Chapter 7: Spectral Density
7-1
7-2
Introduction
Relation of Spectral Density to the Fourier Transform
 Weiner-Khinchine Relationship
7-3
Properties of Spectral Density
7-4
Spectral Density and the Complex Frequency Plane
7-5
Mean-Square Values From Spectral Density
7-6
Relation of Spectral Density to the Autocorrelation Function
7-7
White Noise
 Noise Terminology: White Noise, Black Noise, Pink Noise
Contour Integration – (Appendix I)
7-8
Cross-Spectral Density
7-9
Autocorrelation Function Estimate of Spectral Density
7-10 Periodogram Estimate of Spectral Density
7-11 Examples and Application of Spectral Density
Concepts:










Relation of Spectral Density to the Fourier Transform
o Weiner-Khinchine Relationship
Properties of Spectral Density
Spectral Density and the Complex Frequency Plane
Mean-Square Values From Spectral Density
Relation of Spectral Density to the Autocorrelation Function
White Noise, Black Noise, Pink Noise
Contour Integration – (Appendix I)
Cross-Spectral Density
Autocorrelation Function Estimate of Spectral Density
Periodogram Estimate of Spectral Density
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
1 of 22
ECE 3800
Chapter 7: Spectral Density
Much of engineering employs frequency domain methods for signal and system analysis.
Therefore., we are interested in frequency domain concepts and analysis related to probability
and statistics!
But first, let’s review Fourier Transforms …
Y w 

 yt   exp iwt   dt

If y(t) is in volts, Y(w) is expressed in terms of Volts/(radians per second). Thus, the Y(w)
represents the relative magnitude and phase of steady-state sinusoids that may be summed to
produce the original signal. As such, it describes the amplitude density as a function of
frequency.
In order for the Fourier Transform to exists, two conditions must be met:

 yt   dt  
1)
The integral of yt  over all time exists, or
2)
There are a finite number of discontinuities in y(t).

For sample functions containing random variables resulting in an ensemble of sample functions,
the individual Fourier Transforms may exist, but we cannot readily describe the Fourier
Transform of the entire ensemble.
What can we describe for all sample functions of an ensemble that contains time- or period- or
frequency-based information?
For WSS random processes, the autocorrelation function is time based and, for ergodic
processes, describes all sample functions in the ensemble! In these cases the WienerKhinchine relations is valid that allows us to perform the following.
We can define a power spectral density for the ensemble as:
S XX w  R XX   

 R XX    exp iw   d

If Rxx(t) is a volts-squared (V2) type term, so a Fourier transform would expressed in terms of
V2/(radians per second) or V2/Hz. Thus, the SXX(w) represents the relative magnitude of
sinusoidal components that are present in the auto-correlation.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
2 of 22
ECE 3800
Note that Rxx(t) is symmetric in time so we could also computed the Power Spectral Density as

S XX w  R XX    2   R XX    cosw   d
0
The power spectral density of a real auto-corr3elation function has no phase!
By the way, if we define the Power Spectral Density, we can define the inverse …
S XX w  R XX  
1
R XX t  
2
R XX     1 S XX w

 S XX w  expiwt   dw

More on the Fourier Transform of a time domain signal …

 yt   exp iwt   dt
Y w 

The “Power Spectral Density” of the time domain signal can be described as
S YY w   Y w   conj Y w   Y w
2
Proof:


S YY w     y t   y t     dt   exp iw   d
   


Swapping the order of integration


S YY w   y t     y t     exp iw   d   dt

 


S YY w 

 yt   Y w  expiwt   dt


S YY w  Y w   y t   expiwt   dt

S YY w   Y w   conj Y w   Y w
2
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
3 of 22
ECE 3800
Summary: How to form the power spectral density.
For non-random signals:
take the magnitude squared of the Fourier Transform
Y w   y t  

 yt   exp iwt   dt

S YY w   Y w   conj Y w   Y w
For ergodic, WSS random signals:
2
form the auro-correaltion and take the Fourier Transform
R XX    EX t X t   
or
1
T   2T
 XX    lim
T
 xt   xt     dt 
xt   xt   
T
S XX w  R XX   

 R    exp iw   d
XX

For non-ergodic or non-WSS random signals:
Wiener-Khinchine relations is not valid.
Why this is very important … the Fourier Transform of a “single instantiation” of a random
process may be meaningless or even impossible to generate. But if the random process can be
described in terms of the autocorrelation function (all ergodic, WSS processes), then the power
spectral density can be defined.
I can then know what the expected frequency spectrum output looks like and I can design a
system to keep the required frequencies and filters out the unneeded frequencies (e.g. noise and
interference).
In communications, most of the transmitted waveform is random (changing information content).
But, based on probability, I can still design appropriate transmitting electronics and receiving
systems to send, receive, and detect the information!
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
4 of 22
ECE 3800
Relation of Spectral Density to the Autocorrelation Function
For “the right” random processes, power spectral density is the Fourier Transform of the
autocorrelation:

S XX w  R XX   
 EX t   X t    exp iw   d

For an ergodic process, we can use time-based processing to arrive at an equivalent result …
T
1
 XX    lim
T   2T
 xt   xt     dt 
xt   xt   
T
1
E X t   X t      XX    lim
T   2T

T
 xt   xt     dt
T
T


1

 XX    E  X t   X t    
lim
xt   xt     dt   exp iw   d
T   2T

 
T

T 

1

 XX    lim
xt   xt     exp iw   d   dt


2
T
T 
T  

T 

1

 XX    lim
xt   xt     exp iwt     iwt   d   dt


2
T
T 
T  

T 

1

xt   exp iwt   xt     exp iwt     d   dt
 XX    lim

T   2T
T  

T


1

xt   exp iwt   xt     exp iwt     d   dt
 XX    lim
T   2T
T
 



 
 
 


X    X w
If there exists
1
T   2T
 XX    lim
T
 xt   exp iwt X w  dt
T
1
 XX    X w  lim
T   2T
T
 xt   exp i wt  dt
T
 XX    X w  X  w  X w
2
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
5 of 22
ECE 3800
Properties of the Fourier Transform:
X w  x  

 x   exp iw   d

For x(t) purely real
X w  x  

 x   cosw   i  sinw  d

X w  x  




 x   cosw   d  i   x   sinw   d
X w   E X w   i  O X w  





 x   cosw   d  i   x   sinw   d
 x   cosw   d
E X w 
and O X w 


 x   sinw   d

Notice that:



 x   cosw   d   x   cos w   d  E X  w
E X w 
O X w  





 x   sinw   d    x   sin w   d  O X  w
Therefore, the real part is symmetric and the imaginary part is anti-symmetric!
Note also,
X  w   conj  X w   X w
*
X(w) is conjugate symmetric about the zero axis.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
6 of 22
ECE 3800
Relating this to a real autocorrelation function where R XX    R XX   
R XX    E X w  i  O X w

R XX    
 R XX     cosw   i  sinw  d

R XX    

 R XX t   cos wt   i  sin wt   dt 

R XX    




 R XX t   coswt   dt  i   R XX t   sinwt   dt
R XX     E X w  i  O X w
Since Rxx is symmetric, we must have that
R XX    R XX    and E X w  i  O X w  E X w  i  O X w
For this to be true,  i  O X w  i  O X w , which can only occur if the odd portion of the
Fourier transform is zero! O X w  0 .
This provides information about the power spectral density,
S XX w  R XX    E X w
S XX w  E X w
S XX w  0
The power spectral density necessarily contains no phase information!
This is the quick way; now let’s see how your text got to the same point …
First, investigate the Fourier Transform and see if this makes sense …
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
7 of 22
ECE 3800
Relation of Spectral Density to the Fourier Transform
Basic Fourier Transform
Y w 

 yt   exp iwt   dt


where
 yt   dt  

To make a function “Fourier Transformable” we can apply “time domain windows” to limit
function to be transformed in either time or even magnitude.
More typically, a windowed Spectrum
Y w  lim
T 
T
Windowt   yt   exp iwt   dt
T
The most popular window is a rectangular window from –T to T. This is used to make time
“finite” … an infinite time interval has yet to be completed! Other windows are possible too.


A rectangle Function cuts the signal to a finite time period (that should be integrable).
An exponential scaling can guarantee that a signal waveform goes to zero as time goes to
infinity.
For most applications, we are interested in the spectral magnitude and/or the spectral phase
information, Y w or Y w .
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
8 of 22
ECE 3800
For the time window based on T, the time function can be described as
 y t , t  T  
yT t   
t T
0,
For this case, it can be shown that the integral of the magnitude square is also finite, or


yT t   dt  
2

Based on the transform existing, Parseval’s theorem can provide some useful insight. Parseval’s
theorem states that for two transformable functions with known transforms, the following holds:



1
f t   g t   dt 
2

 F w  G w  dw

For the above time-bounded signal, the results for f t   yt  and g t   y t  are
T

yT t 
2
T
1
 dt 
2


1
Y w  Y  w  dw 
2



Y w  dw
2

As this appear to be the time integral of power, let make it the average power in the time interval
by dividing by 2T:
1
2T
T

yT t 
2
T
1
 dt 
4T


Y w  dw
2

The left hand is now related to the signal’s time averaged second moment, for which the time
average and statistical mean are equivalent if y is ergodic. Using the limit, this may be more
readily seen as:
1
T   2T
lim
T
 yT t   dt  yT t 
2
T
2
1
T   4T
 lim

 Y w 
2
 dw

Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
9 of 22
ECE 3800
Taking an expected value …
T



1
1
2
2
2
E  lim
yT t   dt   Y  lim
E  Y w   dw


T   2T


T   4T
T




Y
1
 lim
T   2T
2

 E yT t  
T
2
T
Y2 Y2 
1
2
1
 dt 
2




2
E  Y w  


 dw
lim
2T
T 

2
E  Y w 


 dw
lim
2T
T 

Observing the right hand side, there appears to be a function related to frequency that describes
the FFT of the time average or letting
Y
2
Y
2
1

2

 SYY w  dw

2
E  Y w 


where SYY w  lim
2T
T 
This function is also defined as the spectral density function (or power-spectral density) and is
defined for both f and w as:
2
E  Y w 


SYY w  lim
2T
T 
2
EY  f  


SYY  f   lim
2T
T 
or
The 2nd moment based on the spectral densities is defined, as:
Y
2
1

2

 SYY w  dw


and
Y
2

 SYY  f   df

Note: The result is a power spectral density (in Watts/Hz), not a voltage spectrum as (in V/Hz)
that you would normally compute for a Fourier transform.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
10 of 22
ECE 3800
Exercise 7-2.2
A stationary random process has a two-sided spectral density given by
S XX w 
a.)
24
w 2  16
Find the mean-square value of the process.
X
1

2


1
S XX w  dw 
2


w

24
2
 16
 dw

24

2
3
 arctan   arctan   


 w 
24 1

 dw 

 arctan
2
2 16
 16   
w  16

2
X
X2 
2
1

3     3

       3
  2  2  
b.)
Find the mean square value of the process in the frequency band of +/- 1 Hz centered on
the origin.
24
X 1Hz 
2
2
2
X 1Hz 
2

 2
2
24 1
 w
 dw 

 arctan  
2
2 16
 4   2
w  16
1
3 
 2 
 2
 arctan
  arctan 
 
 4 
 4
 3
   2  1.004  1.9173
 
This refers to the total signal power and the signal power in a defined part of the frequency band
(that could be extracted or remain after perfect filtering).
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
11 of 22
ECE 3800
Properties of the Power Spectral Density
The power spectral density as a function is always

real,

positive, (never negative as it is a magnitude)

and an even function in w.
As an even function, the PSD may be expected to have a polynomial form as:
S XX w  S 0
w 2n  a 2n  2 w 2n  2  a 2n  4 w 2n  4    a 2 w 2  a0
w 2m  b2m  2 w 2m  2  b2m  4 w 2m  4    b2 w 2  b0
where m>n.
Notice the squared terms, any odd power would define an anti-symmetric element that, by
definition and proof, can not exist!
Finite property in frequency:
The Power Spectral Density must also approach zero as w approached infinity …. Therefore,
w 2n  a 2n2 w 2n2    a 2 w 2  a0
w 2n
1
S
S XX w     lim S 0 2 m

lim
 lim S 0 2m  n   0
0
m
2

2
2
2
m
w 
w 
w 
w  b2 m  2 w
w
w
   b2 w  b0
For m>n, the condition will be met.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
12 of 22
ECE 3800
Example:
Doing it the textbook way … form the PSD given
X t   A  B  cos2f 0 t   
where A, B, and f0 are constant and theta is a uniform R.V. from 0 to 2pi.
Assuming a truncated sequence
FX  f  
T
 A  B  cos2f 0t    exp j 2ft   dt
T


FX  f  



 t 
rect     A  B  cos2f 0 t     exp j 2ft   dt
 2T 


Fourier Transform Property: The FT of a product is the convolution of the FTs. Therefore,
B
B


FX  f   2T  sinc 2T  f    A    f      f  f 0   exp j      f  f 0   exp j 
2
2


FX  f   2 AT  sinc2T  f  
BT  sinc2T   f  f 0   exp j   BT  sinc2T   f  f 0   exp j 
Forming the magnitude squares
F X  f   F X  f   FX  f 
2
4 A 2T 2  sin c2T  f 2  B 2T 2  sin c2T   f  f 0 2  B 2T 2  sin c2T   f  f 0 2  fn 

E FX  f 
2
 4A T
2
2
 sin c2T  f   B 2T 2  sin c2T   f  f 0  
2
2
B 2T 2  sin c2T   f  f 0   E  fn 
2
The expected value of the remaining phase is
E fn   0
Therefore,
2
E  FX  f    4 A 2T 2  sin c2T  f 2  B 2T 2  sin c2T   f  f 0 2 


B 2T 2  sin c2T   f  f 0 2
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
13 of 22
ECE 3800
Then for
2
EY  f  


SYY  f   lim
2T
T 
1
S  f   lim
T   2T
4 A 2T 2  sin c2T  f 2  B 2T 2  sin c2T   f  f 2
0

 B 2T 2  sin c2T   f  f 0 2
2
 2
2 B
2T  sin c2T   f  f 0 2
 A 2T  sin c2T  f  

4
S  f   lim 
T   B 2
2
 4 2T  sin c2T   f  f 0 
 








But this can be defined, in the limit as
B2
B2
S  f   A    f  
  f  f0  
  f  f0 
4
4
2
(what a pain in the ….)
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
14 of 22
ECE 3800
Alternately, using the autocorrelation function
X t   A  B  cos2f 0 t   
R XX    E X t X t     E  A  B  cos2f 0 t    A  B  cos2f 0 t      


 A 2  AB  cos2f 0 t     AB  cos2f 0 t      
R XX    E 
 B 2  cos2f 0 t     cos2f 0 t      

R XX    A 2  E  AB  cos2f 0 t     AB  cos2f 0 t       


E B 2  cos2f 0 t     cos2f 0 t      
R XX    A 2  B 2  E cos2f 0 t     cos2f 0 t      
1
1

R XX    A 2  B 2  E   cos2f 0    cos2f 0 2t     2 
2
2

R XX    A 2 
B2
 cos2f 0 
2
Performing the Fourier transform
 2 B2

S  f   R XX     A 
 cos2f 0 
2


S   f   R XX    A 2    f  
B2
B2
  f  f0  
  f  f0 
4
4
Which do you think is easier?


One derivation attempts to provide a better “physical meaning” but may be confusing.
The other definition may be harder to accept at face value.
(a little math that produces the correct result)
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
15 of 22
ECE 3800
Another Example of a Discrete Spectral Density (p. 267)
X t   5  10  sin2  6  t  1   8  cos2  12  t   2 
where the phase angles are uniformly distributed R.V from 0 to 2π.
With practice, we can see that
1
1

R XX    A 2  B 2  E   cos2  f1    cos2  f1 2t     21 
2
2

1
1

 C 2  E   cos2  f 2    cos2  f 2 2t     2 2 
2
2

which lead to
R XX    25 
100
64
 cos2  6  
 cos2  12 
2
2
And then taking the Fourier transform
S XX  f   25    f  
100  1
1
1
 64  1

     f  6      f  6  
     f  12      f  12 
2 2
2
2
 2 2

S XX  f   25    f   25    f  6    f  6  16    f  12    f  12
We also know from the before
1
X 
2
2


 S w  dw   S  f   df
XX

XX

Therefore, the 2nd moment can be immediately computed as

X2 
 25    f   25    f  6    f  6  16    f  12    f  12  df

X 2  25  25  1  1  16  1  1  25  50  32  107
We can also see that
X  E 5  10  sin 2  6  t   1   8  cos2  12  t   2   5
So,
 2  107  5 2  82
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
16 of 22
ECE 3800
Another example:
Determine the autocorrelation of the binary sequence, assuming p=0.5.
xt  

 pt  t 0  k  T 
A
k
k  
xt   pt  

A
k  
k
  t  t 0  k  T 
Determine the auto correlation of the discrete time sequence
y t  

A
k
k  
  t  t 0  k  T 
 

  
E  y t   y t     E   Ak   t  t 0  k  T     A j   t    t 0  j  T 
  j  

 k  
  

R yy    E  y t   y t     E    A j  Ak   t  t 0  k  T    t    t 0  j  T 

 k   j  
 

2
  Ak   t  t 0  k  T    t    t 0  k  T 

 k  

R yy    E   

   A j  Ak   t  t 0  k  T    t    t 0  j  T 
 k   jj  

k
R yy   

 E A

k
k  
2
 E t  t
  EA


k   j  
j k
j
0
 k  T    t    t 0  k  T 

 Ak  E  t  t 0  k  T    t    t 0  j  T 
  T1      EA 
R yy    E Ak 
2
 
2
k
2
1 
     m  T 
T m  
m0
 T1      EA 
R yy    E Ak  E Ak  
2

2
k

1 
     m  T 
T m  
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
17 of 22
ECE 3800
R yy     A2 
1
1  

      A2        m  T 
T
T m  

S yy  f    A2 
1
1 1  
m 
  A2        f  
T
T T m   
T 
From here, it can be shown that
S xx  f   P  f   S yy  f 
2

m 
1
1
2 

S xx  f   P f    A2    A2  2     f  
T
T 
T m   

S xx  f   P f  
2
 A2
T
 P f  
2
 A2
T
2



m
   f  T 
m  
This is a magnitude scaled version of the power spectral density of the pulse shape and numerous
impulse responses with magnitudes shaped by the pulse at regular frequency intervals based on
the signal periodicity.
The result was picture in the textbook as …
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
18 of 22
ECE 3800
Binary Pulse Amplitude (PAM) signaling formats
I did a series of similar definitions for the ECE4600 Communications course a few years ago …
the results follow
(a) Unipolar RZ & NRZ , (b) Polar RZ & NRZ , (c) Bipolar NRZ , (d) Split-phase Manchester,
and (e) Polar quaternary NRZ.
From: A. Bruce Carlson, P.B. Crilly, Communication Systems, 5th ed., McGraw-Hill, 2010.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
19 of 22
ECE 3800
PAM Power Spectral Density: Polar NRZ
The random signal can be describes as
vt  

a
k  
p Td  
k
 t  Td  k  Tb 

 rect
T
b


 
E an   0, E an   2
2
1
,
Tb
0  Td  Tb and


E a j  ak  0,
  
Rvv    Evt   vt      2  1  ,
 Tb 
for j  k
Tb    Tb
S vv w  E vt   vt      2  Tb  sinc 2  f  Tb 
PAM Power Spectral Density: Arbitrary Pulse – similar to our textbook
vt  

a
k  
p Td  
 t  Td  k  D 
 p

D


 
1
,
D
0  Td  D and E an   ma , E an   a  ma
2
S vv  f  
 a2
D
 P f 
2
n0
n0
and Tb  D, rb 
2
m  
 a   
 D  n  
S vv  f    a  rb  P f   ma  rb  
2
2
2

1
2
 P  f    Ra n   exp j  2  f  D 
D
n  
 a 2  ma 2 ,
Ra n    2
ma ,
S vv  f  
k
2
2
1
D
2
n
n

P     f  
D
D


 Pn  r 
n  
b
2
   f  n  rb 
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
20 of 22
ECE 3800
Power spectrum of Unipolar, binary RZ signal

 t
p t   rect
 Tb
 2
E an  

 f 

1

 sinc
  rect 2  rb  t  where P f  
2  rb

 2  rb 

 
A
A2
2
, E an 
2
2
 2
A2
2

m


,n  0
a
 a
2
and Ra n   
2
m 2  A ,
n0
 a
4
2
2
 f 
A2
A2 
n
 
 sinc
S vv  f  
  sinc     f  n  rb 
16  rb
2
 2  rb  16 n  
Power spectrum of Unipolar, binary NRZ signal
 t 
f
1
pt   rect   rectrb  t  where P f    sinc 
rb
 Tb 
 rb 
E an  
 
A
A2
2
, E an 
2
2
 2
A2
2

m


,n  0
a
 a
2
and Ra n   
2
m 2  A ,
n0
 a
4
2
f 
A2
A2 
2
 sinc  
S vv  f  
  sincn     f  n  rb 
4  rb
4 n  
 rb 
But based in the sinc function equals
2
f 
A2
A2
 sinc  
S vv  f  
  f 
4  rb
4
r
b
 
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
21 of 22
ECE 3800
Power spectrum of Polar, binary RZ signal (+/- A/2)

 t
p t   rect
 Tb
 2

 f 

1

 sinc
  rect 2  rb  t  where P f  
2  rb

 2  rb 

 
E an   0, E an
2
2
A
 2  m 2  A2 , n  0
a
4
and Ra n    a
4
2
ma  0, n  0
 f 
A2

S vv  f  
 sinc
16  rb
 2  rb 
2
Power spectrum of Polar, binary NRZ signal (+/- A/2)
 t 
f
1
pt   rect   rectrb  t  where P f    sinc 
rb
 Tb 
 rb 
 
E an   0, E an
2
 2  m 2  A2 , n  0
a
4
and Ra n    a
A
4
2
ma  0, n  0
2
f 
A2
S vv  f  
 sinc 
4  rb
 rb 
2
Why do we care?
The bandwidth and spectral characteristics of the signals are very important. Issues include
spectral capacity, filter selections, adjacent signal interference, etc.
Notes and figures are based on or taken from materials in the course textbook: Probabilistic Methods of Signal and System
Analysis (3rd ed.) by George R. Cooper and Clare D. McGillem; Oxford Press, 1999. ISBN: 0-19-512354-9.
B.J. Bazuin, Spring 2015
22 of 22
ECE 3800