-
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 8: Response of Linear Systems to Random Inputs
8-1 Introduction
8-2 Analysis in the Time Domain
8-3 Mean and Variance Value of System Output
8-4 Autocorrelation Function of System Output
8-5 Crosscorrelation between Input and Output
8-6 Example of Time-Domain System Analysis
8-7 Analysis in the Frequency Domain
8-8 Spectral Density at the System Output
8-9 Cross-Spectral Densities between Input and Output
8-10 Examples of Frequency-Domain Analysis
8-11 Numerical Computation of System Output
Concepts:
Linear Systems Analysis in the Time Domain Mean and Variance Value of System Output Autocorrelation Function of System Output Crosscorrelation between Input and Output Example of Time-Domain System Analysis Analysis in the Frequency Domain Spectral Density at the System Output Frequency Limited Filtering Bandlimited White Noise SIMO Systems Cross-Spectral Densities between Input and Output Examples of Frequency-Domain Analysis Numerical Computation of System Output Simulating Stochastic System Responses
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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
Chapter 8: Response of Linear Systems to Random Inputs
Linear transformation of signals: convolution in the time domain
txthty
th ty
System
tx
Linear transformation of signals: multiplication in the Laplace domain
sXsHsY
sX sH sY
The convolution Integrals (applying a causal filter)
0
dhtxty or
t
dxthty
Comment: Dr. Bazuin will use causal filters, defined as
tth
tthth
0,0,0
and when doing so typically prefers doing convolutions as
0
dhtxty
-
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
The Mean Value at a System Output
0
dhtXEtYE
Using the expected value as an operator (moving inside the intergral)
0
dhtXEtYE
For a wide-sense stationary process, this result in
00
dhXEdhXEtYE
The coherent gain of a filter is defined as:
0
dtthhgain
Therefore, gainhXEtYE
The expected value of the output is the expected value of the input times the coherent gain!
The Mean Square Value at a System Output
2
0
2 dhtXEtYE
0
2 dhRhtYE XX
Therefore, take the convolution of the autocorrelation function and then sum the filter-weighted result from 0 to infinity.
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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
The Autocorrelation at a System Output txthtxthEtYtYERYY
The expected values of the product of two distinct convolutions:
0222
0111 dhtxdhtxERYY
021212
01 ddhhtxtxERYY
Identifying and forming the input autocorrelation
02121
021 ddhhtxtxERYY
02121
012 ddhhRR XXYY
0122
0211 ddhRhR XXYY
01111 dhRhR XXYY
or
0
1111 dhRhR XXYY
Notice that first, you convolve the input autocorrelation function with the filter function and then you convolve the result with a time inversed version of the filter!
-
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
Example: White Noise passed through an RC filter
0122
0211 ddhRhR XXYY
White Noise Signal: tNtRXX 20
0122
021
01 2
ddhNhRYY
For 0for and 01
0,2 0
1110
fordhhNRYY
For 0for and 02
0,2 0
2220
fordhhNRYY This is a correlation function of the impulse response of the filter with itself!
Now for and RC filter, the computation can continue …
tuCRt
CRth
exp1
0,expexp1
2 02
0
fordCRCRCR
NRYY
0,2expexp1
2 02
0
fordCRCRCR
NRYY
0,exp
41
0
forCRCR
NRYY
The complete autocorrelation of white noise in an RC filter is
0,expexp12
0
fordCRCRCR
NRYY
Leading to
0,exp4
10
for
CRCRNRYY
and finally
CRCRNRYY
exp
41
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 6 of 24 ECE 3800
The Crosscorrelation of Input to Output txthtxEtYtXERXY
0111 dhtxtxERXY
0
111 dhtxtxERXY
0
111 dhRR XXXY
This is the convolution of the autocorrelation with the filter.
What about the other Crosscorrelation?
txtxthEtXtYERYX
0111 dhtxtxERYX
0
111 dhtxtxERYX
0
111 dhRR XXYX
This is the correlation of the autocorrelation with the filter, inherently different than the previous, but equal at =0.
For a white noise process:
tNtRXX 20
0,0
0,2
2
0
0111
0
hNdh
NRXY
0,2
0,0
2 00111
0
hNdh
NRYX
-
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
Exercise Examples Exercise 8-2.1 Filter: tuttth 2exp
Random Process: 120,121, MforfMtX M (a) Write an expression for the output sample function. (b) Find the mean value of the output. (c) Find the variance of the output.
First:
62
12 MEtXE
4861212 222222 MMMEtXE
482 MERtXtXE XX Coherent gain
00
2exp dtttdtthhgain
1expexp 2
xaaxadxxax
411
4112
42exp
0
tthgain
Output Signal
400
MdhMdhtXtY
Therefore, 23
46
41
XEtYE
2
0
2 dhtXEtYE
2
0
2
2
0
2
dhMEdhMEtYE
34148
22
tYE
And finally 43
4912
233
2222
YY tYE
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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
Exercise 8-2.2
Filter:
else
ttth
,010,35
Random Process: 20,21,2cos2 forfttX (a) Write an expression for the output sample function. (b) Find the mean value of the output. (c) Find the variance of the output.
First: 02cos2 tEtXE
224
2222cos142cos2 222
tEtEtXE
ttERtXtXE XX 2cos22cos2
2cos2
2222cos2cos4 tERXX
Coherent gain
835351
00
dttdtthhgain
Output Signal
0
dhtXtY
1
0
352cos2 dttY
1
0
2cos62cos10 dtttY
1
022sin62cos10
tttY
2
2sin22sin62cos10
ttttY
ttY 2cos10
-
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
Then, 0 gaingain hXEhXEtYE
22 2cos10 tEtYE 50
2100
2222cos11002
tEtYE
or
0122
0211
2 ddhRhtYE XX
0122
1
0211
2 352cos235 ddtYE
0
1112 2cos1035 dtYE
502cos30501
011
2 dtYE
And finally 50050 2222 YY tYE
-
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
Exercise 8-3.1
Filter: tuttth 2exp
Random Process: wwSnoisetX XX 42,
Therefore: 4242
,22
00 N
RN
XX
(a) Find the mean value of the output. (b) Find the variance of the output. (c) Find the mean square of the output.
First: 24 XXRtXE
642222 XXXXRtXE 22 X
Coherent gain
00
2exp dtttdtthhgain
1expexp 2
xaaxadxxax
411
4112
42exp
0
tthgain
Output Signal
0
dhtXtY
Therefore, 21
412 gainhXEtYE
2
0
2 dhtXEtYE
0122
0211
2 ddhRhtYE XX
-
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
01222
02111
2 2exp422exp ddtYE
01
02221111
2 2exp42exp22exp ddtYE
011111
2
4142exp22exp dtYE
0
11112
12 2exp4exp2 dtYE
0111
0
12
12 4exp422
44exp2
41 dtYE
0
112 14
164exp1
41 tYE
165
161
412 tYE
And finally
161
1645
21
165 2222
YY tYE
Exercise 8-3.2
Filter: 5.010 tututh
Random Process: 5, wSnoisetX XX
Therefore: 52
,52
00 NRN
XX
(a) Find the mean value of the output. (b) Find the mean square of the output.
First: 0 XXRtXE
5222 XXXXRtXE 52 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 12 of 24 ECE 3800
Coherent gain
00
5.010 dttutudtthhgain
52
10105.0
0
dthgain Output Signal
0
dhtXtY
Therefore, 050 gainhXEtYE
2
0
2 dhtXEtYE
0122
0211
2 ddhRhtYE XX
0122
0211
2 5 ddhhtYE
0
12
12 5 dhtYE
0
222 5.0105 dttututYE
250
2500500
5.0
0
2 dttYE
And finally 2500250 2222 YY tYE
-
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 section 8-4: An Exponential Autocorrelation signal through an RC filter
Signal: tStRXX exp20
The mean value is 0exp2
0
SRXXX
and 2
0 0222S
R XXXXX
Filter: CRbwheretutbbth 1,exp The autocorrealtion
0122
0211 ddhRhR XXYY
0122
021
01 exp2
ddhShRYY
For 0for and 01
012221
01
0122
021
01
1
1
exp2
exp2
ddhSh
ddhShRYY
012221
01
0122
021
01
expexp2
exp
expexp2
exp
1
1
ddbbSbb
ddbbSbbRYY
012211
20
01
02211
20
1
1
expexpexp2
expexpexp2
ddbbbS
ddbbbSRYY
01
111
02
01
111
02
expexpexp2
1expexpexp2
db
bbSb
dbb
bbSbRYY
-
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
01
111
02
01
111
02
expexpexp2
exp1expexp2
db
bbSb
db
bb
bSbRYY
011
02
01111
02
2exp2
2expexp2
dbb
Sb
dbbb
SbRYY
011
02
0111
02
2expexp2
2expexpexpexp2
dbbb
Sb
dbbbb
SbRYY
202expexp
2
202expexp0expexp
2
02
02
bbb
bSb
bbb
bb
bSbRYY
b
bSbb
bSb
bbSbRYY exp4
exp4
exp2
0002
b
bbSb
bbSbRYY exp2
exp2
02
02
bbb
SbRYY expexp2 220
Based on symmetry, we have
bbb
SbRYY expexp2 220
Conversion of input to a white noise case To compare this result to the white noise result, let 200 NS and . Then
tNtRXX 20
and
bNbbb
Sbb
SbRYY exp4exp
2exp
20
220
2
220
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 15 of 24 ECE 3800
For the input signal approached white noise
bbb
SbRYY expexp2 220
becomes (isolating noise based terms prior to the others)
222
220 expexp
2
bb
bbbSbRYY
bbb
bSbRYY exp1exp2 222
0
bbb
bSbRYY exp11
1exp2
22
0
The result consists of a filtered white noise term, a constant based on b and beta, and a term in the exponential.
This suggests that for “wide bandwidth” input noise, it is often possible to approximate the noise as filtered white noise as opposed to filtering an already band-limited 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 16 of 24 ECE 3800
The Power Spectral Density at a System Output
The power spectral density is the Fourier Transform of the autocorrelation:
diwtXtXERwS XXXX exp
diwtYtYERwS YYYY exp
Where
txthty For a WSS, ergodic process,
tYtYERYY
0222
0111 dhtxdhtxERYY
021212
01 hhtxtxddERYY
021212
01 txtxEhhddRYY
021212
01 XXYY RhhddR
This is where we have gotten before. Now we need to perform the Fourier transform to find the PSD!
diwRhhddRwS XXYYYY exp
021212
01
-
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 24 ECE 3800
Taking the Power Spectral Density
diwRhhddRwS XXYYYY exp
021212
01
Isolating the transform to function in time (tau)
021212
01 exp diwRhhddRwS XXYYYY
021212
01 exp jwwShhddRwS XXYYYY
Separating dependent and independent elements based on the integration
022211
01 expexp jwhdjwhdwSRwS XXYYYY
wHjwhdwSRwS XXYYYY
110
1 exp
110
1 exp jwhdwHwSRwS XXYYYY
Therefore
wHwHwSRwS XXYYYY or
2wHwSRwS XXYYYY
Relation of Spectral Density to the Crosscorrelation Function
As you might expect, computation of the cross-spectral only has a single “filter PSD term”.
wHwSwS XXXY and wHwSwS XYYY
wHwSwS XXYX and wHwSwS YXYY
-
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 24 ECE 3800
Revisiting Equivalent Noise Bandwidth
In the previous notes, white noise was passed through an RC, 1st order Butterworth filter as recapped here.
Let tNtRXX 20
00
dhXEdhXEtYE
00
Hdtthhgain
0
2 dhRhtYE XX
BNdhNtYE
00
12
102
2
For the unity coherent gain filter example.
01
21
00
22
dhNBNNE EQB
0
221 dtthBEQ
This is for a normalized filter. If the filter is not a gain of 1 and DC, it must first be normalized before performing the operations.
Note: any filter gain (or loss) equally magnifies the signal and noise. Therefore, it has no effect on the signal to noise ratio (SNR).
-
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 24 ECE 3800
If the filter has coherent gain, normalize the filter first. The normalized is
0
0
Hth
hth
dtth
thtggain
Then
0
2
0
2
021
21 dt
HthdttgBEQ
20
2
02 H
dtthBEQ
From Parseval’s theorem
0
2222
21 dtthdwwHdffHdtth
Frequency Domain Computation
2
2
2
2
0402 H
dwwH
H
dffHBEQ
Time Domain Computation
2
0
0
2
20
2
21
2
dtth
dtth
h
dtthB
gainEQ
Therefore, the Equivalent noise bandwidth can be calculated purely in the time domain or purely in the frequency domain, whichever is easier to compute!
-
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
Example: A rectangular filter time domain BEQ
Ttututh 1
TdtdtthhT
gain
00
1
TT
TT
d
dtth
dh
B
T
EQ
21
22
1
2
220
1
2
0
01
21
The Fourier Filter Domain BEQ
T
dttfidttfithfH0
2exp2exp
fifi
fiTfi
fitfifH
T
202exp
22exp
22exp
0
fi
Tfififi
tfifHT
22exp
21
22exp
0
22exp
22exp
22exp
21 TfiTfiTfi
fifH
22sin2
22exp
21 TfiTfi
fifH
TfTfTTfifH
sin
22exp
TfcTTfifH
sin
22exp or
2sin
2exp TwcTTwifH
TfcTfH sin or
2sin TwcTwH
gainhTH 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 21 of 24 ECE 3800
Notice that the result matches the time domain.
From here, we could solve
TT
dfTfcT
H
dffH
BEQ
2
12
sin
02 2
2
2
2
Notice that the nulls of the sinc function appear at f=1/T but that the Beq=1/2T … half the frequency distance to the first null!
Note: This is easier in the discrete time-frequency domain … summing and summing squares!
-
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 24 ECE 3800
8-6 Examples of Time-Domain Analysis:
It is important to remember both the time domain and frequency domain relationships in order to pick the one that is easier based on the signal type.
When the autocorrelation has a simple form over a finite time interval, the time domain can be easier.
Filter: TtutuT
th 1
Random Process:
TARXX
12
(a) Find the mean value of the output. (b) Find the mean square of the output.
First: 0tXE
2222 0 ARtXE XXXX 22 AX
Coherent gain
T
gain dtTtutuTdtthh
00
1
11
0
TTdt
Th
T
gain
Output Signal
0
dhtXtY
Therefore, 010 gainhXEtYE
0122
0211
2 ddhRhtYE XX
T T
ddTT
AT
tYE0
120
2122 111
T T
ddTT
AtYE0
120
212
22 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 23 of 24 ECE 3800
To eliminate the absolute value, perform two integral for each part,
T TT T
dddT
ddTAtYE
012122
021
012
02
22
1
111
TT
dT
TTAtYE
0121
22
0
22
212
2
22
1
1
221
T
dTTT
TTAtYE
01
21
1
2212
2
22
2221
T
dTTTAAtYE
01
211
2
3
222
2
T
TTTAAtYE
0
31
21
1
2
3
222
322
322
323
3
222 TTTT
TAAtYE
2332
22
32
3AT
TAAtYE
And finally 222222320
32 AAtYE YY
The autocorrelation function (using tables)
2wHwSRwS XXYYYY
2
exp
2
2sin
TwjTw
Tw
wH
dwjT
AwST
TXX
exp12
22
2
2
2sin
Tw
TwTAwS XX
2wHwSwS XXYY
44
22
2
2
2
2
2
2sin
2
2sin
2
2sin
Tw
TwTA
Tw
Tw
Tw
TwTAwSYY
-
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
Power Spectral Density of a filtered white noise process.
0122
0211 ddhRhR XXYY
Let tNtRXX 20
0122
021
01 2
ddhN
hRYY
0
1110
2 dhhNRYY
Taking the PSD
diwRRwS YYYYYY exp
iwdhhNdwSYY exp2 0111
0
0
1110 exp
2 diwhhdNwSYY
0
110 exp
2wHiwhdNwSYY
0
110 exp
2 wihdwHNwSYY
20022
wHN
wHwHN
wSYY