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
Wiener–Khinchin Theorem
For WSS random processes, the autocorrelation function is time based and has a spectral
decomposition given by the power spectral density.
Also see:
http://en.wikipedia.org/wiki/Wiener%E2%80%93Khinchin_theorem
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
Chapter 7: Spectral Density
Section 7-4 and 7-5 Discussion and important information. Setting up Chapter 8 material
on linear system signal processing of a random process …
ECE 3100 and ECE 3710 stuff …
Spectral densities and linear systems Laplace domain Bode plots.
The Laplace Domain descriptor “s” and in fact all transfer functions can be defined in terms of a
frequency response to a real or complex input signal. As such we often describe the relationship:
s  j  w or w   j  s
A natural question is how do the spectral density plots from autocorrelation relate to the
power-spectrum Bode plots of transfer functions.
Power Spectral Density (from Autocorrelation): Inherently a power plot. No phase exists.
Power Spectral Density (from Bode plots): Described the magnitude squared response of a
transfer function or filter. A phase plot in frequency is expected to existed and be performed.
Can we define transfer functions based on the autocorrelation’s PSD? Sure …
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
Substituting for w
S XX s   S 0
 js 2n  a 2 n2  js 2n2  a 2 n4  js 2n4    a 2  js 2  a0
 js 2 m  b2m2  js 2m2  b2 m4  js 2 m4    b2  js 2  b0
n
n 1

 1  s 2 n  a 2 n  2   1  s 2 n  2    a 2   1  s 2  a 0
S XX s   S 0
 1n  s 2m  b2 m2   1m1  s 2 m2    b2   1  s 2  b0
This can be expressed as:
S XX s   H s   H  s  
cs   c s 
d s   d  s 
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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Example: Section 7-4 p. 271.


10  w 2  5
S XX w  4
w  10  w 2  24
S XX s  

10   js   5
 js 
4
2
 10   js   24

S XX s  

2

 10  s 2  5
S XX s   4
s  10  s 2  24






 10  s  5  s  5
 10  s  5  s  5

2
2
s 4  s 6
s  2  s  2  s  6  s  6



S XX s   H s   H  s  



   10  s  5 
6  s  2  s  6 

10  s  5
s  2  s 
The poles and zeros can be readily identified!
20
Pole-Zero Map
1
0
0.8
0.6
Imaginary Axis (seconds-1)
Mag (dB)
-20
-40
-60
-80
0.4
0.2
0
-0.2
-0.4
-0.6
-100
-0.8
-120
-1
10
0
10
1
10
Freq (w)
2
10
3
10
-1
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Real Axis (seconds -1)
Poles
-2.4495
-2.0000
2.4495
2.0000
psd_sys =
-10 s^2 + 50
----------------s^4 - 10 s^2 + 24
Zeros
2.2361
-2.2361
Continuous-time transfer 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
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Example: Section 7-4 p. 272.


w 2  w 2  25
S XX w  6
w  33  w 4  463  w 2  7569
S XX s  


 s 2   s 2  25
 s 6  33  s 4  463  s 2  7569
The poles and zeros can be found using MATLAB!
-20
Pole-Zero Map
6
-30
-40
4
Imaginary Axis (seconds-1)
-50
Mag (dB)
-60
-70
-80
-90
2
0
-2
-100
-4
-110
-120
-1
10
0
10
1
2
10
Freq (w)
10
-6
-6
3
10
-4
-2
0
2
4
6
Real Axis (seconds -1)
Poles
-2.0000 + 5.0000i
-2.0000 - 5.0000i
2.0000 + 5.0000i
2.0000 - 5.0000i
-3.0000 + 0.0000i
3.0000 + 0.0000i
psd_sys =
-s^4 + 25 s^2
----------------------------s^6 + 33 s^4 + 463 s^2 - 7569
Continuous-time transfer function.
Zeros
0
0
5
-5
H s  
H  s  
s  s  5
s  2  j5  s  2  j5  s  3
 s   s  5
 s  s  5 

 s  2  j5   s  2  j5   s  3 s  2  j5  s  2  j5  s  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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Continuing
Once the Laplace form has been defined, we can compute the Autocorrelation function using the
expression
S XX s   H s   H  s  
cs   c s 
d s   d  s 
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 
In this structure, all the s-domain poles in the left half-plane appear in the first term and all the
poles in the right half-plane appear in the second term.
and solve for
1
R XX   
2
j
g s 
 d s   exps   ds
 j
for determining   0 , use the RHP expansion, replace –s with s, perform the Laplace transform
and replace t with –t.
Extension of this concept …
Once we have defined
S XX s   H s   H  s  
cs   c s 
d s   d  s 
A transfer functions that provide the equivalent power spectrum can be defined as
H s  
cs 
d s 
By construction, stability can be guaranteed; all the poles (and zeros) are in the left half-plane.
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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Example: Find the autocorrelation
S XX w  
A2 
 1

 X
2
X
2

  w 2


2  A2  
 2  w2
Substitute for s 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 
2
L
  A  exp t 
   s  
for t  0
Taking the RHP with –s and then –t.


A2
 A 2  exp t   A 2  exp   t   A 2  expt 
L

    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
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ECE 3800
7-5 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

 S XX w  dw
0
  R
EX
2

XX
0   2  S XX  f   df
0
For cases where the PSD is a complicated, the operation can be done in the Laplace Domain, as
shown in the textbook.
Procedure:
(1) Form the PSD in the Laplace Domain.
 1n  s 2 n  a 2 n2   1n1  s 2n2    a 2   1  s 2  a0
S XX s   S 0
 1n  s 2m  b2 m2   1m1  s 2 m2    b2   1  s 2  b0
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
(2) Factor the PSD in s into the transfer function elements
S XX s   H s   H  s  
cs   c s 
d s   d  s 
(3) Perform the following integration (also not easy, based on text approach)
1
1
cs   c s 
  S XX s   ds 
 
 ds
2  j
2  j d s   d  s 
j 
X2 
j 
A Table of Integrals has been provided on p. 276 for d(s) polynomials for n=1, 2, 3.
An example application appear on p. 275-276 with a second on p. 277-278.
One final method – Contour Integrals (Calculus 5? – Complex Variable Theory)
A brief summary of the theory can be found in Appendix I.
I am going to skip this material!
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
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 
N
S XX w   S 0  0
2
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
N0

S 0 
S XX w   
2
0

f W
W  f
The equivalent noise power is then:
  R
EX
2
XX
0 
W
 S 0  dw  2  W  S 0  2  W 
W
N0
 N 0 W
2
For communications, we use kTB where W=B and N0=kT.
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 radio 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.
But notice that your commercial; receiver is in microvolts, where 2.0 uV is very good.
Power into 50 ohms is V^2/R or 8e-14 W = -130.97 dBW  -101 dBm.
An MDS of -103 dBm  1.6 uV
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ft  
 expi 2Wt  exp i 2Wt  
 S0  

R XX t   S 0  


i 2t
i 2t
 i 2t  W

W
R XX t   S 0 
For sinc 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
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ECE 3800
Noise and Filtered Noise Matlab Simulation
Based on HW Problem 6-4.6.
x=randn(N,1); % zero mean, unit power random signal
[b,a] = butter(4,20/500);
y=filter(b,a,x); % applying a digital filter
y=y/std(y); % normalizing the output power
Rxx=xcorr(x)/(N+1);
Ryy=xcorr(y)/(N+1);
DFTx = fftshift(fft(x))/N;
DFTy = fftshift(fft(y))/N;
DFTRxx = fftshift(fft(Rxx,2*N))/N;
DFTRyy = fftshift(fft(Ryy,2*N))/N;
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
Sec7_7_White_Pink_Noise
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
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-correlation is real (for X and Y real functions),
 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 or method of using the Laplace transform to form first the positive (poles in
LHP) and then the negative portions (poles in RHP) of the “time-based” cross-correlation is
required. See example in P. 290-291.
In general, a “correlation receiver” can be used when detecting pulses (radar, sonar, etc.) and
communication symbols (PSK, FSK, QAM, etc.). Later we will see that his has “matched filter”
characteristics … a form of optimal filtering/signal reception.
If a signal plus noise is received and the “correlater” contains a copy of the transmitted signal …
the “signal” is being auto-correlated (with a time delay) and the noise is being cross-correlated or
“filtered” by the transmitted signal. You can estimate the “filtered noise power” if you know the
“equivalent noise bandwidth” of the transmitted signal!
(An M-FSK correlation was demonstrated previously in class without noise added.)
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
Autocorrelation Function Estimation of Spectral Density
An estimate of the autocorrelation was defined as:
1
Rˆ XX   
T 
T 
 xt   xt     dt
0
An unbiased estimate was given for sample data systems as
Rˆ XX kt   Rˆ XX k  
1
N 1 k
N k
 xi   xi  k 
i 0
where it is assumed that values of k<<N are of interest.
For the continuous time estimate, we should insure that there is significant information
integrated; a general restriction on the estimate may be defined as
T 

1

xt   xt     dt
Rˆ XX    T  
0

0

0  m
m 
In fact this can be thought of as windowing the estimate as
T 

 1
xt   xt     dt
ˆ
ˆ
r R XX    wr    R XX     T  
0

0

0  m
m 
where a time window defined as follows is applied
1
wr    
0
0   m
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
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ECE 3800
As might be expected we want to take the Fourier Transform of this window autocorrelation
function … (remember that multiplication in the time-domain is equivalent to a convolution in
frequency-domain)
ˆ
r S XX w 




 r Rˆ XX    exp iw   d   wr    Rˆ XX    exp iw   d
ˆ
r S XX
w  Wr w  Sˆ XX w
While this function appears to be defined for all w, remember that Wr(w) is present and the
estimate of the PSD may be highly inaccurate if sufficient time sample are not available for
integration.
As a note and another potential problem, the Fourier Transform of the rectangular window
function is
Wr  f  

 wr    exp i2f   d  2  m  sinc2  m  f 

The sinc function possesses negative sidelobes that, when convolved, could cause the estimate of
the PSD to become negative. This would violate one of the properties of the PSD! Therefore, the
structure of the window function is of importance when performing this estimate.
Alternate windows are proposed to “adjust” for the sinc sidelobes
Hamming Window
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

  

0.54  0.46  cos
wham    
 m 
0

0   m
m  

Wham  f  
 wham    wr    exp i2f   d



0.46  
1 
1  
    f 
   Wr  f 
Wham  f   0.54    f  
    f 
2  
2  m 
2   m  


Applying this window function would result in
ˆ
h S XX w  

 wh    wr    Rˆ XX    exp iw   d

ˆ
ˆ
h S XX w  Wham w  S XX w



ˆ w  W  f   0.54    f   0.23     f  1     f  1    Sˆ w
S
h XX
r
XX

 
2   m 
2   m  


 




1  ˆ
1  
ˆ
ˆ
ˆ
  S XX w f 
 
h S XX w  Wr  f   0.54  S XX w  0.23   S XX w f 

2


2


m
m



 



1  ˆ 
1  
ˆ
ˆ
ˆ 

 r S XX  f 
h S XX w  0.54 r S XX w  0.23   r S XX  f 

2

2



m
m 



This function will modify the sidelobes present in the initial estimate to a level that may be
acceptable.
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
Whenever the Fourier transform is applied, the use of a window should be considered.
There are entire tables of window functions for FFT’s that show various performance factors!
For example:
For an exhaustive analysis, see Fred Harris
F. J. Harris, “On the use of Windows for harmonic analysis with the Discrete Fourier
Transform,” Proc. IEEE, vol. 66, (no. 1), pp. 51–83, Jan 1978.
http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1455106&isnumber=31261
or for a less exhaustive review see
Rapuano, S.; Harris, F.J., "An introduction to FFT and time domain windows," Instrumentation
& Measurement Magazine, IEEE , vol.10, no.6, pp.32,44, December 2007.
http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4428580&isnumber=4428566
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: Applying a Hamming Window to the previous example
see Sec7_7_FilteredNoise_RxxWin
Rxx=xcorr(x)/(N+1);
Ryy=xcorr(y)/(N+1);
Rxy=xcorr(x,y)/(N+1);
Rxx=Rxx.*hamming(2*N-1);
Ryy=Ryy.*hamming(2*N-1);
Rxy=Rxy.*hamming(2*N-1);
Filtered
Unfiltered
Autocorrelation
Autocorrelation
1.2
1.2
Rxx
Ryy
1
Rxx
Ryy
1
0.8
0.8
Magnitude
Magnitude
0.6
0.4
0.6
0.4
0.2
0.2
0
0
-0.2
-0.4
-2
-1.5
-1
-0.5
0
Lag (sec)
0.5
1
1.5
-0.2
-2
2
-1.5
-1
-0.5
DFT of Autocorrelation
0
Lag (sec)
0.5
1
1.5
2
DFT of Autocorrelation
0
0
-10
-10
-20
-20
PSD
PSD
-30
-30
-40
-40
-50
-50
-60
-60
-70
-500
-70
White Noise
Filtered Noise
-400
-300
-200
-100
0
100
Frequency
200
300
400
500
-80
-500
White Noise
Filtered Noise
-400
-300
-200
-100
0
100
Frequency
200
300
400
500
The estimated autocorrelation has been reduced at larger lags.
The estimated PSD has been smoothed (not as “noisy”).
For a “finite” time estimate (simulation), which one would you prefer working with?
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
Discrete Implementations
The discrete Fourier Transform
X D k  
N 1
 WNnk  xn
n0
1
xˆ k   
N
N 1
 WN nk  X D k 
n0
  j 2nk 
where WNnk  exp

 N

Equivalence of the continuous and discrete transforms …
(1) Discretization of the input
x n   xn  t  for t  1
f sample
 k 
(2) Spectral bins “actual center freq” X C 
  X C k  f   X D k 
 N  t 
f
where k=0, +/-1, … +/- N/2, and f  sample
N
Get it right at the frequency sample points … based on the applied or actual window used.
f sample
(3) Spectral bin concept

1

f

2N
f sample
 k 
XC 
  df  X D k 
 N  t 
2N
This is a concept; reality requires that the discrete point consist of the complete “windowed”
signal response. (see the Matlab windows). This can be thought of as a piecewise linear estimate
of the bin energy.
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
DFT Symmetry for real numbers
X k 
N 1

X N  k  
 WNnk  xn
n 0
N 1

WNn N  k   xn  
n 0
*
N 1
 WNnN  WN nk  xn
n 0
X  N  k   X k 
Y k   i
Y N  k   i
Y N  k   i
N 1
 WNnk  yn
n 0
N 1

WNn N  k  
y n   i
n 0
N 1
N 1
 WNnN  WN nk  yn
n 0
 WN nk  yn
n 0
N 1
Y *  N  k   i
 WNnk  yn
n0
*
Y  N  k   Y k 
Notice the spectral symmetries involved. Purely real inputs are conjugate symmetric about N,
while imaginary inputs are conjugate anti-symmetric about N.
For real input, bins 0 and N/2 are special, while
X  N  1  X * 1
X  N  2  X * 2

N

N

X   1  X *   1
2

2

Note that k=0 to N/2-1 represent the positive spectrum, while N/2+1 to N-1 represent the
negative spectrum.
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
DFT Symmetry for even symmetric sequences
xn   x2  N  2  n , n  0 : N  1
x0  x2  N  2
x1  x2  N  3

x N  2   x N 
x N  1  x N  1
Note that x(N-1) is singular … it is symmetric to itself
X k  
2 N 2
n 0
X k  
N 2
W
nk
2 N 2
n 0
X k  
X k  
 xn 
W
nk
2 N 2
 xn   W2 NN 12k  x N  1 
N 2
N 2
n 0
n 0
2 N 2
W
n N
nk
2 N 2
 xn 
 W2nkN 2  xn   W2k  xN  1   W22NN22n k  x2 N  2  n 
N 2
W
n 0
nk
2 N 2
X k  
N 2
 xn   W2k  x N  1   W22NN2 2 k  W2Nnk 2  x2 N  2  n 
n 0
N 2
N 2
n 0
n 0
 W2nkN 2  xn   W2k  xN  1   W2Nnk2  xn 
X k  
 W
N 2
n 0
X k  
N 2
nk
2 N 2


 W2Nnk 2  xn   W2k  x N  1
nk 
 2  cos 2  2 N  2   xn    1
k
 xN  1
n 0
Therefore, X(k) is purely real … no phase components as we would require.
Since x(n) is real, we must also have … “conjugate symmetry” but there is no imaginary part!
X k   X  N  2  k 
Therefore, only k=0 to N-1 must be computed!
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
Periodograms and Windowing Data
We can window the data prior to autocorrelation.
 this will cause a change in the energy that must be compensated
T
1
2
AvgEnergy    xt   wt   dt
T 0
T





T
1
1
2
2
2
2
E AvgEnergy    E xt   wt   dt  E xt     wt   dt
T 0
T 0
For a rectangular window, this becomes
2
E AvgEnergy   E xt 


If an alternate window is used, a power scaling will be necessary to maintain the same estimated
energy.
For a given window, function we can apply:
T
scale 
1
2
  wt   dt
T 0
So that
wscale t  
wt 
 wt 
scale
T
1
2
  wt   dt
T 0
This also applies to the sampled data computations where
wscale n  
wn 
 wn 
scale
1 N 1
2
  wm 
N m 0
see Sec7_7_Filtered NoiseWindowed.m
Now, multiple smaller results may be processed to generate an average.
The spectral resolution (number of frequency bins) necessarily decreases, but now multiple
spectra and can be averaged together based on “unique” time sample sets of the data.
It is useful for observing time varying phenomena and capturing the values when the desired
signal is present.
See WaterFallDemo.m for a spectral periodogram ….
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
Note: Some homework problems to review …. in class 7-6.3 and 7-5.1
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