4 Characterization of Communication Signals and Systems
4.1 Representation of band-pass signals and systems
(1). 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).
(2). 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.
4.1.1 Representation of band pass signals
(1). A real-valued signal s (t ) is called a band pass signal if it has a frequency content
concentrated in a narrow band of frequencies in the vicinity of a frequency f c .
(2). Consider a signal that contains only the positive frequencies in s (t ) , s (t ) , called the
analytic signal or the pre-envelope of s (t ) .
(Why the positive frequencies are considered? This is that the negative frequency does not
exist in the real world.)
(3). Viewed from the frequency domain, the Fourier transform of s (t )
S ( f )  F  s (t )  2u( f )S ( f )
(4.1-1)
where
1,
S ( f )  F [ s(t )]; u( f )  
0,
f 0
f 0
Then the pre-envelope of s (t ) is expressed by

s (t ) 
 S ( f )e

j 2 ft
df  F 1[2u( f )]  F 1[ S ( f )]
(4.1-2)
j
t
(4.1-3)

But since
F 1[2u( f )]   (t ) 
then
j
j

s (t )   (t )    s(t )  s(t )   s(t )
t 
t

(4.1-4)
Define the Hilbert transform of s (t ) as

sˆ(t ) 
1
1 s( )
 s (t )  
d
t
  t  
(4.1-5)
where
1
, t 
t
The filter h(t ) is called the Hilbert transformer.
The frequency response of this filter is
h (t ) 

H ( f )  F [[h(t )] 
 h (t )e
 j 2 ft

  j, f  0
1  j 2 ft

dt   e
dt   0, f  0
  t
 j, f  0

1
(4.1-6)

(4.1-7)
The observations about this filter are as follows:
(1). H ( f )  1,    f  
 
  ,
(2).  ( f )   2
,
 2
That is, the role of this filter is like a 90
signal.
f 0
f 0
phase shifter for all frequencies in the input
An equivalent low pass representation of band pass analytic signal s (t ) is defined as
Sl ( f )  S ( f  f c )
(4.1-8)
and
sl (t )  s (t )e  j 2 fct   s(t )  jsˆ(t )  e  j 2 fct
(4.1-9)
or equivalently,
s(t )  jsˆ(t )  sl (t )e j 2 fct
In general, sl (t ) is complex-valued
sl (t )  x(t )  jy (t )
(4.1-10)
(4.1-11)
Additional relations are as
s(t )  x(t )cos(2 f ct )  y(t )sin(2 f ct )
(4.1-12)
sˆ(t )  x(t )sin(2 f ct )  y(t )cos(2 f ct )
(4.1-13)
Notes:
(1). x (t ), y (t ) may be viewed as amplitude modulations impressed on the carrier
components, cos(2 f ct ), sin(2 f ct ) .
(2). Since the two carriers are in phase quadrature, x ( t ), y (t ) are called the quadrature
components of s (t ) .
s(t )  Re  x (t )  jy (t )  e j 2 fct   Re  sl (t )e j 2 f ct 
(4.1-14)
The low pass signal sl (t ) is usually called the complex envelope of the real signal s (t ) .
The polar representation of the low pass signal sl (t ) is
sl (t )  a (t )e j ( t )
(4.1-15)
a(t )  x 2 (t )  y 2 (t )  sl (t )
(4.1-16)
where
y (t )
x (t )
 (t )  tan 1
(4.1-17)
Therefore, the signal s (t ) can be expressed as
s(t )  Re sl (t )e j 2 f ct   Re a (t )e j (2 f ct  ( t ))   a (t ) cos(2 f ct   (t ))
(4.1-18)
where
a ( t ) is the envelope of s (t ) .
 ( t ) is the phase of s (t ) .

S ( f )  F [ s(t ) 
 Res (t )e
j 2 f ct
l
e
 j 2 ft
dt
(4.1-19)

Use of the identity
1
Re[ ]  [    ]
2
(4.1-20)
The Fourier transform of s (t ) is

S( f ) 
1
1
 sl (t )e j 2 fct  sl  (t )e  j 2 fct e  j 2 ft dt   Sl ( f  f c )  Sl  (  f  f c )  (4.1-21)

2 
2
Another important item is the energy of the signal s (t )

E


s 2 (t )dt 

 Re[s (t )e
l
j 2 f ct
] dt
2
(4.1-22)

Or

E

1
1
2
2
sl (t ) dt   sl (t ) cos(4 f ct  2 (t ))dt

2 
2 
(4.1-23)
The second term in the right hand side of the above equation approaches zero, therefore, the
energy of s (t )


1
1
2
E   sl (t ) dt   a 2 (t )dt
2 
2 
(4.1-24)
4.1.2 Representation of linear band pass systems
A band pass linear filter or system h(t ) with real value has
H  (  f )  H ( f ),
where
(4.1-25)
H ( f )  F [h(t )]
Define
 H ( f ), f  0
Hl ( f  fc )  
f 0
0,
(4.1-26)
f 0
0,
H l  ( f  fc )   
 H (  f ), f  0
(4.1-27)
then
Hint: from (4-1-26)
H l ( f  f c )  H ( f ), f  0  H l (  f  f c )  H (  f ),  f  0 ; Take complex conjugate both
sides, we have (4.1.27).
From (4.1.25), we have
H ( f )  H l ( f  f c )  H l  ( f  f c )
(4.1-28)
and
h(t )  F [ H ( f )]  hl (t )e j 2 fct  hl  (t )e  j 2 fct
(4.1-29)
 2 Re[hl (t )e j 2 fct ]
Discussion:
You may have seen that (4.1.28) has no “one half” constant, but (4.1.21) does have. In fact,
the difference stems from the definition of the analytic signal in (4.1.1); the amplitude of the
Fourier transform of s (t ) is intentionally doubled that of s (t ) , and that is why when we
try to evaluate S ( f ) in terms of Sl ( f ) we need to add the reverse operation by dividing
the result by two.
4.1.3 Response of a band pass system to a band pass signal
Consider the output of the band pass system is expressed in terms of its corresponding
equivalent low pass from as
r (t )  Re rl (t )e j 2 fct 
(4.1-30)
where

r (t )  s(t )  h(t ) 
 s( )h(t   )d
(4.1-31)

Viewed from the frequency domain
R( f )  S ( f ) H ( f )
From (4.1.21) and (4.1.28), we obtain
R( f ) 
(4.1-32)
1
 Sl ( f  f c )  Sl  (  f  f c )   H l ( f  f c )  H l  (  f  f c ) 
2
(4.1-33)
Using the fact that
Sl ( f  f c )  0, H l ( f  f c )  0, f  0 , since both the input signal and the impulse response
are narrow-band, as a result
Sl ( f  f c ) H l  (  f  f c )  0, and Sl  (  f  f c ) H l ( f  f c )  0
Finally, (4.1.33) can be simplified as
1
R( f )   Sl ( f  f c ) H l ( f  f c )  Sl  (  f  f c ) H l  (  f  f c ) 
2
(4.1-34)
1

  Rl ( f  f c )  Rl (  f  f c ) 
2
where
(4.1-35)
Rl ( f )   Sl ( f ) H l ( f )
The counterpart of Rl ( f ) in the time domain is

rl (t )  sl (t )  hl (t ) 
 s ( )h (t   )d
l
l
(4.1-36)

For the sake of convenience to illustrate the correspondence between band pass and lowpass
representations, we end with the following figure.
Bandpass representation
s (t )
h(t ), H ( f )
r (t )  s (t )  h (t )
Equivalent lowpass representation
sl (t )
Time domain
hl (t ), H l ( f )
rl (t )  sl (t )  hl (t )
Frequency domain
s(t )  Re sl (t )e j 2 fct  (4.1.18)
h(t )  2 Re[hl (t )e j 2 fct ] (4.1.29)
r (t )  Re rl (t )e j 2 fct (4.1.30)

s (t ) 
 S ( f )e

j 2 ft
df (4.1.2)
1
 Sl ( f  f c )  Sl  (  f  f c )  (4.1.21)
2
H ( f )  H l ( f  f c )  H l  (  f  f c ) (4.1.28)
1
R( f )   Rl ( f  f c )  Rl  (  f  f c )  (4.1.34)
2
S( f ) 
S ( f )  F  s (t )  2u( f )S ( f ) (4.1.1)

sl (t )  s (t )e  j 2 fct
Sl ( f )  S ( f  f c )
4.1.4 Representation of band pass stationary stochastic processes
Suppose n(t ) is a sample function of a WSS with zero-mean and PSD  nn ( f ) . It relates
with the equivalent low pass form as
n(t )  a(t )cos[2 f ct   (t )]
 x(t ) cos(2 f ct )  y (t )sin(2 f ct )
(4.1-37)
(4.1-38)
 Re[ z(t )e j 2 fct ]
(4.1-39)
where
(1). z (t ) is the complex envelope of the real-valued R.P. n(t )
(2). x (t ), y (t ) are the quadrature components of n(t ) .
(3). E [n(t )]  0 , therefore, E[ x (t )]  E[ y (t )]  0
Since n(t ) is WSS, we have the following properties:
 xx ( )   yy ( )
 xy ( )   yx ( )
(4.1-40)
(4.1-41)
The autocorrelation of n(t ) is
nn ( )  E[n(t )n(t   )]
 E{[ x (t ) cos 2 f ct  y (t )sin 2 f ct ]
[ x (t   ) cos 2 f c (t   )  y (t   )sin 2 f c (t   )]}
  xx ( ) cos 2 f ct cos 2 f c (t   )
(4.1-42)
 yy ( )sin 2 f ct sin 2 f c (t   )
 xy ( )sin 2 f ct cos 2 f c (t   )   yx ( ) cos 2 f ct sin(2 f c (t   ))
Use of the trigonometric identities
cos A cos B  12 [cos( A  B )  cos( A  B )]
sin A sin B  12 [cos( A  B )  cos( A  B )]
sin A cos B  12 [sin( A  B )  sin( A  B )]
(4.1-43)
Then
nn ( )  12 [ xx ( )   yy ( )]cos 2 f c
 12 [ xx ( )   yy ( )]cos 2 f c (2t   )
(4.1-44)
 12 [ yx ( )   xy ( )]sin 2 f c
 12 [ yx ( )   xy ( )]sin 2 f c (2t   )
nn ( )  E[n(t )n(t   )]   xx ( ) cos(2 f c )   yx ( )sin(2 f c )
(4.1-45)
The equivalent low pass process
z (t )  x(t )  jy (t )
Its autocorrelation function is defined
1
 zz ( )  E[ z  (t ) z (t   )]
2
(4.1-46)
(4.1-47)
Keep in mind that there is no such a “1/2” constant in most books.
 zz ( )  12 E[ xx ( )   yy ( )  j xy ( )  j yx ( )]
Using (4.1.40) and (4.1.41)
 zz ( )   xx ( )  j yx ( )
Incorporate (4.1.49) with (4.1.45)
nn ( )  Re[ zz ( )e j 2 f  ]
and the PSD relationship

 nn ( f ) 
 Re 

zz
(4.1-48)
(4.1-49)
c
(4.1-50)
1
( )e j 2 fc  e j 2 f  d  [ zz ( f  f c )   zz (  f  f c )]
2
(4.1-51)

Note that  zz ( )   zz  (  ) , accordingly,  zz ( f ) is a real-valued function of f.
Band pass
representation
n(t )
Low pass representation
Relationship
z (t )
n(t )  Re[ z(t )e j 2 fct ]
nn ( )
zz ( )
 nn ( f )
 zz ( f )
nn ( )  Re[ zz ( )e j 2 f  ]
1
 nn ( f )  [ zz ( f  f c )   zz (  f  f c )]
2
c
Properties of quadrature components:
Any cross-correlation function satisfies
then from (4.1.41)
 yx ( )   xy (  )
(4.1-52)
 xy ( )   xy (  )
(4.1-53)
That is,  xy ( ) is an odd function of  , and therefore,  xy (0)  0 . In other words,
x (t ), y (t ) are uncorrelated at   0 . Besides,
(1). If  xy ( )  0, for all  , then zz ( )   xx ( ) is real and the PSD satisfies
 zz ( f )   zz (  f )
(4.1-54)
and then  zz ( f ) is symmetric about f  0 .
(2). If n(t ) is Gaussian, then x (t ) and y (t   ) are jointly Gaussian, hence, at   0 , they
are statistically independent with joint pdf as
p ( x, y ) 
1
2 2

e
( x2  y 2 )
2 2
(4.1-55)
with
 2  xx (0)   yy (0)  zz (0)
Representation of white noise
White noise w(t ) is really a virtual concept; it is defined as a stochastic process that has a
flat (constant) power spectral density over the entire frequency range. From the point of view
of statistics
1
1
 ww ( f )  N 0 , or equivalently in the time domain, ww ( )  N 0 ( ) .
2
2
Note: With such a wideband property, white noise cannot be expressed in terms of
quadrature components, which are for narrowband signals or systems only.
A band pass white noise n(t ) with PSD,  nn ( f )
And its equivalent low pass noise z (t )
 N , | f | 12 B
 zz ( f )   0
1
 0, | f | 2 B
(4.1-56)
From (4.1.56), we can obtain the autocorrelation of z (t ) as
 zz ( )  F 1[ zz ( f )]  N 0
and
sin( B )

zz ( )  N 0 ( ), as B  
Here one must keep in mind that the constant is N 0 , not
(4.1-57)
(4.1-58)
1
2
N0 .
Remember PSDs for white noise and bandpass white noise are symmetric about f  0 ,
therefore,  yx ( )  0, for all  , and
 zz ( )   xx ( )   yy ( )
(4.1-59)
Notes:
(1). x (t ), y (t ) are uncorrelated for all time shifts 
(2). The autocorrelation functions of z (t ), x (t ) and y (t ) are all equal.
Equation Section (Next)
4.2 Signal space representation
4.2.1 Vector space concepts
A vector v expressed in terms of a set of unit vectors (basis vectors) is as
n
v   vi ei
(4.2-1)
i 1
The inner product of two n-dimensional vectors v1 , v 2 is defined as
n
v1  v 2   v1i v2i
(4.2-2)
i 1
Two vectors v1 , v 2 are orthogonal if v1  v 2  0 , and a set of m vectors are orthogonal if
v i  v j  0, i  j
(4.2-3)
The norm of a vector v is denoted by v and is defined as
v  ( v  v)1/ 2 
n
v
i 1
2
i
(4.2-4)
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 , v 2 satisfy the triangle inequality
v1  v2  v1  v2
(4.2-5)
with equality if v1 , v 2 are in the same direction, i.e., v1 =av 2 , a  0, a  R
From the triangle inequality, there follows the Cauchy-Schwarz inequality
v1  v2  v1  v2
(4.2-6)
with equality if v1 =av 2 , a  R
The norm square of the sum of two vectors can be expressed as
2
2
2
v1  v 2  v1  v 2  2 v1  v 2
(4.2-7)
If v1 , v 2 are orthogonal, then v1  v 2  0 and hence,
v1  v 2
2
 v1  v 2
2
2
(4.2-8)
A vector v transforms into some vector v is performed by the transformation matrix A
(4.2-9)
v = Av
In the special case, v = v , i.e., two vectors are collinear
Av   v,   R
(4.2-10)
The vector v is said to be an eigenvector of the transformation A and  is the corresponding
eigenvalue.
The Gram-schmidt orthogonalization procedure
Suppose we have m n-dimensional vectors and try to construct a set of orthonormal vectors
from them. By arbitrarily selecting a vector, say v1
v
(4.2-11)
u1 = 1
v1
Next, we select v 2 , and first subtract the projection of onto u1
u2  v 2  ( v 2  u1 )u1
(4.2-12)
After normalizing the vector u2 , we obtain the second unity length vector u 2 , which is
orthogonal to u1 .
u
(4.2-13)
u2 = 2
u2
The procedure can carrier on in the same manner, for example, for the vector v 3
We obtain u3 first
u3  v3  ( v3  u1 )u1  ( v3  u2 )u2
The normalized vector u 3 , which is orthogonal to both u1 and u 2 vectors, is
u
u3 = 3
u3
(4.2-14)
(4.2-15)
4.2.2 Signal space concepts
For a set of signals defined on some interval [a, b].
The inner product of two complex-valued signals, x1 (t ), x2 (t ) is denoted by
 x1 (t ), x2 (t )  , and is defined as
b
 x1 (t ), x2 (t )   x1 (t ) x2 (t )dt
(4.2-16)
a
The signals are orthogonal if their inner product is zero, i.e.,  x1 (t ), x2 (t )  0
The norm of a signal is defined as
1/ 2
b

(4.2-17)
x(t )    | x(t ) |2 dt 
a

A set of m signals is 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.
The triangle inequality for two signals is
(4.2-18)
x1 (t )  x2 (t )  x1 (t )  x2 (t )
and the Cauchy-Schwarz inequality is
b
 x (t ) x
1
a
2

b
(t )dt   | x1 (t ) | dt
2
a
1/ 2 b
1/ 2
 | x (t ) | dt
2
2
a
with equality when x1 (t )  ax2 (t ), a  C (complex number set )
(4.2-19)
4.2.3 orthogonal expansions of signals
s (t ) is a deterministic, real-valued signal with finite energy

Es 
 [s(t )] dt
2
(4.2-20)

Suppose a set of orthogonal functions { f n (t ), n  1,2,..., K} , i.e.,



0 m  n
f n (t ) f m (t )dt  
1 m  n
(4.2-21)
The signal can be approximated by a weighted linear combination of these functions, i.e.,
K
sˆ(t )   sk f k (t ), {sk ,1  k  K }
(4.2-22)
k 1
The approximation error incurred is
e(t )  s(t )  sˆ(t )
(4.2-23)
Our objective is to choose the proper coefficients sk to minimize the error energy Ee , i.e.,


2
K


Ee   [ s(t )  sˆ(t )] dt    s(t )   sk f k (t )  dt
(4.2-24)
k 1


 
Based on the mean-square criterion, the resulting coefficients can be obtained when the error
is orthogonal to each of the functions in the series expansion. Thus,
2

K


sk f k (t )  f n (t )dt  0, n  1,2,..., K
  s(t )  
k 1

(4.2-25)
Since the functions { f n (t ), n  1,2,..., K} are orthonormal, the above equation reduces to

sn 
 s(t ) f (t )dt,
n  1,2,..., K
n
(4.2-26)

The coefficient sn is exactly equal to the projection of s (t ) onto the function f n (t ) .
The minimum mean square approximation error is

Emin 

2
 [s(t )  sˆ(t )] dt 
 [s(t )  sˆ(t )]s(t )dt





K
 [s(t )] dt    s
2
 k 1

k
f k (t )s(t )dt
(4.2-27)
K
 Es   sk 2  0
k 1
(A bug exists for the link between Mathtype 5.0 and MS word when more than one hat are
typed with the same letter or symbol in the equation, for example, when typing two sˆ(t )
twice in a single equation. However, it is correctly shown on the Mathtype canvas.
***Mathtype5.1 is OK)
When the minimum mean square approximation error Emin  0 ,
K
Es   sk2 
k 1

 [s(t )] dt
2
(4.2-28)

and
K
s ( t )   sk f k ( t )
(4.2-29)
k 1
When every finite energy signal can be represented by a series expansion and with Emin  0 ,
the set of orthonormal functions { f n (t ), n  1,2,..., K} is said to be complete.
Example 4.2-1 Trigonometric Fourier series
A finite energy signal s (t ) that is zero everywhere except (0, T ) can be expanded in a
Fourier series as

a
2 kt
2 kt 

s(t )  0    ak cos
 bk sin
(4.2-30)

2 k 1 
T
T 
The coefficients that minimize the MSE are
T
2
2 kt
ak   s(t ) cos
dt , k  0
T 0
T
(4.2-31)
T
2
2 kt
bk   s (t ) sin
dt , k  1
T 0
T
Gram-Schmidt procedure
We can construct a set of orthonormal functions from a set of finite energy signal waveforms
{si (t ), i  1,2,3..., M } .
By arbitrarily choosing a waveform, say, s1 (t ) , we construct the first orthonormal function
as
s (t )
(4.2-32)
f1 (t )  1
E1
The second orthonormal function using f 2 (t ) is first removing the projection of it onto
f1 ( t )

c12 
 s (t ) f (t )dt
2
1
(4.2-33)

f 2 (t )  s2 (t )  c12 f1 (t )
and then its energy is normalized as
f  (t )
f 2 (t )  2
E2
(4.2-34)
(4.2-35)
where E2 represents the energy of f 2 (t ) . Generally, the orthogonalization of the kth
function leads to
f  (t )
f k (t )  k
(4.2-36)
Ek
where
k 1
f k  (t )  sk (t )   cik f i (t )
(4.2-37)
i 1
and

cik 
 s (t ) f (t )dt,
k
i  1,2,..., k  1
i
(4.2-38)

The number of the resulting orthonormal functions will be N  M .
Accordingly, the M signals can be expressed as linear combinations of the { f n (t )} .
N
sk (t )   skn f n (t ), k  1, 2,..., M
(4.2-39)
n 1
and

Ek 
N
2
2
 [sk (t )] dt   skn  sk
2
(4.2-40)
n 1

where s k is the vector form of the signal sk (t )
(4.2-41)
sk  [ sk1 sk 2 ... skN ]
That is, any signal can be represented geometrically as a point space spanned by the
orthonormal functions { f n (t )} .
A bandpass signal can be written, in terms of the equivalent lowpass signal, by
sm (t )  Re[ slm (t )e j 2 fct ], m  1, 2,..., M
Recall that the signal energy is


1
2
Em   sm 2 (t )dt   slm (t ) dt
2 

Measurement of the similarity between any two signals, say, sm (t ), sk (t )
(1). The normalized cross correlation




1
1


s
(
t
)
s
(
t
)
dt

Re
slm (t ) slk  (t )dt 

m
k


Em Ek 
 2 Em Ek 



(4.2-42)
(4.2-43)
(4.2-44)
The complex-valued cross correlation coefficient is  km defined as
km 
1
2 Em Ek

s
lk
(t ) slm (t )dt
(4.2-45)

Then
Re( km ) 
or, equivalently,
Re( km ) 
(2). The Euclidean distance
1
Em E k

 s (t ) s
k
m
(t )dt
(4.2-46)

sm  sk
s s
 m k
sm sk
Em Ek
(4.2-47)
1/ 2


d  sm  sk    [sm (t )  sk (t )]2 dt   Em  Ek  2 Em Ek Re( km )


If all signals have the equal energy, i.e., Ei  E , for all i , then
(e)
km


1/ 2
(e)
d km
 2 E [1  Re(  km )]
1/ 2
(4.2-48)
(4.2-49)
For digitally, linearly modulated signal, it is convenient to express in terms of two
orthonormal basis functions as
2
f1 (t ) 
cos 2 f ct
T
(4.2-50)
2
f 2 (t )  
sin 2 f ct
T
If the equivalent lowpass signal is denoted as slm (t )  xl (t )  jyl (t ) , then the corresponding
bandpass signal, sm (t ) can be expressed in terms of f1 (t ) and f 2 (t ) as
sm (t )  xl (t ) f1 (t )  yl (t ) f 2 (t )
where xl (t ) and yl (t ) are the signal modulations.
.
(4.2-51)
4.3 Representation of digitally modulated signals
Modulator: A modulator is the interface device that maps the digital information into
analog waveforms that match the characteristics of the channel.
Memoryless modulator: When the mapping from the digital sequence {an } to waveforms
{sm (t )} is performed without any constraint on previously waveforms, the
modulator is called memoryless.
Modulator with memory: A modulator is said have memory if the mapping from the digital
sequence {an } to waveforms {sm (t )} is performed under the constraint that a
waveform transmitted in any time interval depends on one or more previously
waveforms.
Linear/nonlinear modulator: A modulator is said to be linear if the mapping of the digital
sequence into successive waveforms satisfies the principle of superposition,
otherwise, the modulator is said to be nonlinear.
Modulator characteristic
Linear, memoryless
Linear, with memory
Nonlinear, memoryless
Nonlinear, with memory
Modulation types
4.3.1 Memoryless modulation methods
Assume that the sequence of the binary digits at the input to the modulator occurs at a rate of
R bits/sec (bit rate).
PAM: Pulse-amplitude-modulated signals
In digital PAM, the signal waveforms may be represented as
sm (t )  Re  Am g (t )e j 2 fct 
(4.3-1)
 Am g (t ) cos 2 f ct , m  1, 2,..., M , 0  t  T
(4.3-2)
Am  (2m  1  M )d , m  1,2,..., M
Notes:
(1). { Am , 1  m  M } denote the set of M possible amplitudes corresponding to M  2k
possible k-bit block of symbols.
(2). The shaping function g (t ) is a real-valued signal pulse whose shape influences the
spectrum of the transmitted signal.
(3). 2d is the distance between adjacent signal amplitudes.
(4). The symbol rate for the PAM is R/k
(5). The time interval Tb  1/ R is called the bit interval.
(6). The time interval T  k / R  kTb is called the symbol interval.
The M PAM signals have symbol energies
T
T
1
1
Em   sm 2 (t )dt  Am 2  g 2 (t )dt  Am 2 Eg
2
2
0
0
E g stands for the energy of the shaping function, g (t ) .
(4.3-3)
PAM signals are one-dimensional, therefore,
sm (t )  sm f (t )
with
2
f (t ) 
g (t ) cos 2 f ct
Eg
(4.3-4)
(4.3-5)
and
sm  Am
1
Eg , m  1,2,..., M
2
(4.3-6)
PAM Digital PAM is also called amplitude-shift keying (ASK).
The mapping of PAM can use the Gray encoding, in which the adjacent amplitudes differ
by only one binary digit.
Gray encoding is important because the most likely errors in the receiver caused by noise
involve the erroneous selection of adjacent amplitude to the transmitted signal amplitude. In
such case, only a single bit error occurs in the k-bit sequence.
The Euclidean distance between any pair of signal points is
1
(4.3-7)
d mn ( e )  ( sm  sn )2 
Eg Am  An  d 2 Eg | m  n |
2
Then the distance between a pair of adjacent signal points, i.e., the minimum Euclidean
distance is
d min ( e )  d 2 Eg
(4.3-8)
The carrier-modulated PAM can be double-sided band (PAM/DSB), or single-sided band
(PAM/SSB). PAM/DSB is as (4.3-1), which needs twice the channel bandwidth of the
equivalent lowpass signal for transmission.
PAM/SSB, requiring half the channel bandwidth of PAM/DSB, is represented by
sm (t )  Re  Am [ g (t )  jgˆ (t )]e j 2 fct , m  1, 2,..., M
(4.3-9)
Remember
g (t )  G ( f ); gˆ (t )  Gˆ ( f )  G ( f ) H ( f )
 g (t )  jgˆ (t )  G ( f )  G ( f ) H ( f )
2G ( f ) f  0
(1)." " : G ( f )  jG ( f ) H ( f )  
, upper
f 0
 0
f 0
 0
(2)." " : G ( f )  jG ( f ) H ( f )  
, lower
2G ( f ) f  0
where the Fourier transform H ( f ) of the Hilbert transformer is
  j,

H ( f )   0,
 j,

f 0
f 0
f 0
The digital PAM signal can also be used for base band transmission, in which the carrier is
not required.
(4.3-10)
sm (t )  Am g (t ), m  1,2,..., M
A special case is M=2, the binary PAM waveforms have the special property that
s1 (t )   s2 (t )
Such signals are called antipodal.
Phase-modulated signals
In digital phase modulation, the M signal waveforms are represented as
sm (t )  Re  g (t )e j 2 ( m 1) / M e j 2 fct  , m  1, 2,..., M , 0  t  T
2 (m  1) 

 g (t ) cos 2 f ct 

M

2 ( m  1)
2 ( m  1)
 g (t ) cos
cos 2 f ct  g (t )sin
sin 2 f ct
M
M
(4.3-11)
m  2 (m  1) / M , m  1,2,..., M are the M possible phases of the carrier that convey the
transmitted information.
Digital phase modulation is usually called phase-shift keying (PSK).
The PSK signal waveforms are two-dimensional and have equal energy,
T
T
1
1
E   sm 2 (t )dt   g 2 (t )dt  Eg
20
2
0
(4.3-12)
sm (t )  sm1 f1 (t )  sm 2 f 2 (t )
(4.3-13)
2
g (t ) cos 2 f ct
Eg
(4.3-14)
2
g (t )sin 2 f ct
Eg
(4.3-15)
where
f1 (t ) 
f 2 (t )  
And the two-dimensional vector s m
sm  [ sm1 sm 2 ]
 E
2 (m  1)
  g cos
M
 2
Eg
2
sin
2 ( m  1) 
 , m  1, 2,..., M
M

(4.3-16)
As mentioned before, the preferred assignment is Gray encoding, so that the mostly likely
errors caused by noise will result in a single bit error in the k-bit symbol.
The Euclidean distance between signal points is
1/ 2
2
 

d mn  sm  sn   Eg 1  cos
(m  n )  
M

 
and the minimum distance is
(e)
(4.3-17)
2 

(4.3-18)
d min ( e )  Eg  1  cos

M

A variant of 4-phase PSK, called  / 4  QPSK , is obtained by introducing an additional
 / 4 phase shift in the carrier phase in each symbol interval. Such a phase shift facilitates
symbol synchronization.
QAM: Quadrature amplitude modulation
The bandwidth efficiency of PAM/SSB can be obtained by simultaneously impressing two
separate k-bit symbols from the information sequence {an } on two quadrature carriers
cos 2 f ct and sin 2 f ct ; such a technique is called QAM or quadrature PAM.
sm (t )  Re[( Amc  jAms ) g (t )e j 2 fct ], m  1,2,..., M , 0  t  T
 Amc g (t )cos 2 f ct  Ams g (t )sin 2 f ct
Amc , Ams are the information-bearing signal amplitudes of the quadrature carriers.
Alternatively, QAM signal can also be written by
sm (t )  Re[Vme jm g (t )e j 2 fct ], m  1,2,..., M , 0  t  T
 Vm g (t ) cos(2 f ct   m )
(4.3-19)
(4.3-20)
Since Vm  Amc 2  Ams 2 and  m  tan 1 ( Ams / Amc ) , QAM can be viewed as combined
amplitude and phase modulation.
QAM is a two-dimensional modulated signal, i.e.,
sm (t )  sm1 f1 (t )  sm 2 f 2 (t )
where
2
f1 ( t ) 
g (t ) cos 2 f ct
Eg
2
f 2 (t )  
g (t ) sin 2 f ct
Eg
and
(4.3-21)
(4.3-22)

Eg
Eg 
sm  [ sm1 sm 2 ]   Amc
Ams
 , m  1, 2,..., M
2
2 

The Euclidean distance between signal points is
Eg
( Amc  Anc )2  ( Ams  Ans )2 
d mn ( e )  sm  sn 
2
(4.3-23)
(4.3-24)
A special case is that the amplitudes take the set of discrete values,
{(2m  1  M )d , m  1, 2,..., M } and then the signal space diagram is rectangular. In such a
case, the minimum Euclidean distance is the same as that for PAM,
d min ( e )  d 2 E g
(4.3-25)
Multidimensional signals
Orthogonal multidimensional signals
Suppose there are M equal-energy orthogonal signal waveforms that differ in frequency, and
are represented as
sm (t )  Re  slm (t )e j 2 f ct  , m  1, 2,..., M , 0  t  T
(4.3-26)
2E

cos  2 f ct  2 mf t 
T
where
2 E j 2 mf t
(4.3-27)
slm (t ) 
e
, m  1,2,..., M , 0  t  T
T
This type of frequency modulation is called frequency-shift keying (FSK).
The cross correlation coefficients are
T
2 E / T j 2 ( mk ) f t
sin  T (m  k )f j T ( mk ) f
km 
e
dt 
e

2E 0
 T (m  k )f
The real part of  km is
(4.3-28)
sin  T ( m  k ) f
cos  T ( m  k ) f 
 T ( m  k ) f
(4.3-29)
sin 2 T ( m  k ) f

2 T ( m  k ) f
1
and m  k . Since | m  k | 1 corresponds to
Note that r  Re( km )  0 when f 
2T
1
adjacent frequency slots, f 
represents the minimum frequency separation between
2T
adjacent signals for orthogonality of the M signals.
 r  Re(  km ) 
Comparison: | km | 0 at k / T , k  1,2,...
Hint: The cross-correlations of baseband signals and the cross-correlations of bandpass
signals are listed in (4.2.44)-(4.2.46). As for the bandpass signal, the most important is
r  Re( km ) .
The vector representation for M n-dimensional FSK signals (where M=N) is
s1  [ E 0 0 ... 0 0]
s2  [ 0 E 0 ... 0 0]
(4.3-30)
s M  [ 0 0 0 ... 0 E ]
The Euclidean distance between pairs of signals (also the minimum distance) is
dkm( e)  2E ,for all m, k
(4.3-31)
Biorthogonal signals
A set of M biorthogonal signals can be constructed from 12 M orthogonal signals by
including the negatives of the orthogonal signals. Hence, N  12 M are needed, and the
correlation is either  r  0 or  1 . The Euclidean distances are
d  2 E or 2 E (the minimum distance) .
Simplex signals
Suppose we have a set of M orthogonal waveforms {sm (t )} with vectors {sm } . Their mean
is
1 M
s
(4.3-32)
 sm
M m 1
By subtracting the mean from each of the M orthogonal signals, we obtain another set of M
signals, called simplex signals.
sm  sm  s
(4.3-33)
Thus,
The effect of the subtraction is to translate the origin of the M orthogonal signals to the point
s.
Note:
(1). Simplex signals have equal energy, i.e.,
2
1
1 
2
2

sm  sm  s  E 
E
E  E 1  
(4.3-34)
M
M
M

(2). The cross correlation of any pair of signals is equal and requires less energy, by the
factor 1-1/M.
mn
1
sm  sn

(4.3-35)
Re  mn  
  1/ M
1
sm sn


m

n

M 1
1  1/ M
(3). Since only the origin was translated, the distance between any pair of signal points is
maintained at d  2 E .
(4). The signal dimensionality is N  M  1.
Signal waveforms from binary codes
A set of M signaling waveforms can be generated from a set of M binary code words
(4.3-36)
Cm  [cm1 cm 2 ...cmN ], m  1,2,..., M
where cmj  0 or 1 for all m and j . Each code word is mapped into an elementary binary PSK
waveform as
2 Ec
cmj  1  smj (t ) 
cos 2 f ct , 0  t  Tc
Tc
(4.3-37)
2 Ec
cmj  0  smj (t )  
cos 2 f c t , 0  t  Tc
Tc
where Tc  T / N , Ec  E / N . N is called the block length of the code, and it is the
dimension of the M waveforms. Thus, the M code words {Cm } are mapped into a set of M
waveforms {sm (t )} .
The vector form of the waveform is
(4.3-38)
sm  [ sm1 sm 2 ...smN ], m  1,2,..., M
where smj   E / N for all m and j.
Notes:
(1). There are 2 N possible waveforms that can be constructed from the 2 N possible binary
code words. We may select a subset of M< 2 N signal waveforms for transmission of
the information.
(2). The 2 N possible signal points correspond to the vertices of an N-dimensional
hypercube with its center at the origin.
(3). Each of the M waveforms has energy E. The cross correlation between any pair of
waveforms depends on how we select the M waveforms from the 2 N possible waveforms.
Clearly any adjacent signal points have a cross correlation coefficient (using (4.2.47) and
(4.2.49))
E (1  2 / N ) N  2
r 

(4.3-39)
E
N
and a corresponding distance of
(4.3-40)
d ( e )  2 E (1  r )  4 E / N .
4.3.2 Linear modulation with memory
In modulation with memory, dependence is intentionally introduced to the signals
transmitted in successive symbol intervals; the purpose is to shape the spectrum of the
transmitted signal so that it matches the spectral characteristics of the channel.
Signal dependence is usually accomplished by encoding data sequence at the input to the
modulator by means of a modulation code.
The memory characteristic will be confined to be of Markov chain, and the signals to be base
band.
In Figure 4.3-12, NRZ (non-return-to-zero) is memoryless and is equivalent to a binary PAM
or a binary PSK signal in a carrier-modulated system; NRZI (non-return-to-zero I) or
NRZ-M (non-return-to-zero Mark, mark stands for one), in which the 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 process is
(4.3-41)
bk  ak  bk 1
where
{ak } is the input to the encoder, {bk } is the output, and  is the addition modulo 2
operator.
Such an encoding operation can be manifested in three ways:
(1). By a state diagram (a Markov chain) as
Hint: left side of slash represents the input in digit; the right represents the encoder’s output
in waveforms.
(2). By transition matrices: The state diagram can be described by two transition matrices
corresponding to the two possible input bits {0,1} .
(i). ak  0 : The encoder stays in the same state, so the transition matrix is
Note: tij  1 if a k
1 0
T1  

0 1
results in a transition from state i to state j, i, j  1, 2 .
(4.3-42)
(ii). ak  1 : The encoder goes to the other state, so the transition matrix is
0 1
T2  

1 0
(4.3-43)
(3). By the trellis diagram: The trellis diagram provides exactly the same information
concerning the signal dependence as the state diagram, but also depicts a time evolution of
the state transitions.
Another base band modulation with memory is the delay modulation, in which the data
sequence is encoded by the run-length-limited code, called a Miller code, and then uses
NRZI to transmit the encoded data.
This type of digital modulation has been used extensively for digital magnetic recording
and in carrier modulation systems employing binary PSK.
The signal may be described by a state diagram that has four states as below:
Besides, the two transition matrices are: (because the input is binary, the number of matrices
is two)
(i). When ak  0
0
0
T1  
1

1
0 0 1
0 0 1

0 0 0

0 0 0
(4.3-44)
0
0
T2  
0

0
1 0 0
0 1 0

1 0 0

0 1 0
(4.3-45)
(ii) When ak  1
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 s j (t ), j  1, 2,..., K .
pij denotes the probability that s j ( t ) is transmitted in a given signaling interval after the
transmission of the signal waveform si (t ) in the previous signaling interval.
The transition probability matrix P is as
 p11 p12
p
p22
P   21


 pK 1 pK 2
p1K 
p2 K 



pKK 
(4.3-46)
P can be obtained from the transition matrices, {Ti } and the stationary state probabilities,
{ pi , i  1,2,..., K} as
2
P   qi Ti  p(ak  0) T1  p( ak  1) T2
i 1
q1
(4.3-47)
q2
For NRZI signal with equal state probabilities p1  p2  1/ 2 and T1 , T2 as in (4.3-42) and
(4.3-43), the transition probability matrix is
 12 12 
P  1 1
2 2
For Miller-coded NRZI with equal likely symbols ( p1  p2  p3  p4  1/ 4 or
p1  p2  1/ 2 , equivalently), the transition probability matrix is
(4.3-48)
 0 12 0 12 
0 0 1 1 
P  1 1 2 2
 2 2 0 0
1

1
 2 0 2 0
(4.3-49)
The transition probability matrix is useful in the determination of the spectral characteristics
of digital modulation techniques with memory.
4.3.3 Non-linear modulation methods with memory---CPFSK and CPM
In this section, 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.
Continuous-phase FSK (CPFSK)
A conventional FSK signal is generated by shifting the carrier by an amount
f n   12 f I n , I n  1, 3,..., ( M  1) to reflect the digital information that is being
transmitted, and it is memoryless.
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 drawback is that such abrupt switching from one oscillator output to another oscillator in
successive signaling intervals results in relatively large spectral side lobes outside of the
main lobe of the main spectral band of the signal, i.e., it needs a large frequency band for
transmission of the signal.
To avoid such frequency band dissipation, the information-bearing signal
frequency-modulates a single oscillator whose frequency is changed continuously. The
resulting frequency-modulated signal is phase-continuous and, hence, it is called CPFSK.
This type of FSK signal has memory because the phase of the carrier is constrained to be
continuous.
Begin with a PAM signal, which is used to frequency-modulate the carrier
d (t )   I n g (t  nT )
(4.3-50)
n
where
{I n } : The sequence of amplitude obtained by mapping k-bit blocks of binary digits from the
information sequence {an } into the amplitude levels 1, 3,..., ( M  1) ;
g (t ) : A rectangular pulse of amplitude 2T1 and duration T seconds.
The equivalent low pass waveform at the output of the modulator v (t ) is
t
 
 
2E
exp  j  4 Tf d  d )d  0  
T
 

 
f d  the peak frequency deviation
v (t ) 
(4.3-51)
0  the initial phase of the carrier.
The carrier-modulated signal corresponding to (4.3-51) may be expressed as
2E
s (t ) 
cos[2 f ct   (t; I)  0 ]
T
 (t; I ) represents the time-varying phase of the carrier and defined as
 (t; I)  4 Tf d
(4.3-52)
t
 d ( )d

(4.3-53)


 4 Tf d    I n g (  nT )  d

  n
Note: although d ( t ) contains discontinuities, the integral of s (t ) is continuous, hence
v (t ) is continuous-phase.
t
The phase of the carrier in the interval nT  t  (n  1)T using (4.3-53) is
 (t; I)  2 f d T
n 1
I
k 
k
 2 f d (t  nT ) I n   n  2 hI n q(t  nT )
(4.3-54)
where
h  2 fdT
(4.3-55)
represents the modulation index
n 1
n   h  Ik
(4.3-56)
k 
represents the accumulation (memory) of all symbols up to time (n-1)T.
(t  0)
0
 t
q(t )  
(0  t  T )
2
T

(t  T )
 12
(4.3-57)
Continuous-phase modulation (CPM)
In CPM, the phase of the carrier is
 (t; I)  2
n
 I h q(t  kT ),
k 
k k
nT  t  ( n  1)T
{I k } : the sequence of the M-ary information symbols selected from the alphabet
1, 3,..., ( M  1) ;
{hk } : a sequence of modulation indices;
q(t ) : some normalized waveform shape.
It is easy to see that CPFSK is a special case of CPM.
(i) hk  h for all k, the modulation index is fixed for all symbols.
(4.3-58)
(ii). When the modulation index 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.
q(t ) may be expressed as the integral of some pulse g(t):
t
q(t )   g ( )d
0
(a). If g (t )  0 for t  T , the CPM signal is called full response CPM.
(b). If g (t )  0 for t  T , the CPM signal is called partial response CPM.
Three popular pulse shapes are given as
(4.3-59)
LREC denotes a rectangular pulse of duration LT, and L is a positive integer.
(a): L=1CPFSK, the pulse as shown in Figure 4.3-16a.
(b): L=2 the pulse as shown in Figure 4.3-16c.
LRC denotes a raised cosine pulse of duration LT. The pulse shapes for L=1,2 are shown in
Figures 4.3-16b, 4.3-16d.
GMSK denotes a Gaussian minimum-shift keying pulse with bandwidth parameter B, which
represents the –3 dB bandwidth of the Gaussian pulse. Figure 4.3-16e illustrates a set of
GMSK pulses with time-bandwidth productions BT ranging from 0.1 to 1.
In GMSK, the pulse duration increases as the bandwidth of the pulse decreases, as expected.
In practical applications, the pulse is usually truncated to some specified fixed duration.
GMSK with BT=0.3 is used in the European digital cellular communication system, called
GSM. When BT=0.3, the GMSK pulse may be truncated at | t | 1.5T with a relatively
small error incurred for t  1.5T .
It is instructive to sketch the set of phase trajectories  (t; I ) generated by all possible
values of the information sequence {I n } . The phase diagram is called phase trees.
(i). Figure 4.3-17, binary CPFSK with I n  1 , demonstrates the set of phase trajectories
beginning at time t  0 .
(ii). Figure 4.3-18, quaternary CPFSK with I n  1, 3 , demonstrates the set of phase
trajectories beginning at time t  0 .
Observations:
(a). The phase trees for CPFSK are piecewise linear because of the fact that the pulse g (t )
is rectangular.
(b). Smoother phase trajectories and phase trees are obtained by using pulses that do not
contain discontinuities, such as the class of raised cosine pulses.
(c). For comparison, a phase trajectory generated by the sequence (1, -1, -1, -1, 1, 1, -1, 1)
for a partial response, raised cosine pulse of the length 3T is shown in Fig. 4.3-19.
  m 2 m
s   0,
,
,
p
 p
,
( p  1) m 

p

(4.3-60)
  m 2 m
s   0,
,
,
p
 p
,
(2 p  1) m 

p

(4.3-61)
 pM L1 even m
St  
L 1
odd m
2 pM
(4.3-62)
Minimum-shift keying (MSK)
MSK is a special form of binary CFPSK (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
1
2
 (t ; I)  
n 1
I
k 
k
  I n q(t  nT )
1
 t  nT 
 n   In 
 , T  t  (n  1)T
2
 T 
(4.3-63)
The modulated carrier signal is
1

 t  nT  
s(t )  A cos  2 f ct   n   I n 

2
 T 

(4.3-64)
1  1
 

 A cos  2  f c 
I n  t  n I n   n  , nT  t  (n  1)T
4T  2
 

Define two frequencies as
1
f1  f c 
4T
(4.3-65)
1
f2  fc 
4T
The binary CPFSK signal can be expressed as a sinusoid having one of the two possible
frequencies in the interval T  t  (n  1)T as
1


si (t )  A cos  2 f i t   n  n ( 1)i 1  , i  1, 2
(4.3-66)
2


The frequency separation is f  f 2  f1  1/ 2T . From (4.3-29), 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. That is the reason why BCPFSK with
h  1/ 2 is called MSK.
MSK may also be represented as a form of four-phase PSK. Its equivalent lowpass digitally
modulated signal as
v (t ) 

 I
n 
2n
g (t  2nT )  jI 2 n 1 g (t  2nT  T ) 
(4.3-67)
and g(t) is a sinusoidal pulse as
 t
0  t  2T
sin
(4.3-68)
g (t )   2T

otherwise
0
This type of signal is viewed as a four-phase PSK signal in which the pulse shape is one-half
cycle of a sinusoid:
(a). The even-numbered binary symbols, i.e., {I 2 n } are transmitted via the cosine of the
carrier.
(b). The odd-numbered binary symbols, i.e., {I 2 n 1} are transmitted via the sine of the
carrier.
The transmission rate on the two orthogonal carrier components is 1/2T bits/sec, so the total
transmission rate is 1/T bits/sec.
Note: The bit transitions on the sine and cosine carrier components are staggered or offset in
time by T seconds. For this reason, the signal
 


 

s(t )  A    I 2 n g (t  2nT )  cos 2 f ct    I 2 n1g (t  2nT  T )  sin 2 f ct 

 n 

  n 

is called offset quadrature PSK (OQPSK) or staggered quadrature PSK (SQPSK).
Figure 4.3-23 illustrates the representation of an MSK signal as two staggered
quadrature-modulated binary PSK signals.
Note: The corresponding sum of the two quadrature signals is a constant amplitude,
frequency-modulated signal.
It is interesting to compare the waveforms among: (1) MSK, (2) OQPSK with rectangular
g (t ) over 0  t  nT , and (3) QPSK with rectangular g (t ) over 0  t  nT .
(a). All three methods have the identical data rate.
(b). The MSK signal has continuous phase.
(c). The OQPSK signal with a rectangular pulse is two binary signals for which the phase
transitions are staggered in time by T seconds. Thus is, the signal contains phase jumps of
 90 that may occur as often as every T seconds.
(d). The conventional QPSK signal with constant amplitude will contain phase jumps of
 180 or  90 every 2T seconds. See Fig. 4.3-224 for the illustration of these three signal
types.
4.4 Spectral characteristics of digitally modulated signals
Equation Chapter 4 Section 4
(1). The available channel bandwidth is limited for digital communications. From the point
of view of channel bandwidth efficiency, the required channel bandwidth for
transmission should be as small as possible.
(2). Hence, it is important to evaluate the spectral content of the digitally modulated signals.
(3). Since the information sequence is random, a digitally modulated signal is a stochastic
process.
4.4.1 Power spectra of linearly modulated signals
Remember that the expression of a bandpass signal and its corresponding equivalent lowpass
signal is
s(t )  Re  v (t )e j 2 f ct 
The autocorrelation function of s (t ) is
ss ( )  Re vv ( )e j 2 f  
(4.4-1)
c
This equation can be derived as follows.

ss ( )  E[ s(t   ) s(t )]  E Re  v(t   )e j 2 f
c ( t 
)

 Re  v( t ) e j 2 f ct 

1
E v (t   )e j 2 fc ( t  ) v (t )e j 2 f ct  v  (t   )e  j 2 f c ( t  )v (t )e j 2 f ct
4
 v (t   )e j 2 f c ( t  ) v  (t )e  j 2 f ct  v  (t   )e  j 2 f c ( t  )v  (t )e  j 2 f ct 

1
E v (t   )v (t )e j 2 f c (2 t  )  v  (t   )v (t )e  j 2 f c
4
 v (t   )v  (t )e j 2 f c  v  (t   )v  (t )e  j 2 f c (2 t  ) 
Assume s (t ) is WSS, then E[v(t   )v(t )]  E[v (t   )v (t )]  0 , and use the definition of
the autocorrelation function of v (t ) , vv ( )  12 E[v(t   )v  (t )] ,
1
c
2
The power density spectrum  ss ( f ) is
ss ( )  vv ( )e j 2 f   vv  ( )e  j 2 f    Re vv ( )e j 2 f  
c
c
 ss ( f )  12  vv ( f  f c )   vv (  f  f c )
, which can be derived as below:
1
 ss ( f )  F [ss ( )]  F vv ( )e j 2 fc   F vv  ( )e  j 2 fc 
2

1
1
  vv ( f  f c )   vv  ( )e  j 2 fc e  j 2 f  d
2
2 



1
1
  vv ( f  f c )    vv ( )e j 2 ( f  f c ) d 
2
2  



1
1
1
1
  vv ( f  f c )   vv  (  f  f c )   vv ( f  f c )   vv (  f  f c )
2
2
2
2
(4.4-2)
Here we use the fact that vv ( )  vv  (  )   vv ( f )   vv  ( f ) , i.e.,  vv ( f ) is
real-valued.
Now it suffices to determine the autocorrelation function and PSD of v (t ) .
For the linear digital modulation methods,
v (t ) 

I
n 
n
g (t  nT )
(4.4-3)
Transmission rate is 1/ T  R / k . {I n } may be real-valued (PAM) or complex-valued (PSK,
QAM).

1  

vv (t   ; t )  12 E[v  (t )v(t   )]  E   I n  g  (t  nT )  I m g (t    mT ) 
2  n 
m 
 (4.4-4)


1
   E  I n  I m  g  (t  nT ) g (t    mT )
2 n  m 
Assume {I n } is WSS, mean is i and the autocorrelation function is
1
ii (m)  E  I n  I n m 
(4.4-5)
2
Then (4.4-4) can be expressed by
vv (t   ; t ) 



   (m  n) g (t  nT ) g (t    mT )
n  m 


m

ii
ii (m)


g  (t  nT )g  (t    nT  mT )
(4.4-6)
n 
period in t with period T
Thus, vv (t   ; t ) is also periodic in t with period T.
vv (t  T   ; t  T )  vv (t   ; t )
The mean of v (t ) is
(4.4-7)


 

(4.4-8)
E[v(t )]  E   I n g (t  nT )    i g (t  nT )  i  g (t  nT )
n 
 n 
 n 
That is, the mean of v (t ) is also periodic in t with period T.
Both of the mean and the autocorrelation function of v (t ) are periodic in t with period T,
therefore, v (t ) is called a cyclostationary process or a periodically stationary process in
the wide sense.
To make the PSD meaningful, the dependence on t in vv (t   ; t ) must be removed out
(averaged out) first. Since it is periodic, only a single period is needed.
T /2
1
vv ( ) 
vv (t   ; t )dt
T T/ 2

1 
  ii ( m)   g  (t  nT ) g (t    nT  mT )dt
T n  T / 2
m 


T /2

m 
T / 2  nT
1
g  (t ) g (t    mT )dt

n  T  T / 2  nT
 ii (m) 
Define the time-autocorrelation function of g (t ) as
(4.4-9)

 g (t ) g (t   )dt
gg ( ) 

(4.4-10)

Note: here the definition differs from those of other books, for example,
T /2
1
R f ( )  lim
f  (t ) f (t   )dt
T  T 
T / 2
in the books: F. G. Stremler, p. 180 and L.W. Couch II, p.63.
vv ( ) 
1 
 ii (m)gg (  mT )
T m 
(4.4-11)
Observe that the average autocorrelation function vv ( ) depends on the properties of both
the information sequence {I n } and the shaping function g (t ) .
By taking the Fourier transform both sides, we obtain the “averaged” PSD as


1 
 vv ( f )   vv ( )e  j 2 f  d    ii ( m) gg (  mT )e  j 2 f  d
T m 



1 
  ii (m)   gg ( )e  j 2 f ( mT )d
T m 


1 
ii (m)e  j 2 fmT

T m 


gg
( )e  j 2 f  d

 ii ( f )
To obtain  vv ( f ) , we need to evaluate the integral in the above equation as

 

 
 j 2 f 
 gg ( )e d 



g

g  (t ) g (t   )e  j 2 f  dtd 

 
 g

(t ) g (t   )e  j 2 f (t  t ) dtd
 

(t )e j 2 ft dt   g (t   )e  j 2 f (t  ) d


 G ( f )G ( f )  G ( f )
As a result, we obtain the average PSD of v (t ) as
2
 vv ( f ) 
1
2
G ( f )  ii ( f )
T
(4.4-12)
and
 ii ( f ) 

  ( m )e
m 
ii
 j 2 fmT
(4.4-13)
is the PSD of the information sequence {I n } .
------------------------------------------------------------------------------------------------------------F 1
Time average autocorrelation- an example: because G ( f )G( f ) 
g * (t )  g (t ) , if the
shaping function is a rectangular function, then the resulting autocorrelation function is
triangular( g * (t )  g (t ) , g (t ) is real). (If further assume the unit data sequence is white, then
1 
1
ii (m)gg (  mT )  gg ( ) ). Write a program as a project to confirm this

T m 
T
conclusion!
--------------------------------------------------------------------------------------------------------------
vv ( ) 
From the previous discussions, we know the PSD of v (t ) depends only on (i) the
information sequence, {I n } and (ii) the shaping function, g (t ) . Therefore, by designing
the shape of g (t ) and the autocorrelation characteristic of {I n } , we can control the PSD
of v (t ) .
The PSD  ii ( f ) related to the autocorrelation function ii ( m) is in the form of an
exponential Fourier series with the {ii (m)} as the Fourier coefficients.
1/ 2T
ii (m)  T

ii ( f )e j 2 fmT df
(4.4-14)
1/ 2T
{I n } is assumed to be real-valued and mutually uncorrelated, then
 2  i 2 m  0
ii (m)   i 2
m0
 i
(4.4-15)
where  i 2 denotes the variance of an information symbol. When (4.4-15) is used to
substitute for ii ( m) in (4.4-13), we obtain
 ii ( f )   i 2  i 2

e
 j 2 fmT
(4.4-16)
m = 
Note that the summation in (4.4-16) is periodic with period 1/ T . It may also be viewed as
the exponential Fourier series of a periodic train of impulses with each impulse having an
area 1/ T . Thus,
i 2  
m
2
(4.4-17)
ii ( f )   i 
f  

T m =  
T
Substitution of (4.4-17) into (4.4-12) yields the desired result for the PSD of v (t ) when
{I n } is uncorrelated. That is,
 i2
i 2

2
m
m 
 vv ( f ) 
G( f ) 
G    f  

T
T m =   T  
T
2
depend only on
g ( t ),cont . spectrum
(4.4-18)
discrete spectrum separated by 1/ T
(1). The first term: continuous spectrum, its shape depending on g (t ) only.
(2). The second term: discrete spectrum, each spectral line having a power that is
2
proportional to G ( f ) that is evaluated at f  m / T . This term vanishes as i  0 ,
which is usually desirable for digital modulation techniques and it is satisfied when the
information symbols are (i) equally likely, (ii) symmetrically positioned in the complex
plane.
 The most important is that we can control the spectral characteristics of the digitally
modulated signal by proper selection of the characteristics of the information sequence to be
transmitted.
Example 4.4-1 Demonstration of the (energy) spectral shape of the rectangular pulse, g (t )
G ( f )  F [ g (t )]  AT
sin  fT  j fT
e
 fT
Hence
G ( f )   AT 
2
2
 sin  fT 
  fT 


2
(4.4-19)
It is worthy to indicate that:
(i). It contains zeros at multiples of 1/T.
(ii). It decays inversely as f 2 .
(iii). Because of the zeros in G ( f ) , all but one of the discrete spectral components in
(4.4-18) vanish. Therefore, (4.4-18) reduces to
2
 sin  fT 
(4.4-20)
 vv ( f )   i A T 
 A2 i 2 ( f )

  fT 
Example 4.4-2. Demonstration of the (energy) spectral shape of the raised cosine pulse,
g (t ) .
2
g (t ) 
2
A
2
1  cos

2
T
 T 
 t   , 0  t  T
2 

(4.4-21)
AT
sin  fT
(4.4-22)
e  j fT
2  fT (1  f 2T 2 )
Comparing with the rectangular pulse case:
(i). Its spectrum has zeros at f  n / T , n  2,  3,  4, . As a result, all the discrete spectra
components in (4.4-18) except the one at f  0 and f  1/ T vanish.
(ii). It has a broader main lobe but the tails decay inversely as f 6 .
G( f )  F [ g (t )] 
Example 4.4-3. The previous two examples concern the effect of the shaping function. This
example discusses the influence of the information sequence.
Consider a binary sequence {bn } from which we form the symbols
(4.4-23)
I n  bn  bn 1
{bn } are assumed to be uncorrelated and with zero mean and unit variance. The correlation
function of {I n } is
2 m  0

ii (m)  E ( I n I n m )  1 m  1
0 otherwise

Accordingly, the PSD of {I n } is
(4.4-24)
 ii ( f )  2(1  cos 2 fT )  4 cos 2  fT
The corresponding PSD of the lowpass modulated signal is
4
2
 vv ( f )  G ( f ) cos2  fT
T
(4.4-25)
(4.4-26)
In the following, we give a supplementary example (see Couch II, Digital and analog
communication systems, 6th ed., pp. 410-413). The result can be applied to the example 2.1,
p.56 in “Introduction to spread spectrum communications,” by Peterson, Ziemer and Borth.
¥
Let v (t ) ( x (t ) in the following figures, x (t ) =
å
I n g(t - nT ) )
be a polar signal with
n=- ¥
random binary data. The data is assumed independent from bit to bit and has equal
probability in any bit interval (that is, P (I n = 1) = P (I n = - 1) = 1/ 2 ).
To examine the autocorrelation characteristic of v (t ) , we use the result obtained above.
1, m  0
Using (4.4-6) and ii (m)  
, we obtain
0, m  0
vv (t   ; t ) 


m 
n 
 ii (m)  g * (t  nT ) g (t    nT  mT ) 

 g * (t  nT ) g (t    nT )
n 
To obtain the averaged autocorrelation, we need to take out the parameter, t, and since
vv (t   ; t ) is periodic with the period T,
 vv ( ) 

1
1

(
t


;
t
)
dt

 g * (t  nT ) g (t    nT )dt
vv
T T/ 2
T T/ 2 n 
T /2

T /2
T /2
1
 
 g * (t  nT ) g (t    nT )dt
n  T  T / 2
1, | t | T / 2
Besides, g (t )  
, then we first assume that the offset 0    T
0, elsewhere
1, nT  T / 2  t  nT  T / 2  
.
g *(t  nT ) g (t    nT )  
elsewhere
0,
If   T , there is no overlapping, therefore, the product is zero. Since the integral interval is
( T / 2, T / 2) , then only the n  0 term makes the integral having value and as below.
T / 2
1
1
T 
 vv ( )    g *(t  nT ) g (t    nT )dt 
1dt 

T T / 2
T
n  T  T / 2
In the same manner, we can get the result for T    0 . (Alternatively, may use the fact
 vv ( )   vv ( ) ) Accordingly, we conclude
T  | |
 vv ( ) 
, |  | T
T

T /2
It has a triangular shape and its PSD is the Fourier transform of  vv ( ) .
It is very interesting to observe that a random information sequence with rectangular pulse
has a triangular-shape autocorrelation function and that is exactly the convolution of this
pulse with itself! Therefore, the PSD, the Fourier transform of the autocorrelation function is
just the square of the absolute value of the Fourier transform of such pulse function.
F
If g (t ) 
 G( f ) , then  vv ( f )  F{Rvv ( )}  F{g ( )  g ( )} | G ( f ) |2
Correction: Review the example 2.1, P. 56 in Peterson, etc. book.
Since the data is assumed to be purely random, therefore, the adjacent bits (1, 1), (1, -1), (-1,
-1) and (-1, -1) are equally probable. If Tc    Tc , the resulting area will be Tc   as
above. On the other hand, if |  | Tc , c(t )c(t   ) is another purely random data waveform,
hence, its average is just zero.
However, Figure 2.8 was wrongly illustrated. See the following explanation.
(1). The fractional area is the shadow area above zero.
(2). The area is 1  (Tc   )  Tc   .
Then the autocorrelation function is obtained from the definition.
T
A
1
1 c
T 
Rc ( )  lim
c(t )c(t   )dt   c(t )c(t   )dt  c
, 0    Tc

A 2 A
Tc 0
Tc
A
T 
,  Tc    0
Similarly, Rc ( )  c
Tc
T  | |
, |  | Tc
Accordingly, Rc ( )  c
Tc
Also, note that this is not the case for the m-sequence, which is periodic and is used in the
spread spectrum communication system for spreading the transmission bandwidth.
4.4.2 Power spectra of CPFSK and CPM signals
The constant amplitude CPM is expressed as
s(t; I)  A cos 2 f ct   (t; I) 
where
(4.4-27)

 (t; I)  2 h  I k q(t  kT )
(4.4-28)
k 
Assume:
(1). I n {1,  3,
,  ( M  1)} .
(2). I n are iid, and are with prior probabilities
Pn  P( I k  n), n  1,  3,
where

n
,  ( M  1)
Pn  1 .
 1 t  LT
 q(t ) 0  t  LT
(3). g (t )  
, and q(t )   2
.
otherwise
 0
0 t  0
(4.4-29)
The autocorrelation function of the equivalent lowpass signal v(t )  e j ( t ;I ) is

1 


vv (t   ; t )  E exp  j 2 h  I k [q(t    kT )  q(t  kT )] 
2 
k 


(4.4-30)
(i). Expressing the sum in the exponent as a product of exponents, we rewrite (4.4-30) as
1  

(4.4-31)
vv (t   ; t )  E   exp{ j 2 hI k [q(t    kT )  q(t  kT )]}
2  k 

(ii). Performing the expectation over the data symbols {I k } . Since these symbols in {I k }
are statistically independent


1   M 1
(4.4-32)
vv (t   ; t )    Pn exp{ j 2 hn[ q(t    kT )  q(t  kT )]} 

2 k   n  ( M 1)
 n odd

The derivation is as follows.
1
2
1
2






k 

vv (t   ; t )  E[v(t   )v  (t )]  E exp  j 2 h  I k [q(t    kT )  q(t  kT )] 


1
E  exp  j 2 hI k [q(t    kT )  q(t  kT )]
2 k 


1  
  E exp  I k j 2 h[q(t    kT )  q(t  kT )]  


2 k  



1  M 1
   Pn exp  nj 2 h[ q(t    kT )  q(t  kT ) 
2 k  n  ( M 1)
n odd
Hint: E[ X ]   xi P( xi ) , and E[ g ( X )]   g ( xi )P( xi ) 1.
The average autocorrelation function is
T
1
(4.4-33)
vv ( )   vv (t   ; )dt
T 0
To further simplify (4.4-32), we know the fact that if both q(t    kT ) and q(t  kT ) are
either 0 or ½, the exponent terms will be zero, and  Pn  1 . Accordingly, only the nonzero
exponent terms (finite) are needed to evaluate.
Suppose     mT  0, m  0, 0    T
0  q(t  kT )  1/ 2  0  t  kT  LT 
t
t
Lk 
and
T
T
t 
t 
mL k 
m
T
T
To exclude the situation that both q(t    kT ) and q(t  kT ) are either 0 or ½, we take
0  q(t    mT  kT )  1/ 2  0  t    mT  kT  LT 
1
See A. Papoulis and S. U. Pillai, Probability, Random variables and Stochastic Processes, 4 th ed., pages
142-143 about the expected value of function of a random variable for both continuous- and discrete-type
random variables.
the maximum for the upper limit and minimum for the lower limit. Remember
t
t 
m
0    T (not "  T ") and 0  t  T .   L  k 
T
T
From the above inequality,  L  k  2  m (since 0   / T  1)
Thus, k is confined in the interval [1  L,1  m] , or 1  L  k  1  m .
vv (  mT ) 
1
2T
 M 1


P exp{ j 2 hn[ q(t    ( k  m)T )  q(t  kT )]}  dt
0 k
  n

1 L n  ( M 1)
 n odd

T m 1
Consider the case when   mT  LT , which implies m  L
written as
vv (  mT )   ( jh)
m L
(4.4-34)
0    T . (4.4-34) may be
 ( ), m  L, 0    T
(4.4-35)
where  ( jh ) is the characteristic function of the random sequence {I n } , defined as
 ( jh )  E e j hI  
n
M 1

n  ( M 1)
Pn e j hn
(4.4-36)
n odd
and
1
 ( ) 
2T
 M 1

1

Pn exp{ j 2 hn[ 2  q(t  kT )]} 

0 k


1 L n  ( M 1)
 n odd

T
0
 M 1



Pn exp[ j 2 hnq(t    kT )]  dt , m  L



k 1 L n  ( M 1)
 n odd

(4.4-37)
1
(why choose LT? Recall the interval of g (t ) )
(a). Since 0  q(t  kT )  1/ 2  0  t  kT  LT 
t  0,  L  k  0
t
t
 L  k  , we obtain
T
T

t  T, 1 L  k  1
Henceforth, the range of k that q(t  kT ) with 0  q(t  kT )  1/ 2 is L  k  1 , or
1  L  k  0 (k integer).
Similarly,
(b).Since
t 
t 
0  q(t    mT  kT )  1/ 2  0  t    mT  kT  LT 
mL k 
m
T
T
t  0,

T
mLk 

T
m

t  T,


1 m
T
T
Henceforth, the range of k that q(t    mT  kT ) with 0  q(t    mT  kT )  1/ 2 is

1 m  L  k 
mLk 

 1  m , or 1  m  L  k  1  m (k integer and m  L ).
T
T
Their relationship can be manifest in graphic as
1/ 2
The shapes in these two
intervals depend on
function q(t)
q(t  kT )
k
q(t    mT  kT )
1/ 2
k
1 m
1 L
0
1 m  L
Consequently, the integration can be separated into three subintervals:
 vv (  mT ) m L

T
0  M 1
1
1


dt 
 Pn exp  j 2 hn  2  q(t  kT )  
2T 0 k  L1  n  ( M 1)
 n odd

m  L 1

k 1
 M 1

  P exp  j 2 hn  1  0  
2


 n  ( M 1) n
 n odd

 M 1


 
Pn exp  j 2 hn  q(t    ( m  k )T )  0   



k  m  L 1 n  ( M 1)
 n odd

1 m
Hence,
 vv (  mT ) m L 

T
0  M 1
1
1

   ( jh ) m  L
dt
P
exp
j
2

hn

q
(
t

kT
)






n
2




2T 0 k  L1  n  ( M 1)
 n odd

 M 1

1
    Pn exp  j 2 hnq(t    k T ) 

k 1 L  n  ( M 1)
 n odd

( Let k   k  m )
The Fourier transform of vv ( ) yields the average power density spectrum  vv ( f ) as




0

 j 2 f 
 j 2 f 
 vv ( )e d  2 Re   vv ( )e d 
 vv ( f ) 
(4.4-38)
(4.4-38) can be derived as below:
 vv ( f )  F vv ( )  

  vv ( )e
 j 2 f 


0
 j 2 f 
 j 2 f 
 j 2 f 
 vv ( )e d   vv ( )e d   vv ( )e d


0

d   vv (  )e
0
j 2 f 

d   vv ( )e
0
 j 2 f 
0
 

d     vv (  )e  j 2 f  d 
0


But
vv ( )  E  (t   )  (t )  , vv (  )  E  (t   )  (t ) 



 vv (  )  E  (t   )  (t )   E   (t   ) (t )   vv ( )


  vv ( f )  2 Re   vv ( )e  j 2 f  d 
0

But


vv
( )e
 j 2 f 
LT

d 

vv
( )e
 j 2 f 

d   vv ( )e  j 2 f  d
0
(4.4-39)
LT
With the aid of (4.4-35), the integral in the range LT     may be written as


vv
( )e
 j 2 f 
 ( m 1)T
d  
m L
LT

vv ( )e j 2 f  d
(4.4-40)
mT
Let     mT


vv
( )e
 j 2 f 
 T
d    vv (  mT )e  j 2 f (  mT )d
m L 0
LT
 T
    ( )  ( jh ) 
m L
e  j 2 f (  mT ) d
(4.4-41)
m L 0

T
   ( jh ) e  j 2 fnT   ( )e  j 2 f (  LT ) d
n
n 0
0
Note that the right side of the above equation is separable, which mean the summation term
and the integral term may be evaluated respectively. Since  ( jh)  1 , depending on the
absolute value of the characteristic function, two cases are discussed:
(A). When  ( jh)  1 for some h

 ( jh)
n 0
n
e  j 2 fnT 
1
1   ( jh )e  j 2 fT
(4.4-42)
In this case, (4.4-41) reduces to

 j 2 f 
 vv ( )e d 
LT
T
1
 ( )e  j 2 f (  LT )d
 j 2 fT 
1   ( jh )e
0
(4.4-43)
T
1

vv (  LT )e  j 2 f (  LT )d
1   ( jh )e  j 2 fT 0
To further simplify the integral term in (4.4-43), we use the equation (4.4-35)
m L
vv (  mT )   ( jh)  ( ),   mT  LT , and at m  L (remember the exponential
term is L), we have
 ( jh)mL  ( )   ( )  vv (  LT ),
0  T
By combining (4.4-38), (4.4-39), and (4.4-43), we can obtain the PSD of the CPM signal as
 LT
1
 vv ( f )  2 Re   vv ( )e  j 2 f  d 
1   ( jh )e  j 2 fT
0
( L 1) T


vv ( )e  j 2 f  d 

LT
(4.4-44)
Note:
(1). The PSD can be evaluated from (4.4-44) by the numerical method.
(2). The average autocorrelation function for 0    ( L  1)T can be computed from
(4.4-34) numerically.
(B). For some h fthat  ( jh)  1 , e.g., h  K , K  I , we can set
 ( jh)  e j 2 , 0    1
(4.4-45)
Then the sum in (4.4-41) becomes

e
 j 2 T ( f  / T ) n
n 0

1 1

2 2T



n
1


   f  T  T   j 2 cot  T  f  T 
(4.4-46)
n 
Thus, the PSD now contains impulse located at frequencies
fn 
n 
, 0    1, n  1, 2,
T
(4.4-47)
(4.4-46) can be derived as below:


 e j 2 T ( f  / T )n   e j 2 f  n with f   T ( f   / T )
n 0

e
n 0
n 0
 j 2 f  n

=
 u ( n )e
 j 2 f  n
n 
Thus, we have viewed the sum term as the discrete-time Fourier transform of a unit step
1 n  0,1, 2,
sequence, i.e., u(n )  
0 n  1, 2,
It is worthy to say that u(n ) cannot be evaluated its DTFT directly because it does not
converged. Let
 1 n  0,1, 2,
1 1
u(n )   sgn(n ) with sgn(n )  
. Then
2 2
 1 n  1, 2,
1 
1
 ( f   k) 
. Also see Oppenheim, et al.,

2 k 
1  e  j
Discrete-time signal processing 2nd ed., pp. 53-54, Examples. 2.23-24, and the equation
2.153)
(Hint; F [u(n )]  F [ 12  12 sgn(n )] 
To find DTFT of sgn( n ) , we let
x1 (n)  a nu(n), x2 (n)  a nu(n  1), a  1
Therefore
sgn( n )  x1 ( n )  x2 ( n )   a nu(n )  a  nu( n  1) 

(i).
a e
n  j n
n 0
Then (i)+(ii)=

   ae
 j
n 0

n
a 1
1
and (ii).

1  ae  j

 a e
 n  j n
n 1

    ae
n 1
j

n

ae j
1  ae j
1  a 2  2ae j
(1  ae j )(1  ae j )
The DTFT of sgn( n ) is
2
e j / 2
F [sgn(n)]  lim[(i)  (ii)] 

 1  j cot( / 2)
a 1
1  e j j sin( / 2)
Finally
1  

 k
 1 j
  1  
U ( f )     T ( f  )  k    cot   
f  

2 k  
T
T T
 2 2
 2  2T k  
 1 j
 
   cot  
 2 2
2
Back to the case for which  ( jh)  1 , when symbols are equally probable, i.e.,
1
for all n
M
The characteristic function simplifies to the form
1 M 1 j hn 1 sin M  h
 ( jh) 
 e  M sin  h
M n  ( M 1)
Pn 
(4.4-48)
n odd
Note that in this case  ( jh ) is real. The average autocorrelation function given by (4.4-34)
also simplifies in this case to
1
vv ( ) 
2T

1
2T
 M 1

j 2 hn [ q ( t  ( k  m ) T )  q ( t kT )] 

Pe
dt
0 k
  n

1 L n  ( M 1)
 n odd

T m 1
T [ / T ]
(4.4-49)
1 sin{2 hM [ q(t    kT )  q(t  kT )]}
dt
sin{2 h[q(t    kT )  q(t  kT )]}
 M
0 k 1 L
It is also real.
N
Where we have used the expansion
e
j (  n )
n 0
  0,  2 h .

sin[( N  1) / 2] j (  N / 2)
e
with
sin( / 2)
The PSD reduces to
 LT
 vv ( f )  2   vv ( ) cos 2 f  d
0


1   ( jh ) cos 2 fT
2
1   ( jh )  2 ( jh ) cos 2 fT
( L 1) T
 ( jh )sin 2 fT
1   ( jh )  2 ( jh ) cos 2 fT
( L 1) T
2

vv ( ) cos 2 f  d
(4.4-50)
LT


vv ( )sin 2 f  d 

LT
(4.4-50) can be derived as below:
 LT
1
 vv ( f )  2 Re   vv ( )e  j 2 f  d 
1   ( jh )e  j 2 fT
0
( L 1) T

LT

vv ( )e  j 2 f  d 


1
 2  vv ( ) cos 2 f  d  2 Re 
 1   ( jh ) cos 2 fT  j ( jh )sin 2 fT
0
LT
( L 1) T



vv ( )  cos 2 f   j sin 2 f    d

LT
 1   ( jh ) cos 2 fT  j ( jh )sin 2 fT
 2  vv ( ) cos 2 f  d  2 Re 
2
 1   ( jh ) cos 2 fT  2 ( jh ) cos 2 fT
0
LT
( L 1) T



vv ( )  cos 2 f   j sin 2 f    d

LT
LT
 2  vv ( ) cos 2 f  d 
0
( L 1) T


1   ( jh ) cos 2 fT
1   ( jh ) cos 2 fT  2 ( jh ) cos 2 fT
vv ( ) cos 2 f  d 
LT
( L 1) T


vv ( )sin 2 f  d
LT
Power density spectrum of CPFSK
2
 ( jh )sin 2 fT
1   ( jh ) cos 2 fT  2 ( jh ) cos 2 fT
2
When the pulse shape of g (t ) is rectangular and zero outside the interval [0,T], q(t ) is
linear for 0  t  T . The resulting PSD may be expressed as
2 M M
1 M

(4.4-51)
 vv ( f )  T   An 2 ( f )  2  Bnm ( f ) An ( f ) Am ( f ) 
M n 1 m1
 M n 1

where
An ( f ) 
sin   fT  12 (2n  1  M )h 
  fT  12 (2n  1  M )h 
cos(2 fT   nm )   cos  nm
1   2  2 cos 2 fT
  h(m  n  1  M )
Bnm ( f ) 
 nm
(4.4-52)
sin M  h
M sin  h
The PSD of CPFSK for M=2,4,and 8 is plotted in Figures 4.4-3-4.4-5 as a function of the
normalized frequency fT with the modulation index h  2 f d T as a parameter.
   ( jh ) 
Notes:
(1). Only one-half of the bandwidth occupancy is shown in these graph. The origin
corresponds to the carrier, f c .
(2). For h  1 , the graphs illustrate that the spectrum of CPFSK is relatively smooth and
well confined.
(3). As h approaches unity, the spectra become very peaked and for h  1 when |  | 1 , we
find that impulses occur at M frequencies.
(3). When h  1 , the spectrum becomes broader. Therefore, in communication systems
where CPFSK is used, the modulation index is designed to conserve bandwidth, so that
h  1.
The special case of binary CPFSK with h 
MSK. The spectrum of such a signal is
1
2
(or f d  1/ 4T ) and   0 corresponds to
2
16 A2T  cos 2 fT 
(4.4-53)
 vv ( f ) 
 2  1  16 f 2T 2 
Where the signal amplitude A  1 in (4.4-52). In contrast the spectrum of OQPSK with a
rectangular pulse g (t ) of duration T is
2
 sin  fT 
(4.4-54)
 vv ( f )  A T 

  fT 
When we compare these spectral characteristics, we should normalize the frequency variable
by the bit rate or the bit interval Tb . Since MSK is binary FSK, it follows that T  Tb in
2
(4.4-53). On the other hand, in OQPSK, T  2Tb so that (4.4-54) becomes
2
 sin 2 fTb 
 vv ( f )  2 A Tb 

 2 fTb 
The spectra of the MSK and OQPSK signals are illustrated in Figure 4.4-6.
2
(4.4-55)
Observe that
(1). The main lobe of MSK is 50% wider than that for OQPSK. However, the side lobes in
MSK fall off considerably faster.
(2). If we take the bandwidth W containing 99% of the total power, we find that W  1.2 / Tb
for MSK and W  8 / Tb .Accordingly, MSK has a narrower spectral occupancy when
viewed in terms of fractional out-of-band power above fTb  1 . The fractional out of
band power for OQPSK and MSK are shown in Figure 4.4-7.
(3). MSK is significantly more bandwidth-efficient that QPSK. This efficiency accounts for
the popularity of MSK in many digital communication systems.
(4). Even greater bandwidth efficiency than MSK can be achieved by reducing the
modulation index. However, the FSK signals will no longer be orthogonal and there
will be an increase in the error probability.
Spectral characteristics of CPM
(1). The bandwidth efficiency of CPM depends on the choice of the modulation index h, the
pulse shape g (t ) , and the number of signals M.
(2). In CPFSK, small indexes h result in CPM signals with relatively small bandwidth
occupancy, while large indexes h result in CPM signals with relatively large bandwidth
occupancy. This is also the case for the more general CPM signals.
(3). The use of smooth pulses such as raised cosine pulses of the form
2 t 
 1 
1  cos


 0  t  LT
(4.4-56)
g (t )   2 LT 
LT 
0
otherwise

where L  1 for full response and L  1 for partial response, result in smaller
bandwidth occupancy and, hence, greater bandwidth efficiency than the use of
rectangular pulses.
(4). Figure 4.4-8 illustrates the PSD for binary CPM with different partial response raised
cosine (LRC) pulses when h  12 . The PSD of CPFSK is also shown for comparison.
Note that as L increases the pulse, g (t ) becomes smoother and the corresponding spectral
occupancy of the signal is reduced.
Figure 4.4-9 illustrates the effect of the modulation indexes h in a CPM signal, here M=4 and
a raised cosine pulse of the form (4.4-56) with L=3. The spectral characteristics are similar to
the ones illustrated previously for CPFSK, except that these spectra are narrower due to the
use of a smaller pulse shape.
Figure 4.4-10 illustrates the fractional out of band power for two-amplitude CPFSK with
several different values of h.
4.4.3 Power spectra of modulated signals with memory
In last two sections, we have determined the spectral characteristics for the class of linearly
modulated signals without memory and for the class of angle-modulated signals such as
CPFSK and CPM, which are nonlinear and possess memory. In this section, we consider the
spectral characteristics of linearly modulated signals that have memory that can be
modeled by a Markov chain.
For signals that are generated by a Markov chain with transition probability matrix P, the
PSD of the modulated signal may be expressed in the general form (see Titsworth and Welch,
1961)
2
1  K
n 1 K
n 
 ( f )  2   pi Si     f     pi Si ( f )
T n  i 1
T  T i 1
T  
2
K K

2
 Re   pi Si ( f )S j ( f ) Pij ( f ) 
T
 i 1 j 1

(4.4-57)
where Si ( f ) is the Fourier transform of the signal waveform si (t )
K
si (t )  si (t )   pk sk (t )
i 1
Pij ( f ) is the Fourier transform of the discrete-time sequence pij (n ) , defined as

Pij ( f )   pij (n )e  j 2 nfT
n 1
(4.4-58)
and K is the number of states of the modulator. The term pij (n ) denotes the probability that
the signal s j ( t ) is transmitted n signaling intervals after the transmission of si (t ) . Hence,
{ pij ( n )} are the transition probabilities in the transition probability matrix P n . Note that
pij (1)  pij .
Considering the case that there is no memory in the modulation methods, the signal
waveform transmitted on each signaling interval is independent of the waveforms
transmitted in previous signaling intervals. The PSD of the resulting signal may still be
expressed in the form of (4.4-57), if the transition probability matrix is replaced by
 p1
p
P 1


 p1
p2
p2
p2
pK 
pK 



pK 
(4.4-59)
and we impose the condition that Pn  P for all n  1 . Consequently, The PSD becomes a
function of the stationary state probabilities { pi } only, and it reduces to the simpler form.
1
( f )  2
T


n 
2
n 1 K
n 
p
S

f


i i
 
   pi (1  pi ) Si ( f )
T  T i 1
T  
i 1
K
2
(4.4-60)
2 K K
  pi p j Re  Si ( f ) S j  ( f ) 
T i 1 j 1
We observe that
(1). The PSD of memoryless linear modulation given by (4.4-18) may be viewed as a special
case of (4.4-60) in which all waveforms are identical except for a set of scale factors that
convey the digital information.
(2). The first term in the expression for the PSD given by either (4.4-57) and (4.4-60)
consists of discrete frequency components. This line spectrum vanishes when
n
K
 p S  T   0
i 1
i
(4.4-61)
i
The condition (4.4-61) is usually imposed in the design of practical digital communication
systems and is easily satisfied by an appropriate choice of signaling waveforms.
In the following, the PSD of base band modulated signals will be examined.
(i). The NRZ signal is characterized by the two waveforms s1 (t )  g (t ) and s2 (t )  g (t ) ,
where g (t ) is a rectangular pulse of amplitude of A. For K  2 (4.4-60) reduces to
2
(2 p  1)2 
n  4 p(1  p) K
n 
( f ) 
G

f

pi G  f 
  T   T   T 
T2
n 
i 1
where
2
(4.4-62)
 sin  fT 
G( f )  ( AT ) 

  fT 
2
2
2
(4.4-63)
Observe that when p  1/ 2 , the line spectrum vanishes and  ( f ) reduces to
1
2
( f )  G( f )
(4.4-64)
T
(ii). The NRZI signal is characterized by the transition probability matrix
1 1
(4.4-65)
P   12 12 
2 2
Note that in this case Pn  P for all n  1 . Hence, the special form for the PSD given by
(4.4-62) applies to this modulation format as well. Accordingly, The PSDs of NRZ and
NRZI are identical.
(iii). Delay modulation has a transition probability matrix
 0 12 0 12 
0 0 1 1 
P  1 1 2 2
(4.4-66)
 2 2 0 0
1

1
 2 0 2 0
and stationary state probabilities pi  1/ 4 for i  1,2,3,4 . Powers of P may be obtained by
use of the relation
P4ρ   14 ρ
(4.4-67)
where ρ is the signal correlation matrix with elements
ij =
T
1
si (t ) s j (t )
T 0
(4.4-68)
and where the four signals {si (t ), i  1,2,3,4} are shown in Figure 4.3-15. It is easily seen
that
 1 0 0 1
 0 1 1 0 

(4.4-69)
ρ
 0 1 1 0 


 1 0 0 1 
Consequently, powers of P can be generated from the relation
Pk 4ρ   14 Pk ρ, k  1
(4.4-70)
Use of (4.4-66), (4.4-69), and (4.4-70) in Equation (4.4-57) yields the PSD of delay
modulation and is shown as below:
1
(23  2 cos  22 cos 2  12 cos 3  5cos 4
2 (17  8cos8 )
12 cos5  2 cos 6  8cos 7  2 cos8 )
where    fT .
The spectra of these base band signals are illustrated in Figure 4.4-11.
( f ) 
2
(4.4-71)
Notes:
(1). The spectra of the NRZ and NRZI signals peak at f  0 .
(2). Delay modulation has a narrower spectrum and a relatively small zero-frequency content.
Its bandwidth occupancy is significantly smaller than that of the NRZ signal. These two
characteristics make delay modulation an attractive choice for channels that do not pass
DC, such as magnetic recording media.