Signal & system

Wireless Information Transmission System Lab.
National Sun Yat-sen UniversityInstitute of Communications Engineering
Chapter 4
Characterization of Communication
Signals and Systems
Reference: Chapter 6, Communication Systems, 4th Edition.
by Simon Haykin

2
Table of Contents
4.1 Representation of Band-Pass Signals and Systems
4.1.1 Representation of Band-Pass Signals
4.1.2 Representation of Linear Band-Pass Systems
4.1.3 Response of a Band-Pass System to a Band-Pass
Signal
4.1.4 Representation of Band-Pass Stationary Stochastic
Processes
4.2 Signal Space Representations
4.2.1 Vector Space Concepts
4.2.2 Signal Space Concepts
4.2.3 Orthogonal Expansions of Signals

3
Table of Contents
4.3 Representation of Digitally Modulated Signals
4.3.1 Memoryless Modulation Methods
4.3.2 Linear Modulation with Memory
4.3.3 Non-linear Modulation Methods with Memory –
CPFSK and CPM
4.4 Spectral Characteristics of Digitally Modulated
Signals
4.4.1 Power Spectra of Linearly Modulated Signals
4.4.2 Power Spectra of CPFSK and CPM Signals (*)
4.4.3 Power Spectra of Modulated Signals with Memory (*)

4
4.1 Representation of Band-Pass Signals
and Systems
The channel over which the signal is transmitted is
limited in bandwidth to an interval of frequencies
centered about the carrier.
Signals and channels (systems) that satisfy the condition
that their bandwidth is much smaller than the carrier
frequency are termed narrowband band-pass signals
and channels (systems).
With no loss of generality and for mathematical
convenience, it is desirable to reduce all band-pass
signals and channels to equivalent low-pass signals and
channels.

5
4.1.1 Representation of Band-Pass
Signals
Suppose that a real-valued signal s(t) has a frequency content
concentrated in a narrow band of frequencies in the vicinity of
a frequency fc, as shown in the following figure:
Our object is to develop a mathematical representation of such
signals.
Spectrum of a band-pass signal.

6
A signal that contains only the positive frequencies in s(t) may
be expressed as:
where S( f ) is the Fourier transform of s(t) and u( f ) is the unit
step function, and the signal s+(t) is called the analytic signal or
the pre-envelope of s(t).
( ) ( ) ( )
( ) ( )
( ) ( )
2
1 1
2
2
j ft
S f u f S f
s t S f e df
F u f F S f
π
+
∞
+ +−∞
− −
=
= ⋅
= ∗⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
∫
( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
1
2
1
j
F u f t
t
j
s t t s t s t j s t s t js t
t t
δ
π
δ
π π
−
+
⎡ ⎤ = +⎣ ⎦
⎡ ⎤
= + ∗ = + ∗ ≡ +⎢ ⎥
⎣ ⎦
Signals

7
Define:
A filter, called a Hilbert transformer, is defined as:
The signal may be viewed as the output of the Hilbert
transformer when excited by the input signal s(t).
The frequency response of this filter is:
( ) ( )
( )1 1 s
s t s t d
t t
τ
τ
π π τ
∧ ∞
−∞
= ∗ =
−∫
( )
1
,h t t
tπ
= − ∞ < < ∞
( )s t
( ) ( )
( )
( )
( )
2 2
0
1 1
0 0
0
j ft j ft
j f
H f h t e dt e dt f
t
j f
π π
π
∞ ∞
− −
−∞ −∞
⎧− >
⎪
= = = =⎨
⎪
<⎩
∫ ∫
Signals

8
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
cos 2 sin 2
sin 2 cos 2
l
c c
c c
s t x t jy t
s t x t f t y t f t
s t x t f t y t f t
π π
π π
∧
= +
= −
= +
We observe that |H( f )|=1 and the phase response Θ( f )=-π/2
for f >0 and Θ( f )=π/2 for f <0. Thus, this filter is basically a
90 degrees phase shifter for all frequencies in the input signal.
The analytic signal s+(t) is a band-pass signal. To obtain an
equivalent low-pass representation, we define:
In general, sl(t) is complex-valued:
( ) ( )
( ) ( ) ( ) ( )2 2c c
l c
j f t j f t
l
S f S f f
s t s t e s t j s t eπ π
+
∧
− −
+
= +
⎡ ⎤
= = +⎢ ⎥⎣ ⎦
( ) ( ) ( ) 2 cj f t
ls t j s t s t e π
∧
⎡ ⎤
+ =⎢ ⎥⎣ ⎦
Signals

9
s(t)=x(t)cos2πfct-y(t)sin2πfct is the desired form for the
representation of a band-pass signal. The low-frequency signal
components x(t) and y(t) may be viewed as amplitude
modulations impressed on the carrier components cos2πfct and
sin2πfct, respectively.
x(t) and y(t) are called the quadrature components of the band-
pass signal s(t).
s(t) can also be written as:
The low pass signal sl(t) is usually called the complex envelope
of the real signal s(t) and is basically the equivalent low-pass
signal.
( ) ( ) ( ){ } ( )2 2
Re Rec cj f t j f t
ls t x t jy t e s t eπ π
⎡ ⎤⎡ ⎤= + =⎣ ⎦ ⎣ ⎦
( ) ( ) ( ) 2 cj f t
ls t j s t s t e π
∧
⎡ ⎤
+ =⎢ ⎥⎣ ⎦
Signals

10
sl(t) can be also be written as:
s(t) can be represented as:
a(t) is called the envelope of s(t), and θ(t) is called the phase
of s(t).
( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( )
2 2 1
where and tan
j t
ls t a t e
y t
a t x t y t t
x t
θ
θ −
=
= + =
( ) ( ) ( ) ( )
( ) ( )
22
Re Re
cos 2
cc
j f t tj f t
l
c
s t s t e a t e
a t f t t
π θπ
π θ
⎡ + ⎤⎣ ⎦⎡ ⎤⎡ ⎤= =⎣ ⎦ ⎣ ⎦
⎡ ⎤= +⎣ ⎦
Signals

11
Three equivalent representations of band-pass signals:
The Fourier transform of s(t) is:
( ) ( ) ( )
( )
( ) ( )
2
cos2 sin 2
Re
cos 2
c
c c
j f t
l
c
s t x t f t y t f t
s t e
a t f t t
π
π π
π θ
= −
⎡ ⎤= ⎣ ⎦
⎡ ⎤= +⎣ ⎦
( ) ( ) ( ){ }22 2
Re cj f tj ft j ft
lS f s t e dt s t e e dtππ π∞ ∞
− −
−∞ −∞
⎡ ⎤= = ⎣ ⎦∫ ∫
( ) ( ) ( )
( ) ( )
2 2 21
2
1
2
c cj f t j f t j ft
l l
l c l c
S f s t e s t e e dt
S f f S f f
π π π∞
−∗ −
−∞
∗
⎡ ⎤= +⎣ ⎦
⎡ ⎤= − + − −⎣ ⎦
∫
( ) ( )1
Re
2
ξ ξ ξ∗
= +
Signals

12
The energy in the signal s(t) is defined as:
( ) ( ){ }
2
22
Re cj f t
lε s t dt s t e dtπ∞ ∞
−∞ −∞
⎡ ⎤= = ⎣ ⎦∫ ∫
( )
( ) ( )
( )( ) ( )( )
( ) ( ) ( )
( ) ( )( )
24 42 * *
22 4 24 22 *
2 2
22 2 *
1
2
4
1 1
( )
2 4
1 1
cos 4 2
2 2
where ( )
c c
cc
j f t j f t
l l l l
j f t tj f t t
l
l l c
l
s e s s s e dt
s t dt a t e a t e dt
s t dt s t f t t dt
s t a t a t
π π
π θπ θ
ε
π θ
∞
−
−∞
∞ ∞ − ++
−∞ −∞
∞ ∞
−∞ −∞
⎡ ⎤= + +
⎢ ⎥⎣ ⎦
⎡ ⎤= + +
⎢ ⎥⎣ ⎦
⎡ ⎤= + +⎣ ⎦
= =
∫
∫ ∫
∫ ∫
( ) ( )1
R e
2
ξ ξ ξ ∗
= +
( ) ( ) ( )j t
ls t a t e θ
=
Signals

13
Since the signal s(t) is narrow-band, the real envelope a(t)=|sl(t)|
or, equivalently, a2(t) varies slowly relative to the rapid
variations exhibited by the cosine function.
The net area contributed by the second integral is very small
relative to the value of the first integral, hence, it can be
neglected. ( )
21
2
ls t dtε
∞
−∞
= ∫
Signals
(4.1-24)

14
A linear filter or system may be described either by its impulse
response h(t) or by its frequency response H( f ), which is the
Fourier transform of h(t). Since h(t) is real, H*(-f )=H( f ),
because:
( )
( ) ( )
( )
( )
( )
( ) ( ) ( )
*
*
0
Define
0 0
0 0
then
0
l c
l c
H f f
H f f
f
f
H f f
H f H f f
⎧ >⎪
− = ⎨
<⎪⎩
⎧ >⎪
− − = ⎨
− = <⎪⎩
( ) ( ) ( )
( )
( ) ( )
( ) ( )
*
2*
* 2
2
Note that is real.
j f t
j ft
j ft
H f h t e dt
h t e dt h t
h t e dt H f
π
π
π
∞ − −
−∞
∞
−
−∞
∞
−
−∞
− =
=
= =
∫
∫
∫
4.1.2 Representation of Linear Band-
Pass Systems

15
In general, the impulse response hl(t) of the equivalent low-pass
system is complex-valued.
( ) ( ) ( )
( ) ( )
( )
( )
( )
( )( )
*
2 2*
2
* 2
-
2* 2
*
2
As a result
thus
2Re
where using
c c
c
c
l c l c
j f t j f t
l l
j f t
l
j ft
l c c
j f tj xt
l
j xt
l
H f H f f H f f
h t h t e h e
h t e
H f f e df x f f
H x e dx e
H x e dx e
π π
π
π
ππ
π
−
∞
∞
∞
−−
−∞
∞
−∞
= − + − −
= +
⎡ ⎤= ⎣ ⎦
− − = − −
= ⋅
= ⋅
∫
∫
∫ ( )2 2*c cj f t j f t
lh t eπ π− −
= ⋅
4.1.2 Representation of Linear Band-
Pass Systems

16
We have shown in Sections 4.1.1 and 4.1.2 that narrow band
band-pass signals and systems can be represented by equivalent
low-pass signals and systems.
We demonstrate in this section that the output of a band-pass
system to a band-pass input signal is simply obtained from the
equivalent low-pass input signal and the equivalent low-pass
impulse response of the system.
The output of the band-pass system is also a band-pass signal,
and, therefore, it can be expressed in the form:
where r(t) is related to the input signal s(t) and the impulse
response h(t) by the convolution integral.
( ) ( ) 2
Re cj f t
lr t r t e π
⎡ ⎤= ⎣ ⎦
( ) ( ) ( )r t s h t dτ τ τ
∞
−∞
= −∫
4.1.3 Response of a Band-Pass
System to a Band-Pass Signal

17
( ) ( ) ( ) ( ) ( )
( ) ( )
1
2
1
2
l c l c l c l c
l c l c
R f S f f H f f S f f H f f
R f f R f f
∗ ∗
∗
⎡ ⎤= − − + − − − −⎣ ⎦
⎡ ⎤= − + − −⎣ ⎦
The output of the system in the frequency domain is:
For a narrow band signal, Sl( f-fc)≈0 for f <0 and H*
l(-f-fc)=0 for f >0.
For a narrow band signal, S*
l(-f-fc)≈0 for f >0 and Hl( f-fc)=0 for f <0.
( ) ( ) ( )
( ) ( ) ( ) ( )
1
2
l c l c l c l c
R f S f H f
S f f S f f H f f H f f∗ ∗
=
⎡ ⎤ ⎡ ⎤= − + − − − + − −⎣ ⎦ ⎣ ⎦
( ) ( ) 0l c l cS f f H f f∗
− − − =
( ) ( ) 0l c l cS f f H f f∗
− − − =
( ) ( ) ( )
( ) ( ) ( )
l l l
l l l
R f S f H f
r t s h t dτ τ τ
∞
−∞
≡
= −∫
4.1.3 Response of a Band-Pass
System to a Band-Pass Signal
From 4.1-21 and 4.1-28.
4.1-27
4.1-26
Compared
with 4.1-21

18
In this section, we extend the representation to sample
functions of a band-pass stationary stochastic process. In
particular, we derive the relations between the correlation
functions and power spectra of the band-pass signal and the
correlation function and power spectra of the equivalent low-
pass signal.
Suppose that n(t) is a sample function of a wide-sense
stationary stochastic process with zero mean and power spectral
density Φnn(f ). The power spectral density is assumed to be
zero outside of an interval of frequencies centered around fc,
where fc is termed the carrier frequency. The stochastic
process n(t) is said to be a narrowband band-pass process if the
width of the spectral density is much smaller than fc.
Stationary Stochastic Processes

19
Under this condition, a sample function of the process n(t) can
be represented by the following equations from Section 4.1.1:
a(t) is the envelope and θ(t) is the phase of the real-valued signal.
x(t) and y(t) are the quadrature components of n(t).
z(t) is called the complex envelope of n(t).
If n(t) is zero mean, then x(t) and y(t) must also have zero mean values.
The stationarity of n(t) implies that:
( ) ( ) ( )
( ) ( )
( ) 2
cos 2
cos 2 sin 2
Re c
c
c c
j f t
n t a t f t t
x t f t y t f t
z t e π
π θ
π π
⎡ ⎤= +⎣ ⎦
= −
⎡ ⎤= ⎣ ⎦
( ) ( )
( ) ( )
xx yy
xy yx
φ τ φ τ
φ τ φ τ
=
= −
Proved next.

20
Proof of
Autocorrelation function of n(t) is:
by using:
( ) ( ) ( )
( ) ( ){
( ) ( ) ( ) ( ) }
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
cos 2 sin 2
cos 2 sin 2
cos 2 cos 2 sin 2 sin 2
sin 2 cos 2 cos 2 sin 2
nn
c c
c c
xx c c yy c c
xy c c yx c c
E n t n t
E x t f t y t f t
x t f t y t f t
f t f t f t f t
f t f t f t f t
φ τ τ
π π
τ π τ τ π τ
φ τ π π τ φ τ π π τ
φ τ π π τ φ τ π π τ
⎡ ⎤= +⎣ ⎦
⎡ ⎤= −⎣ ⎦
⎡ ⎤× + + − + +⎣ ⎦
= + + +
− + − +
( ) ( )
( ) ( )
( ) ( )
1cos cos cos cos
2
1sin sin cos cos
2
1sin cos sin sin
2
A B A B A B
A B A B A B
A B A B A B
= − + +⎡ ⎤⎣ ⎦
= − − +⎡ ⎤⎣ ⎦
= − + +⎡ ⎤⎣ ⎦
( ) ( ) ( ) ( )andxx yy xy yxφ τ φ τ φ τ φ τ= = −

21
We can obtain:
Since n(t) is stationary, the right-hand side must be independent of t.
As a result,
Therefore,
Note that this equation is identical in form to:
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
1 1
cos2 cos2 2
2 2
1 1
sin 2 sin 2 2
2 2
nn
xx yy c xx yy c
yx xy c yx xy c
E n t n t
f f t
f f t
φ τ τ
φ τ φ τ π τ φ τ φ τ π τ
φ τ φ τ π τ φ τ φ τ π τ
⎡ ⎤= +⎣ ⎦
⎡ ⎤ ⎡ ⎤= + + − +⎣ ⎦ ⎣ ⎦
⎡ ⎤ ⎡ ⎤− − − + +⎣ ⎦ ⎣ ⎦
( ) ( ) ( ) ( )andxx yy xy yxφ τ φ τ φ τ φ τ= = −
( ) ( ) ( )cos2 sin 2nn xx c yx cf fφ τ φ τ π τ φ τ π τ= −
( ) ( ) ( )cos2 sin 2c cn t x t f t y t f tπ π= −
Q.E.D.

22
The autocorrelation function of the equivalent low-pass process
z(t)=x(t)+jy(t) is defined as:
Since φxx(τ)=φyy(τ) and φxy(τ)= -φyx(τ)
we obtain: φzz(τ)=φxx(τ)+ jφyx(τ)
This equation relates the autocorrelation function of the
complex envelope to the autocorrelation and cross-correlation
functions of the quadrature components.
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
1
2
1
2
zz
zz xx yy xy yx
E z t z t
j j
φ τ τ
φ τ φ τ φ τ φ τ φ τ
∗
⎡ ⎤= +⎣ ⎦
⎡ ⎤= + − +⎣ ⎦

23
By combining
and
we can obtain: φnn(τ)=Re[φzz(τ)e j2πfcτ]
Therefore, the autocorrelation function φnn(τ) of the band-
pass stochastic process is uniquely determined from the
autocorrelation function φzz(τ) of the equivalent low-pass
process z(t) and the carrier frequency fc.
The power density spectrum of the stochastic process n(t) is:
( ) ( ) ( )
( ) ( ) ( )cos 2 sin 2
zz xx yx
nn xx c yx c
j
f f
φ τ φ τ φ τ
φ τ φ τ π τ φ τ π τ
= +
= −
( ) ( ){ }
( ) ( )
2 2
Re
1
2
cj f j f
nn zz
zz c zz c
f e e d
f f f f
π τ π τ
φ τ τ
∞
−
−∞
⎡ ⎤Φ = ⎣ ⎦
⎡ ⎤= Φ − + Φ − −⎣ ⎦
∫

24
Properties of the quadrature components
If n(t) is Gaussian, x(t) and y(t +τ) are jointly Gaussian. For
τ=0, they are statistically independent, and the joint PDF is:
( ) ( )
( ) ( )
( ) ( )
( ) ( )
(2.2-10)
(4.1-41)
is an odd function of and 0 0.
yx xy
xy yx
xy xy
xy xy
φ τ φ τ
φ τ φ τ
φ τ φ τ
φ τ τ φ
= −
= −
⇒ = − −
⇒ =
( ) ( )
( ) ( )
( )
If 0 for all , then is real (from 4.1-48) and
the power spectral density satisfies
(i.e. is symmetric about 0).
xy zz
zz zz
zz
f f
f f
φ τ τ φ τ=
Φ = Φ −
Φ =
( ) ( )
( ) ( ) ( )
2 2 2
2 2
2
1, , 0 0 0
2
x y
xx yy nnp x y e
σ
σ φ φ φ
πσ
− +
= = = =
( ) ( ) ( ) ( ) ( )
1
2
zz xx yy xy yxj jφ τ φ τ φ τ φ τ φ τ⎡ ⎤= + − +⎣ ⎦

25
Representation of white noise
White noise is a stochastic process that is defined to have a flat
(constant) power spectral density over the entire frequency
range. This type of noise can’t be expressed in terms of
quadrature components, as a result of its wideband character.
In the demodulation of narrowband signals in noise, it is
mathematically convenient to model the additive noise process
as white and to represent the noise in terms of quadrature
components. This can be accomplished by postulating that the
signals and noise at the receiving terminal have passed through
an ideal band-pass filter.

26
Representation of white noise (cont.)
The noise resulting from passing the white noise process
through a spectrally band-pass filter is termed band-pass
white noise and has the power spectral density:
The band-pass white noise can be represented by:
( ) ( ) ( )
( ) ( )
( ) 2
cos 2
cos 2 sin 2
Re c
c
c c
j f t
n t a t f t t
x t f t y t f t
z t e π
π θ
π π
⎡ ⎤= +⎣ ⎦
= −
⎡ ⎤= ⎣ ⎦

27
Representation of white noise (cont.)
The equivalent low-pass noise z(t) has a power spectral
density:
The power spectral density for white noise and band-pass
white noise is symmetric about f =0, so φyx(τ)=0 for all τ.
( )
( ) ( ) ( )
0
0 0
1
2
1
0
2
sin
and
zz
zz zz
B
N f B
f
f B
B
N N
π τ
φ τ φ τ δ τ
πτ →∞
⎧ ⎛ ⎞
≤⎜ ⎟⎪
⎪ ⎝ ⎠
Φ = ⎨
⎛ ⎞⎪ >⎜ ⎟⎪ ⎝ ⎠⎩
= =
( ) ( ) ( ) (from 4.1-48)zz xx yyφ τ φ τ φ τ= =
4.1-54

28
4.2 Signal Space Representations
We will demonstrate that signals have characteristics that are
similar to vectors and develop a vector representation for signal
waveforms.
A vector v in an n-dimensional space is characterized by its n
components [v1 v2 … vn] and may also be represented as a
linear combination of unit vectors or basis vectors ei, 1≤i≤n,
The inner product of two n-dimensional vectors is defined as:
1
n
i i
i
v e
=
= ∑ν
1 2 1 2
1
n
i i
i
v v
=
⋅ = ∑ν ν

29
A set of m vectors vk, 1≤k≤m are orthogonal if:
The norm of a vector v is denoted by ||v|| and is defined as:
A set of m vectors is said to be orthonormal if the vectors are
orthogonal and each vector has a unit norm.
A set of m vectors is said to be linearly independent if no one
vector can be represented as a linear combination of the
remaining vectors.
Two n-dimensional vectors v1 and v2 satisfy the triangle
inequality:
0 for all 1 , , and .i j i j m i j⋅ = ≤ ≤ ≠ν ν
( )
1 2 2
1
n
i
i
v
=
= ⋅ = ∑ν ν ν
1 2 1 2+ ≤ +ν ν ν ν

30
Cauchy-Schwarz inequality:
The norm square of the sum of two vectors may be expressed as:
Linear transformation in an n-dimensional vector space:
In the special case where v’=λv,
the vector v is called an eigenvector and λ is the corresponding
eigenvalue.
1 2 1 2⋅ ≤ +ν ν ν ν
2 2 2
1 2 1 2 1 22+ = + + ⋅ν ν ν ν ν ν
'
A=ν ν
A λ=ν ν

31
Gram-Schmidt procedure for constructing a set of
orthonormal vectors.
Arbitrarily selecting a vector v1 and normalizing its length:
Select v2 and subtract the projection of v2 onto u1.
Normalize the vector u2’ to unit length.
Selecting v3:
By continuing this procedure, we construct a set of
orthonormal vectors.
1
1
1
=
ν
u
ν
( )'
2 2 2 1 1= − ⋅u ν ν u u
'
2
2 '
2
=
u
u
u
( ) ( )'
3 3 3 1 1 3 2 2= - -⋅ ⋅u ν v u u v u u
'
3
3 '
3
=
u
u
u

32
The inner product of two generally complex-valued signals x1(t)
and x2(t) is denote by < x1(t),x2(t)> and defined as:
The signals are orthogonal if their inner product is zero.
The norm of a signal is defined as:
A set of m signals are orthonormal if they are orthogonal and
their norms are all unity.
A set of m signals is linearly independent, if no signal can be
represented as a linear combination of the remaining signals.
( ) ( ) ( ) ( )1 2 1 2,
b
a
x t x t x t x t dt∗
= ∫
( ) ( )( )
1 2
2b
a
x t x t dt= ∫

33
The triangle inequality for two signals is:
The Cauchy-Schwarz inequality is:
with equality when x2(t)=ax1(t), where a is any complex
number.
( ) ( ) ( ) ( )1 2 1 2x t x t x t x t+ ≤ +
( ) ( ) ( ) ( )
1 2 1 2
2 2
1 2 1 2
b b b
a a a
x t x t dt x t dt x t dt∗
≤∫ ∫ ∫

34
Suppose that s(t) is a deterministic, real-valued signal with finite
energy:
Suppose that there exists a set of functions {fn(t), n=1,2,…,K}
that are orthonormal in the sense that:
We may approximate the signal s(t) by a weighted linear
combination of these functions, i.e.,
( )
2
s s t dtε
∞
−∞
⎡ ⎤= ⎣ ⎦∫
( ) ( )
( )
( )
0
1
n m
m n
f t f t dt
m n
∞
−∞
⎧ ≠⎪
= ⎨
=⎪⎩
∫
( ) ( )
1
K
k k
k
s t s f t
∧
=
= ∑
4.2.3 Orthogonal Expansions of
Signals

35
The approximation error incurred is:
The energy of the approximation error:
To minimize the energy of the approximation error, the
optimum coefficients in the series expansion of s(t) may be
found by:
Differentiating Equation 4.2-24 with respect to each of the coefficients
{sk} and setting the first derivatives to zero.
Use a well-known result from estimation theory based on the mean-
square-error criterion, which is that the minimum of εe with respect to
the {sk} is obtained when the error is orthogonal to each of the functions
in the series expansion.
( ) ( ) ( )e t s t s t
∧
= −
( ) ( ) ( ) ( )
22
1
K
e k k
k
s t s t dt s t s f t dtε
∧∞ ∞
−∞ −∞
=
⎡ ⎤⎡ ⎤
= − = −⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦
∑∫ ∫
Signals
(4.2-24)

36
Using the second approach, we have:
Since the functions {fn(t)} are orthonormal, we have:
Thus, the coefficients are obtained by projecting the signals s(t)
onto each of the functions.
The minimum mean square approximation error is:
( ) ( ) ( )
1
0, 1,2,...,
K
k k n
k
s t s f t f t dt n K
∞
−∞
−
⎡ ⎤
− = =⎢ ⎥
⎣ ⎦
∑∫
( ) ( ) , 1,2,...,n ns s t f t dt n K
∞
−∞
= =∫
( ) ( ) ( ) ( ) ( )
2
min
1
2
1
K
k k
k
K
s k
k
e t s t dt s t dt s f t s t dt
s
ε
ε
∞ ∞ ∞
−∞ −∞ −∞
=
=
⎡ ⎤= = −⎣ ⎦
= −
∑∫ ∫ ∫
∑
Signals

37
When the minimum mean square approximation error εmin=0,
Under such condition, we may express s(t) as:
When every finite energy signal can be represented by a series
expansion of the form for which εmin=0, the set of orthonormal
functions {fn(t)} is said to be complete.
( )
22
1
K
s k
k
s s t dtε
∞
−∞
=
⎡ ⎤= = ⎣ ⎦∑ ∫
( ) ( )
1
K
k k
k
s t s f t
=
= ∑
Signals

38
Example 4.2-1 Trigonometric Fourier Series:
Consider a finite energy signal s(t) that is zero everywhere
except in the range 0≤ t ≤T and has a finite number of
discontinuities in this interval. Its periodic extension can be
represented in a Fourier series as:
where the coefficients {ak, bk} that minimize the mean
square error are given by:
( )
0
2 2
cos sink k
k
kt kt
s t a b
T T
π π∞
=
⎛ ⎞
= +⎜ ⎟
⎝ ⎠
∑
( ) ( )0 0
1 2 1 2
cos , sin
T T
k k
kt kt
a s t dt b s t dt
T TT T
π π
= =∫ ∫
{ }The set of trigonometric functions 2/ cos2 / , 2/ sin 2 /
is complete, and the series expansion results in zero mean square error.
T kt T T kt Tπ π
Signals

39
Gram-Schmidt procedure
Constructing a set of orthonormal waveforms from a set of
finite energy signal waveforms {si(t), i=1,2,…,M}.
Begin with the first waveform s1(t) which has energy ε1.
The first orthonormal waveform is:
The 2nd waveform is constructed from s2(t) by first
computing the projection of f1(t) onto s2(t):
Then c12 f1(t) is subtracted from s2(t):
( )
( )1
1
1
s t
f t
ε
=
( ) ( )12 2 1c s t f t dt
∞
−∞
= ∫
( ) ( ) ( )'
2 2 12 1f t s t c f t= −
Signals

40
Gram-Schmidt procedure (cont.)
If ε2 denotes the energy of f2＇(t), the normalized
waveform that is orthogonal to f1(t) is:
In general, the orthogonalization of the kth function leads to
The orthogonalization process is continued until all the M
signal waveforms have been exhausted and N≤ M
orthonormal waveforms have been constructed.
( )
( )'
2
2
2
f t
f t
ε
=
( )
( )
( ) ( ) ( )
( ) ( )
' 1
'
1
where
, 1,2,..., 1
k
k
k k k ik i
ik
ik k i
f t
f t f t s T c f t
c s t f t dt i k
ε
−
=
∞
−∞
= = −
= = −
∑
∫
Signals

41
Once we have constructed the set of orthonormal waveforms
{fn(t)}, we can express the M signals {sn(t)} as linear
combinations of the {fn(t)}:
Each signal may be represented as a point in the N-
dimensional signal space with coordinates {ski, i=1,2,…,N}.
( ) ( )
1
, 1,2,...,
N
k kn n
n
s t s f t k M
=
= =∑
( )
2 22
1
N
k k kn k
n
s t dt s sε
∞
−∞
=
⎡ ⎤= = =⎣ ⎦ ∑∫
[ ]1 2 ...k k k kNs s s s=
Signals

42
The energy in the kth signal is simply the square of the
length of the vector or, equivalently, the square of the
Euclidean distance from the origin to the point in the N-
dimensional space.
Any signal can be represented geometrically as a point in
the signal space spanned by the {fn(t)}.
The functions {fn(t)} obtained from the Gram-Schmidt
procedure are not unique.
If we alter the order in which the orthogonalization of the
signals {sn(t)} is performed, the orthonormal waveforms
will be different.
Nevertheless, the vectors {sn(t)} will retain their geometrical
configuration and their lengths will be invariant to the
choice of orthonormal functions {fn(t)}.
Signals

43
Consider the case in which the signal waveforms are band-pass
and represented as:
Similarity between any pair of signal waveforms is measured
by the normalized cross correlation:
Complex-valued cross-correlation coefficient ρkm is defined as:
( ) ( )
( ) ( )
2
22
Re , 1,2,...,
1
(from 4.1-24)
2
cj f t
m lm
m m lm
s t s t e m M
s t dt s t dt
π
ε
∞ ∞
−∞ −∞
⎡ ⎤= =⎣ ⎦
= =∫ ∫
( ) ( ) ( ) ( )
1 1
Re
2
m k lm lk
m k m k
s t s t dt s t s t dt
ε ε ε ε
∞ ∞
∗
−∞ −∞
⎧ ⎫⎪ ⎪
= ⎨ ⎬
⎪ ⎪⎩ ⎭
∫ ∫
( ) ( )
1
2
km lm lk
m k
s t s t dtρ
ε ε
∞
∗
−∞
= ∫
Signals
(4.2-44)
(4.2-45)

44
The Euclidean distance between a pair of signals is defined as:
When εm=εk=εfor all m and k, this expression simplifies to:
( ) ( ) ( )
1
Re km m k
m k
s t s t dtρ
ε ε
∞
−∞
= ∫
( )Re m k m k
km
m k m k
s s s s
s s
ρ
ε ε
⋅ ⋅
= =
( )
( ) ( ){ }
( ){ }
1 2
2
1 2
2 Re
e
km m k m k
m k m k km
d s s s t s t dt
ε ε ε ε ρ
∞
−∞
⎡ ⎤= − = −⎣ ⎦
= + −
∫
( )
( ){ }
1 2
2 1 Ree
km kmd ε ρ⎡ ⎤= −⎣ ⎦
Signals
From 4.2-44 & 4.2-45.

45
4.3 Representation of Digitally
Modulated Signals
Digitally modulated signals, which are classified as linear, are
conveniently expanded in terms of two orthonormal basis
functions of the form:
If slm(t) is expressed as slm(t)=xl(t)+jyl(t), sm(t) may be expressed
as:
In the transmission of digital information over a communication
channel, the modulator is the interface device that maps the
digital information into analog waveforms that match the
characteristics of the channel.
( ) ( )1 2
2 2
cos 2 sin 2c cf t f t f t f t
T T
π π= = −
( ) ( ) ( ) ( ) ( )1 2m l ls t x t f t y t f t= + From 4.2-42.

46
The mapping is generally performed by taking blocks of k=log2M
binary digits at a time from the information sequence {an} and
selecting one of M=2k deterministic, finite energy waveforms
{sm(t), m=1,2,…,M} for transmission over the channel.
When the mapping is performed under the constraint that a
waveform transmitted in any time interval depends on one or
more previously transmitted waveforms, the modulator is said to
have memory. Otherwise, the modulator is called memoryless.
Functional model of passband data transmission system
Modulated Signals

47
The digital data transmits over a band-pass channel that can be
linear or nonlinear.
This mode of data transmission relies on the use of a sinusoidal
carrier wave modulated by the data stream.
In digital passband transmission, the incoming data stream is
modulated onto a carrier (usually sinusoidal) with fixed frequency
limits imposed by a band-pass channel of interest.
The modulation process making the transmission possible
involves switching (keying) the amplitude, frequency, or phase of
a sinusoidal carrier in some fashion in accordance with the
incoming data.
There are three basic signaling schemes: amplitude-shift keying
(ASK), frequency-shift Keying (FSK), and phase-shift keying
(PSK).
Modulated Signals

48
Illustrative waveforms for the three basic forms of signaling
binary information. (a) ASK (b) PSK (c) FSK.
Modulated Signals

49
Unlike ASK signals, both PSK and FSK signals have a constant
envelope. This property makes PSK and FSK signals impervious
to amplitude nonlinearities.
In practice, we find that PSK and FSK signals are preferred to
ASK signals for passband data transmission over nonlinear
channels.
Digital modulation techniques may be classified into coherent and
noncoherent techniques, depending on whether the receiver is
equipped with a phase-recovery circuit or not.
The phase-recovery circuit ensures that the oscillator supplying
the locally generated carrier wave in the receiver is synchronized
(in both frequency and phase) to the oscillator supplying the
carrier wave used to originally modulated the incoming data
stream in the transmitter.
Modulated Signals

50
Modulated Signals
M-ary signaling scheme:
For almost all applications, the number of possible signals
M=2n.
Symbol duration T=nTb, where Tb is the bit duration.
We have M-ary ASK, M-ary PSK, and M-ary FSK.
We can also combine different methods of modulation into a
hybrid form. For example, M-ary amplitude-phase keying
(APK) and M-ary quadrature-amplitude modulation (QAM).
M-ary PSK and M-ary QAM are examples of linear
modulation.
An M-ary PSK signal has a constant envelope, whereas an M-
ary QAM signal involves changes in the carrier amplitude.

51
Modulated Signals
M-ary signaling scheme (cont.):
M-ary PSK can be used to transmit digital data over a
nonlinear band-pass channel, whereas M-ary QAM requires
the use of a linear channel.
M-ary PSK, and M-ary QAM are commonly used in coherent
systems.
ASK and FSK lend themselves naturally to use in non-
coherent systems whenever it is impractical to maintain carrier
phase synchronization.
We can’t have noncoherent PSK.

52
Pulse-amplitude-modulated (PAM) signals
Double-sideband (DSB) signal waveform may be
represented as:
where Am denote the set of M possible amplitudes corresponding to
M=2k possible k-bit blocks of symbols.
The signal amplitudes Am take the discrete values:
2d is the distance between adjacent signal amplitudes.
g(t) is a real-valued signal pulse whose shape influences the
spectrum of the transmitted signal.
The symbol rate is R/k, Tb=1/R is the bit interval, and T=k/R=kTb is
the symbol interval.
( ) ( )
( )
2
Re
cos 2 , 1,2,..., , 0
cj f t
m m
m c
s t A g t e
A g t f t m M t T
π
π
⎡ ⎤= ⎣ ⎦
= = ≤ ≤
( ) ( ) ( )( )2 1 , 1,2,..., 1 ... 1mA m M d m M M d M d= − − = − − −
Bandpass
Signal

53
Pulse-amplitude-modulated (PAM) signals (cont.)
The M PAM signals have energies:
These signals are one-dimensional and are represented by:
f(t) is defined as the unit-energy signal waveform given as:
Digital PAM is also called amplitude-shift keying (ASK).
( ) ( )2 2 2 2
0 0
1 1
2 2
T T
m m m m gs t dt A g t dt Aε ε= = =∫ ∫
( ) ( )m ms t s f t=
( ) ( )
MmAs
tftgtf
gmm
c
g
,...,2,1,
2
1
2cos
2
==
=
ε
π
ε

54
Signal space diagram for digital PAM signals:

55
Gray encoding: The mapping of k information bits to the
M=2k possible signal amplitudes may be done in a number
of ways. The preferred assignment is one in which the
adjacent signal amplitudes differ by one binary digit.
The Euclidean distance between any pair of signal points is:
The minimum Euclidean distance between any pair of
signals is:
( )
( )
2 1
2
2
e
mn m n g m n gd s s A A d m nε ε= − = − = −
( )
min 2e
gd d ε=

56
Single Sideband (SSB) PAM is represented by:
where is the Hilbert transform of g(t).
The digital PAM signal is also appropriate for transmission
over a channel that does not require carrier modulation and
is called baseband signal:
If M=2, the signals are called antipodal and have the special
property that:
( )g t
( ) ( ) ( ) 2
Re , 1,2,...,cj f t
m ms t A g t j g t e m Mπ
∧
⎧ ⎫⎡ ⎤
= ± =⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
( ) ( ), 1,2,...,m ms t A g t m M= =
( ) ( )1 2s t s t= −
4.1-9

57
Four-amplitude level baseband and band-pass PAM signals

58
Phase-modulated signals (Binary Phase-Shift Keying)
In a coherent binary PSK system, the pair of signals s1(t) and
s2(t) used to represent binary symbols 1 and 0, respectively is
defined by
where 0≤t≤Tb and Eb is the transmitted signal energy per bit.
A pair of sinusoidal waves that differ only in a relative phase-
shift of 180 degrees are referred to as antipodal signals.
( ) ( )
( ) ( ) ( )
1
2
2
cos 2
2 2
cos 2 cos 2
b
c
b
b b
c c
b b
E
s t f t
T
E E
s t f t f t
T T
π
π π π
=
= + = −

59
To ensure that each transmitted bit contains an integral
number of cycles of the carrier wave, the carrier frequency fc
is chosen equal to nc/Tb for some fixed integer nc.
In the case of binary PSK, there is only one basis function of
unit energy:
The coordinates of the message points are:
( ) ( ) bc
b
Tttf
T
t <≤= 0,2cos
2
1 πφ
( ) ( ) ( ) ( )1 1 2 1, , 0b b bs t E t s t E t t Tφ φ= = − ≤ <
( ) ( )11 1 10
bT
bs s t t dt Eφ= = +∫ ( ) ( )21 2 10
bT
bs s t t dt Eφ= = −∫

60

61
Phase-modulated signals (Quadriphase-Shift Keying)
Quadriphase-Shift Keying (QPSK)
where i = 1, 2, 3, 4; E is the transmitted signal energy per
symbol, and T is the symbol duration.
Equivalently
( )
( )c
2
cos 2 f 2 1 , 0
4
0, elsewhere
i
E
t i t T
s t T
π
π
⎧ ⎡ ⎤
+ − ≤ ≤⎪ ⎢ ⎥= ⎨ ⎣ ⎦
⎪
⎩
( ) ( ) ( ) ( ) ( )c c
2 2
cos 2 1 cos 2 f sin 2 1 sin 2 f
4 4
i
E E
s t i t i t
T T
π π
π π
⎡ ⎤ ⎡ ⎤
= − − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

62
Defined a pair of quadrature carriers:
There are four message points, and the associated signal
vectors are defined by
( ) ( ) Tttf
T
t c ≤≤= 0,2cos
2
1 πφ
( ) ( ) Tttf
T
t c ≤≤= 0,2sin
2
2 πφ
( )
( )
( )
4,3,2,1,
4
12sin
4
12cos
=
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−−
⎟
⎠
⎞
⎜
⎝
⎛
−
= i
iE
iE
tsi
π
π

63
Each possible value of the phase corresponds to a unique dibit.
For example, we may choose the Gray coding.

64
Signal space diagram of coherent QPSK system

65

66
Phase-modulated signals (M-ary PSK)
The M signal waveforms are represented as:
Digital phase modulation is usually called phase-shift
keying (PSK).
( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
2 1 2
Re , 1,2,..., , 0
2
cos 2 1
2 2
cos 1 cos 2 sin 1 sin 2
c
m
j m M j f t
c
c c
s t
g t e e m M t T
g t f t m
M
g t m f t g t m f t
M M
π π
π
π
π π
π π
−
⎡ ⎤= = ≤ ≤⎣ ⎦
⎡ ⎤
= + −⎢ ⎥
⎣ ⎦
= − − −

67
Signal space diagram for octaphase shift keying (i.e., M=8)

68
The signal waveforms have equal energy:
The signal waveforms may be represented as a linear
combination of two orthonormal signal waveforms:
The two-dimensional vectors sm=[sm1 sm2] are given by:
( ) ( )2 2
0 0
1 1
2 2
T T
m gs t dt g t dtε ε= = =∫ ∫
( ) ( ) ( )1 1 2 2m m ms t s f t s f t= +
( ) ( ) ( ) ( )1 2
2 2
cos2 and sin 2c c
g g
f t g t f t f t g t f tπ π
ε ε
= = −
( ) ( )
2 2 2 2
cos 1 sin 1 , 1,2,...,m
g g
s m m m M
M M
π π
ε ε
⎡ ⎤
= − − =⎢ ⎥
⎢ ⎥⎣ ⎦

69
The Euclidean distance between signal points is:
The minimum Euclidean distance corresponds to the case in
which |m-n|=1, i.e., adjacent signal phases.
Average probability of symbol error for coherent M-ary PSK:
( )
( ) ( ) ( )
1 2
2 2
1 1 2 2
2
1 cos
e
mn m n
m n m n g
d s s
s s s s m n
M
π
ε
= −
⎧ ⎫⎡ ⎤
= − + − = − −⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
( )
min
2
1 cose
gd
M
π
ε
⎛ ⎞
= −⎜ ⎟
⎝ ⎠
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
≈
MN
E
Pe
π
sinerfc
0

70
Phase-modulated signals (Offset QPSK)
The carrier phase changes by ±180 degrees whenever both the
in-phase and quadrature components of the QPSK signal
changes sign.
This can result in problems for power amplifiers.
The problem may be reduced by using offset QPSK.
In offset QPSk, the bit stream responsible for generating the
quadrature component is delayed (i.e. offset) by half a symbol
interval with respect to the bit stream responsible for
generating the in-phase component.
( ) ( ) Tttf
T
t c ≤≤= 0,2cos
2
1 πφ

71
The two basis functions of offset QPSK are defined by
The phase transitions likely to occur in offset QPSK are
confined to ±90 degrees.
However, ±90 degrees phase transitions in offset QPSK
occur twice as frequently.
( ) ( ) Tttf
T
t c ≤≤= 0,2cos
2
1 πφ
( ) ( )
2
3
2
,2sin
2
2
T
t
T
tf
T
t c ≤≤= πφ

72
Amplitude fluctuations in offset QPSK due to filtering have
a smaller amplitude than in the case of QPSK.
The offset QPSK has exactly the same probability of symbol
error in an AWGN channel as QPSK.
The reason for the equivalence is that the statistical
independence of the in-phase and quadrature components
applies to both QPSK and offset QPSK.

73

74
Phase-modulated signals (π/4-Shifted QPSK)
The carrier phase used for the transmission of successive
symbols is alternately picked from one of the two QPSK
constellations in the following figure and then the other.

75
It follows that a π/4-shifted QPSK signal may reside in any
one of eight possible phase states:

76
Attractive features of the π/4-shifted QPSK scheme
The phase transitions from one symbol to the next are
restricted to ±π/4 and ±3π/4.
Envelope variations due to filtering are significantly
reduced.
π/4-shifted QPSK signals can be noncoherently detected,
thereby considerably simplifying the receiver design.
Like QPSK signals, π/4-shifted QPSK can be differently
encoded, in which case we should really speak of π/4-
shifted DQPSK .

77
Dual-Carrier Modulation (DCM)
Adopted in Multi-band OFDM (Ultra Wideband)
The coded and interleaved binary serial input data, b[i] where
i = 0, 1, 2, …, shall be divided into groups of 200 bits and
converted into 100 complex numbers using a technique called
dual-carrier modulation.
The conversion shall be performed as follows:
1.The 200 coded bits are grouped into 50 groups of 4 bits.
Each group is represented as (b[g(k)], b[g(k)+1], b[g(k) +
50)], b[g(k) + 51]), where k ∈ [0, 49] and
( )
[ ]
[ ]
0,242
25,492 50
kk
g k
kk
⎧ ∈⎪
= ⎨
∈+⎪⎩

78
2. Each group of 4 bits (b[g(k)], b[g(k)+1], b[g(k) + 50)], b[g(k)
+ 51]) shall be mapped onto a four-dimensional constellation,
and converted into two complex numbers (d[k], d[k + 50]).
3. The complex numbers shall be normalized using a
normalization factor KMOD.
The normalization factor KMOD = 10-1/2 is used for the
dual-carrier modulation.
An approximate value of the normalization factor may
be used, as long as the device conforms to the
modulation accuracy requirements.

79

80
Dual-Carrier Modulation (DCM) Encoding Table

81
Quadrature amplitude modulation (QAM)
Quadrature PAM or QAM: The bandwidth efficiency of
PAM/SSB can also be obtained by simultaneously
impressing two separate k-bit symbols from the information
sequence {an} on two quadrature carriers cos2πfct and
sin2πfct.
The signal waveforms may be expressed as:
( ) ( ) ( )
( ) ( )
2
Re , 1,2,..., , 0
cos2 sin 2
cj f t
m mc ms
mc c ms c
s t A jA g t e m M t T
A g t f t A g t f t
π
π π
⎡ ⎤= + = ≤ ≤⎣ ⎦
= −
( ) ( ) ( ) ( )
( )
2
2 2 1
Re cos 2
and tan
m cj j f t
m m m c m
m mc ms m ms mc
s t V e g t e V g t f t
V A A A A
θ π
π θ
θ −
⎡ ⎤= = +⎣ ⎦
= + =

82
Quadrature amplitude modulation (QAM) (cont.)
We may select a combination of M1-level PAM and M2-
phase PSK to construct an M=M1M2 combined PAM-PSK
signal constellation.

83
As in the case of PSK signals, the QAM signal waveforms
may be represented as a linear combination of two
orthonormal signal waveforms f1(t) and f2(t):
The Euclidean distance between any pair of signal vectors is:
( ) ( ) ( )1 1 2 2m m ms t s f t s f t= +
( ) ( ) ( ) ( )1 2
2 2
cos 2 and sin 2c c
g g
f t g t f t f t g t f tπ π
ε ε
= = −
[ ]1 2
1 1
2 2
m m m mc g ms gs s s A Aε ε
⎡ ⎤
= = ⎢ ⎥
⎣ ⎦
( )
( ) ( )
2 21
2
e
mn m n g mc nc ms nsd s s A A A Aε ⎡ ⎤= − = − + −
⎣ ⎦

84
Several signal space diagrams for rectangular QAM:
( )
min 2e
gd d ε=

85
Multidimensional signals:
We may use either the time domain or the frequency domain
or both in order to increase the number of dimensions.
Subdivision of time and frequency axes into distinct slots:

86
Orthogonal multidimensional signals
Consider the construction of M equal-energy orthogonal
signal waveforms that differ in frequency:
This type of frequency modulation is called frequency-shift
keying (FSK).
( ) ( )
[ ]
( )
2
2
Re , 1,2,..., , 0
2
cos 2 2
2
, 1,2,..., , 0
cj f t
m lm
c
j m f t
lm
s t s t e m M t T
f t m f t
T
s t e m M t T
T
π
π
ε
π π
ε Δ
⎡ ⎤= = ≤ ≤⎣ ⎦
= + Δ
= = ≤ ≤

87
Orthogonal multidimensional signals (cont.)
These waveforms have equal cross-correlation coefficients:
Note that Re(ρkm)=0 when Δf=1/2T and m≠k.
| ρkm |=0 for multiple of 1/T.
( ) ( )
( )
( )2
0
sin2
2
T j m k ft j T m k f
km
T m k fT
e dt e
T m k f
π ππε
ρ
ε π
− Δ − Δ− Δ
= =
− Δ∫
( ) ( )[ ]
( )
( )[ ]
( )[ ]
( ) fkmT
fkmT
fkmT
fkmT
fkmT
kmr
Δ−
Δ−
=
Δ−
Δ−
Δ−
=≡
π
π
π
π
π
ρρ
2
2sin
cos
sin
Re
4.2-45

88
Orthogonal
multidimensional signals
(cont.)
Cross-correlation
coefficient as a function
of frequency separation
for FSK signals:

89
For Δf=1/2T, the M-FSK signals are equivalent to the N-
dimensional vectors:
The distance between pairs of signals is:
[ ]
[ ]
[ ]ε
ε
ε
0000
0000
0000
2
1 …
=
=
=
Ns
s
s
( )
2 , for all ,e
kmd m kε=

90
Orthogonal signals for M=N=3 and M=N=2.

91
Binary Frequency-Shift Keying
In a binary FSK system, symbols 1 and 0 are distinguished
from each other by transmitting one of two sinusoidal waves
that differ in frequency by a fixed amount.
The transmitted frequency is
( )
( )
2
cos 2 , 0
0, elsewhere
b
i b
i b
E
f t t T
s t T
π
⎧
≤ ≤⎪
= ⎨
⎪
⎩
2,1andintegerfixedsomefor =
+
= in
T
in
f c
b
c
i

92
The FSK signal is a continuous phase signal in the sense that
phase continuity is always maintained, including the inter-bit
switching times.
This form of digital modulation is an example of continuous-
phase frequency-shift keying (CPFSK).
The signal s1(t) and s2(t) are orthogonal
( ) ( )
⎪
⎩
⎪
⎨
⎧
≤≤
=
elsewhere,0
0,2cos
2
ti
bi
b
Tttf
T
π
φ

93
( ) ( )
( ) ( )
⎩
⎨
⎧
≠
=
=
=
=
∫
∫
ji
jiE
dttf
T
tf
T
E
dttss
b
i
b
T
i
b
b
T
iij
b
b
,0
,
2cos
2
2cos
2
t
0
j
0
ππ
φ
⎥
⎦
⎤
⎢
⎣
⎡
=
0
1
bE
s ⎥
⎦
⎤
⎢
⎣
⎡
=
bE
s
0
2

94
Biorthogonal signals
A set of M biorthogonal signals can be constructed from M/2
orthogonal signals by simply including the negatives of the
orthogonal signals.
The correlation between any pair of waveforms is either
ρr=-1 or 0.
Signal space diagrams for M=4 and M=6 biorthogonal signals.

95
Simplex signals
For a set of M orthogonal waveforms {sm(t)} or their vector
representation {sm} with mean of:
Simplex signals are obtained by translating the origin of the
m orthogonal signals to the point .
The energy per waveform is:
1
1 M
m
m
s s
M =
= ∑
'
, 1,2,...,m ms s s m M= − =
s
22' 2 1 1
1m ms s s
M M M
ε ε ε ε
⎛ ⎞
= − = − + = −⎜ ⎟
⎝ ⎠

96
Simplex signals (cont.)
The cross correlation of any
pair of signals is (4.2-27):
The set of simplex
waveforms is equally
correlated and requires less
energy, by the factor 1-1/M,
than the set of orthogonal
waveforms.
( )
' '
' '
1
Re
1 1
1
1
m n
mn
m n
s s M
Ms s
M
ρ
⋅ −
= =
−
= −
−
Signal space diagrams for
M-ary simplex signals.

97
Signal waveforms from binary codes
A set of M signaling waveforms can be generated from a set
of M binary code words of the form:
Each component of a code word is mapped into an
elementary binary PSK waveform:
where Tc=T/N and εc=ε/N.
[ ]1 2 , 1,2,..., , 0 or 1.m m m mN mjC c c c m M c= = =
( )
( )
2
1 cos 2 , 0
2
0 cos 2 , 0
c
mj mj c c
c
c
mj mj c c
c
c s t f t t T
T
c s t f t t T
T
ε
π
ε
π
= ⇒ = ≤ ≤
= ⇒ = − ≤ ≤

98
Signal waveforms from binary codes (cont.)
The M code words {Cm} are mapped into a set of M
waveforms {sm(t)}.
The waveforms can be represented in vector form as:
N is called the block length of the code and is also the
dimension of the M waveforms.
[ ]1 2 , 1,2,..., , where / .m m m mN mjs s s s m M s Nε= = = ±
Signal space diagrams for signals
generated from binary codes.

99
Signal waveforms from binary codes (cont.)
Each of the M waveforms has energy ε.
Any adjacent signal points have a cross-correlation coefficient:
The corresponding distance is:
( )
N
NN
r
221 −
=
−
=
ε
ε
ρ
( )
( )2 1
4 /
e
rd
N
ε ρ
ε
= −
=

100
There are some modulation signals with dependence between
the signals transmitted in successive symbol intervals.
This signal dependence is usually introduced for the purpose of
shaping the spectrum of the transmitted signal so that it matches
the spectral characteristics of the channel.
Examples of baseband signals and the corresponding data sequence

101
NRZ : the binary information digit 1 is represented by a
rectangular pulse of polarity A and the binary digit 0 is
represented by a rectangular pulse of polarity –A.
The NRZ modulation is memoryless and is equivalent to a
binary PAM or a binary PSK signal in a carrier-modulated
system.
NRZI: the signal is different from the NRZ signal in that
transitions from one amplitude level to another occur only when
a 1 is transmitted.
This type of signal encoding is called differential encoding.
The encoding operation is described by the relation:
where {ak} is the binary information sequence into the
encoder, {bk} is the output sequence of the encoder.
1−⊕= kkk bab

102
NRZI (cont.)
When bk=1, the transmitted waveform is a rectangular pulse
of amplitude A, and when bk=0, the transmitted waveform is
a rectangular pulse of amplitude –A.
The differential encoding operation introduces memory in
the signal.
The combination of the encoder and the modulator
operations may be represented by a state diagram (Markov
chain):
Input bit
輸入0不會改變狀態，輸入1才會改變

103
NRZI (cont.)
The state diagram may be described by two transition
matrices corresponding to the two possible input bits {0,1}.
When ak=0, the encoder stays in the same state and the state
transition matrix for a zero is:
where tij=1 if ak results in a transition from state i to state j,
i=1,2, and j=1,2; otherwise tij=0.
The state transition matrix for ak=1 is:
⎥
⎦
⎤
⎢
⎣
⎡
=
10
01
1T
⎥
⎦
⎤
⎢
⎣
⎡
=
01
10
2T

104
NRZI (cont.)
The trellis diagram for the NRZI signal:
Delay modulation is equivalent to encoding the data
sequence by a run-length-limited code called a Miller
code and using NRZI to transmit the encoded data
(will be shown in Chapter 9).

105
Delay modulation (cont.):
The signal of delay modulation may be described by a state
diagram that has four states:
There are two elementary waveforms s1(t) and s2(t) and their
negatives -s1(t) and -s2(t), which are used for transmitting
the binary information.

106
Delay modulation
The state transition matrices can be obtained from the figure
in previous page:
When ak=0:
When ak=1:
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
0001
0001
1000
1000
1T
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
0100
0010
0100
0010
2T
1 2
3 2
2 3
4 3
s s
s s
s s
s s
→
→
→
→
3 1
4 1
1 4
2 4
s s
s s
s s
s s
→
→
→
→

107
Modulation techniques with memory such as NRZI and Miller
coding are generally characterized by a K-state Markov chain
with stationary state probabilities {pi, i=1,2,…,K} and
transition probabilities {pij, i,j=1,2,…,K}. Associated with
each transition is a signal waveform sj(t), j=1,2,…,K.
Transition probability matrix (P) can be arranged in matrix
form as:
11 12 1
21 22 2
1 2
K
K
K K KK
p p p
p p p
p p p
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦
P

108
The transition probability matrix can be obtained from the
transition matrices {Ti} and the corresponding probabilities of
occurrence of the input bits:
where q1=P(ak=0) and q2=P(ak=1).
For the NRZI signal with equal state probabilities p1=p2=0.5,
2
1
i i
i
q T
=
= ∑P
1 1
1 0 0 11 12 2
1 1 0 1 1 02 2
2 2
⎡ ⎤
⎢ ⎥ ⎡ ⎤ ⎡ ⎤
= = ⋅ + ⋅⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦⎢ ⎥
⎢ ⎥⎣ ⎦
P

109
The transition probability matrix for the Miller-coded signal
with equally likely symbols (q1=q2=0.5 or, equivalently,
p1=p2=p3=p4=0.25) is:
The transition probability matrix is useful in the determination
of the spectral characteristics of digital modulation techniques
with memory.
1 1
0 0
2 2
1 1
0 0
2 2
1 1
0 0
2 2
1 1
0 0
2 2
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
= ⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦
P

110
Introduction
In this section, we consider a class of digital modulation
methods in which the phase of the signal is constrained to be
continuous.
This constraint results in a phase or frequency modulator
that has memory.
The modulation method is also non-linear.
Continuous-phase FSK (CPFSK)
A conventional FSK signal is generated by shifting the
carrier by an amount
to reflect the digital information that is being transmitted.
This type (conventional type) of FSK signal is memoryless.
1
2 , 1, 3, , ( 1),n n nf f I I M= Δ = ± ± ± −…
4.3.3 Non-linear Modulation Methods
with Memory---CPFSK and CPM

111
Continuous-phase FSK (CPFSK) (cont.)
The switching from one frequency to another may be
accomplished by having M=2k separate oscillators tuned to
the desired frequencies and selecting one of the M
frequencies according to the particular k-bit symbol that is to
be transmitted in a signal interval of duration T=k/R seconds.
The reasons why we have CPFSK: (or the defects of
conventional FSK)
Such abrupt switching from one oscillator output to
another in successive signaling intervals results in
relatively large spectral side lobes outside of the main
spectral band of the signal.
Consequently, this method requires a large frequency band
for transmission of the signal.

112
Solution:
To avoid the use of signals having large spectral side lobes,
the information-bearing signal frequency modulates a
single carrier whose frequency is changed continuously.
The resulting frequency-modulated signal is phase-
continuous and, hence, it is called continuous-phase FSK
(CPFSK) .
This type (continuous-phase type) of FSK signal has memory
because the phase of the carrier is constrained to be continuous.

113
In order to represent a CPFSK signal, we begin with a PAM
signal:
d(t) is used to frequency-modulate the carrier.
{In} denotes the sequence of amplitudes obtained by mapping
k-bit blocks of binary digits from the information sequence
{an} into the amplitude levels ±1,±3,…,±(M-1).
g(t) is a rectangular pulse of amplitude 1/2T and duration T
seconds.
( ) ( )∑ −=
n
n nTtgItd

114
Equivalent low-pass waveform v(t) is expressed as
fd is the peak frequency deviation, φ0 is the initial phase
of the carrier.
The carrier-modulated signal may be expressed as
where φ(t;I) represents the time-varying phase of the carrier.
( ) ( )
⎭
⎬
⎫
⎩
⎨
⎧
⎥⎦
⎤
⎢⎣
⎡ += ∫ ∞−
04exp
2
φττπ
ε t
d ddTfj
T
tv
( ) ( ) 0
2
cos 2 ;Ics t f t t
T
ε
π φ φ= + +⎡ ⎤⎣ ⎦

115
Note that, although d(t) contains discontinuities, the integral of d(t) is
continuous. Hence, we have a continuous-phase signal.
θn represents the accumulation (memory) of all symbols up to time
nT ?.
Parameter h is called the modulation index.
( ) ( ) ( )
( )
( )
1
;I 4 4
2 2
2
t t
d d n
n
n
d k d n
k
n n
t Tf d d Tf I g nT d
f T I f t nT I
hI q t nT
φ π τ τ π τ τ
π π
θ π
−∞ −∞
−
=−∞
⎡ ⎤
= = −⎢ ⎥
⎣ ⎦
= + −
= + −
∑∫ ∫
∑
Tfh d2=
∑
−
−∞=
=
1n
k
kn Ihπθ ( )
( )
( )
( )
0 0
2 0
1 2
t
q t t T t T
t T
<⎧
⎪
= ≤ ≤⎨
⎪
>⎩
nT≤t≤(n+1)T

116
Continuous-phase modulation (CPM)
CPFSK becomes a special case of a general class of
continuous-phase modulated (CPM) signals in which the
carrier phase is
when hk=h for all k, the modulation index is fixed for all symbols.
when hk varies from one symbol to another, the CPM signal is called
multi-h. (In such a case, the {hk} are made to vary in a cyclic manner
through a set of indices.)
The waveform q(t) may be represented in general as the
integral of some pulse g(t), i.e.,
( ) ( ) ( ); 2 , 1
n
k k
k
t I I h q t kT nT t n Tφ π
=−∞
= − ≤ ≤ +∑
( ) ( )∫=
t
dgtq
0
ττ

117
Continuous-phase modulation (CPM) (cont.)
If g(t)=0 for t >T, the CPM signal is called full response CPM. (Fig a. b.)
If g(t)≠0 for t >T, the modulated signal is called partial response CPM.
(Fig c. d.)
LREC, L=1,
results in
CPFSK
LRC, L=1
LREC, L=2
LRC, L=2

118
The CPM signal has memory that is introduced through the phase
continuity.
For L>1, additional memory is introduced in the CPM signal by
the pulse g(t).
Three popular pulse shapes are given in the following table.
LREC denotes a rectangular pulse of duration LT.
LRC denotes a raised cosine pulse of duration LT.
Gaussian minimum-shift keying (GMSK) pulse with
bandwidth parameter B, which represents the -3 dB
bandwidth of the Gaussian pulse.

119
Some commonly used CPM pulse shapes
LREC
LRC
GMSK
( )
( )
( )
1
0 1
2
0 otherwise
LT
g t LT
⎧
≤ ≤⎪
= ⎨
⎪
⎩
( )
( )
( )
1 2
1 cos 0 1
2
0 otherwise
t
LT
LT LTg t
π⎧ ⎛ ⎞
− ≤ ≤⎪ ⎜ ⎟
= ⎝ ⎠⎨
⎪
⎩
( ) ( ) ( )
( )
2
1/ 2 1/2
/ 2
2 ln 2 2 ln 2
2 2
1
2
x
t
T T
g t Q B t Q B t
Q t e dt
π π
π
∞
−
⎧ ⎫⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞
= − − +⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥
⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎩ ⎭
= ∫

120
Minimum-shift keying (MSK).
MSK is a special form of binary CPFSK (and, therefore, CPM) in which
the modulation index h=1/2.
The phase of the carrier in the interval nT ≤ t ≤ (n+1)T is
The modulated carrier signal is
1
1
( ;I) ( )
2
1
, ( 1)
2
n
k n
k
n n
t I I q t nT
t nT
I nT t n T
T
φ π π
θ π
−
=−∞
= + −
−⎛ ⎞
= + ≤ ≤ +⎜ ⎟
⎝ ⎠
∑
1
( ) cos 2
2
1 1
cos 2 , ( 1)
4 2
c n n
c n n n
t nT
s t A f t I
T
A f I t n I nT t n T
T
π θ π
π π θ
⎡ − ⎤⎛ ⎞
= + + ⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞
= + − + ≤ ≤ +⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
From 4.3-54
From 4.3-52

121
Minimum-shift keying (MSK) (cont.)
The expression indicates that the binary CPFSK signal can be
expressed as a sinusoid having one of two possible frequencies
in the interval nT ≤ t ≤ (n+1)T. If we define these frequencies
as
Then the binary CPFSK signal may be written in the form
1
2
1
4
1
4
c
c
f f
T
f f
T
= +
= −
11
( ) cos 2 ( 1) , 1,2
2
i
i i ns t A f t n iπ θ π −⎡ ⎤
= + + − =⎢ ⎥
⎣ ⎦

122
Why binary CPFSK with h=1/2 is called minimum-shift
keying (MSK)?
Because the frequency separation ∆f =f2−f1=1/2T, and ∆f =1/2T
is the minimum frequency separation that is necessary to
ensure the orthogonality of the signals s1(t) and s2(t) over a
signaling interval of length T.
The phase in the nth signaling interval is the phase state of the
signal that results in phase continuity between adjacent
interval.

123
Minimum-shift keying (MSK) (Haykin)
Consider a continuous-phase frequency-shift keying (CPFSK)
signal, which is defined for the interval 0≤t≤Tb as follows:
Eb is the transmitted signal energy per bit.
Tb is the bit duration.
The phase θ(0), denoting the value of the phase at time t = 0,
sums up the past history of the modulation process up to time
t = 0.
( )
( )[ ]
( )[ ]⎪
⎪
⎩
⎪
⎪
⎨
⎧
+
+
=
0symbolfor02cos
2
1symbolfor02cos
2
2
1
θπ
θπ
tf
T
E
tf
T
E
ts
b
b
b
b

124
Another useful way of representing the CPFSK signal s(t) is
to express it in the conventional form of an angle-modulated
signal as follows:
θ(t) is the phase of s(t).
When the phase θ(t) is a continuous function of time, we find
that the modulated signal s(t) itself is also continuous at all
times, including the inter-bit switching times.
( ) ( )[ ]ttf
T
E
ts c
b
b
θπ += 2cos
2
(*)

125
The phase θ(t) of a CPFSK signal increases or decreases
linearly with time during each bit duration of Tb seconds
The plus (minus) sign corresponds to sending symbol 1 (0).
We can find that
We thus get
( ) ( )0 , 0 b
b
h
t t t T
T
π
θ θ= ± ≤ ≤
1
2
f
T
h
f
b
c =+
2
2
f
T
h
f
b
c =−
( )21
2
1
fffc += ( )21 ffTh b +=

126
h is referred to as the deviation ratio.
With h=1/2, the frequency deviation equals half the bit rate.
This is the minimum frequency spacing that allows the two
FSK signals representing symbols 1 and 0 to be coherently
orthogonal.
At time t = Tb:
This is to say, the sending of symbol 1 increases the phase of
a CPFSK signal s(t) by πh radians, whereas the sending of
symbol 0 reduces it by an equal amount.
( ) ( )
for symbol 1
0
for symbol 0
b
h
T
h
π
θ θ
π
⎧
− = ⎨
−⎩

127
Phase Trellis

128
Phase trellis of the sequence 1101000 (h =1/2)
From (*), we may express the CPFSK signal s(t):
( ) ( ) ( ) ( ) ( )
( ) ( )
2 2
cos cos 2 sin sin 2
cos 2 sin 2
b b
c c
b b
I c Q c
E E
s t t f t t f t
T T
S f t S f t
θ π θ π
π π
= −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
≡ − (**)

129
Consider first the in-phase component SI
With the deviation ratio h=1/2,
where the plus (minus) sign corresponds to symbol 1 (0).
A similar result holds for θ(t) in the interval –Tb ≤ t ≤ 0,
except that the algebraic sign is not necessarily the same
in both intervals.
Since the phase θ(0) is 0 or π, depending on the past
history of the modulation process, we find that, in the
interval –Tb ≤ t ≤ Tb, the polarity of cos[θ(t)] depends
only on θ(0), regardless of the sequence of 1s and 0s
transmitted before or after t = 0.
( ) ( ) b
b
Ttt
T
t ≤≤±= 0,
2
0
π
θθ

130
Thus, for –Tb ≤ t ≤ Tb, the in-phase component sI(t) consists
of a half-cycle cosine pulse defined as follows:
where the plus (minus) sign corresponds to θ(0) = 0
(θ(0)=π).
( ) ( ) ( )
2 2
cos cos 0 cos
2
2
cos ,
2
b b
I
b b b
b
b b
b b
E E
s t t t
T T T
E
t T t T
T T
π
θ θ
π
⎛ ⎞
≡ =⎡ ⎤ ⎡ ⎤ ⎜ ⎟⎣ ⎦ ⎣ ⎦
⎝ ⎠
⎛ ⎞
= ± − ≤ ≤⎜ ⎟
⎝ ⎠

131
Similarly, for 0 ≤ t ≤ 2Tb, the quadrature component sQ(t)
consists of a half-cycle sine pulse, whose polarity depends
only on θ(Tb), defined as follows:
where the plus (minus) sign corresponds to θ(Tb) = π/2
(θ(Tb) = -π/2).
( ) ( ) ( )
2 2
sin sin sin
2
2
sin , 0 2
2
b b
Q b
b b b
b
b
b b
E E
s t t T t
T T T
E
t t T
T T
π
θ θ
π
⎛ ⎞
≡ =⎡ ⎤ ⎡ ⎤ ⎜ ⎟⎣ ⎦ ⎣ ⎦
⎝ ⎠
⎛ ⎞
= ± ≤ ≤⎜ ⎟
⎝ ⎠

132
MSK signal may assume any one of four possible forms:
The phaseθ(0)=0 and θ(Tb)=π/2, corresponding to the
transmission of symbol 1.
The phaseθ(0)=π and θ(Tb)=π/2, corresponding to
the transmission of symbol 0.
The phaseθ(0)=π and θ(Tb)=-π/2, corresponding to
the transmission of symbol 1.
The phaseθ(0)=0 and θ(Tb)=-π/2, corresponding to the
transmission of symbol 0.

133
Define the orthonormal basis functions:
From (**), we have:
where the coefficients s1 and s2 are related to the phase states
θ(0) andθ(Tb), respectively.
( ) ( ) bc
bb
Tttft
TT
t ≤≤⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
= 0,2cos
2
cos
2
1 π
π
φ
( ) ( ) bc
bb
Tttft
TT
t ≤≤⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
= 0,2sin
2
sin
2
2 π
π
φ
( ) ( ) ( ) bTttststs ≤≤+= 0,2211 φφ

134
Both integrals are evaluated for a time interval equal to twice
the bit duration.
Both the lower and upper limits of the product integration
used to evaluate the coefficient s1 are shifted by the bit
duration Tb with respect to those used to evaluate the
coefficient s2.
The time interval 0≤t≤Tb, for which the phase states θ(0)
andθ(Tb) are defined, is common to both integrals.
( ) ( ) ( )1 1 cos 0 ,
b
b
T
b b bT
s s t t dt E T t Tφ θ
−
= = − ≤ ≤⎡ ⎤⎣ ⎦∫
( ) ( ) ( )
2
2 20
sin , 0 2
bT
b b bs s t t dt E T t Tφ θ= = − ≤ ≤⎡ ⎤⎣ ⎦∫

135
Signal space diagram for MSK system

136
In QPSK the transmitted symbol is represented by any one of
the four message points, whereas in MSK one of two
message points is used to represent the transmitted symbol at
any one time.

137

138
Coherent detection of the MSK signal
In the case of an AWGN channel, the received signal is
given by
where s(t) is the transmitted MSK signal, and w(t) is the
sample function of a white Gaussian noise process of zero
mean and power spectral density N0/2.
For the optimal decision of θ(0),
( ) ( ) ( )x t s t w t= +
( ) ( )1 1 1 1,
b
b
T
b bT
x x t t dt s w T t Tφ
−
= = + − ≤ ≤∫
( )
( )
1
1
ˆIf 0 0 0
ˆIf 0 0
x
x
θ
θ π
> ⇒ =
< ⇒ =

139
Coherent detection of the MSK signal (cont.)
For the optimum detection of θ(Tb),
Error probability of MSK
which is exactly the same as that for binary PSK and QPSK.
( ) ( )
2
2 2 2 20
, 0 2
bT
bx x t t dt s w t Tφ= = + ≤ ≤∫
( )
( )
2
2
ˆIf 0 / 2
ˆIf 0 / 2
b
b
x T
x T
θ π
θ π
> ⇒ = −
< ⇒ =
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
02
1
N
E
erfcP b
e

140
Block diagrams for MSK transmitter

141
Block diagrams for coherent MSK receiver

142
Power spectra of MSK signals
( )
( ) ( )
2
2 2 2
cos 232
2
2 16 1
g bb
B
b b
f T fE
S f
T T f
ψ π
π
⎡ ⎤ ⎡ ⎤
= =⎢ ⎥ ⎢ ⎥
−⎣ ⎦⎣ ⎦
MSK does not produce as much interference
outside the signal band of interest as QPSK.

143
MSK may also be represented as a form of four-phase PSK.
The equivalent low-pass digitally modulated signal in the form
g(t) is a sinusoidal pulse
[ ]2 2 1( ) ( 2 ) ( 2 )n n
n
v t I g t nT jI g t nT T
∞
+
=−∞
= − − − −∑
sin (0 2 )
( ) 2
0 (otherwise)
t
t T
g t T
π⎧
≤ ≤⎪
= ⎨
⎪⎩
Problem 4.14

144
This type of signal is viewed as a four-phase PSK signal in
which the pulse shape is one-half cycle of a sinusoid (0 ~ π).
The even-numbered binary-valued (±1) symbols {I2n} of the
information sequence {In} are transmitted via the cosine of the
carrier, while the odd-numbered symbols {I2n+1} are
transmitted via the sine of the carrier.
The transmission rate on the two orthogonal carrier
components is 1/2T bits/s so that the combined transmission
rate is 1/T bits/s.
Note that the bit transitions on the sine and cosine carrier
components are staggered or offset in time by T seconds.

145
For this reason, the signal
is called offset quadrature PSK (OQPSK) or staggered
quadrature PSK (SQPSK).
Figure in next page illustrates the representation of an MSK
signal as two staggered quadrature-modulated binary PSK
signals. The corresponding sum of the two quadrature
signals is a constant amplitude, frequency-modulated signal.
2
2 1
( ) ( 2 ) cos2
( 2 ) sin 2
n c
n
n c
n
s t A I g t nT f t
I g t nT T f t
π
π
∞
=−∞
∞
+
=−∞
⎧⎡ ⎤
= −⎨⎢ ⎥
⎣ ⎦⎩
⎫⎡ ⎤
+ − − ⎬⎢ ⎥
⎣ ⎦ ⎭
∑
∑

146

147
Compare the waveforms for MSK with OQPSK (pulse g(t)
is rectangular for 0≤t≤2T) and with conventional QPSK
(pulse g(t) is rectangular for 0≤t≤2T).
All three of the modulation methods result in identical
data rates.
The MSK signal has continuous phase.
The OQPSK signal with a rectangular pulse is basically
two binary PSK signals for which the phase transitions
are staggered in time by T seconds. Thus, the signal
contains phase jumps of ±90º.
The conventional four-phase PSK (QPSK) signal with
constant amplitude will contain phase jumps of ±180º or
±90º every 2T seconds.

148
Compare the waveforms for MSK with OQPSK and QPSK (cont.)

149

150
4.4 Spectral Characteristics of Digitally
Modulated Signals
In most digital communication systems, the available
channel bandwidth is limited.
The system designer must consider the constraints
imposed by the channel bandwidth limitation in the
selection of the modulation technique used to transmit
the information.
From the power density spectrum, we can determine the
channel bandwidth required to transmit the information-
bearing signal.

151
Beginning with the form
where ν(t) is the equivalent low-pass signal.
Autocorrelation function (from 4.1-50)
Power density spectrum (from 4.1-51)
First we consider the general form
where the transmission rate is 1/T = R/k symbols/s and {In}
represents the sequence of symbols.
2
( ) Re ( ) cj f
ss e π τ
υυφ τ φ τ⎡ ⎤= ⎣ ⎦
[ ]
1
( ) ( ) ( )
2
ss c cf f f f fυυ υυΦ = Φ − + Φ − −
( ) ( ) 2
Re cj f t
s t v t e π
⎡ ⎤= ⎣ ⎦
( ) ( )n
n
t I g t nTυ
∞
=−∞
= −∑
4.4.1 Power Spectra of Linearly
Modulation Signals
Random Variable
Deterministic

152
Autocorrelation function
We assume the {In} is WSS with mean μi and the autocorrelation
function
[ ]
[ ]∑ ∑
∞
−∞=
∞
−∞=
∗∗
∗
−+−=
+=+
n m
mn mTtgnTtgIIE
ttEtt
)()(
2
1
)()(
2
1
);(
τ
τυυτφυυ
1( )
2ii n n mm E I Iφ ∗
+
⎡ ⎤= ⎣ ⎦
( )( )
'
( ; ) ( ) ( ) ( ) let '
( ') ( ) ' let '
( ) ( ) ( )
ii
n m
ii
m n
ii
m n
t t m n g t nT g t mT m m n
m g t nT g t m n T m m
m g t nT g t nT mT
υυφ τ φ τ
φ τ
φ τ
∞ ∞
∗
=−∞ =−∞
∞ ∞
∗
=−∞ =−∞
∞ ∞
∗
=−∞ =−∞
+ = − − + − = −
= − + − + =
= − + − −
∑ ∑
∑ ∑
∑ ∑
Modulation Signals

153
The second summation
is periodic in the t variable with period T.
Consequently, φνν(t+τ;t) is also periodic in the t variable with
period T. That is
In addition, the mean value of v(t), which is
is periodic with period T.
);();( ttTtTt τφτφ υυυυ +=+++
( ) ( )
n
g t nT g t nT mTτ
∞
∗
=−∞
− + − −∑
[ ]( ) ( ) ( )n i
n n
E t E I g t nT g t nTυ μ
∞ ∞
=−∞ =−∞
⎡ ⎤
= − = −⎢ ⎥
⎣ ⎦
∑ ∑
Modulation Signals

154
Therefore v(t) is a stochastic process having a periodic mean and
autocorrelation function. Such a process is called a cyclostationary
process or a periodically stationary process in the wide sense.
In order to compute the power density spectrum of a cyclo-
stationary process, the dependence of φνν(t+τ;t) on the t
variable must be eliminated. Thus,
( )
2
2
2
2
2
2
1
( ) ( ; )
1
( ) ( ) ( )
1
( ) ( ) ( )
T
T
T
ii T
m n
T nT
ii T nT
m n
t t dt
T
m g t nT g t nT mT dt
T
m g t g t mT dt t t nT
T
υυ υυφ τ φ τ
φ τ
φ τ
−
∞ ∞
∗
−
=−∞ =−∞
∞ ∞ −
∗
− −
=−∞ =−∞
= +
= − + − −
′ ′ ′ ′= + − = −
∫
∑ ∑ ∫
∑ ∑ ∫
Modulation Signals

155
We interpret the integral as the time-autocorrelation function of
g(t) and define it as
Consequently,
The (average) power density spectrum of v(t) is in the form
where G( f ) is the Fourier transform of g(t), and Φii( f ) denotes
the power density spectrum of the information sequence
∫
∞
∞−
∗
+= dttgtggg )()()( ττφ
∑
∞
−∞=
−=
m
ggii mTm
T
)()(
1
)( τφφτφυυ
21
( ) ( ) ( )iif G f f
T
υυΦ = Φ
∑
∞
−∞=
−
=Φ
m
fmTj
iiii emf π
φ 2
)()(
convolution
Applying equation 2.2-27
Modulation Signals

156
The result illustrates the dependence of the power density
spectrum of v(t) on the spectral characteristics of the pulse g(t)
and the information sequence {In}.
That is, the spectral characteristics of v(t) can be controlled by (1)
design of the pulse shape g(t) and by (2) design of the correlation
characteristics of the information sequence.
Whereas the dependence of Φνν( f ) on G( f ) is easily
understood upon observation of equation, the effect of the
correlation properties of the information sequence is more subtle.
First of all, we note that for an arbitrary autocorrelation φii(m)
the corresponding power density spectrum Φii( f ) is periodic in
frequency with period 1/T. (see next page)
Modulation Signals

157
In fact, the expression relating the spectrum Φii(f ) to the
autocorrelation φii(m) is in the form of an exponential Fourier
series with the {φii(m)} as the Fourier coefficients.
Second, let us consider the case in which the information symbols
in the sequence are real and mutually uncorrelated. In this case,
the autocorrelation function φii(m) can be expressed as (applying
2.2-5 and 2.2-6)
where denotes the variance of an information symbol.
∫−
Φ=
T
T
fmTj
iiii dfefTm
21
21
2
)()( π
φ
⎪⎩
⎪
⎨
⎧
≠
=+
=
)0(
)0(
)( 2
22
m
m
m
i
ii
ii
μ
μσ
φ
2
iσ
Modulation Signals

158
Substitute for φii(m) in equation, we obtain
The desired result for the power density spectrum of v(t) when
the sequence of information symbols is uncorrelated.
2
2 2 2
2
2
( ) ( )
( )
m
j fmT
ii ii
m
j f T
i i
m
i
i
m
f m e
e
m
f
T T
π
π
φ
σ μ
μ
σ δ
∞
−
=−∞
∞
−
=−∞
∞
=−∞
Φ =
= +
= + −
∑
∑
∑
It may be viewed as
the exponential Fourier
series of a periodic
train of impulses with
each impulse having an
area 1/T.
∑
∞
−∞=
−+=Φ
m
ii
T
m
f
T
m
G
T
fG
T
f )()()()(
2
2
2
2
2
δ
μσ
υυ
( )
1
2
where
sjn t
s
n ns
s
s
t nT e
T
T
ω
δ
π
ω
∞ ∞
=−∞ =−∞
− =
=
∑ ∑
Modulation Signals

159
The expression for the power density spectrum is purposely
separated into two terms to emphasize the two different types of
spectral components.
The first term is the continuous spectrum, and its shape depends
only on the spectral characteristic of the signal pulse g(t).
The second term consists of discrete frequency components
spaced 1/T apart in frequency. Each spectral line has a power that
is proportional to |G( f )|2 evaluated at f = m/T.
Note that the discrete frequency components vanish when the
information symbols have zero mean, i.e., μi=0. This condition
is usually desirable for the digital modulation techniques under
consideration, and it is satisfied when the information symbols are
equally likely and symmetrically positioned in the complex plane.
Modulation Signals

160
Example 4.4-1 To illustrate the spectral shaping resulting from
g(t), consider the rectangular pulse shown in figure. The Fourier
transform of g(t) is
Hence
Thus
fTj
e
fT
fT
ATfG π
π
π −
=
sin
)(
2
22 sin
)()( ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
fT
fT
ATfG
π
π
)(
sin
)( 22
2
22
fA
fT
fT
TAf ii δμ
π
π
συυ +⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=Φ Decays inversely as the
square of the frequency
Modulation Signals
zero

161
Example 4.4-2 As a second illustration of the spectral shaping
resulting from g(t), we consider the raised cosine pulse
its Fourier transform is:
Tt
T
t
T
A
tg ≤≤⎥
⎦
⎤
⎢
⎣
⎡
⎟
⎠
⎞
⎜
⎝
⎛
−+= 0,
2
2
cos1
2
)(
π
fTj
e
TffT
fTA
fG π
π
π −
−
=
)1(
sin
2
)( 22
nonzero
Decays inversely as the f 6
Modulation Signals

162
Example 4.4-3 To illustrate that spectral shaping can also be
accomplished by operations performed on the input information
sequence, we consider a binary sequence {bn} from which we
form the symbols
The {bn} are assumed to be uncorrelated random variables,
each having zero mean and unit variance. Then the
autocorrelation function of the sequence {In} is
1−+= nnn bbI
( )( )
( )
( )
1 1
2 2
1 1
2
1 1 1 1
2
1 2 1 1 2
( ) ( )
2 0
1
ii n n m n n n m n m
n n n n
n n n n n n n
n n n n n n n
m E I I E b b b b
E b b b b m
E b b b b b b b m
E b b b b b b b
φ + − + + −
− −
+ − + −
− − − − −
⎡ ⎤= = + +⎣ ⎦
⎡ ⎤+ + =⎣ ⎦
⎡ ⎤+ + + = +⎣ ⎦=
⎡ ⎤+ + +⎣ ⎦ ( )
[ ] ( )1 1 1 1
2 ( 0)
1 ( 1)
-1 0 (otherwise)
otherwisen n m n n m n n m n n m
m
m
m
E b b b b b b b b+ + − − + − + −
⎧
⎪ =⎧⎪⎪ ⎪
= = ±⎨ ⎨
⎪ ⎪=
⎩⎪
+ + +⎪⎩
E[XiXj]=E[Xi]E[Xj]
E[(Xi-0)2]=1
Modulation Signals

163
Example 4.4-3 (cont.)
Hence, the power density spectrum of the input sequence is (from
4.4-13)
and the corresponding power density spectrum for the (low-pass)
modulated signal is (from 4.4-12)
2
2
( ) ( )
2(1 cos2 )
4cos
j fmT
ii ii
m
f m e
fT
fT
π
φ
π
π
∞
−
=−∞
Φ =
= +
=
∑
fTfG
T
f πυυ
22
cos)(
4
)( =Φ
Modulation Signals

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