Mobile_Lec6

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1
Lecture 6Digital Modulation I
Mobile Communication Systems
Dr Charan Litchfield C.litchfield@yahoo.co.uk
5th November 2009
Resources:
Mobile Communication Systems Nov09
2Digital Modulation I
Reading:
Chapter 5 + 6 of Goldsmith: “Wireless Communications”.
Proakis: “Digital Communications”
References:
“The choice of a digital modulation scheme in a mobile radio system”, Mundra, 1993.
“An overview of modulation and coding for wireless communications”, Ziemer, IEEE
1993.
“Further results in the unified analysis of digital communication systems”, M. Fitz,
IEEE Trans. on Communications, March 1992.
Game Plan:
3
6.1 Sampling and Bandwidth.
6.2 Modulation Theory.
Digital Modulation I
Key Outcomes:
4
1. Understand principles of sampling and
digital modulation.
2. Understand BPSK and QPSK modulation
principles.
3. Understand Bit Error Probability.
4. Understand Eye diagram and Spectrum
Analyzer and how we use them in
modulation analysis.
Introduction
Generic Wireless Communication Modulation System.
( )tkψ
Generic View.
-Source Coding: Trimming out redundancy maximizing entropy.
-Channel Coding: Systematically placing redundancy into the information
symbols enabling detector to separate out different message sets with small
probability of error.
-Signal Mapping: Defining a signal basis where coded message sets are mapped
to unique signals. Signals should have good decorrelation properties thus
minimizing probability of decision error at receiver. Example: 4 – bit uncoded
message mapped to 16 signals. If two bits redundancy, 6 – bit to 64 signals.
Introduction
Typical Wireless Communication Modulation System.
• Typical wireless communication system utilizes scalar modulation (i.e. BPSK,
MQAM ). Modulation typically maps sequence of bits to a symbol. Symbol could
be phase, amplitude, or both. Symbol is discrete.
• Channel coding typically on {1,0} binary digits. Difficult to find codes for given
modulation scheme. Joint approaches – Trellis Coded Modulation.
• Pulse Shaping: Discrete symbols passed through shaping filter for transmission.

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Introduction
•Analogue Waveform Vs Digital.
For SNR = 10dB, Bit error probability ~ 10-6.
If we regenerated the analogue signal from
digital one, the effective gain from a digital
modulation like BPSK is about 50 – 60dB SNR
(with bit error probability locked at 1 bit
error per million bits average). Proof:
Compare digital TV picture to Analogue.
Continuous noise limits information content –
hence only few bits information for analogue.
Introduction
Encode
Transmit
Pulse
modulateSample Quantize
Demodulate/
Detect
Channel
Receive
Low-pass
filter
Decode
Pulse
waveformsBit stream
Format
Format
Digital info.
Textual
info.
Analog
info.
Textual
info.
Analog
info.
Digital info.
source
sink
Introduction
Filter /
Filter Bank
Encoder
Message
RF Carrier
i.e. 101
Pulse Sequence
Introduction
m1
m2
m3
mM
Digital Communication System
channel
Probability
Computer
Select
Most
Likely
Represent source as a finite state
machine, and receiver is an estimator (to
determine the state / which message
sent).
Introduction
Likelihood
or MAP
Calculator
Decoder
Symbols
Message bit or
Sequence
Input from
MF
Output bit
or
Sequence
Example of Demodulation Process.
6.1 Sampling and Bandwidth
Key Outcomes:
1. Concept of Signal and Symbol power and energy.
2. Concept of Fourier Transform and PSD.
3. Concept of Sampling Theorem.
4. Concept of Bandwidth.
5. Concept of Matched Filtering.
6. Concept of Pulse shaping filters and why they are
used.

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6.1.1 Energy and Power
13
We send finite energy signals in BPSK – i.e. Pulses whose
energy is finite and Power = 0. Such pulses are: Square,
RRC, SINC etc.
Example:
0
( )














±∈
sT
t
recttx
t
2
Ts
2
Ts
−
sT
A
∫−∞→
∂=
2/
2/
2
)(
1 T
TT
ttx
T
LimP ∫
∞
∞−
∂= ttxE
2
)(
Energy of Signal:Power of Signal:
•For Single Pulse (not continuous
signal):
Sqrt(Ts) ensures signal energy is
unitary for signal with A = 1.
•For random Pulse train with x(t) as
signal:
2
2
At
T
A
E
s
=∂= ∫
∞
∞−
s
T
T
s
T T
A
t
T
A
T
LimP
2
2/
2/
2
1
=∂= ∫−∞→
14
•Power Spectral Density:
Concept of Autocorrelation Function:
Usually when we have a
nondeterministic signal, w(t), we can
generate a statistical measure called
its autocorrelation function which is
the average correlation with itself for
different time offsets. Since such
signals represent the mapping from a
probability space to a real or complex
time series, the expectation (average)
is used to define an average energy
measure as a function of time offset,
τ. This Autocorrelation function is
given as:
( ) ( ) ( )∫−
∞→
∂−=
T/2
T/2
T
tτtwtw
T
1
LimτA
15
The Wiener Khinchin Theorem is given as:
( ) ( )∫
∞
∞−
∂⋅= ωeωW
2π
1
τA τjω
IFT
i.e. The PSD is the Fourier Transform of
the ACF.
In other words, the Power spectral Density is:
( ) ( )∫
∞
∞−
∂⋅= τeτAωW τjω-
Units:
(dB/Hz) or
(WHz-1) = (J)
0
( )ωW
ω
16
The PSD gives the distribution of signal energy in terms of
its basis components – i.e. The energy spectrum. The area of
PSD function is a measure of power.
( )∫
∞
∞−
∂= ωωWP
0
( )ωW
ω
Example: SINC Pulses has
Rectangular PSD. Assume Pulse
has Amplitude = A and Symbol
Duration = T . SINC Pulse shape
normalized such that unit energy
(scaled by amplitude).
( )
T
A
2
B
EωωWP
2
w
0 ==∂= ∫
∞
∞−
π
2
Bw
0E
2
Bw
−
2
0 AE =
17
Example: Random Rectangular Pulse Train.
Has infinite Energy, but finite power.
Amplitude = B. (Normalize so energy depend on B).
Autocorrelation function is a Triangular function.
The Fourier Transform of the autocorrelation
Function is the PSD:
( ) ( ) ( )∫−
∞→
∂−=
T/2
T/2
T
tτtxtx
T
1
LimτA
0
( )














±∈
sT
t
rectAtx
t
2
Ts
2
Ts
−
-1.5 -1 -0.5 0 0.5 1 1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time
Autocorrelation(Normalized)
s
2
T
A
π
sTτ =sTτ −=
sT
A
( )τA
2
s
s
22
)(fT
)fT(sin
π
A
P(f)=
Power over whole bandwidth:
( )
s
2
T
A
ffPP =∂= ∫
∞
∞− Area = π/Ts
6.1.2 Sampling Theorem
18
•Sampling:
Simple concept. Process of turning a continuous time signal
into a discrete time signal. Goal of this section is to find
the minimum sampling rate where we can recreate the
signal with no loss of fidelity.
In reality, we will always have some loss due to quantization
of a continuous value (where each sample is turned into
binary digits). If a sample has infinite number of bits
(realistic), we will quantize to a certain number of bits, i.e.
for audio 8 bits or 16 bits may be represented per sample
leading to quantization noise. The maximum quantization
error (assuming constant quantization steps) is 2-(m+1) with m
the number of bits. Usually it is a modelled as a uniform RV
with limits -2-(m+1) ≤ Q ≤ 2-(m+1).

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19
•Sampling:
Most basic sampler is a Dirac impulse function. Dirac function is viewed
as the limit where a pulse centred about zero (i.e. Rectangular or
Gaussian), of unit area, becomes infinite height and zero width. The
Dirac impulse has unique property that the integral of a continuous
function multiplied by the Dirac function returns the instantaneous value
of that function at a specified point.
t
f(t) Getting
towards impulse
by keeping pulse
area = 1 but
making narrower
and higher.
1)()(
)()(
==
=
∫
∞
∞−
−
dtetjX
ttx
tjω
δω
δ
ω
X(jω)
20
Impulse function is interesting: It only exists at t=0, but
has infinite height. Its area is defined as “1”.
1)()()(
2/
2/
=== ∫∫ −
−
∞
∞−
−
dt
dt
tjtj
dtetdtetjX ωω
δδω
Over such small limits, the area under impulse is 1. An
impulse is a sampling operator – i.e. returning the value of
a function at t = 0 with
2/,0
2/2/,
)(
tt
ttt
t
∂>
∂<<∂−∞
=δ
∫
∞
∞−
∂= tδ(t)f(t)f(0)
21
•Sampling:
The concept works equally well for a shifted Dirac function, hence the
sampling theorem centre’s on modelling the sampling function as a train of
Dirac impulse functions, where time distance between each shifted Dirac
impulse has an important meaning.
Tt
Tt
Tt
>
=∞
=−
,0
,
)(δ
∫
∞
∞−
∂−= tT)f(t)δ(tf(T)
Example: Train of Dirac Functions sampling a continuous function.
∑
∞
−∞=
∂−⇒∆
i
i t)f(t)Tδ(tt)f(
t
f(t)
22
•Sampling:
The basic principle is: Sample the signal at regular intervals, Ts where Ts
is the sampling period. Based on the sampling period, Ts, how well can we
reconstruct the signal from the samples using a zero order hold and
filtering. Sampling frequency fs = 1/Ts.
B
sT
ω
π
≤
( )tX ( )t∆X
( )tXˆ
Sampling: Consider that the signal X(t) is sampled where:
( ) ( ) ( ) ( ) ( )tWtXkTtTtXtX
k
ss∆ =





−∂= ∑
∞
−∞=
( ) ( )∑
∞
−∞=
−∂=
k
ss kTtTtW
23
The Fourier Transform of the Sampled sequence
is:
BωBω−
( )ω∆X
Bω2
( ) ( ) ( )ωωω jW*jXjX ∆ =
( ) ( ) ( )tWtXtX ∆ ×=
( ) ∑
∞
−∞=






−∂=
k
2
jW k
Ts
π
ωω
Hence:
( ) ∑
∞
−∞=












−=
k
∆
2
jXjX k
Ts
π
ωω
with
24
The continuous time variable can be reconstructed from
the samples if and only if:
( )ωjX
( )



>
≤
=
Aω0,
Aω1,
ωWA
with the frequency for which is non – zero
(usually use Anti Aliasing filter).
Proof:
B
sT
ω
π
≤
Bω
Define a spectral window,
Subject to: ( )
s
B
T
π
ωω0jωX =>= for
B
s
f
T
2
1
≤

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25
Hence:
( ) ( ) ( )








⋅= −
jωWjωXFTtX
sT
π∆
1ˆ
( ) ( ) ( )
( ) ( )


























=








= −
s
ss
∆
T
π
1
∆
T
πt
T
πt
sin
2
T
*tXtX
jωWFT*tXtX
s
ˆ
ˆ
26
Hence:
( )
( )
( )
∑
∞
−∞=
−






−
=
k
s
s
s
s
k
kTt
T
π
kTt
T
π
sin
XtXˆ
kXwhere are the coefficients of the sampled data
record.
Called Shannon Interpolation. Technique requires
infinite memory in the samples and infinite delay.
Usually only practical for very long data records.
Often can approximate applying zero order hold and
FIR low pass filter with sharp cut off frequency.
27
Key Learning:
1) When sampling a signal, sampling frequency must be at
least twice (or more) the maximum frequency
component of the signal. This is to ensure aliasing is
not an issue – i.e. where it is impossible to recreate
the original continuous signal from the samples. In
practice, we usually bandlimit a signal with anti-
aliasing filter to ensure the maximum frequency
components of signal not infinite (as it would usually
be so in practice).
2) Formulae to remember:
B
s
f
T
2
1
≤ or B
s
s f
T
f 2
1
≥=
6.1.3 Bandwidth
28
Baseband versus bandpass:
Bandwidth dilemma:
Bandlimited signals are not realizable! Anticausal and
of infinite duration.
Realizable signals have infinite bandwidth!
Baseband
signal
Bandpass
signal
Local oscillator
6.1.3 Bandwidth
29
Bandwidth: Usually defined by -3dB cut off in frequency response of
system or filter / signal. Usually consider “channel bandwidth” in
analogue communication or Nyquist Channel Bandwidth in Digital.
SSBW = Single Sided Bandwidth, DSBW = Double Sided Bandwidth
(which is also the Noise bandwidth at the receiver filter).
.B2ωBW =
.B2ω=
.Bω=
6.1.3 Bandwidth
30
Rectangular Pulses:
The Fourier transform is
In time domain, no overlap between p(t) and adjacent
pulses p(t - Ts) and p(t + Ts)
In frequency domain, sinc has infinite two-sided extent;
hence, the spectrum is not bandlimited




≤<−
=





=
otherwise0
2
1
2
1
if1
rect)( ss
s
TtT
T
t
tp
( ) ( )
x
x
x
Tf
Tf
TTfTfP
s
s
sss
sin
)(sincwhere
)sin(
sinc)( ===
π
π
π
t
1
p(t)
-½ Ts ½ Ts

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6.1.3 Bandwidth
31
Rectangular Pulses:
-4 -3 -2 -1 0 1 2 3 4
-50
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Normalized Frequency (Hz)
Amplitude(dB)
-1.5 -1 -0.5 0 0.5 1 1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
time
Autocorrelation(Normalized)
•Power between -1/Ts and 1/Ts ~ 95% of total.
•Power at 3dB cut off ~ 90% (roughly region between -1/2Ts and 1/2Ts).
•Using the SINC interpretation of Double Sided Bandwidth B = 1/Ts gives
only approximation of Signal Power. If 95% Bandwidth definition, Double
Sided Bandwidth of rectangular pulse is about double the SINC Pulse!
ACF
PSD
6.1.3 Bandwidth
32
Sinc Pulses:
sT2
1
sT2
1−
sT
)( fH
f t
)/sinc()( sTtth =
1
0 sT sT2
sT−
sT2−0
sT
W
2
1
=
Ideal Nyquist filter Ideal Nyquist pulse
s
W
T
WB
1
2 ==
Ts = Symbol Duration
Bw = Double Sided Bandwidth
6.1.3 Bandwidth
33
Sinc Pulses: Bandwidth Efficiency
sT2
1
sT2
1−
sT
)( fH
f0
sT
W
2
1
=
Nyquist bandwidth constraint (on
equivalent system):
The theoretical minimum required
system bandwidth to detect Rs
[symbols/s] without ISI is:
Proof simple:
For sequence of digital SINC pulses,
the minimum time separation for no
ISI between two pulses must be T =
Ts. Could space pulses further apart,
i.e. T > Ts resulting in drop of symbol
rate Rs = 1/T. Hence:
[Hz]W
2
R
BR s
Ws ≤⇒≤
2WRBR
T
1
R
T
1
T
1
sWs
s
s
s
≤⇒≤⇒≤⇒≤
s
W
T
B
1
=
6.1.3 Bandwidth
34
Sinc Pulses:
Nyquist bandwidth constraint (on equivalent system):
Equivalently, a system with bandwidth W=1/2T=Rs/2 [Hz] can
support a maximum transmission rate of 2W=1/T=Rs [symbols/s]
without ISI. Hence “symbol rate” is appropriate Nyquist Bandwidth
measure. This is only true for SINC pulses (strictly bandlimited).
If not, bandwidth efficiency will be less.
Bandwidth efficiency, R/W [bits/s/Hz] :
An important measure in DCs representing data throughput per
hertz of bandwidth.
Showing how efficiently the bandwidth resources are used by
signaling techniques.
Hz][symbol/s/2
22
1
≤⇒≤=
W
R
W
R
T
ss
I.e. 1 symbol or less
information can be
transmitted per
double sideband,
2W.
6.1.3 Bandwidth
35
Sinc Pulses:
Discrete symbol x[m] is the mth sample of the transmitted signal;
there are W samples per second.
Continuous time signal x(t), 1 sec ≡ W discrete symbols
Each discrete symbol is a complex number;
It represents one (complex) dimension or degree of freedom.
Bandlimited x(t) has W degrees of freedom per second.
Signal space of complex continuous time signals of duration T
which have most of their energy within the frequency band
[−W/2,W/2] has dimension approximately WT.
Continuous time signal with bandwidth W can be represented by
W complex dimensions per second.
Degrees of freedom of the channel to be the dimension of the
received signal space of y[m].
We will coin the term Degree of Freedom (DOF) regularly in this course. DOF
indicates how many independent dimensions exist in a system or observation.
In a statistical sense, it implies the number of independent variables or
choices. In a system sense, it indicates the number independent outputs – i.e.
the number of non-zero eigenvalues in the system. In communication systems,
it often means the number of independent channels or independent
symbols/sec.
6.1.3 Bandwidth
36
Sinc Pulses:
In time domain, infinite overlap between other pulses
Fourier transform has extent f ∈ [-W, W], where P(f) is
ideal lowpass frequency response with bandwidth W . In
frequency domain, sinc pulse is strictly bandlimited.
Very significant practical problems – infinite impulse response of
SINC pulse (anticausal nature is combatable but only with infinite
time delay!). Solution: Truncate sinc pulse by multiplying it by
rectangular pulse. Causes smearing in frequency domain
(multiplication in time domain is convolution in frequency domain).
Solution – find alternative pulses that have similar bandlimiting
properties but have better robustness to sampling jitter (causing
ISI) and spectral characteristics so time limiting not such a
problem.
t
T
t
T
t
T
tp
s
s
s
π
π
π 





=





=
sin
sinc)( 





=
ss T
f
T
fP rect
1
)(
sT
W
2
1
=

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6.1.4 Raised Cosine Pulses
37
Pulse shaping used in communication systems
W is bandwidth of an
ideal lowpass response
α ∈ [0, 1] rolloff factor
Zero crossings at
t = ± Ts , ± 2 Ts , …
( )
222
161
2cos
sinc)(
tW
tW
T
t
tp
s α
απ
−





=
ideal lowpass filter
impulse response
Attenuation by 1/t2 for
large t to reduce tail
38
Pulse shaping used in communication systems
Bandwidth:
(1 + α) W = 2 W – f1
f1 transition begins
from ideal lowpass
response to zero
( )









−<≤













−
−
−
<≤
=
otherwise0
2||if
22
||
sin1
4W
1
||0if
2W
1
)( 11
1
1
fWff
fW
Wf
ff
fP
π
sT
W
2
1
=
W
f1
1−=α
39
2
)1(Baseband sSB
sR
rW +=
|)(||)(| fHfH RC=
0=r
5.0=r
1=r
1=r
5.0=r
0=r
)()( thth RC=
T2
1
T4
3
T
1
T4
3−
T2
1−
T
1−
1
0.5
0
1
0.5
0 T T2 T3T−T2−T3−
sRrW )1(Passband DSB +=
Practical Aspects
•RRC Filters are practically implemented digitally (i.e. DSP processor or FPGA).
Means we usually window the transfer function of the filter (remember it is infinite)
and hence define a finite number of taps (usually an odd number – i.e. 127). A
rectangular window quite practical and close to optimum.
•At transmitter, we usually modulate symbols by delta modulator (a single delta
function d[n-k] scaled by symbol amplitude at desired sampling instant and rest
zero padded). To get desired pulse shaping sequence, next symbol sample should be
separated by time distance of zero crossings (T). Hence number of zero’s padded
between symbol spikes = number of samples of RRC filter in this range.
•At receiver, usually apply oversampling but often not as simple to form RRC filter
order matched to sampling rate. In this case, samples input to the receiver may
require regression for sample rate matching purposes (i.e linear regression or SINC
interpolation common techniques). Downside = correlated noise formed.
Symbol Sequence
To DAC
Sampled
Filter IR
Example:
t
•Each RRC pulse overlaps by
one symbol interval where at
the sampling instant (at
peaks of each pulse), there is
no ISI. Sum of all delayed
pulses = convolution – this
example is just showing the
overlap of each pulse by
symbol delay, T. Reality to
the eye = add the whole lot
together!
•Less ISI than SINC pulses
if sampling jitter present.
T
Sequence: 1,1,-1,-1,1,-1,1
Example:
Take just single
sample, and pad
with 4 zeros.
Repeat and get
desired Pulse
sequence.
Filter delay

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6.1.5 Matched Filter
43
Key Concepts: Noise
bandwidth very high,
where noise power is
related to bandwidth. To
ensure noise power not too
high, must bandlimit the
noise before detecting the
symbol received. A very
special filter can be
designed to maximize the
received SNR, and this
filter is identical to the
transmitter filter shaping
the symbols. This filter
introduces Processing Gain
in the system.
44
Key Concept: If the transmitter pulse shaping filter has
impulse response f(t), the receiver matched filter is a
filter with impulse response f(t). For convenience we
write the matched filter impulse response f(T-t) due to
time direction and causality. Proof:
Ah(t)
v(t)
r(t)=Ah(t)+v(t) y(t)= ho(t)+n(t)
Filter,
f(T-t)
Sample at t = T
45
Schwartz Inequality: Two vectors in inner product space
satisfy:
Equality if:
Optimum detector in white noise is the matched filter.
Given an input signal waveform h(t), the matched filter is
F(f)=H*(f), (i.e. f(t)=h(-t)) that maximizes the SNR.
Maximized if:
Why?
Syntax Absolute: Syntax Norm:
= length
∑ •=•
i
2
( ) ( ) 22
bajbajba +=−×+=•
46
Let the finite dimension vector model be
where is the sampled
received vector with hi the channel envelope, d the data
symbol and vi the sampled noise. The output SNR in inner
product form is given as:
The Cauchy – Schwartz inequality, , equates to
unity if both h and f are linearly dependent. If this
condition is met,
( ) vfhfrf HHH
l dy +==
[ ]LL2211l vdh,,vdh,vdhr +++= L L
C∈
( ) 1hffh ≤⋅
− H1
2
2
v
2
d
max σ
σ
N
S
h=





2
H
2
2
v
2
d
2
H
2
v
2
d
h
σ
σ
σ
σ
N
S








⋅
=








=
fh
hf
f
hf
∝provided f h.
47
Key Concept: Processing Gain of Matched Filter.
Indicates gain in SNR by using matched filter where
processing gain is the ratio of input noise bandwidth and
output noise bandwidth (which is always greater than one):
But Noise bandwidth is infinite? Not true, RF
components in link bandlimited – after mixing, we usually
apply bandpass filter at Front end of receiver and use
low pass filter (anti aliasing filter) to rid unwanted
mixing components. Even without filtering, noise is not
with infinite bandwidth – think of blackbody radiation
from cavity! If noise bandwidth was same as signal
bandwidth would we achieve a gain in SNR?
out
N
in
N
B
B
G =
48
Link between SNR and Processing Gain: Simplest proof if
we assume SINC pulses (Bandwidth definition clear).
Define the SINC pulse as h(t), where A•h(t) is the
transmitted symbol, A = symbol. Double Sided Signal
Bandwidth as 1/T = BN
out. Symbol Energy symbol^2.
( ) 0
2
0
2
2
)(2
2/
)(
GN
tthE
BN
tth
T
A
s
in
N
in
SNR



 ∂
=
∂
=
∫∫
∞
∞−
∞
∞−
γ
0
2
22
2
2
2
2
2
)(2
)(
)(
)()(
1
)(
N
tthE
tth
tthE
tthtn
T
tth
T
E
s
n
s
s
out
SNR



 ∂
=
∂



 ∂
=
∂∗



 ∂
=
∫
∫
∫
∫
∫
∞
∞−
∞
∞−
∞
∞−
∞
∞−
∞
∞−
σ
γ
in
SNR
out
SNR Gγγ =∴
Matched Filter SNR:

3/23/2010
9
6.1.6 Signal to Noise Ratio
49
Previous slide made assumption – noise IID, and time sample at peak
of convolution (where both filter waveforms overlap) hence allowing
convolution notation to be removed. Previous slide linked in Noise PSD
and noise variance. Comes from Gaussian statistics – where n(t) is the
random independently distributed (IID) Gaussian noise waveform.
[ ] { } 0,)()()()( 2
=∂=±= ττστtntnEtnA
ACF gives us Energy (Assume Noise Voltage).
∫∫ −
∞→
∞
∞−
∂−=∂
2/
2/
)()(
1
)(
2
1
T
T
T
j
ttntn
T
LimeN τωω
π
ωτ Power.
⇔=
2
)( 0N
N ω
2
02 N
=σ
[ ]{ } 2
)()( σω == NtnAFT
Statistical ACF
From Lecture 2, Noise energy was shown to be N0 /2, with N0 = kT.
50
η+=ψ iAˆ
iA
η
iA
η
ψˆ
In digital communication, always consider SNR after
Matched Filter. Bw = single sided bandwidth. 2Bw =
double sided.
Filter,
g(T- t)
Sample at t = T
Filter,
g(t)
iA
)t(w
)t(r )t(ˆψ ψˆ
W
2/T
2/T
2
i
)i(
I
P
t)t(S
T
1
∫−
∂
=γ
)t(gA)t(S ii =
W0W BNP =
51
)t(w)t(gA)t(r i +=
)t()t(fA)t(ˆ i η+=ψ
Sample at t = T
Components of Model:
Equivalent Model:
)t(fAi
)t(η
)t(ˆψ
ψˆ
)t(g)t(g)t(f ∗=with
Noise w(t) transformed by MF.
N
2
i)i(
0
P
)T(fA
=γ
Convolution
Def: )T(f
2
N
f)f(G
2
N
P 020
N =∂= ∫
∞
∞−
)t(w)t(g)t( ∗=η
52
1)T(f = Unit energy assumption.Def:
0
2
i
N
2
i)i(
0
N
)T(fA2
P
)T(fA
==γ
Def: 2
i
)i(
s AE = Hence SNR for symbol Ai is:
0
)i(
s)i(
0
N
E2
=γ
The average SNR is:
{ }
0
i
)i(
si
0
)i(
s
0
N
E)ss(p2
N
EE2 ∑ ⋅=
==γ
{ }=•E Expectation Operator
Our analysis seems ambiguous – but these are important tricks to
learn.
6.1.7 Summary
53
Information (data- or bit-) rate:
Symbol rate :
Sampling at rate
(sampling time=Ts)
Quantizing each sampled
value to one of the
L levels in quantizer.
Encoding each q. value to
bits
(Data bit duration Tb=Ts/l)
Encode
Pulse
modulateSample Quantize
Pulse waveforms
(baseband signals)
Bit stream
(Data bits)
Format
Digital info.
Textual
info.
Analog
info.
source
Mapping every data bits to a
symbol out of M symbols and transmitting
a baseband waveform with duration T
ss Tf /1= Ll 2log=
Mm 2log=
[bits/sec]/1 bb TR =
ec][symbols/s/1 TR = mRRb =
6.2 Modulation Theory
Key Outcomes:
1. Concept of FSK, PSK, and QAM Modulation.

3/23/2010
10
6.2.2 General Theory
• Digital modulation, as we shall see, is primarily concerned
with the mapping of bits to signals. This mapping can happen
in a very direct way, i.e. a “1” bit maps to positive pulse, a “0”
bit mapping to a negative going pulse. This mapping can also
happen where we group bits to form a message, and thus map
a message to a unique symbol and unique waveform.
Ψ1
Ψ2
Ψ3
ΨM
Symbols or waveforms
Bits or message
• What is a Symbol? What is a Waveform?
Message = simple concept. We can send 1 bit per channel use (i.e.
transmitting 1 bit at a time or 1 bit per unit of bandwidth), or “M” bits
per channel use (i.e. transmitting > 1 bits at a time). A message is simply
a sequence of bits (or even 1 bit, i.e. 1 bit message) we transmit in any
defined time interval. This time interval, Ts, is related to the rate we
transmit messages by:
This time interval is not necessarily the width of a pulse – rather it is the
interval between sending pulses (The width of a pulse has a big role in
bandwidth, but we could transmit lower bandwidth / wider in time pulses and
overlap them – some continuous phase modulations do this). The receiver
must be synchronized to this rate. We also call this “symbol” rate
or “signal rate”:
where a message can be represented as a symbol.
sec)/(
1
messages
T
R
s
m =
sec)/(sec)/(
1
baudorSymbols
T
R
s
s =
• Waveform?
1 waveform is transmitted per channel use (1 waveform every Ts
seconds). Each waveform is a “representation of digital
information” – our receiver will decode each waveform by
comparing it with others (using distance measures) and inferring
the sequence of bits received once a decision is made on the
transmitted waveform.
Ts
2Ts 3Ts
00 0110 11
In this example, 2 bit message map to unique signal.
• Waveform?
It takes a certain amount of bandwidth to send one signal. This leads us
towards the concept of signal dimensions. Once we have dimensions, we
can then define what a symbol means. A symbol is different to a
waveform in principle, although the symbol rate and number of
waveforms/sec is identical. Lets start by considering the total signal
bandwidth (i.e. power spectral density of all possible combinations of
individual waveforms than can be transmitted):
Total 95% Double Sided Bandwidth = BW
Number of Waveforms/Sec = TS
Hence BWTS indicates the number of
signal basis components (dimensions).
sWTBN ≅ is the number of basis functions (dimensions) of modulation.
It is only approximate, since the signal bandwidth not normally strictly
bandlimited.
• Waveform?
The real number of basis functions of a signal are known beforehand. In
our previous example, had 4 orthogonal frequencies (basis functions),
thus number of dimensions = 4.
•Symbol?
Is a real/complex vector (or scalar) representing the weighted
contribution of all the basis components to a particular waveform – this
waveform is built from all the basis signals defined for the modulation.
Think about this: Lets say we have a Sine wave, of certain frequency, f and
amplitude, A. We could in fact build a signal with this wave alone – i.e. we can
trigger information into its amplitude – e.g. a ‘1’ we send positive amplitude A, a
‘0’ we send negative amplitude -A (this is also an example of “phase modulation”
since the minus sign alters the phase by π radians). Hence would be two symbols
(+A, -A). Lets now add a Cosine wave into the mix. We could now assign two bits
to a symbol, where each symbol represents a weighted sum of Cosine and Sine
waves – i.e. could send Acos(ωt) + Asin(ωt), Acos(ωt) - Asin(ωt),
-Acos(ωt) + Asin(ωt), -Acos(ωt) - Asin(ωt). Our symbols would be a vector now:
[A,A], [A,-A], [-A,A], [-A,-A] OR (A+jA, A-jA, -A+jA, -A-jA).
• Waveform and Symbol? Lets look at an example of BPSK
modulation. Basis function is the cosine normalized to unit energy. The
waveform has two domains of representation – at baseband, and at RF.
bits Select
Pulse
RF Up
Convert
BPSK: Information keyed as 180o phase shift – this is the RF
version. At Baseband, BPSK signal is a pulse (either positive or
negative). Symbol represented as (+B, -B). The transmitter power
amplifier will scale signal power (and thus energy of the symbols).
P AMP
( )t
T
c
s
ωcos
2
0
AA−
Meaning: For every channel use, transmit Acos(ωt) or -Acos(ωt). The
symbols are the constellation point on the “X” axis, where the amplitude of
the symbol is proportional to the square root of symbol/bit energy.

3/23/2010
11
Preferred viewpoint is the baseband energy received rather than RF
transmitted. Assuming our receiver includes Automatic Gain Control,
after carrier down conversion, the signal will be the baseband pulses
we originally transmitted (and any noise picked up at the receiver).
( )tp
0
bEA =bEA −=−
( ) ( ) ( )tptt
T
tp s
s
T
T
c
s
=∂∫−
2/
2/
2
cos
2
ω
The energy per bit is: ( ) 2
2/
2/
2
AttpE
s
s
T
T
b =∂= ∫−
so pulse area is unitary (with amplitude, A) then:
2
AEb =
From carrier down conversion:
Pulse
• Waveform and Symbol?
Two examples of BPSK waveforms – one with square pulses, one with Root
Raised Cosine Pulses.
0000 1111 2222 3333 4444 5555
-2-2-2-2
-1-1-1-1
0000
1111
2222
time (s)
p(t)
Baseband Signal
0000 1111 2222 3333 4444 5555
-2-2-2-2
-1-1-1-1
0000
1111
2222
RF Waveform
time (s)
S(t)
0000 0.50.50.50.5 1111 1.51.51.51.5 2222
-1-1-1-1
-0.5-0.5-0.5-0.5
0000
0.50.50.50.5
1111
time (s)
Baseband waveform
0000 0.50.50.50.5 1111 1.51.51.51.5 2222
-1-1-1-1
-0.5-0.5-0.5-0.5
0000
0.50.50.50.5
1111
time (s)
RF signal
In this case, the amplitude of B/B signal is A = 1.
•Can also refer to messaging in a channel in terms of number
of bits/dimension and bits/degree of freedom (DOF).
The DOF represents the number of unique/independent waveforms
transmitted in 1 second (we assume each waveform is independent of
future or past waveform). Have “W” DOF where W = 1×Rs.
( ) ( )M
W
MR
DOFBits s
2
2
log
log
/ ==
•It should hopefully be quite clear that the message bits must be
represented in the channel as a “physical” continuous waveform.
In a digital modulation, each unique message maps to a uniquely
decodable waveform (these waveforms don’t have to be
independent or orthogonal, but they must contain some
independent parameter our receiver “looks out for” – i.e. a unique
phase, amplitude etc…).
M= Number of bits/signal. The information rate (bits/sec) is
related to bits/DOF by:
In mobile communication, not all bits are information and we often
send Pilot signals multiplexed into the signals. We reduce the
information rate, thus the DOF. Thus DOF is analogous to the
usage of the bandwidth for transmitting independent signals.
Number of bits/dimension something different. If we are
signaling at rate Rs signals per second, the number of bits per
second is:
The dimensionality of the signals is:
This represents the number of real dimensions – if complex
dimensions, then
( ) sec)/(log2 bitsMRR sb =
( ) sec)/(
log2
bits
T
M
R
s
b =
sWTBN ≅
sWTBN 2≅
•If the speed of transmission is Rb (bits/sec), where an N –
dimensional signal is transmitted every Ts seconds, then have
N/ Ts dimensions transmitted per second. Link to Rb is
therefore:
Bandwidth needed is B Hz per dimension per second, where
bandwidth requirement is: B typically = 1 for real
signals, B = ½ for complex.
BW = double sided bandwidth. Hence the number of bits per
dimension is ( )
N
M
p 2log
=
( )
( ) sec)/(
log
sec)/(
log
2
2
bits
T
M
bits
T
N
N
M
R
s
s
b
=
×=
s
W
T
BN
B =
Signal Basis: Vector Space concepts
We have already seen that in the I-Q example there were two basis
functions – both of them sinusoidal and orthogonal. These two basis
functions serve as the bedrock for all complex scalar modulation
signal representations. By scalar, we mean signal has a two
dimensional (one dimensional if space of complex numbers) state
space. As an example, the signal space is:
( )tcωcos
( )tcωsin
X and Y axis are our basis functions. All signals in space are linear
combinations of basis functions – we say signals “span” this space. The
constellation point is conveniently seen as complex in scalar modulation – i.e.
the space is a complex one with real (I) and imaginary (Q) parts.
We call this point a constellation
point – it is actually a vector of real
or complex numbers. The length
squared of the vector (i.e, norm^2)
is the “energy” of the point.

3/23/2010
12
Example: Constellation Points
We will look at two examples: One example will be the soon to be
familiar QPSK modulation, the other will be some generic three
dimensional modulation scheme. Here we will show how the symbol is in
fact the constellation point, and how this symbols length is related to
energy.
( )tcωcos
( )tcωsin
( )t1ψ
( )t2ψ
( )t3ψ
QPSK is simple – it is made from two BPSK signals. For brevity, we
assume the baseband sequence (the pulses) has unit area scaled by just
the amplitude, A. We show the basis functions as cosine and sine waves
for completeness – assume the carriers at receiver are scaled by 2
leaving us with just the original baseband sequence with no extra
scalars.
( )tcωcos
( )tcωsin
A
A
2
AEb =
X
bEAX 22 2
==
bs EXE 2
2
==
It takes more power to transmit QPSK than BPSK.
The symbol is a complex scalar (A+jA), or to be consistent
with vector terminology, our symbol is X = [A,A] which is
the amplitude of the two BPSK signals.
Energy/bit
Energy/symbol
Length:
Learning Outcome: The constellation point is a symbol which is a
vector/scalar representing the weight of the linear combination over
the signals basis functions (in example of QPSK, these were the sine and
cosine waves). The length (norm) of the symbol is the square root of the
symbol energy. The norm^2 is the symbol energy. We assume waveform
energy normalized to 1.
( )t1ψ
( )t2ψ
( )t3ψ
In this Modulation example, the dimensions of
all signals is 3, i.e. Number of basis functions
= 3. These basis functions are independent and
orthogonal. The number of bits per
waveform/symbol is assumed to be M. The
signal in this example is:
X=[x1,x2,x3]
2
X=sE
( ) ( )∑=
=
3
1k
kks txt ψψ
The Symbol Energy:
The Bit Energy:
MLog
E
E s
b
2
=
• The concept of inner product of two vectors can be used for
defining such concepts as length (or norm) of a vector, and the
angle between two vectors.
• Hilbert space is an inner product space (i.e., a space where an
inner product is defined) with certain additional completeness
property.
Discrete-Time Signal Space:
Continuous-Time Signal Space:
Othogonality
• The angle between two vectors can be defined as:
Subspace is a subset of the vectors satisfying the
properties of a vector space, and can be obtained from
an arbitrary set of vector by including all the linear
combinations of those vectors.
Example: H is Hilbert
space. Let M be a subspace
and X a vector of H. There
is a Unique Vector of M that
has property:
where is the
projection of X in M.
Say the Subspace can form
an orthonormal basis as in
vector modulation.
Vector space concepts give us two main types of modulation:
1. Vector Modulation: Where the signal is represented as vector
(or rather, its constellation point is a vector of dimensionality >
2). In this form of modulation, there is more than one basis
signal – there can be several basis functions. Examples: M-Ary
Modulation, M-FSK. These modulations are usually considered
for power limited channels, where we map message blocks over
signals of increasing bandwidth (low B/W efficiency).
2. Scalar Modulation: The signal is a complex (or real) scalar
defined over one real or complex basis function (single
frequency). Such modulations could key information using
phase, amplitude, or both. Examples are: M-QAM, M-PSK.
Modulation considered for bandwidth limited channels. The set
of symbols increases in size for a fixed channel bandwidth
meaning distance is reduced for increasing alphabet.

3/23/2010
13
Concept of distance: Distance, D, between waveforms defines
the probability of decoding error when noise interrupts the
signal.
In scalar modulation (fixed bandwidth, no channel coding) – distance
between uniformly spaced symbols depends on the transmitted/received
power. In vector modulation, distance depends on bandwidth (power
assumed fixed).
Example below is QPSK – where distance is simply calculated as:
( )tcωcos
( )tcωsin
D
sE
sED 2=
sE
Proof: Use Trigonometry!
2
2
4
cos
2
ss EE
D
=





=
π
Example: Two orthogonal basis functions – where modulation
in this example is simply the mapping of a bit/message to a
particular basis function (an orthogonal modulation technique).
Means constellation point for first waveform is [√Es,0,0,…,0].
Second waveform, constellation point is [0,√Es,0,…,0], and so
on for other waveforms. We always define the length of the vector
as its RMS value – its square value is energy. Hence the RMS is square
root of energy. For historical reasons, we coin the notation Es as being the
energy of a symbol (the symbol is a vector or scalar representing the
constellation point). ( )t1ψ
sED 2=
( )t2ψ
sE
sE
Proof: Use Pythagoras!
22
ss EED +=
Example: We shall use simple vector definitions and prove
some very important concepts for modulation. We want to be
able to define distance in a very arbitrary/abstract manner.
( )tz1 212121, xxyyzz +=
( )tz2
x1x2
y1
y2
Want to find distance, D:
D Inner Product between two vectors:
write 222111 sin,sin θθ zyzy ==
1θ 2θ
hence 


 +⋅= 21212121 sinsincoscos, θθθθzzzz
222111 cos,cos θθ zxzx ==
Since cos(A-B) = cosAcosB + sinAsinB, then
( ) 


 −⋅= 122121 cos, θθzzzz
Since is the angle between the two vectors, the
inner product is simply:
The distance, D, is (from Pythagoras):
θcos, 2121 zzzz ⋅=
12 θθθ −=
( ) ( )
( )2121
2
2
2
2
2
1
2
1
2
21
2
21
2 xxyyyxyx
xxyyD
+−+++=
−+−=
Since: 212121
2
2
2
2
2
2
2
1
2
1
2
1 ,,, xxyyzzxyzxyz +=+=+=
then:
21
2
2
2
1 ,2 zzzzD −+=
Importance of result: Distance is maximized if, and only if,
the inner (scalar) product equates to 0. This occurs when the
signal waveforms are orthogonal!
i=1:M where the wij are real
Set may be orthonormal
Orthogonality: ∫fj(t) fi(t) ∂t = 0 i ≠ j
Normality: E = ∫fi(t) fi(t) ∂t = 1
Basis signals form coordinate system of vector space
N (number of basis functions) <= M is dimension of
signal set
If N = M signals in set are all linearly independent
(i.e., none of the signals can be written as a linear
combination of the other signals in the set)
∑=
=
N
j
jiji twts
1
)()( φ
Basis Functions:
)1,0,2(
)1,2,0(
)0,2,0(
)0,0,2(
4
3
2
1
=
−=
=
=
s
s
s
s
Example:

3/23/2010
14
Filters can be matched to
signals si(t), i=1,2, ... ,M,
instead to basis functions ψi(t),
i=1,2, ... , N. Usually, use of
basis functions give less
complexity.
Matched Filters:
Gram-Schmidt Procedure:
Gram-Schmidt orthogonalization determines a set of
orthonormal basis functions of {si(t)}
1
)(
)( 1
1
sE
ts
t =φ
2
)(
)()(])()([)()( 2
2
0
11222
d
T
E
td
ttdtttststd
s
=−= ∫ φφφ
i
s
d
i
i
i
j
T
jjiii
E
td
ttdtttststd
)(
)()(])()([)()(
1
1 0
=−= ∑ ∫
−
=
φφφ
Similar to projection theorem except with functions!
Orthonormal Basis Functions:
Determining weights wij
ij
T
jk
T N
k
ikj
N
k
kik
T
ji wdtttwdtttwdttts ∫∫ ∑∑∫ ===
== 00 110
)()()()]([)()( φφφφφ
∫=
T
jiij dtttsw
0
)()( φ
∫ ∑∫ ∑∑
∞
∞− ===
===
N
j
ij
T N
k
kik
N
j
jijis wdttwtwdttsE i
1
2
0 11
2
)]([)]([)( φφ
Signal Constellations:
Each signal si(t) can be represented by a vector wi =
(wi1,wi2, …, wiN) or by a point in N-dimensional signal
space where {fj} defines the space and wi represents
the coordinates in this space
Set of basis functions is not unique, but different
basis functions will not change the lengths (energies)
of the signal vectors
It would be interesting to see what happens if we increase the number
of basis functions. As an intuitive example, if we wish to send 1 bit at a
time through the channel (i.e. 1 bit per channel use), we would need at
least one basis function where we will decode the bit based on received
signal (i.e. The bit scales the basis function in a unique way such as
shifting its phase or adjusting its amplitude). We could also use two
basis functions to send one bit – i.e. A “1” maps to one unique signal, a “0”
to another unique signal. Would we need more bandwidth to send this
signal? Yes! We increase the bandwidth by increasing the dimensions of
the system.
If we send 2 bits at once through a channel (2 bits per channel use),
where the 2 bits map to a unique signal, we have two options:
1. Use two basis functions where the bits will map to a unique symbol.
2. Use four basis functions where each two tuple will map to a unique
signal that is uniquely decodable (remember, the basis functions
should be orthogonal).
In practice, would apply channel coding to the message bits to expand the
bandwidth (or reduce the signal rate to append channel codes in available
bandwidth) – in an ideal sense, this channel coding would be used to increase the
distance between the message blocks (and waveforms representing these blocks)
so as to minimize the probability of confusing one signal with another in noise. In
many ways, modulation and channel coding are complementary activities, although
linking the two is often illusive and difficult.
Example:
Consider a modulation with 8 basis functions (all orthogonal). We could design
the modulation such that 3 bits map to a unique basis function. If we considered
a simple channel coding scheme where the 3 bits get an additional 2 bits added
(rate 3/5 code), we could have now a total of 32 orthogonal basis functions. If
we design the encoder such that only 8 out of 32 waveforms are selected, we can
maximize the distance between the waveforms. If this distance maximization
results in a bigger SNR (remember noise is bigger over larger bandwidth) relative
to uncoded system, we would get an improvement in the bit error rate. In the
case where 500bits map to 2500 waveforms, would get much closer to capacity
limit for given code rate.

3/23/2010
15
6.2.3 Theory for AWGN Channels
The limit imposed on modulation for given channel rate
determined by Shannon Capacity.
Most important indicators of channel: Probability of
error and Channel Capacity.
Fundamental = Noisy Channel Coding Theorem: There exists a
code at certain rate that allows the amount of transmitted
information per channel use to approach information rate measure
(Capacity) of channel with arbitrary reliability in detecting the
transmitted information bits.
Binary Symmetric Channel: A model for BPSK.
( )
















−
−+







−=
p1
1
Logp1
p
1
pLog1C 22 Bits/Channel Use
Discrete Channel – Number of bits per single channel
use.
N uses of channel with fixed Bandwidth results in
famous Capacity formula.
( ) 





+==≤
0
b
22
N
2E
ν1log
2
1
MlogCR Bits/Channel Use
Introduce the concept of capacity in AWGN
channel.
If channel is continuous in time, the capacity
gives the upper bound to the rate of information
where:
0
2
2
2
2
2 1log
N
E
W
C
WC
b
n
s
n
s
=






+⋅=
σ
σ
σ
σ
Bits/Sec
SNR
Eb = energy per bit
The Transmission Rate for a continuous source
for a continuous time channel is given as:
The Transmission Rate for a continuous source
for a discrete channel is:






σ
σ
+⋅=≤ 2
n
2
s
2 1logWCR






σ
σ
+=≤ 2
n
2
s
2 1log
2
1
CR
Bits/sec
Bits/Symbol/Real
Dimension
*In both cases, R = C if and only if the source is Continuous Gaussian
RV and an appropriate error control code is applied.

3/23/2010
16
Information I(X|Y) conveyed by a complex-valued
channel with additive white Gaussian noise, for
different input alphabets, with all symbols in the
alphabet equally likely:
Information I(X|Y) conveyed by a real-valued channel
with additive white Gaussian noise, for different input
alphabets, with all symbols in the alphabet equally likely:
2
2
n
x
SNR
σ
σ
=












σ
σ
+⋅≤ 2
n
2
s
2 1logWR






σ
σ
+≤=ν 2
n
2
s
2 1log
W
R
Comparison to Shannon Capacity:






+≤⇒ νν
0
2 1log
N
Eb
Region where reliable
communication
impossible
W = one sided bandwidth
From Previous side, it is possible to derive lower bound:
dB
N
E
N
E
e
Lim
b
Lim
b
6.1~2log
12
12
000
0
−≈
−
=
−
≥
→→
ν
ν
ν
νν
ν
where
Lim υ → 0 is same as setting W → ∞
High Spectral
Efficiency region.
Low Spectral
Efficiency region.

3/23/2010
17
Example: M – Ary FSK
Comparison of different
modulation methods at
10-5 symbol probability
of error is given.
Bandwidth limited region
is for Rb/W>1.
Power limited region is
for Rb/W<1.
Signal to noise ratio is
equal to energy per bit
divided by the noise
PSD.
Power versus bandwidth
Reverse IS-95 Channel Modulation Process
99Digital Modulation I 100
(b) Comparison of M-ary orthogonal
modulation against the ideal system for Pe
= 10 −5 and increasing M.
3dB
B
bR
12
N
E B
R
0
b
b
−
=
6.2.4 Theory for Fading Channels

3/23/2010
18
Defined as the maximum MI of channel
Maximum error-free data rate a channel can
support.
Theoretical limit (not achievable)
Channel characteristic
Not dependent on design techniques
Capacity defines theoretical rate limit
Maximum error free rate a channel can
support
Depends on what is known about channel
Fading Statistics Known
Hard to find capacity
Fading Known at Receiver Only:
( ) ( )
( )SNR2
SNRSNRSNR
0
2
γ1BlogC
dγγpγ1BlogC
+≤⇒
+= ∫
∞
Fading Known at Transmitter and Receiver:
For fixed transmit power, same as with only
receiver knowledge of fading
Transmit power S(g) can also be adapted
Leads to optimization problem
γγ
γγ
γγ
dp
S
S
B
SSES
C )(
)(
1log
)]([:)(
max
0
2 





+
=
= ∫
∞
Optimal Adaptive Scheme:
Power Adaptation
Capacity


 ≥−
=
else0
)( 0
11
0
γγγ γγ
S
S 1
0
γ
1
γ
γ0 γ
.)(log
0
2
0
γγ
γ
γ
γ
dp
B
R






= ∫
∞
Waterfilling
106
Channel Inversion:
Fading inverted to maintain constant SNR
Simplifies design (fixed rate)
Greatly reduces capacity
Capacity is zero in Rayleigh fading
Truncated inversion
Invert channel above cutoff fade depth
Constant SNR (fixed rate) above cutoff
Cutoff greatly increases capacity
Close to optimal
Fundamental capacity of flat-fading channels
depends on what is known at TX and RX.
Capacity when only RX knows fading same as when
TX and RX know fading but power fixed.
Capacity with TX/RX knowledge: variable-rate
variable-power transmission (water filling) optimal.
Almost same capacity as with RX knowledge only.
Channel inversion practical, but should truncate.

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