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 24
ECE 3800
Chapter 7: Spectral Density
Wiener–Khinchin Theorem
For WSS random processes, the autocorrelation function is time based and has a spectral
decomposition given by the power spectral density.
We can define a power spectral density as the Fourier transform of the autocorrelation function:

S XX w  R XX   
 R XX    exp iw   d

Based on this definition, we also have
S XX w  R XX  
1
R XX t  
2
R XX     1 S XX w

 S XX w  expiwt   dw

Also see:
http://en.wikipedia.org/wiki/Wiener%E2%80%93Khinchin_theorem
Alternate textbook form,
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 
or
2
EY  f  


SYY  f   lim
2T
T 
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 24
ECE 3800
Properties: 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.
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
 lim S 0 2 m  lim S 0 2m  n   0
S XX w     lim S 0 2 m
2m2
2
w 
w


w


w  b2 m  2 w
   b2 w  b0
w
w
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
3 of 24
ECE 3800
Generic Example of a Discrete Spectral Density (p. 267)
X t   A  B  sin2  f1  t  1   C  cos2  f 2  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    A 2 
B2
C2
 cos2  f 1    
 cos2  f 2   
2
2
And then taking the Fourier transform
S XX  f   A 2    f  
B2
2
2
1
1
 C
     f  f 1      f  f 1  
2
2
 2
1
1

     f  f 2      f  f 2 
2
2

B2
C2
   f  f1     f  f1  
   f  f 2     f  f 2 
S XX  f   A    f  
4
4
2
We also know from the before
X2 
1
2




 S XX w  dw 
 S  f   df
XX
Therefore, the 2nd moment can be immediately computed as

X2 
 2

B2
C2









A


f



f

f


f

f

   f  f 2     f  f 2   df
1
1

4
4

X 2  A2 
B2
C2
B2 C 2
 2 
 2  A 2 

4
4
2
2
We can also see that
X  E A  B  sin 2  f 1  t   1   C  cos2  f 2  t   2   A
So,
B2 C 2
B2 C 2
2
 A 

A 

2
2
2
2
2
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
4 of 24
ECE 3800
Another example: p. 267-269 more “periodic symbol transmission stuff”
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 
Notice that there is a random part and a deterministic part that can be separated. Determine the
auto correlation of the random 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
 E A

R yy   
k
k  

2
 E t  t
  EA


j
0
 k  T    t    t 0  k  T 

 Ak  E  t  t 0  k  T    t    t 0  j  T 
k   j  
jk
Taking the time average of the expected values based on T periods,

1
2
2 1
R yy    E Ak       E  Ak        m  T 
T
T m  
 
m0
 
 T1      EA 
R yy    E Ak  E Ak  
2
2
2
k

1 
     m  T 
T m  
 

      m  T 
m  


1
1 1
m 

S yy  f    A2    A2        f  
T
T T m   
T 
R yy     A2 
1
1
      A2 
T
T
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 24
ECE 3800
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 …
A note on real signals, the “pulse” can be longer than the symbol period T. For symbo9l
transmissions using “Square-Root Nyquist Filters” the pulse may actually be 3-10 symbols long
with very specific properties so that “inter-symbol interference” will not occur when a
correlation signal detector samples the received symbol at the “perfect moment in time”.
- see MRSP course example of exam #2 test signal.
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 24
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
7 of 24
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
8 of 24
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     f  n  rb 
S vv  f  
 sinc
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
S vv  f  
  sincn     f  n  rb 
 sinc  
4  rb
4 n  
 rb 
But based in the sinc function equals
2
f 
A2
A2
S vv  f  
  f 
 sinc  
r
4  rb
4
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
9 of 24
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
10 of 24
ECE 3800
Autocorrelation to Power Spectral Density (p. 283-284.)
Example 1: Random Pulse in Time - analysis 1
Let X(t) be an asynchronous bipolar signal with a random pulse width, T, and amplitude +/-A.
The pulse width, T, is exponentially distributed!
 T

 t   t0 
2

X t   A  rect 
T






Reminders for the exponential distribution
 1
 1 
 t 
  exp 
f T t     T

T


0


t0
else

 1   1

 1 
1
1




E T    t 
t
t







exp
1
 exp 
 t   dt 

2
 
  



 1 
T
T
T
T
T

 
0


0
  
T 

 1
  1

E T     T  exp 
 0    
 0  1    T  1   1   T
 T   T


 1 
1
 exp 
 t   dt
E T 2  t2 
T
 T 
0
1
 
 
ET
2


 1   1 2 2
1
 



 t    
t  2
 t  2 
exp

3
T 
T  1 
T
 T   
0
  
T 

 1

2
2
E T 2    T  exp 
 0   2  2   T
 T 
2
2
 T  2   T   T 2   T 2
1
1
  

And returning to the aut0-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
11 of 24
ECE 3800
Now for the autocorrelation:

T

 
 T
 
 t     t 0  
 t   t0   

2
2
 
    A  rect 
R XX t , t     E X t   X t     E  A  rect 
T
T

 

 

 


 


 

 

T
 T



 t   t0 
 t     t 0 

2
2
  rect 

R XX t , t     E A 2  E rect 
T
T














 
For τ > 0 we have
 
 

R XX    E A  Pr  T   E A   1  1 
2
2

 1

 exp 
 x   dx
T
 T 
1



 1

1
1
2

R XX    E A 

 exp 
 x 
 T 1

T


T


 
 1

  
R XX    A 2  exp 
 T 
For a real symmetric autocorrelation, the negative portion must be correctly included as
 1

  
R XX    A 2  exp 
 T

The power spectral density based on the auto-correlation becomes ….
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 24
ECE 3800
And the power spectral density becomes
S XX w  R XX   

 R XX    exp iw   d

S XX w 

A

2
 1
 exp 

 T

  exp iw   d

0


 1
 1
 
 




 iw      d   exp 
 iw      d 
S XX w  A    exp  
 
 

  T
  T
 0


2



0

 1
 1
 
  
1
1
S XX w  A 2  
 exp  
 iw     
 exp 
 iw     
1
  1  iw
 0
   
  T
  T
 iw

 T

T


 1

1


iw
iw






T
T
1 
1
2
2 

A 
S XX w  A  


1
1

  1
 1

 iw
 iw 
 





iw
iw


 T

 

T


   T
 
  T
2
A2 
T
S XX w 
2
 1 

  w 2

 T
For graphical example of these functions, see Figure 7-8 in the text.
One of the favorite signals used as an example in the textbook ….
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 24
ECE 3800
Example 2 Bipolar signal with a random pulse-width – uniform dist.
Let X(t) be an asynchronous bipolar signal with a random pulse width, T, and amplitude +/-A.
The pulse width uniformly distributed.
 t  t  TW  kT 


0
2
X t  
Ak  Re ct 

TW


k



where A(k) are zero mean, identically distributed, independent R.V., t0 is a R.V. uniformly
distributed over possible bit period T, and Tw is a random variable uniformly distributed from 0
to T. k describes the bit positions.
For the autocorrelation of one of the pulse periods:


TW
 t    t  TW  kT  
 kT  
 

0
 t  t 0 
2
2
R XX t , t     E 
 
   A  rect 
TW
TW
 

 

 

 


 t  t  TW  kT 
 t    t  TW  kT 




0
0

2
2
R XX t , t     E A  E rect 
  rect 

TW
TW









 
2

 t    t  TW  kT 
 t  t  TW  kT 




0
0

2
2
R XX t , t     E A  E rect 

  rect 
TW
TW









 
2
For tau>0 we have
 
   T1  dx

R XX    E A  Pr  Tw   E A 
2
T
2
 
T
x
R XX    E A   
 T 
 T  
R XX    A 2  

 T 
2
For a real symmetric autocorrelation, the negative portion must be correctly included as
  
R XX    A 2  1  
 T
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 24
ECE 3800
And the power spectral density becomes
S XX w  R XX   

 R XX    exp iw   d

S XX w 

 
A 2  1 
 T



  exp iw   d


0
T



 


S XX w  A  1    exp iw   d  1    exp iw   d 
  T

 T
T
0

2


 1
1  iw    1 
1  iw    1 
T  1
0 
 
 exp iw    0  
 
 exp  iw     T 
S XX w   A 2  
  iw T  iw 2 
  iw T  iw 2 



 1
 1
 1  
1  iw  T  1 
1

 
 exp  iw  T   
 

2 
  iw T
  iw T  iw 2  




iw




S XX w   A 2  





 1
1   1
1
1  iw   T   1 
 

 
 exp iw  T  

 iw T  iw 2    iw T
 iw 2 



 1
 1
1
1 
1
1 
  exp iw  T   
  exp  iw  T   





 iw  iw w 2T 
  iw  iw w 2T 

2 
S XX w   A  




 1  1   1  1  1   1 

  iw T  iw 2  iw T  iw 2 




 1 
 2 
1 
  exp iw  T   
  expiw  T   

S XX w  A 2   
2
 w 2T 
 T  w 2 
 w T 
1
S XX w  A 2 
w 2T
S XX w  A 2 
 
T
T 

 exp  iw    exp iw  
2
2 

 
2
2
4
1 
 T
 T 
 sin  w  
  2i  sin  w    A 2 
 2
 2 
w 2T 
w 2T
2
2
2
T
 T

sin  w  
sin  2f  
2
2
2
 A2  T  
S XX w  A 2  T  
 A 2  T  sinc f  T 
2
2
T
 T

w 
 2f  
2
 2

Note: sinc x   sin x  this follows Matlab’s definitions of the sinc function.
x
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 24
ECE 3800
Note: what happens if the amplitude is not a binary value between +/- A.
(1) If it remains zero mean,
The magnitude of the autocorrelation and power spectral density is multiplied by the second
moment of A.
(2) If it is not zero mean …
The cross-correlation of bits to other bits is no longer zero, as
E  Ai  Ak   E Ai   E  Ak   0
to make a long story short, the cross-correlation operation is identical to that shown for the
autocorrelation at periodic locations in time tau, except that the “amplitude” is now the square of
the mean instead of the second moment!
From chapter 6 …. for T rect functions
  
2
 XX    E A 2  1    E A
 T
 
S XX w  R XX   

 R XX    exp iw   d

S XX w   A  T  sinc f  T    A    f 
2
2
2
From chapter 6 …. for T < T rect functions

   n T 
 
2
 XX    E A 2    1 
  E  A    1 

T 
 T 

 
Then, the power spectral density consists of the Fourier Transform of the superposition of two
functions, (1) periodic triangles of magnitude E A2 and (2) a single triangle at the origin of
 
magnitude  A 2  E A 2  E  A2 .
Note that the periodic triangle is equivalent to the convolution of a single triangle function with a
comb function in time with spacing T. Thus, the PSD consists of a continuous sinc^2 function (2
above) summed with a “line-spectrum” sinc^2 function (1 above). This is similar to figure 7-3!
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
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ECE 3800
Deriving the Mean-Square Values from the Power Spectral Density
Using the Fourier transform relation between the Autocorrelation and PSD
S XX w 

 R XX    exp iw   d

1
R XX t  
2

 S XX w  expiwt   dw

The mean squared value of a random process is equal to the 0th lag of the autocorrelation
 
EX
2
1
 R XX 0  
2
  R
EX
2
XX
0 


1
S XX w  expiw  0   dw 
2


 S XX w  dw





 S XX  f   expi2f  0  dw   S XX  f   df
Therefore, to find the second moment, integrate the PSD over all frequencies.
As a note, since the PSD is real and symmetric, the integral can be performed as
 
EX
2
1
 R XX 0   2 
2
  R
EX
2

 S XX w  dw
0

XX
0   2  S XX  f   df
0
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
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ECE 3800
Converting between Autocorrelation and Power Spectral Density
Using the properties of the functions
The power spectral density as a function is always
 real,
 positive,
 and an even function in w/f.
You can convert between the domains using:
The Fourier Transform in w
S XX w 

 R XX    exp iw   d

1
R XX t  
2

 S XX w  expiwt   dw

The Fourier Transform in f

S XX  f  
 R XX    exp i2f   d

R XX t  

 S XX  f   expi2ft   df

The 2-sided Laplace Transform
S XX s  

 R XX    exp s   d

1
R XX t  
j 2
j
 S XX s   expst   ds
 j
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
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ECE 3800
Notes on using the Laplace Transform
(1) When converting from the s-domain to the frequency domain use:
s  jw or w   js
(2) 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
This can be expressed as:
cs   c s 
d s   d  s 
S XX w 
To compute the autocorrelation function for 0   use a partial fraction expansion such that
S XX w 
g s  g  s 

d s  d  s 
and solve for
1
R XX t  
2
j

 j
g s 
 expst   ds
d s 
for determining 0   , use the RHP expansion, replace –s with s, perform the Laplace transform
and replace t with –t.
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
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ECE 3800
Example:
S XX w 
A2 
 1

 X
2
X
2

  w 2


2  A2  
 2  w2
Substitute for w
S XX s  
2  A2  
 2  s2

2  A2  
  s     s 
Partial fraction expansion
k0
k 0    s   k1    s 
k1
2  A2  
S XX s  



  s     s 
  s     s 
  s     s 
 k 0  k1   s  0
k 0  k1     2  A 2  
S XX s  

k 0  k1

2k 0  2  A 2
A2
A2

  s     s 
Taking the LHP Laplace Transform
 A2 
L
  A  exp t 
   s  
for t  0
Taking the RHP with –s and then –t.


A2
2
2
2
L
  A  exp t   A  exp   t   A  expt 
    s  
for t  0
Combining we have
R XX    A 2  exp    t 
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 24
ECE 3800
White Noise
Noise is inherently defined as a random process.
You may be familiar with “thermal” noise, based on the energy of an atom and the mean-free
path that it can travel.
As a random process, whenever “white noise” is measured, the values are uncorrelated with each
other, not matter how close together the samples are taken in time.
Further, we envision “white noise” as containing all spectral content, with no explicit peaks or
valleys in the power spectral density.
As a result, we define “White Noise” as
R XX    S 0   t 
S XX w   S 0
This is an approximation or simplification because the area of the power spectral density is
infinite!
Nominally, noise is defined within a bandwidth to describe the power. For example,
Thermal noise at the input of a receiver is defined in terms of kT, Boltzmann’s constant times
absolute temperature, in terms of Watts/Hz. Thus there is kT Watts of noise power in every Hz
of bandwidth.
For communications, this is equivalent to –174 dBm/Hz or –144 dBW/Hz.
For typical applications, we are interested in Band-Limited White Noise where
 S 0
S XX w   
0
f W
W  f
The equivalent noise power is then:
  R
EX
2
XX
0 
W
 S 0  dw  2  W  S 0
W
For communications, we use kTB (yes, we are off by a factor of 2 based on +/- freq).
How much noise power, in dBm, would I say that there is in a 1 MHz bandwidth?
dBkTB   dBkT   dBB   174  60  114 dBm
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
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ECE 3800
Receiver Sensitivity
What does it mean when you buy a receiver?
For a great receiver (spectrum analyzer grade), assume a 200 kHz FM radio bandwidth.
Noise Power
kT
-174. dBm/Hz
Equivalent Noise Bandwidth
B
53. dB Hz
Receiver Noise Figure
NF
10. dB
Signal Detection Threshold
D
8. dB
Minimum Detectable Signal
MDS
-103. dBm
FM radio stations can transmit up to 1 Megawatt  +90 dBm
Why doesn’t your receiver get blasted? Path loss, distance, higher noise figure, receiving
antenna inefficiency, 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
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ECE 3800
Band Limited White Noise
 S 0
S XX w   
0
f W
W  f
The equivalent noise power is then:
  R
EX
2
XX
0 
W
 S 0  dw  2  W  S 0
W
but what about the autocorrelation?
W
R XX t  
 S 0  expi2ft   df
W
 expi 2Wt  exp i 2Wt  
 expi 2ft  
R XX t   S 0  
 S0  



i 2t
i 2t

 i 2t  W
W
R XX t   S 0 
For sin c xt  
2  i  sin i 2Wt 
i 2t
xt 
xt
R XX t   2  W  S 0  sinc2Wt 
Using the concept of correlation, for what values will the autocorrelation be zero?
(At these delays in time, sampled data would be uncorrelated with previous samples!)
2Wt  k
k
t
2W
for k  1,  2,
Sampling at 1/2W seems to be a good idea, but isn’t that the Nyquist rate!!
Also note, noise passed through a filter becomes band-limited, and the narrower the filter the
smaller the noise power … but the wider is the sinc autocorrelation function.
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
23 of 24
ECE 3800
The Cross-Spectral Density
Why not form the power spectral response of the cross-correlation function?
The Fourier Transform in w
S XY w 

 R XY    exp iw   d
and SYX w 

1
R XY t  
2



 RYX    exp iw   d

1
S XY w  expiwt   dw and RYX t  
2


 SYX w  expiwt   dw

Properties of the functions
S XY w  conjSYX w
Since the cross-correaltion is real,
 the real portion of the spectrum is even
 the imaginary portion of the spectrum is odd
There are no other important (assumed) properties to describe
Note: the trick using the Laplace transform to form the positive and negative portions of the
“time-based” cross-correlation is required to determine the correct “inverse 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
24 of 24
ECE 3800