf of received signal, high noise integrate this part for error probability 0 e f (e) E (a) pdf of received signal, low noise shifted f (e) E e e −A −A Figure 2.12. (a) The pdf of the error e (assumed Gaussian). (b) and (c) The pdf of the received signal s(n) when the transmitted symbol is −A and the noise has pdf fE(e). (b) Large noise variance, and (c) small noise variance. 2.3.2 Error probability for QAM signals Suppose the symbols s(n) in Fig. 2.1 are drawn from a QAM constellation such as the one in Fig. 2.4. Once again the receiver constructs an approximation s(n)
- 45. 24 Review ofdigital communications of the transmitted symbol stream s(n), of the form s(n) = s(n) + e(n), (2.20) where e(n) is the reconstruction error. Recall here that s(n), and hence s(n), are complex numbers. It is usually reasonable to assume that the error e(n) is complex and has the form e(n) = er(n) + jei(n), where er(n) and ei(n) are independent zero-mean Gaussian random variables with identical variance 0.5σ2 e . In this case the total variance of the complex error e(n) is 0.5σ2 e + 0.5σ2 e = σ2 e . The joint pdf of the variables [er(n), ei(n)] is given by fE(er, ei) = e−e2 r/σ2 e πσ2 e × e−e2 i /σ2 e πσ2 e = e−(e2 r+e2 i )/σ2 e πσ2 e . (2.21) Figure 2.13 demonstrates this for σ2 e = 0.01. This is a special case of a so-called circularly symmetric complex Gaussian random variable.4 Next, the complex symbol s(n) is of the form sr(n) + jsi(n), where sr(n) and si(n) are PAM symbols. Since |s(n)|2 = s2 r(n) + s2 i (n), it follows that the average energy of the constellation is the sum of the average energies of the real and imaginary parts. Thus, for a b-bit QAM constellation with average energy Eave, the real and imaginary parts are (b/2)-bit PAM con- stellations with average energy Eave/2, and each of these PAM constellations sees an error source with variance σ2 e /2. For the real-part PAM the probability of error can be obtained from Eq. (2.17) by replacing b, Eave, and σ2 e with half their values: Pe,re = 2(1 − 2−b/2 )Q 3Eave (2b − 1)σ2 e . (2.22) Since the factor of one-half cancels out in the ratio Eave/σ2 e , this is nothing but the error probability for a (b/2)-bit PAM constellation with energy Eave and noise variance σ2 e . Similarly for the imaginary part Pe,im = 2(1 − 2−b/2 )Q 3Eave (2b − 1)σ2 e . (2.23) The QAM symbol is detected correctly if the real part and imaginary part are both detected correctly. The probability for this is (1 − Pe,re)2 . 4A detailed discussion of circularly symmetric complex random variables can be found in Sec. 6.6.
- 46. 2.3 Error probability25 -0.5 0 0.5 -0.5 0 0.5 0 5 10 15 20 25 Figure 2.13. The pdf of Eq. (2.21) plotted for σ2 e = 0.01. The probability of error in detection of the QAM symbol is therefore Pe,QAM (b) = 1 − (1 − Pe,P AM (b/2))2 , (2.24) where we have used the functional arguments to indicate the number of bits. Thus Pe,P AM (b/2) is the error probability for a (b/2)-bit PAM constellation with energy Eave and noise variance σ2 e . For small errors the preceding equation can be approximated as Pe,QAM (b) = 1 − 1 − 2Pe,P AM (b/2) + P2 e,P AM (b/2) ≈ 2Pe,P AM (b/2), (2.25) where we have neglected P2 e,P AM (b/2). This approximation is quite reasonable in practice. For example, even if Pe,P AM (b/2) = 10−3 (a rather large value), its square is 10−6 , which can be neglected. Thus the error probability for the QAM constellation can be approximated by Pe,QAM (b) ≈ 2Pe,P AM (b/2) = 4(1 − 2−b/2 )Q 3Eave (2b − 1)σ2 e , (2.26) where b is the number of bits, Eave is the average energy of the constellation, and σ2 e is the variance of the complex Gaussian error term e(n) at the input of the detector. For comparsion, recall that a b-bit PAM constellation with the same energy Eave and noise variance σ2 e would have error probability Pe,P AM (b) = 2(1 − 2−b )Q 3Eave (22b − 1)σ2 e . (2.27)
- 47. 26 Review ofdigital communications Example 2.2: Two-bit QAM or QPSK Consider the case of 2-bit QAM, also known as a QPSK (quadrature phase- shift keying) constellation. This is shown in Fig. 2.14. Setting b = 2 in Eq. (2.26), the average error probability is given by Pe,QP SK = 2Q Eave σ2 e . (2.28) For both the PAM and QAM systems, note that the error probability depends on the ratio Eave/σ2 e , rather than the individual values of the energy Eave and the error variance σ2 e . This ratio is called the signal-to-error ratio or signal-to-noise ratio SNR at the input of the detector: SNR = Eave σ2 e (2.29) Figure 2.15 shows plots of the symbol error probability Pe,P AM as a function of this SNR for PAM systems, for various values of the number of bits b. Figure 2.16 shows similar plots for QAM systems. To compare PAM and QAM systems, it is useful to introduce the bit error rates (BER), which are related to the symbol error rates. Before doing this we have to introduce a binary coding system called the Gray code. 2.3.3 Gray codes In digital communication systems we are often required to transmit binary streams. These streams can be converted to PAM or QAM symbols by ap- propriately grouping the bits. This is called the symbol modulation process. For example, if the binary stream is divided into blocks of size 3: . . . 010 001 111 011 100 . . . then each 3-bit block can be turned into a 3-bit PAM symbol. More generally, a b-bit block can be translated into an M-word constellation with M = 2b . There are many ways to define the mapping from the binary representation to the constellation words. Figure 2.17 shows an example of such a representation for a 3-bit PAM constellation. Thus the preceding binary sequence is converted to . . . − A, −5A, 3A, −3A, 7A, . . . In this example the binary words are assigned such that adjacent symbols in the constellation (Fig. 2.17) differ only in one of the bit locations. Such a representation is called a Gray code [Proakis, 1995].
- 48. 2.3 Error probability27 Re Im Figure 2.14. The 2-bit QAM constellation, also known as a QPSK constellation. An important property of Gray codes is that, for reasonably high SNR, the symbol error rate can be related to the bit error rate in a simple manner. Thus, assume the SNR at the input of the detector is large enough, so that, when there is a symbol error, the estimated symbol is an adjacent symbol (rather than a symbol that is far away). In this case, only one bit is in error. Thus, in a symbol stream with N symbols, if there are K symbol errors then there are K bit errors as well. The symbol error rate (SER) is K/N, but since N symbols have Nb bits, the bit error rate (BER) is K/Nb. Thus, for a Gray coded system with b bits, BER = SER b . (2.30) From Eq. (2.25) we know that the symbol error probabilities for the PAM and QAM systems (for fixed SNR) are related approximately by Pe,QAM (b) ≈ 2Pe,P AM (b/2). (2.31) Using this, a simple relation between the bit error rates can be derived. Thus, dividing both sides of the preceding equation by b we get Pe,QAM (b) b ≈ 2Pe,P AM (b/2) b = Pe,P AM (b/2) b/2 , which shows that the BER for b-bit QAM is identical to BER for b/2 bit PAM: BERQAM (b) ≈ BERP AM (b/2). (2.32) Thus, for a given error rate, the QAM system can transmit twice as many bits. However, QAM also requires twice as much bandwidth compared to PAM, as we shall see in Sec. 2.4.3.
- 49. Discovering Diverse ContentThrough Random Scribd Documents
- 50. Adler broke outinto a coarse laugh. “Why, the little fellow is feeling his oats,” he cried; “he looks like a bantam rooster.” “Never mind what I look like,” retorted Herbert hotly. “I want to know whether you’ll take that word back.” “Don’t get excited, little chap.” “Will you take it back? Say yes or no!” demanded Herbert. “I say no,” drawled Adler. “Then I say take that!” As he spoke, Herbert reached up and gave the fellow a resounding slap on the cheek. Adler was so dazed at the unexpected assault that he stood still gazing stupidly at his assailant. The small boys in the group were secretly delighted at the indignity put upon their worthless companion, but were discreetly silent. Herbert walked off tingling with delight at having satisfied his outraged feelings.
- 51. CHAPTER IV IN WHICHFORTUNE UNEXPECTEDLY FAVORS DAVID HARKINS Herbert Harkins prepared to go to bed that night with a very heavy heart. He could not rid himself of the notion that he was the cause of the troubles that were gathering so rapidly about their home. Sleep is said to be the best medicine for a troubled mind; but unfortunately Herbert was not able to go to sleep. Usually he was in the land of dreams as soon as his head touched the pillow, but this night he was afflicted with a peculiar nervousness that could not be overcome. More than this he was greatly disturbed over the agitated condition of his father. He knew that he was sitting at his desk in the front room downstairs. He had spoken to him when he came home, and now from the light that was shining up the stairway he knew that his father was still awake. Presently he heard the movement of a chair, and then the steady tramping of feet indicating that Mr. Harkins was walking up and down the room. Suddenly this monotonous sound was broken by a sharp rap on the front door. Herbert heard his father respond to the summons. The bolt was drawn back, the door opened, and then came a sound like the cry of recognition from two men. The door was softly closed again, and then came the steady mumbling of voices. This continued so long that Herbert became frightened. He got out of bed in the dark, and going into the hallway crept downstairs silently, step by step, until he had reached the doorway leading into the parlor. The light was turned down and the room was quite dim; but he could see his father and another man seated at a table engaged in earnest conversation. The stranger wore a full beard, and his head was covered with a great shock of red hair, in much disorder. The two men were so much engaged that they did not notice the half frightened boy standing near the doorway. Herbert on his part was
- 52. so much interestedin what he saw that for the time being he forgot the situation in which he had placed himself. At times the two men were so close together that it would hardly have been possible to have drawn a sheet of paper between them. The stranger, in order to illustrate some point that he was making in his talk, threw his arm violently in the air, and in doing so overturned a little China ornament that was on the table, sending it crashing to the floor. Both men started violently at this unexpected happening, and then glanced nervously around the room as if to see whether anyone were listening. At the first sound of the falling ornament, Herbert started to run upstairs; but when the conversation was resumed some strange power seemed to draw him back to the doorway again. His intention was to take one last look and go away. He knew that he had no right there, and that his father might be very angry if he thought that he was out of bed and listening to the conversation; but some strange will over which he appeared to be powerless, kept him rooted to the spot. The two men talked in such a low tone at first that all he could hear was the mumbling of voices. Presently, however, his father becoming more earnest, said excitedly to the other man in a louder voice: “I won’t do it. I tell you I can’t do it. It’s not right to you.” “Don’t be a fool,” responded the red-haired man in a deep bass voice. “This will save you, and it cannot do me any harm. I’ll never miss it, I can assure you.” “But it seems so unjust,” urged his father; “it doesn’t seem quite square to act with you in this way. After all these years I should not be placed in the position of taking this from you.” “I am the best judge of that,” growled the other man in his heavy voice; “take it and say no more about it.” As he spoke he pushed a package in the direction of Mr. Harkins, who still with reluctance, picked it up and placed it in his pocket. This act seemed to relieve his feelings, because he said right away in a voice that sounded lighter and more contented:
- 53. “Well, I guessit is all for the best. I’ll take it, and you can rest assured that you’ll lose nothing by your kindness.” Their voices became lower again at this point, and Herbert, sorry for having remained so long, hurried back to bed and was soon in the land of slumber. Father, mother and son met at the breakfast table the next morning, and all seemed to be in a more cheerful frame of mind than they had been for some days. Mr. Harkins was bubbling over with good spirits. He turned to his wife in a laughing manner, and said: “I’ve got a surprise for you this morning—a bit of good news that will make you feel good.” “What is it?” asked the wife curiously. “Simply that I have the money and I am going to pay off that obligation to John Black before the clock strikes another hour.” The poor woman was so overjoyed at this unexpected news that she ran over and gave her husband a hearty kiss. “This is good news, David,” she said. “How on earth did you manage to raise the money in such a short time?” “Oh ho!” he replied merrily; “it’s news you are after, is it? Well you can’t have it just now. This money came from a gentleman who is a very good friend of mine. His name will have to remain a secret for the present at least.” Herbert sat and listened to this conversation with a feeling of dismay. He felt like crying out and telling his father that he had been present at the mysterious midnight interview and had heard things that were not intended for his ears; but his lips refused to frame the words, and he sat there feeling very mean and very guilty. Finally both conscience and curiosity got the better of him. He made up his mind to confess his little indiscretion—for it was not anything more serious than an indiscretion—and then to ask his father to tell him the name of the strange man who had appeared at such an unusual hour and under such unusual circumstances. Mr. Harkins had his hat
- 54. and coat onpreparing to leave the house when Herbert arose from the table and said to him in a voice that quivered with nervousness: “Father, I could not sleep last night.” “I am very sorry to hear that, my son,” was the kindly reply. “Probably you are not feeling well. You had better stop in and see Dr. Smith on your way from school this afternoon.” “No, no; it’s not that,” stammered Herbert; “it’s something I want to tell you. When I found that I could not sleep I got out of bed—” “I am in a hurry now, Herbert,” exclaimed his father, talking very rapidly and moving towards the door. “I must get down and see Mr. Coke. You can tell me this story when you come home from school this afternoon.” And the next moment the street door closed with a bang and Mr. Harkness was on his way to the bank. Herbert sat down in a chair feeling very much disappointed. He felt somehow or other that his father had become involved, and if he had been able to speak, that much mystery might have been dissipated.
- 55. CHAPTER V IN WHICHDAVID HARKINS BECOMES THE VICTIM OF PECULIAR CIRCUMSTANCES David Harkins left his home that morning, walking rapidly and gaily humming a tune to himself. He felt better and happier than he had for many weeks before. The thought of canceling the note and freeing himself from the obligation which he was under to John Black lifted an immense weight from his mind and enabled him to take a cheerful view of life. As he walked along he mentally matured plans for increasing his income during the year to come and placing his family in a position where they would not be compelled to feel concerned regarding the future. In a few minutes he reached the office of Horace Coke, the lawyer, who was installed in a little second story room of a modest house on the main street. The apartment was very much like the lawyer— simple and old-fashioned, but entirely adequate for the needs of the law. There was a plain, flat-top desk, littered with legal papers. An office boy who hoped eventually to become a member of the bar, sat copying a deed; and the silence in the room was broken by the steady scratching of his pen. The shelves about the room were filled with law books covered with calfskin and bearing their titles in little gold letters on a slip of black over what might be called their backbones. Mr. Coke himself was puffing away at a big black cigar— which, by the way, was his only dissipation. He was looking over some papers when David Harkins entered the room, but jumped from his chair immediately and greeted the newcomer with a hearty: “Hello there, Dave! What’s bringing you out so early in the morning?” “Some legal business, Horace,” replied the other laughingly.
- 56. “I am sorryto hear that,” said the venerable attorney, shaking his head in a doubtful manner. “I always advise my friends to keep out of the law. It’s a bad business. It takes up all your money, and rarely gives you any good results.” “That sounds like queer talk for a man who depends on the law for his livelihood.” Horace Coke laughed heartily at this retort, and said: “It does sound queer, doesn’t it? But I don’t talk that way to everybody. Of course, if people will get into trouble and will invoke the law, I might as well take their money and attend to their business as the next one; but I satisfy my conscience by advising all of my friends to keep out of the law, because as I said before, it’s a mighty bad business.” Then the good-natured counsellor dropped into his chair and indulged in another hearty laugh. It was one of the oddities of his nature that he should be continually berating the profession of which he was such an ornament and for which he really had a deep reverence. “But not to get off the subject,” added Mr. Harkins, “I would like to inform you that I have come here to pay off that note to John Black. Under ordinary circumstances I would go to the bank to transact this business; but as long as Mr. Black has found it necessary to employ a lawyer to secure his money, I felt that it was proper to come here and pay you.” The lawyer looked at David Harkins searchingly through his eye- glasses. He was silent for a moment, and then said in a low voice, in marked contrast with his jolly manner of a few minutes before: “See here, Dave, can you spare this money? I don’t believe you can, and I hate to see a man pressed. If you say the word, I’ll go over to old Black and try to get an extension on the note.” “Not at all,” was the cheerful rejoinder. “I do not desire an extension; I want to pay it and get it off my mind forever.”
- 57. Mr. Coke walkedover to Harkins and taking him by the hand, exclaimed in his cheery voice: “Congratulations, old man! I am glad to hear you talk in that way, and I am mighty glad to know that you were able to raise the money in such a short time. It will not only be a good thing to pay off the note, but it will be the means of establishing your credit in Cleverly. There’s nothing like a reputation for a man, and if you can get a good one it is liable to stick to you just as well as a bad one.” The two men sat down at the desk together, and after the necessary papers had been prepared and signed, Mr. Harkins handed over one thousand dollars in fresh banknotes. Half an hour later the lawyer put his hat and coat on and started towards the bank where he had an appointment with John Black. The door was closed when he arrived; but following his usual custom he entered without knocking. The banker’s back was turned to him at the time, and when he heard the door open and close, Mr. Black cried out in a harsh voice: “Who’s that? What are you doing there?” “It is only I, John,” said the lawyer. “I came here to attend to a little matter of business.” “Oh!” exclaimed the banker, changing his tone slightly at the sight of the lawyer. “I thought it was one of those impudent clerks coming in here without being civil enough to knock at the door.” After this he started to walk up and down the office, stamping his feet and frowning in a very ugly manner. His expression was forbidding, and Mr. Coke looked at him in astonishment. “What’s the matter, Black?” asked the lawyer. “You don’t seem to be in a very good humor this morning.” “Good humor? I should say not. I’ve got a good notion to leave this town. A man’s property isn’t safe over night. You get no protection. You pay big taxes and put up with all sorts of inconveniences, and
- 58. what do youget in return? That’s what I would like to know; what do you get in return?” “Why what in the world are you driving at?” asked the lawyer; “what has happened?” “Happened? Why everything’s happened. Some thief entered my house last night, got into the library, broke open my desk and stole a package of money that I had put there for safe keeping over night. What do you think of that? Wouldn’t you say that something had happened if your house had been broken into and your desk had been rifled? Wouldn’t you, I say? Wouldn’t you?” “Why, yes,” said the lawyer, staring at his client. “I suppose I should say that something had happened under those circumstances. But have you any clue to the robbery?” “Clue! Clue!” retorted the banker, with his habit of repeating words. “Certainly not. How could you expect me to have a clue in a town like this? The police officials are no good, never were any good, and never will be any good.” “But have you any hope of recovering your money?” “Hope? Certainly I have hope. I am going to recover that money if it costs every other cent that I have in the world. I don’t propose to sit down like a lamb and be fleeced. Do you think that I am that kind of a man? Do you?” “No,” said the lawyer, “I do not. I am very sorry to hear about your loss; but I don’t suppose there is any use crying over spilt milk.” “Spilt milk! What do you mean by that? How can you talk about a large amount of money as if it were spilt milk? What do you mean anyhow?” “Oh,” said the lawyer, “that was simply a little illustration of mine. You see the moral is a good one.” “Hump! I don’t think it’s good at all, and I don’t like to hear you talk in that way.” Then after a momentary pause, “But what is it you
- 59. want? Why didyou come here?” “I came with some good news,” said the lawyer. “David Harkins called on me this morning and paid off that note of a thousand dollars, and I have brought the money to you.” The crafty face of the banker lighted up with surprise at this announcement. It was so unexpected that he hardly knew what to say in reply. Finally he managed to remark: “Paid you? Paid you this morning, did he? I wonder where he got the money.” “I am sure I do not know,” said the lawyer, “and really I don’t think it makes much difference as long as you get the amount of your note.” The two men sat down at the desk together, and the lawyer, after some preliminary remarks, handed over the money to the banker. The minute it was laid before him he jumped with a start. “Why, this is all new money,” he exclaimed. “That’s just the kind of money that was taken from me last night. I don’t believe Dave Harkins came by that money honestly. It makes him look like a thief. It was probably done by that smart boy of his.” “I wouldn’t say that,” cried the lawyer, trying to pacify the banker. “But I will say it. Both father and son have a grudge against me, and I don’t believe they would hesitate at anything to get even.” “But my dear sir,” remarked the lawyer in a soothing tone, “you have made a very rash assertion, and you have absolutely nothing to base it upon.” John Black was silent for a moment, and then suddenly turning around, he said in a harsh tone: “Did you get that money direct from David Harkins?” “I did,” was the response. “Then,” exclaimed the banker in a tone of triumph, “that proves my suspicion. The money that was taken out of my desk consisted of
- 60. ten $100 bills,and the money you have just given me is made up exactly of ten $100 bills. That satisfies me.” “It is a coincidence,” admitted the lawyer. “Coincidence,” snorted the banker, “it’s sufficient to convict the man. It satisfies me, and it ought to be enough to satisfy any other man with brains.” “I wouldn’t be too hasty,” suggested the lawyer. “There is nothing to be gained by acting in that manner.” “Hasty? Don’t talk about being hasty. I am going to have justice no matter who is injured; and I don’t want to be soft-soaped out of doing the right thing. I am going to act, and I am going to act quickly.” “But, my dear sir,” said the lawyer, persisting in his objections, “you must have proof; don’t you understand that? You must have proof before you can accuse a man.” John Black was in a terrible rage by this time. He paced up and down the office rapidly, and then standing in front of the lawyer and raising his finger in a threatening way, exclaimed: “I’ll have proof all right. The proof will be a warrant for the arrest of David Harkins on the charge of stealing my money.” “I am sorry to hear you talk that way,” said the lawyer, “I think you are making a mistake. But, however, you are master of your own actions. When do you propose to do this?” “Within twenty-four hours,” replied the other solemnly. “If you want to, you can serve a warning on Dave Harkins, and if he will restore my money at once I may be merciful to him; but if not, he must take the consequences. In any event he will have to make up his mind within the next twenty-four hours.”
- 61. CHAPTER VI IN WHICHDAVID HARKINS QUITS THIS LIFE AND TAKES HIS SECRET WITH HIM News travels quickly in a small town. Before breakfast the following morning it was very generally reported that John Black had been robbed, and that he was going to issue a warrant for the arrest of David Harkins. The report shocked most of those who heard it. John Black was a hard man, and more than one of the citizens of Cleverly had felt the force of his iron hand. He worked incessantly, and never spent a penny unless it was absolutely necessary. Such a man may be considered just; but he is bound to be unpopular. David Harkins, on the contrary, was well liked by all who knew him. He was on the best of terms with his neighbors, and always had time for a kind word to everyone he met —man, woman and child. The people therefore were disposed to suspend judgment until they had heard both sides of the story. While David Harkins was at the table Horace Coke drove up, and asked to have a minute’s conversation. As soon as they were alone he said hastily: “Have you heard the rumors?” “I have,” responded Harkins, “and I consider them scandalous. I wonder where such malicious stories could originate?” “That’s easily told,” replied the lawyer. “They come from no less a person than John Black.” “How dare he say such things!” exclaimed Harkins with passion. For answer the lawyer told him the details of his interview with the banker and the singular likeness between the banknotes that had
- 62. been stolen andthe money which had been used to pay off the note. David Harkins listened in astonishment, and when Coke had concluded, said: “But even that doesn’t justify Black in slandering me.” “Certainly not; but you must agree that the coincidence is not only remarkable, but could be construed as suspicious.” “But my part of the transaction was perfectly straight.” “I’m sure of that,” responded Coke with fervor, “and that’s why I’m here this morning. Let me state the case in a nutshell. You have been foolish enough to make an enemy of a powerful and wealthy man. You have borrowed money of him. He demands the payment of the money from you in the belief that you are penniless and cannot comply with his demand. His house is entered and robbed of a thousand dollars. The next morning you pay him a thousand dollars in bills identical to those stolen from him.” “But there are thousands of such bills in circulation.” “True; but the thing for you to do is to shut the mouth of gossip at once. That can be done in a very simple manner. All you have to do is to prove what is known in the law as an alibi. Tell where you got the money and produce the man who gave it to you.” Harkins shook his head sadly at this. “Your suggestion seems simple enough; but I fear I cannot comply with it.” “Why not?” in manifest astonishment. “Because it was given to me in confidence and with the understanding that the name of the donor should not be divulged.” “But it came from a friend?” “One of the best I have in the world.”
- 63. “Well, he wouldsurely not permit you to rest under a shadow for the sake of a foolish promise. Go to him at once and get a release from your pledge to silence.” “I’m afraid it’s too late,” said Harkins gravely. “He was to start for England this very day. However, your advice is good. I’ll hire a team and try to reach him. If I succeed I will report to you this afternoon.” As soon as Mr. Coke departed, Herbert made an effort to tell his father the story of his indiscretion in listening at the doorway on the occasion of the midnight visit of the mysterious stranger. But once again Mr. Harkins was too busy to stop and listen, and father and son parted without that exchange of confidence which would have done so much to clear up an embarrassing situation. Mr. Harkins went to the nearest livery stable and soon had a one-horse buggy harnessed and ready for the road. He told no one his destination, but whipping up the horse, passed down the main streets, out into the outskirts of the town and was soon lost to view. It was late in the afternoon when he returned, and then the wheels of the carriage were covered with mud and the horse was covered with lather as if he had traveled far and fast that day. There was a careworn look about David Harkins’ eyes and a drooping of the lips that betokened disappointment. He drove back over the same streets whence he had taken his departure in the morning, nodding pleasantly to several acquaintances he passed on the way. Just when he was in sight of the livery stable, a sudden gust of wind raised a cloud of dust that blinded animals and pedestrians alike. This was followed by another, and the second squall carried in its wake a batch of old newspapers and sent them eddying about in the air like some strange craft in a whirlpool. One of the papers struck the horse square in the eye. The animal, already frightened by the wind and dust, raised up on its haunches and gave a shrill neigh. Harkins grasping the reins tightly, pulled it down to earth again. But the moment the horse’s feet struck the ground it darted off like a flash and went tearing down the street at an insane gait. The driver kept cool and self contained. Standing on the floor of the carriage
- 64. and leaning overthe dashboard he pulled at the lines with all his strength. Just when he felt that the animal was being brought into subjection, the lines gave a snap and broke, leaving him thrown back on the seat with two useless bits of leather in his hand. He was as helpless as a seaman without a rudder, or more so. The horse released from the grasp of the driver, redoubled its speed and kept on its way like mad. Harkins, now alarmed, considered the advisability of jumping out of the vehicle in order to avert a worse fate. But while he was debating the situation the horse solved it for him. Coming to a cross street it swerved in its furious career and turned the corner. The suddenness of the move swung the buggy from one side of the street to the other, and on its rebound it struck an iron lamp-post, smashing the frail vehicle to pieces and throwing David Harkins head first on to the sidewalk. A crowd collected immediately and several men hurried to the assistance of the stricken man. He was insensible, and his breath came in short, sharp gasps. A stretcher was procured, and he was carried to his home. A physician was telephoned for, and he arrived at the home simultaneously with the men who were carrying the prostrate form. The doctor worked unceasingly for nearly an hour, and at the end of that time announced that his patient must have absolute quiet and that no one must attempt to speak to him for the present. Horace Coke, who had arrived at the house, was very much distressed over the accident and showed especial pain over the doctor’s order. “Doctor,” he said, “couldn’t I ask him one question?” “My dear sir,” answered the physician pityingly, “you can do as you please; but the instant you or anyone else disobeys my orders I will give this case up and will not answer for the consequences.” “Is it that bad?” asked the lawyer.
- 65. “It couldn’t beworse,” replied the doctor; “he only regained consciousness a few minutes ago. I succeeded in putting him into a light slumber. If he rests undisturbed for an hour I may save his life.” Herbert slipped quietly out of the room while the two men were speaking. “He is still sleeping,” he said to the doctor. The doctor shot a sharp glance at the boy. “I hope you didn’t attempt to speak to him,” rather sternly. “Certainly not,” replied Herbert, flushing up at this reflection upon his good sense. Slowly, slowly, the minutes ticked by. A few of the neighbors remained in the parlor. The doctor and Mrs. Harkins alone remained in the sick room. A half hour elapsed. It began to look as if the life might be saved. Presently the door opened and a young girl attired in a dark suit entered the room. Although youthful, she had the air of restfulness usually found only in persons of more mature years. She had great black eyes now full of sympathy with those in the room. Her dark, glossy hair parted in the middle, emphasized the extreme whiteness of her broad forehead. This was Mary Black, daughter of the banker, and sister of Arthur Black. She glanced about the apartment until her glance rested upon Herbert, and going up to him, put her hand in his with such frankness and tenderness as to bring tears to his eyes. He stepped to one side of the room. She was the first to speak. “Herbert, I feel for you very, very much,” she said in a low, melodious voice. “Mother would not rest until I had come over here to inquire how your father was getting on. Indeed we all feel for you and your mother very much. Father was anxious also.” She was quick to see that Herbert’s face clouded up at the mention of her father, and hastened to add:
- 66. “That is whatI wished to speak about particularly. I know that your father and my father had words; but I can assure you that there is no ill feeling on father’s part now. I talked with him long and earnestly, and he finally consented to permit me to come over here and say this to your father. The moment he is able to see anyone, I want to tell him this.” “You are an angel,” murmured Herbert. “I don’t thank your father for this visit, but I am very, very grateful to you.” Just then Mrs. Harkins stepped out of the room, and Mary made haste to repeat to her what she had already told Herbert. The face of the older woman softened at the kind words that were poured into her ears, and in a moment the girl and the mother were in each other’s arms, indulging in one of those crys which do so much to relieve the tension of grief and sorrow. But Mary Black did not waste much time in useless tears. She quickly dried her eyes, and turning to Mrs. Harkins, said with energy: “Now, I’m going to make myself useful; tell me what to do first.” Mrs. Harkins smiled through her tears at this manifestation of industry. But she felt relieved to know that feminine hands and feminine eyes would be in charge of her house while she remained at the bedside of her stricken husband. Mary Black, during that hour of anxiety and for many days afterward, proved herself a genuine angel of mercy. Those who gazed at her knew that while her nature was kind and gentle she was yet resolute and determined. The minutes went by and those who were assembled in the outer room kept anxious watch on the door leading to the sick chamber. All instinctively realized that a crisis was at hand, and that it was to be decided very shortly. Presently there was a movement within and the doctor came out, supporting Mrs. Harkins on his shoulder. A hush went over the little circle. “What is it, doctor?” asked Mr. Coke, voicing the question that hung unspoken on the lips of all the others.
- 67. The doctor lookedat his questioner in silence for a moment, and then said impressively: “He is dead!” A convulsive sob from the newly made widow brought Mary Black and some of the neighbors to her side in an instant. While they were leading the weeping woman up to her room, the doctor noted the questioning look in Mr. Coke’s eyes. “It came very suddenly,” he said; “all was over in an instant. He died without opening his lips.” Herbert, who was standing in the rear of the room unobserved, heard this with blanched face and parched throat. He realized that the death of his father marked an epoch in his life. He felt that he had lost his dearest friend. Yet the tears would not come to his strained, glassy eyes. He was amazed that his heart beat on as before. All that he was conscious of was a strange, unnatural feeling of numbness.
- 68. CHAPTER VII IN WHICHHERBERT MEETS ADVERSITY AND LEARNS THE MEANING OF HARD WORK The Harkins home was a very desolate place for many days after the funeral. Mary Black remained with the family for several days, moving about noiselessly and attending to the multitude of details which would otherwise go neglected at such a sad period. After the first sharp grief had worn away, Herbert and his mother sat down and talked over their prospects for the future. Mr. Harkins had been prudent enough to leave a small insurance policy, made out to the order of Mrs. Harkins, and this money proved to be of immediate assistance to the widow. Mrs. Harkins was a firm believer in the value of education, and felt that it was her duty to give Herbert all the schooling that was possible even if it was necessary to make a personal sacrifice to do so. She insisted upon his going to school for at least a year after the death of his father. He did so and made gratifying progress; but he was now old enough to appreciate the responsibility that rested upon him as an only son, so just before the close of the school term he went to his mother and said: “See here, mother, I’ve got to help you. There is no possible way out of it. If I can do so and continue going to school, all right; if not, I will never return to the school.” “What you say is probably true, my boy,” replied his mother; “but the question is what to do and how to do it.” “Well, suppose we settle it now,” said Herbert resolutely. “Can’t we postpone the thing for a day or so?” asked Mrs. Harkins anxiously.
- 69. “Yes,” responded Herbert,“we could; but there is nothing like doing to-day, instead of postponing until to-morrow.” “In that case,” said his mother, “I think you had better continue going to school until the close of the present term, at least.” “That’s bully!” exclaimed Herbert heartily. “I am going to put my mind on my studies, and I don’t think I’ll be a blockhead when the term is over.” “That’s true,” responded his mother sadly. “But there is another feature of the case that gives me great sorrow.” “What is it?” asked Herbert. “Your college education,” replied his mother. “You know it was your father’s fondest wish, as well as my own, that after leaving the Cleverly School you should take the four year course at St. Joseph’s College. I don’t see how it can be done now.” Herbert hung his head and said nothing. The necessity of abandoning this cherished project was a severer blow to him than he was willing to admit to his mother. He had dreamed of a professional career and often thought that if he were able to go through the College he would be fitted to take the necessary examination for either the legal or the medical profession. But now his dream was over; he was an only son, and his duty to his mother was clear. Mr. and Mrs. Harkins were the parents of three other children; but each of these had died in early infancy; and now the great heap of earth which covered the remains of the lamented father of the house was in close proximity to the three little mounds which were watered and kept green by the tender care and love which only a mother can understand and give. Herbert thought of all these things as he sat silent that day. Presently he lifted his head and spoke to his mother. “Mother, I am old enough to understand my duty. I wanted to go to the College very, very much; but now I know that it is impossible. We must meet adversity, and meet it bravely.”
- 70. Her only answerwas to embrace the boy who was acquiring manliness at such a rapid rate. The school question for the term having been settled, the next question was to consider what steps could be taken to increase their very small income. The subject having been opened, was discussed at various times during the next two weeks. There was a twenty acre farm adjoining the little home of the Harkins. It came up against the little vegetable garden which Mr. Harkins had cultivated with care and profit during his lifetime. The tenant of the large tract had been unfortunate, and he was anxious to sub-let his lease for a very modest sum of money. Herbert consulted with Mr. Coke, the lawyer, regarding the matter, and after some days it was decided to purchase the lease, which had about two years to run. The first step in the new life was the engaging of a farmhand to do the heavy work on the twenty acre tract. A reliable, industrious man was secured for a very reasonable amount of wages; but with the understanding that he would be kept for at least two years. The work was begun under pleasant auspices. After it had proceeded a few weeks, it was decided that Herbert should get as much schooling as he could in the meantime. It must be admitted that he attended school rather irregularly during this period. It was at this time of his life that he learned in a manner never to be forgotten that this is a world of hard work. Often he got out of bed before dawn in order to ride the horse to plough among the growing corn, potatoes and hops. The program was to get as much ploughed by ten o’clock in the morning as could be hoed during the remainder of that day. After this Herbert would start for school, where he sometimes arrived as the afternoon session was half through. In winter his work was lighter, but the snow was often deep and drifted. The cold was intense, the north wind piercing and his clothing so thin that he felt real discomfort. At night, when his work was over and he had a spare hour, he made it a habit to study the art of debating. The first book he ever owned was the “Columbian Orator,” which was given to him by his uncle
- 71. one winter ashe lay very sick with the measles. In the natural order of things Herbert soon became recognized as the head of the house, and his mother leaned on him for advice and accepted his decisions without question. At the end of the first year, when Herbert balanced his carefully kept accounts, he found that they had come out just even. It was a little bit discouraging to find that they had made no profit from their hard work; but it was a real consolation to know that there had been no further drain upon the small amount of money which Mrs. Harkins had laid aside from her husband’s insurance policy. At the beginning of the second year of farming, Herbert learned to his amazement that the man from whom they had purchased the unexpired lease owed money to a number of tradesmen for implements and supplies. These men came to him and demanded the payment of their claims; but he was neither able nor willing to satisfy them. Herbert and his man had finished their summer tilling and their haying when a heavy rain set in near the end of August. The dreary character of the weather seemed to fill him with a foreboding of approaching calamity. One night Mr. Coke came to him with tidings that their ill fortune was about to culminate. The following morning the sheriff and some other officials, with two or three of the principal creditors, appeared and after formally demanding payment of their claims, proceeded to levy on the farm stock, implements, household effects and other worldly possessions, coupled with a threat of arrest and imprisonment for the original tenant who was invisible for some days. Herbert and his mother stopped with a friendly neighbor while the work of levying went on. In the meantime Mr. Coke had not been idle. He denounced the proceedings as an outrage, saying that it was wrong both in law and morals to hold Herbert and his mother responsible for the faults or crimes of another. He did more than protest, however. He acted and acted promptly. He went into court, explained the matter very clearly to the Judge, and succeeded in obtaining an order by which the levy was stopped. Herbert and his mother immediately resumed their old life; but at the end of the
- 72. year both decidedthat it would be advisable to quit farming, which in their circumstances offered little return for the hard labor involved. The hired man, who had proven himself to be an unusually efficient and industrious man, still had two months of his time to run. He generously offered to release Herbert from this obligation; but the boy had inherited his father’s trait of pluck and manliness, declined to accept the offer. He had heard that one of the merchants in the town who had purchased a large amount of ground on the other side of the railroad, was anxious to have someone undertake the job of clearing up fifty acres of the wildest land. Herbert informed his assistant of that fact, and said that if he was willing to undertake the work he would guarantee to give him all that they had contracted to pay in the beginning. It was in November, and when the man and boy started to work the snow was just going and the water and slush in some places were knee deep. Both were resolute, but they were indifferent choppers compared with those who usually grapple with forests, and the job looked so formidable that farmers and others passing along the turnpike were accustomed to halt and predict that Herbert would be a grown man before he saw the end of the job. But his fighting blood was up and he determined to plod along without rest until the work was accomplished. So they continued cutting trees and bushes, chopping up grown trunks into small lengths, digging out rotten pines from the soil where they had imbedded themselves, burning the brush and worthless sticks, and carting home such wood as served for fuel. So they persevered until the job was finally completed. Herbert received $200 for the work; and after paying the hired man the $60 that was his due he had $140 left to put in the family fund. There was still a balance to their credit. Herbert was very glad the work was finished. At times he felt that he would give way under the strain, but pluckily refused to do so. Frequently at night the sharp lances of the Canadian thistles had to be dug out of his festered feet with needles; but he had the stuff in him of which successful men are made. However, two years of this sort of toil were sufficient, and
- 73. at the endof that time he cheerfully marked “the end” at the conclusion of his experience at farming.
- 74. CHAPTER VIII HERBERT BECOMESAMBITIOUS AND IS FASCINATED BY THE SMELL OF PRINTERS’ INK From the time that he was first able to spell and connect one word with another, Herbert was fascinated by the sight of a printed page. If he saw a circular or a fragment of newspaper on the sidewalk he was impelled to pick it up and read its contents. The weekly paper was a rare treat to him and he perused its columns from the first page to the last, until he knew the contents almost by heart. The sight of a book of fiction or adventure or biography was one of the greatest joys on earth to him, and he eagerly devoured everything of that kind that came in his way. Early in his school-days he had written little essays which after being read in secret, many times, were finally consigned to the flames as being unworthy of publicity. The town, among its other places and things of interest, possessed a weekly newspaper known as the Cleverly Banner. Herbert never passed the office of this newspaper without being filled with a wild desire to be on the inside instead of the outside of the building. Frequently he stood looking in the window watching the old- fashioned press as it slowly ground out the regular weekly edition. Once or twice he had occasion to call at the office of the Banner with reference to some printing that was being done there, and on such occasions he was thrown into transports of delight. The smell of the ink, the sound of the presses, and the sight of the freshly printed pages sent him into an ecstacy that was almost heavenly in its pleasure. When he decided to quit farming his eye and heart unconsciously turned towards the little newspaper office. One morning he heard that an apprentice was needed there, he hastened to make application for the position. The building occupied by the Banner set
- 75. back on alittle lot facing the main street of the town. It was a two story and a half dwelling, and an old faded wooden sign over the doorway announced the name of the paper and informed the residents that “Job printing of all kinds could be furnished on short notice.” The building itself was half rotted away from age and want of paint. One editor and one owner after another had succeeded to the Banner; but it had never occurred to any of them that it would be a good stroke of business policy to repair or at least paint the exterior of the building. The first floor of the Banner office was taken up with a little counter where such business as was transacted with the public might be cared for. The remainder of the room was occupied by a very large old-fashioned printing press. It worked very slowly, and as a consequence had to go steadily two or three days a week in order to turn out the edition of the paper. The second floor, which resembled a hay loft more than a place of business, was utilized as the editorial and composing room. An old-fashioned stove in the centre of the room threw out a heat that made the apartment decidedly uncomfortable at times. A big, sleek cat dozing placidly beneath this stove was one of the permanent fixtures of the room. It was quite early in the morning when Herbert called at the Banner office, and he did not find anyone on the first floor. He rapped on the counter to attract attention, and presently a voice from upstairs called out in clear, loud tones: “Come upstairs.” He climbed up the rude stairway slowly, and finally emerged into the editorial and composing room. An elderly man sat in an old- fashioned armchair in front of a little desk with its top sloping very much like the desks used in some schools. He was writing rapidly and pausing every now and then to dip his pen into a big ink-pot which stood by his side. Visitors to the Banner office were well acquainted with that enormous ink-stand. It had been used by the various editors from the time of the foundation of the Banner and went back so far that its origin must finally have been lost in the
- 76. mists of antiquity.When the industrious writer had finished a sentence or a paragraph to his satisfaction he wheeled about in his chair and expectorated a mouthful of tobacco juice into an ample cuspidor which stood on the other side of the desk. He had a shock of snow white hair, very much in disorder, caused no doubt from his habit of running his fingers through his hair when in search of a fugitive thought. He was in his shirt sleeves, which was his usual habit, for he always protested that it was not possible for a man to do his best work harnessed up in a coat and vest. Such was Noah Brooks, the editor of the Cleverly Banner, and one of the characters of the town. He looked up from his work as Herbert entered, and said: “Hello there, young man! What can I do for you?” “I want you to give me a job,” said Herbert simply. This reply seemed to amaze the editor, for he laid down his pen, pushed back his chair, and placing his feet on the desk before him, looked at Herbert with a good-natured smile. It seemed almost a minute before he spoke. When he did it was to say: “So you want a job, do you? Well, that’s a laudable ambition; but I am afraid you have come to the wrong place.” “I am sorry to hear that,” said Herbert. Noah Brooks looked at Herbert again before replying, and then moving slightly and raising the index finger of his right hand, he pointed to the rear of the room and said: “Do you see those fellows over there?” Herbert looked around and saw a man engaged in setting type, while a boy with a great big ink roller in one hand was engaged in taking a proof of a circular that was about to be printed. “Yes sir,” he answered obediently; “I see them.” “Well, do you know,” said the old gentleman with a chuckle, “that about all those two fellows do is to sit around and wait for Saturday
- 77. night in orderto draw their salaries.” The humor of this seemed to appeal to the speaker so strongly that he had to pause and engage in a hearty laugh before proceeding. The man and the boy did not appear to be offended. On the contrary, they laughed too, as if they were accustomed to the good- natured jests of their employer. “I am very sorry,” said Herbert, breaking the silence; “but I was really anxious to get employment on this paper—I have long wished to enter the newspaper business.” “Is that so? Do you know anything about the business?” “No,” said Herbert; “I am entirely ignorant of it; but I felt that I could learn.” “That’s the way to talk,” was the hearty reply. “The only way to learn to do a thing is to do it. I think you would pan out all right in an office of this kind; but I am sorry to say we have no opening at the present time.” Herbert said “Good-by” quietly; but once out of the building he felt very much depressed at his failure to secure a situation. He did not tell his mother of his adventure, not wishing to annoy her with anything that was not of a cheerful nature. During the next few months he managed to earn a small amount of money by odd bits of employment that were furnished to him through Horace Coke, the lawyer; but as he had no taste for the law he did not feel very much encouraged over this occasional work. His mind still dwelt upon the newspaper business. One evening he wrote a little item describing an entertainment given at the Cleverly High School, and mailed it to the office of the Banner, without indicating the name or address of the writer. After he had sent this little message on its way, he was figuratively speaking, on pins and needles until the next issue of the Cleverly Banner should appear. On the date of its regular issue, he hurried home in order to get the paper as soon as possible. He was disappointed. It had not arrived. Unable to wait, he rushed to the post office, and securing
- 78. the paper, heeagerly tore off the wrapping and opened the page which contained the local news. What he found there caused his face to flush scarlet. The little item that he had written with such care was reproduced, word for word, as he had penned it, without a change of any kind. He felt so glad that he could have shouted for joy. Several other persons were in the post office, and he looked around at them as if to see whether they had read his secret; but apparently no one was paying any attention to him. He walked home in a fever of happiness, and it was only by the strongest effort on his part that he refrained from telling his mother about the incident. Naturally he continued to send little items to the paper from week to week. Sometimes they failed to appear. On such occasions he felt a sense of loss and disappointment that was far out of proportion to the importance of the subject. But when the paragraphs did appear that feeling of elation and joy returned to him on each occasion. Finally he determined to call at the office of the Banner once more. It was just possible that there might be an opening, and he made up his mind not to miss it merely for the sake of asking. The venerable editor with the snow white hair was in his place as usual. He recognized Herbert immediately, and cried out: “Hello young man! I see you are here again.” “Yes sir,” replied Herbert. “I do not want to be a bore, but I felt that it would be all right to inquire whether an opportunity had arisen by which I could secure employment on the Banner.” Once again the old man looked at him in that quizzical manner. “Perseverance wins, boy,” he said, “and you have won. I do need somebody. My apprentice has left me very suddenly, and I think I can make use of you. He only got four dollars a week. I know that will be pretty small for you; but I can afford to give you six dollars, and if you are willing to take it the job is yours.” Herbert could not conceal the pleasure that he felt.
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