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Part 5Response of Linear Systems
6. Linear Filtering of a Random Signals
7. Power Spectrum Analysis
8. Linear Estimation and Prediction Filters
9. Mean-Square Estimation
2
6. Linear Filtering of a Random Signal
Linear System
Our goal is to study the output process statistics in terms of the input process statistics and the system function.
)}({)( tXLtY
)}.({)}({)}()({ 22112211 tXLatXLatXatXaL
][L )(tX )(tY
t t
),( istX),( istY
3
Deterministic System
Deterministic Systems
Systems with MemoryMemoryless Systems)]([)( tXgtY
Linear-Time Invariant (LTI) systems
Time-Invariant systems
Linear systems)]([)( tXLtY
Time-varying systems
)()(.)()(
)()()(
thtXdtXh
dXthtY
( )h t( )X t
LTI system
4
Memoryless SystemsThe output Y(t) in this case depends only on the present value of the input X(t). i.e., .)}({)( tXgtY
Memorylesssystem
Strict-sense stationary input
Strict-sense stationary output.
Memorylesssystem
Wide-sense stationary input
Need not bestationary in any sense.
Memorylesssystem
X(t) stationary Gaussian with
)(XX
R
Y(t) stationary,butnot Gaussian with
).()( XXXY
RR
5
Linear Time-Invariant SystemsTime-Invariant SystemShift in the input results in the same shift in the output.
Linear Time-Invariant SystemA linear system with time-invariant property.
)()}({)}({)( 00 ttYttXLtXLtY
LTI)(t )(th
Impulse
Impulseresponse ofthe system
t
)(th
Impulseresponse
Fig. 14.5
)}({)(1)( tLth dtt
6
Linear Filtering of a Random Signal
LTI
)()(
)()()(
dtXh
dXthtYarbitrary input
t
)(tX
t
)(tY
)(tX )(tY
)()()( dtXtX
.)()()()(
)}({)(
})()({
})()({)}({)(
dtXhdthX
dtLX
dtXL
dtXLtXLtY
By Linearity
By Time-invariance
7
Theorem 6.1
Pf ：
)()]([
)]([)()()()]([
thtXE
dtXEhdtXhEtYE
8
Theorem 6.2
dvduvuRvhuhR XY )()()()(
If the input to an LTI filter with impulse response h(t) is a
wide sense stationary process X(t), the output Y(t) has the
following properties:(a) Y(t) is a WSS process with expected value
autocorrelation function
(b) X(t) and Y(t) are jointly WSS and have I/O cross- correlation by
(c) The output autocorrelation is related to the I/O cross-correlation by
dhtYE XY )()]([
duuRuhR XXY )()()(
dwwRwhR XYY )()()(
)()( hRX
)()( hRXY
9
Theorem 6.2 (Cont’d)Pf:
10
Example 6.1X(t), a WSS stochastic process with expected value X = 10 volts, is the input to an LTI filter with
What is the expected value of the filter output process Y(t) ?Sol：
.otherwise0
sec, 1.00)(
5 teth
t
Ans: 2(e0.51) V
11
Example 6.2A white Gaussian noise process X(t) with autocorrelation function RW ( ) = 0 ( ) is passed through the moving-average filter
For the output Y(t), find the expected value E[Y(t)], the I/O cross-correlation RWY ( ) and the autocorrelation RY ( ). Sol：
.otherwise0
,0/1)(
TtTth
otherwise.0
,/)()(
otherwise.0
,0/)(:
200 TTT
RTT
RAns YWY
12
Theorem 6.3If a stationary Gaussian process X(t) is the input to an LTI Filter h(t) , the output Y(t) is a stationary Gaussian process with expected value and autocorrelation given by Theorem 6.2.
Pf ： Omit it.
13
Example 6.3For the white noise moving-average process Y(t) in Example 6.2, let 0 = 1015 W/Hz and T = 103 s. For an arbitrary time t0, find P[Y(t0) > 4106].Sol：
Ans: Q(4) = 3.17105
14
Theorem 6.4The random sequence Xn is obtained by sampling the continuous-time process X(t) at a rate of 1/Ts samples per second. If X(t) is a WSS process with expected valueE[X(t)] = X and autocorrelation RX ( ), then Xn is a WSS random sequence with expected value E[Xn] = X and autocorrelation function RX [k] = RX (kTs).
Pf：
15
Example 6.4Continuing Example 6.3, the random sequence Yn is obtained by sampling the white noise moving-average process Y(t) at a rate of fs = 104 samples per second. Derive the autocorrelation function RY [n] of Yn.Sol：
otherwise.0
,10)1.01(10][ :
6 nnnRAns Y
16
Theorem 6.5If the input to a discrete-time LTI filter with impulse response hn is a WSS random sequence, Xn, the output Yn
has the following properties. (a) Yn is a WSS random sequence with expected value and autocorrelation function (b) Yn and Xn are jointly WSS with I/O cross-correlation
(c) The output autocorrelation is related to the I/O cross- correlation by
n
nXnY hYE .][
i j
XjiY jinRhhnR ].[][
.][][
i
XiXY inRhnR
iXYiY inRhnR ].[][
17
Example 6.5A WSS random sequence, Xn, with X = 1 and auto-correlation function RX[n] is the input to the order M1 discrete-time moving-average filter hn where
For the case M = 2, find the following properties of the output random sequence Yn : the expected value Y, the autocorrelation RY[n], and the variance Var[Yn].Sol：
.20
,12
,04
][ and ,otherwise0
,1,,1,0/1
n
n
n
nRMnM
h Xn
otherwise.0
,22/1
,12
,03
][:n
n
n
nRAns Y
18
Example 6.6
.otherwise0
,1,,1,0/1 MnMhn
A WSS random sequence, Xn, with X = 0 and auto-correlation function RX[n] = 2n is passed through the orderM1 discrete-time moving-average filter hn where
Find the output autocorrelation RY[n].Sol：
otherwise.0
),1(/)(][ :
22 MnMnMnRAns Y
19
Example 6.7A first-order discrete-time integrator with WSS inputsequence Xn has output Yn = Xn + 0.8Yn-1 . What is theimpulse response hn ?Sol：
otherwise.0
,2,1,08.0][ :
nnRAns
n
Y
20
Example 6.8Continuing Example 6.7, suppose the WSS input Xn with expected value X = 0 and autocorrelation function
is the input to the first-order integrator hn . Find the second moment, E[Yn
2] , of the output.Sol：
.20
,15.0
,01
][
n
n
n
nRX
21
Theorem 6.6
.
1
1
][ ,
]0[]1[]1[
]1[
]0[]1[
]1[]1[]0[
n
XXX
X
XX
XXX
X XE
RRMR
R
RR
MRRR
Rn
If Xn is a WSS process with expected value and auto-correlation function RX[k], then the vector has correlationmatrix and expected value given by
nX
nXR
][ nXE
TnnX
TnMnMnn
XXER
MXXXX
n
. vectorldimensiona- theis ] [ where 21
22
Example 6.9The WSS sequence Xn has autocorrelation function RX[n] asgiven in Example 6.5. Find the correlation matrix of
Sol：
. 3332313033 XXXXX
.20
,12
,04
][
n
n
n
nRX
23
Example 6.10
. 1111 T
Mh
The order M1 averaging filter hn given in Example 6.6 can be represented by the M element vector
The input is
The output vector , then .
. 110T
LXXXX
TMLYYYY 210
XHY
X
L
ML
M
H
M
M
M
Y
ML
L
M
X
X
X
X
h
hh
hh
h
Y
Y
Y
Y
1
1
0
1
01
01
0
2
1
1
0
.otherwise0
,1,,1,0/1 MnMhn
24
6. Linear Filtering of a Random Signals
7. Power Spectrum Analysis
8. Linear Estimation and Prediction Filters
9. Mean-Square Estimation
25
7. Power Spectrum Analysis Definition: Fourier Transform
Definition: Power Spectral Density
dfefGtgdtetgfG
fGtg
ftjftj 22 )()( ,)()(
ifpair ansformFourier tr a are )( and )( Functions
.)(
2
1lim
)( 2
1lim)( is )( process
stochastic WSS theoffunction density spectralpower The
22
2
T
T
ftj
T
TT
X
dtetXET
fXET
fStX
26
Theorem 7.1
dfefSRdeRfS
fSRtX
fjXX
fjXX
XX
22 )()( ,)()(
pair ansformFourier tr
theare )( and )( process, stochastic WSSa is )( If
Pf：
27
Theorem 7.2
)()( (c)
)0()()( (b)
allfor 0)( (a)
:properties following ith thefunction w value-real a is )(
density spectralpower the,)( process stochastic WSSaFor
2
fSfS
RtXEdffS
ffS
fS
tX
XX
XX
X
X
Pf：
28
Example 7.1
.)(power average thecalculate and )(function
density spectralpower theDerive .0 where)(
function ation autocorrel has )( process stochastic A WSS
2 tXEfS
bAeR
tX
X
bX
Sol：
AtXEfb
AbfSAns X
)(,
)2(
2)(: 2
22
29
Example 7.2A white Gaussian noise process X(t) with autocorrelation function RW ( ) = 0 ( ) is passed through the moving-average filter
For the output Y(t), find the power spectral density SY (f ).
Sol：
.otherwise0
,0/1)(
TtTth
2
0
)2(
)2cos(12)(:
fT
fTfSAns Y
30
Discrete-Time Fourier Transform (DTFT)
Definition :
2/1
2/1
22
21012
)( ,)(
ifpair (DTFT) ansformFourier tr time-discrete a
are )(function theand },,,,,,{ sequence The
deXxexX
Xxxxxx
njn
n
njn
Example 7.3 : Calculate the DTFT H() of the order M1 moving-average filter hn of Example 6.6. Sol：
.otherwise0
1,,0/1 MnMhn
2
2
1
11)(:
j
Mj
e
e
MHAns
31
Power Spectral Density of a Random Sequence
Definition :
. 12
1lim)( is
sequence random WSS theoffunction density spectralpower The2
2
L
Ln
njn
LXn eXE
LSX
Theorem 7.3 : Discrete-Time Winer-Khintchine
2/1
2/1
22 )(][ ,][)(
:pair ansformFourier tr time-discrete a
are )( and ][ process, stochastic WSSa is If
deSkRekRS
SkRX
kjXX
k
kjXX
XXn
32
Theorem 7.4
).()( ,integer any for (d)
),()( (c)
. [0])( (b)
, allfor 0)( (a)
:properties following the
has )(density spectralpower the, sequence random WSSaFor
221
21
XX
XX
Xn
/
/- X
X
Xn
SnSn
SS
RXEdS
S
SX
33
Example 7.4
.otherwise0
,1,0,14/)2(][
. offunction density spectral
power theDerive follows. as ][function ation autocorrel
and valueexpected zero has sequence random WSSThe
2 nnkR
X
kR
X
X
n
X
n
Sol：
)2cos(12
)(:2
XSAns
34
Example 7.5
.2/10 where),(2
1)(
2
1)(
].[function
ation autocorrel theis What follows. as )(density spectral
power andmean zero has sequence random WSSThe
000
X
X
X
n
S
kR
S
X
Sol：
)2cos(][: 0kkRAns X
35
Cross Spectral DensityDefinition :
.][)(
theyieldsn correlatio-cross theof transform
Fourier the, and sequences random Sjointly WSFor
.)()(
theyieldsn correlatio-cross theof transform
Fourier the,)( and )( processes random Sjointly WSFor
2
2
k
kjXYXY
nn
fjXYXY
ekRS
tytral densicross spec
YX
deRfS
tytral densicross spec
tYtX
36
Example 7.6
).(output theofdensity spectralpower theFind
).()()()()(
thatfound weS,jointly WS are )( and )( when case, In this
0. with process noise WSSa is )( where)()()(Let
tY
RRRRR
tNtX
tNtNtXtY
NNXXNXY
N
Sol：
)()()()()(: fSfSfSfSfSAns NNXXNXY
37
Example 7.7
).(n observatio theofdensity spectralpower and
ation autocorrel thefind t,independen are )( and )( that Suppose
).()()()()(
thatfound weS,jointly WS are )( and )( when case, In this
0. with process noise WSSa is )( where)()()(Let
tY
tNtX
RRRRR
tNtX
tNtNtXtY
NNXXNXY
N
Sol：
)()()(: fSfSfSAns NXY
38
Frequency Domain Filter Relationships
Time Domain ： Y(t) = X(t)h(t)Frequency Domain : W(f) = V(f)H(f)
where V(f) = F{X(t)}, W(f) = F{Y(t)}, and H(f) = F{h(t)}.
( )h t
LTI system
x(t)
)()(.)()(
)()()(
thtxdtxh
dxthty
39
Theorem 7.5
).()()( is output the
ofdensity spectralpower the),(function sfer with tran
filter LTI a input to theis sequence random WSSaWhen
).()()( is )(output the
ofdensity spectralpower the),(function sfer with tran
filter LTI a input to theis )( process stochastic WSSaWhen
2
2
XYn
n
XY
SHSY
H
X
fSfHfStY
fH
tX
Pf：
40
Example 7.8
process? stochasticoutput theofpower average theisWhat
).(output filter theofation autocorrel anddensity spectral
power the),( and )( find ,/1 and 0 Assume
,otherwise.0
,0)1()(
response impulseh filter wit RCan input to theis )(
function ation autocorrel with )( process stochastic A WSS
tY
RfSRCbb
te/RCth
eR
tX
YY
-t/RC
bX
Sol：
RCb
RCtYE
bfRCf
RCbfSAns Y /1
/1)]([ ,
])2][()/1()2[(
)/1(2)(: 2
2222
2
41
Example 7.9
]?[ is What . sequence
output theoffunction density spectralpower the, )( Derive
otherwise.0
,1 ,11
,01
response impulseh filter wit a input to theis sequence This
).2cos(22)(density sepctralpower has sequence random The
2nn
Y
n
Xn
YEY
S
n
n
h
SX
Sol：
2][ ),6cos(22)(: 2 nYESAns Y
42
Example 7.10
.output filter average
-moving time-discrete for the )(density spectralpower theFind
otherwise.0
,10/1
filter average-moving time-discrete 1order the
throughpassed is ][function ation autocorrel and
0 valueexpected with sequence random WSSThe2
n
Y
n
nX
Xn
Y
S
MnMh
M
δnR
X
Sol：
)2cos(1
)2cos(1)(:
2
2
M
MSAns Y
43
Theorem 7.6
).()()( ),()()(
arefunction density spectralpower output theandfunction density spectral
power crossoutput -input theoutput,filter theis and ),(function
h transferfilter wit LTI a input to theis process stochastic WSS theIf
).()()( ),()()(
arefunction density spectralpower output theandfunction density spectral
power crossoutput -input theoutput,filter theis )( and ),(function
h transferfilter wit LTIan input to theis )( process stochastic WSS theIf
*
*
XYYXXY
n
n
XYYXXY
SHSSHS
YH
X
fSfHfSfSfHfS
tYfH
tX
Pf：
44
I/O Correlation and Spectral Density Functions
h(t)hn
h(-t)h-n
RX()RX[k]
RXY()RXY[k]
RY()RY[k]
H(f)H()
H*(f)H*()
SX(f)SX()
SXY(f)SXY()
SY(f)SY()
Time Domain
Frequency Domain
45
6. Linear Filtering of a Random Signals
7. Power Spectrum Analysis
8. Linear Estimation and Prediction Filters
9. Mean-Square Estimation
46
8. Linear Estimation and Prediction Filters Linear Predictor1. Used in cellular telephones as part of a speech
compression algorithm.2. A speech waveform is considered to be a sample
function of WSS process X(t).3. The waveform is sampled with rate 8000 samples/sec
to produce the random sequence Xn = X(nT).4. The prediction problem is to estimate a future
speech sample, Xn+k using N previous samples Xn-
M+1 , Xn-M+2 , …, Xn.5. Need to minimize the cost, complexity, and power
consumption of the predictor.
47
Linear Prediction Filters
TnMnn XXXY
1Use
to estimate a future sample X=Xn+k.
We wish to construct an LTI FIR filter hn with input Xn such that the desired filter output at time n, is the linear minimum mean square error estimate
Then we have
The predictor can be implemented by choosing .
. where, )( 1XYYn
TnL RRaXaXX
. where
, )(
01T
M
nT
nL
hhh
XhXX
B
B
ahB
48
Theorem 8.1
h
][
]1[
]1[
1
1
kR
kR
kMR
X
X
X
X
ERR
X
X
X
kn
n
n
Mn
XXXY knn
Let Xn be a WSS random process with expected value E[Xn] = 0 and autocorrelation function RX[k]. The minimum mean square error linear filter of order M1, for predicting Xn+k at time n is the filter such that
where
is called as the cross-correlation matrix.
.1
knnn XXX RRh
B
49
Example 8.1 Xn be a WSS random sequence with E[Xn] = 0 and autocorrelation function RX[k]= ( 0.9)|k|. For M = 2 samples, find , the coefficients of the optimum linear predictor for X = Xn+1, given . What is the optimum linear predictor of Xn+1 , given Xn 1 and Xn.What is the mean square error of the optimal predictor?Sol：
Thhh 10
Tnn XXY 1
50
Theorem 8.2
, , , 11 nMnMn XXX
If the random sequence Xn has a autocorrelation function RX[n]= b|k| RX[0], the optimum linear predictor of Xn+k, given the M previous samples is
and the minimum mean square error is .Pf：
nk
kn XbX ˆ
)1](0[ 2* kXL bRe
51
Linear Estimation Filters
TnMnn YYYY
1Estimate X=Xn based on the noisy observations Yn=Xn+Wn.We use the vector of the M most recent observations.Our estimates will be the output resulting from passing the sequence Yn through the LTI FIR filter hn. Xn and Wn are assumed independent WSS with E[Xn]=E[Wn]=0 and autocorrelation function RX[n] and RW[n]. The linear minimum mean square error estimate of X given the observation Yn is
Vector From :
. where, )( 1XYYn
TnL nn
RRaYaYX
.
,
where,
1
1
TnMnn
TnMnn
nnn
WWW
XXX
WXY
52
Theorem 8.3
h
Let Xn and Wn be independent WSS random processes with E[Xn]=E[Wn]=0 and autocorrelation function RX[k] and RW[k]. Let Yn=Xn+Wn. The minimum mean square error linear estimation filter of Xn of order M1 given the input Yn is given by such that
by given is where
)( 101
nn
nnnn
XX
XXWX
TM
R
RRRhhh
B
]0[
]1[
]1[
1
1
X
X
X
n
n
n
Mn
XXXY
R
R
MR
X
X
X
X
ERRnnn
53
Example 8.2 The independent random sequences Xn and Wn have expected zero value and autocorrelation function RX[k]= (0.9)|k| and RW[k]= (0.2)k. Use M = 2 samples of the noisy observation sequence Yn = Xn+Wn to estimate Xn. Find the linear minimum mean square error prediction filter
Sol：
Thhh 10
54
6. Linear Filtering of a Random Signals
7. Power Spectrum Analysis
8. Linear Estimation and Prediction Filters
9. Mean-Square Estimation
55
9. Mean Square Estimation
]))()([( 2tXtXE
Linear Estimation：Observe a sample function of a WSS random process Y(t) and design a linear filter to estimate a sample function of another WSS process X(t), where Y(t) = X(t) + N(t).
Wiener Filter：The linear filter that minimizes the mean square error.
Mean Square Error：
( )h t
LTI system
Y(t) )(tX
56
Theorem 9.1 ： Linear Estimation
.
is minimum The
otherwise.,0
,0),)(
)()(
is ]))()([(
theminimizeshat function t transfer The ).(
function transfer and )(input h filter witlinear a ofoutput theis )(
).(function density spectral cross and ),( and )(function
density spectralpower with process stochastic WSSare )( and )(
2
2
X
-Y
XYX
*L
YY
XY
XYY
df(f)S
(f)S(f)Se
re error mean squa
fSfS
fSfH
tXtXE
e errormean squarfH
tYtX
fSfSfS
tYtX
57
Example 9.1
filter? estimation optimum theoferror squaremean theis What (b)
?)(given )(
estimatingfor filter linear optimum theoffunction transfer theis What (a)
t.independenmutually are )( and )( .10)(function
density spectralpower with WSSa is )( where),()()( Observe
.50002
)50002sin()(
function ation autocorrel and 0 with process stochastic WSSa is )(
5
tYtX
tNtXfS
tNtNtXtY
R
tX
N
X
X
Sol：
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