1
Chapter 6
Random Processes and
Spectral Analysis
2
Introduction
(chapter objectives)
? Power spectral density
? Matched filters
Recall former Chapter that random signals are
used to convey information,Noise is also
described in terms of statistics,Thus,
knowledge of random signals and noise is
fundamental to an understanding of
communication systems,
3
Introduction
? Signals with random parameter are random singals ;
? All noise that can not be predictable are called random
noise or noise ;
? Random signals and noise are called random process ;
? Random process (stochastic process) is an indexed set of
function of some parameter( usually time) that has
certain statistical properties,
? A random process may be described by an indexed set of
random variables,
? A random variable maps events into constants,whereas
a random process maps events into functions of the
parameter t,
4
Introduction
? Random process can be classified as strictly stationary
or wide-sense stationary;
? Definition,A random process x(t) is said to be
stationary to the order N if,for any t1,t2,…,t N,,
3)-(6 ))+(),.,,,+(),+((=))(),.,,,(),(( 0020121 ttxttxttxftxtxtxf NxNx
? Where t0 si any arbitrary real constant,Furthermore,
the process is said to be strictly stationary if it is
stationary to the order N→infinite
? Definition,A random process is said to be wide-sense
stationary if
15b)-(6 )τ(=),( 2
1 5 a)-(6 an dco n s t an t = )( 1
21 xx RttR
tx
? Where τ=t2-t1,
5
Introduction
? Definition,A random process is said to be ergodic if all
time averages of any sample function are equal to the
corresponding ensemble averages(expectations)
? Note,if a process is ergodic,all time and ensemble
averages are interchangeable,Because time average
cannot be a function of time,the ergodic process must
be stationary,otherwise the ensemble averages would be
a function of time,But not all stationary processes are
ergodic,
7)-(6 +σ=>)(<=
6 c )-(6 = )(][=])([
6b)-(6 ])([
1
l i m=)]([
6 a )-(6 ][][
222
∞
∞-
T / 2
T / 2-
∫
∫
xxr m s
xx
T →→
xdc
mtxX
mdxxfxtx
dttx
T
tx
=mx ( t )=x ( t )=x
6
Introduction
? Definition, the autocorrelation function of a real process
x(t) is,
1 3 )-(6 ),()()(),( ∫ ∫∞ ∞- 21∞ ∞- 21212121 dxdxxxfxxtxtxttR xx ??
? Where x1=x(t1),and x2=x(t2),if the process is a second-
order stationary,the autocorrelation function is a
function only of the time difference τ=t2-t1,
1 4 )-(6 )()(=)τ( 21 txtxR x? Properties of the autocorrelation function of a real wide-
sense stationary process are as follows,
2
2
22
σ=)∞(-)0( )5(
p o w e r dc=)]([=)∞( )4(
18)-(6 )0(≤|)τ(| )3(
17)-(6 )τ(=)τ-( )2(
16)-(6 a=( t ) }{=)(=)0( )1(
xx
x
xx
xx
x
RR
txER
RR
RR
p o w erv e r a g exEtxR
7
Introduction
? Definition, the cross-correlation function for two real
process x(t) and y(t) is,
1 9 )-(6 ),()()(),( ∫ ∫∞ ∞- ∞ ∞- 212121 d x d yyxx y ftytxttR xxy ??
? if x=x(t1),and y=x(t2) are jointly stationary,the cross-
correlation function is a function only of the time
difference τ=t2-t1,
)(),( 21 ?xyxy RttR ?? Properties of the cross-correlation function of two real
jointly stationary process are as follows,
22)-(6 )]0()0([
2
1
|)(| )3(
21)-(6 )0()0(|)(| )2(
20)-(6 )()( )1(
yxx
yxxy
yxxy
RRR
RRR
RR
??
?
??
?
?
??
8
? Two random processes x(t) and y(t) are said to be
uncorrelated if,
2 7 )-(6 )]([)]([)( yxxy mmtytxR ???
? For all value of τ,similarly,two random processes x(t)
and y(t) are said to be orthogonal if
2 8 )-(6 0)( ??xyR
? For all value of τ,If the random processes x(t) and y(t)
are jointly ergodic,the time average may be used to
replace the ensemble average,For correlation
functions,this becomes,
2 9 )-(6 )]()][([)]()][([)( tytxtytxR xy ???
Introduction
9
Introduction
? Definition,a complex random process is,
31)-(6 )()()( tjytxtg ??Where x(t) and y(t) are real random processes,
? Definition,the autocorrelation for complex random
process is,
3 3 )-(6 )()(),( 21*21 tgtgttR g ?
Where the asterisk denotes the complex conjugate,the
autocorrelation for a wide-sense stationary complex
random process has the Hermitian symmetry property,
3 4 )-(6 )()( * ?? gg RR ??
10
Introduction
? For a Gaussian process,the one-dimension PDF can
be represented by,
]σ2 )m-(-ex p [σπ21=)( 2
2
xxxf
? some properties of f(x) are,
? (1) f(x) is a symmetry function about x=mx;
? (2) f(x) is a monotony increasing function at(-
infinite,mx) and a monotony decreasing funciton at
(mx,),the maximum value at mx is 1/[(2π)(1/2)σ];
5.0=)(=)( a n d 1=)( ∫∫∫ ∞mm∞-∞ ∞-
x
x dxxfdxxfdxxf
11
Introduction
? The cumulative distribution function (CDF) for the Gaussian
distribution is,
)σ2-()2/1(=)σ-(=]σ2 )-(-ex p [σπ2 1=)( ∫ ∞- 2 2 xxx x mxer f cmxQdzmzxF
? Where the Q function is defined by,
λ)2λ-(e x pπ21=)( 2∞∫ dzQ z
? And the error function (erf) defined as,
λ)λ-(e x pπ21=)( 2∞∫ dze r f c z
? And the complementary error function (erfc) defined as,
λ)λ-(e x pπ21=)( 2z0∫ dze r f
? And
1-22=)(o r 222=)(
1=)(
z)Q(ze r fz)Q(-ze r f c
- e r f ( z )ze r f c
12
6.2 Power Spectral Density
(definition)
? The definition of the PSD for the case of deterministic
waveform is Eq.(2-66),
6 6 )-(2 )(l i m=)(
2
∞→ω T
fWf
TP
? Definition,The power spectral density (PSD) for a
random process x(t) is given by,
42)-(6 )])([(l i m=)(
2
∞→ T
fXf T
Tx
P
? where
43)-(6 )()( ∫ T / 2T / 2 π2－ －＝ dtetxfX ftjT
13
6.2 Power Spectral Density
(Wiener-Khintchine Theorem)
? When x(t) is a wide-sense stationary process,the PSD
can be obtained from the Fourier transform of the
autocorrelation function,
4 4 )-(6 τ)τ(=)]τ([=)( ∫ ∞ ∞- τπ2-ω deRRf fjxxFP? Conversely,
4 5 )-(6 )(=)]([=)τ( ∫ ∞ ∞- τπ21 dfeffR fjxxx PP－F? Provided that R(τ) becomes sufficiently small for large
values of τ,so that
4 6 )-(6 ∞<τ|)τ(τ|∫ ∞ ∞- dR x? This theorem is also valid for a nonstationary process,
provided that we replace R(τ) by < R(t,t+τ) >,
? Proof,(notebook p)
14
6.2 Power Spectral Density
(Wiener-Khintchine Theorem)
? There are two different methods that may be used to
evaluate the PSD of a random process,
42)-(6 )])([(l i m=)( 1
2
∞→ T
fXf T
TxP m e t h o dd i r e c t ? 2 using the indirect method by evaluating the Fourier
transform of Rx(τ),where Rx(τ) has to obtained first
? Properties of the PSD,
? (1) Px(f) is always real;
? (2) Px(f)>=0;
? (3) When x(t) is real,Px(-f)= Px(f);
? (4) When x(t) is wide-sense stationary,
5 4 )-(6 ( 0 ) xP)( 2∞ ∞-∫ xx Rdff ??＝P
55)-(6 )()0( ∫ ∞ ∞- ?? dR xx ?P ( 5 )
15
6.2 Power Spectral Density
? Example 6-3,(notebook p)
16
6.2 Power Spectral Density
? summary,the general expression for the PSD of a digital
signal can obtained by starting from,
56)-(6 )()( ?
?
???
??
n
sn nTtfatx
? Where f(t) is the signaling pulse shape,and Ts is the
duration of one symbol,{an} is a set of random variables
that represent the data,The autocorrelation of data is,
6 8 )-(6 mnknn aaaaR ( k ) ?? ?? By truncating x(t) we get,
)()( ?
??
??
N
Nn
snT nTtfatx? Where T/2=(N+1/2)T
s,its Fourier transform is,
5 7 )-(6 )()]([)}({)( ??
??
?
??
????
N
Nn
nTj
n
N
Nn
snTT seafFnTtfatxfX
?FF
17
6.2 Power Spectral Density
? According to the definition of PSD,we get,
58)-(6 )(
)12(
12
l i m
|)(|
)(
)12(
1
l i m|)(|
1
l i m|)(|
|)(|
1
l i m )(
1
l i m)(
∞→
∞→
2
)(
∞→
2
2
2
∞→
2
∞→
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
? ?
?
?
???
??
?
???
?? ??
?
??
?
nN
nNk
Tjk
s
N
s
N
Nn
nN
nNk
Tjk
s
N
N
Nn
N
Nm
Tnmj
mn
T
N
Nn
nTj
n
T
T
T
x
s
s
s
s
ekR
TN
N
T
fF
ekR
TN
fF
eaa
T
fF
eafF
T
fX
T
f
?
?
?
?
P
? Thus,
7 0 b )-(6 )( |)(|)( ??
?
?
???
?? ??
???k
Tjk
s
x sekRT
fFf ?P
18
6.2 Power Spectral Density
? furthermore
70b)-(6 )()()0(
|)(|
)()()0(
|)(|
)(
|)(|
)(
11
1
1
?
?
?
?
?
?
?
?
????
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
?
?
?
??
??
?
?
?
?
?
?
?
?
?
???
?
???
k
Tjk
k
Tjk
s
k
Tjk
k
Tjk
s
k
Tjk
s
x
ss
ss
s
ekRekRR
T
fF
ekRekRR
T
fF
ekR
T
fF
f
??
??
?
P
7 0 a )-(6 )2c o s ()(2)0( |)(| )(
1 ?
?
?
?
???
? ?? ??
?k
s
s
x k f TkRRT
fFf ?P
? Thus an equivalent expression of PSD is,
? Where the autocorrelation of the data is,
7 0 c )-(6 )(
1
?
?
?? ??
I
i
iiknnknn PaaaaR ( k )? In which P
i is the probability of getting the product
(anan+k),of which there are I possible value
19
6.2 Power Spectral Density
? Note that the quantity in brackets in Eq.(6.70b) is
similar to the discrete Fourier transform of the data
autocorrelation function R(k),except that the frequency
variable ω is continuous; that the PSD of the baseband
digtial signal is influenced by both the,spectrum” of the
data and the spectrum of the pulse shape used for the
line code; that spectrum may contain delta functions if
the mean value of data,an,is nonzero,that is,
0
0
0
0
2
222
n
??
?
?
?
?
???
??
?
?
?
?
???
?
? k
k
m
m
k
k
aa
aaaR( k )
a
aa
knn
knn
?
? this is the case that the data symbols are uncorrelated,
20
6.2 Power Spectral Density
70d)-(6 )(|)(|)(|)(|
)(
|)(|
)|(|
)(
s p e c t r u m
22
s p e c t r u m
22
22
2
22
2
????? ?????? ??
?? ??? ??
d i s cr e t e
n
a
c o n t i n u o u s
a
n
aa
s
k
Tjk
aa
s
x
nDfnDFDmfFD
nDfDm
T
fF
em
T
fF
f
s
?
?
?
?
???
?
???
?
???
???
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
?
?
??
?
??
?
?
P
? Where D=1/Ts,And the Poisson sum formula is used,For
the general case where there is correlation between the
data,let the data autocorrelation function R(k) be
expressed in terms of the normalized–data autocorrelation
function ρ(k),the PSD of the digital signal is
? thus
21
6.2 Power Spectral Density
? where
7 0 e )-(6 )(|)(|)()(|)(|)(
s p ec t r u m d i s c r et e
22
s p ec t r u m co n t i n u o u s
22
????? ?????? ????? ???? ??
?
?
???
???
n
aax nDfnDFDmffFDf ?? WP
7 0 f )-(6 )()( 2??
???
??
k
k f Tj sekf ?
? ?W
? is a spectral weight function obtained form the Fourier
transform of the normalized autocorrelation impulse train
)()(?
?
???
?
k
skTk ???
22
6.2 Power Spectral Density
? White noise processes,
? Definition,A random process x(t) is said to be a white-
noise process if the PSD is constant over all frequencies;
that is,
7 1 )-(6 2)( 0Nfx ?P
? Where N0 is a positive constant,
? The autocorrelation function for the white-noise process is
obtained by taking the inverse Fourier transform of eq,
Above,The result is,
)(2)( 0 ??? NR x ?
23
6.2 Power Spectral Density
? White Guassian Noise,n(t) is a random process (random
signal)
? Gaussian – Gaussian PDF(probability-density-function)
? White -- a flat PSD (Power-Spectrum-Density) or a
impulse-like auto-correlation
2v a r;0 ???m e a n
22 2/
2
1)( ?
??
tetf ??
)(
2
)()(
)(
2
)(
02
20
??????
?
N
R
f
N
f
n
n
??
????????P
24
6.2 Power Spectral Density
? Bandpass White Gaussian Noise,n(t) is a (narrow)
bandpass random process (random signal) of 2BHz,
while the baseband signal is BHz)
00122 2)()0(v a r;0 BNNffRm e a n n ?????? ?
22 2/
2
1)( ?
?
nenf ??
)
2
2c o s (
2
2s i n
2)(
,0
,2/
)(
12
0
210
??
??
??
?
ff
B
B
BNR
o t h er w i s e
fffN
f
n
n
?
?
?
?
?
?
??
??
?
?
?
?P
?Gaussian – Gaussian PDF (probability-density-function)
?White -- a flat PSD (Power-Spectrum-Density) in a band
of BHz or a sinc-like auto-correlation
25
6.2 Power Spectral Density
? Measurement of PSD
? Analog techniques
? Numerical computation of the PSD
? Note,in either case the measurement can only
approximate the true PSD,because the measurement is
carried out over a finite time interval instead of the infinite
interval,
4 2 )-(6 )
])([
(l i m)(
2
∞→ T
fX
f T
Tx
?P
??
?
??
??? )()()(o r ])([1)( 2
TRFffXTf xTTT
???PP
26
Input-Output Relationships for
Linear System
? Theorem,if a wide-sense stationary random process x(t) is
applied to the input of a time-invariant linear network
with impulse response h(t) the output autocorrelation is,
8 2 b )-(6 )(*)(*)()(
o r
8 2 a )-(6 )()()()( 211221
????
????????
xy
xy
RhhR
ddRhhR
??
??? ? ??
??
?
??
? The output PSD is,
83)-(6 )( |)(| )( 2 ffHf xy PP ?
? Where H(f)=F{h(t)},Linear network h(t)
H(f)
Input x(t)
output y(t)
X(f)
Rx(τ)
Px(f)
Y(f)
Ry(τ)
Py(f) Fig.6-6 Linear system
27
6.8 Matched Filters
? Matched filtering is a technique for designing a linear
filter to minimize the effect of noise while maximize the
signal,
? A general representation for a matched filter is illustrated
as follows,
Matched filter
h(t)
H(f)
r(t)=s(t)+n(t)
Fig.6-15 matched filter
r0(t)=s0(t)+n0(t)
The input signal is denoted by s(t) and the output signal by
s0(t),Similar notation is used for the noise,The signal is
assumed to be (absolutely) time limited to the interval (0,T)
and is zero otherwise,The PSD,Pn(f),of the additive input
noise n(t) is known,if signal is present,its waveform is also
known,
28
6.8 Matched Filters
? The matched-filter design criterion,
? Finding a h(t) or,equivalently H(f),so that the
instantaneous output signal power is maximized at a
sampling time t0,that is,
1 5 4 )-(6 )( )(2
0
2
0
tn
ts
N
S
o u t
???
?
?
???
?
? Is a maximum at t=t0,
? Note,the matched filter does not preserve the input signal
waveshape,Its objective is to distort the input signal
waveshape and filter the noise so that at the sampling time
t0,the output signal level will be as large as possible with
respect to the rms output noise level,
29
6.8 Matched Filters
? Theorem,the matched filter is the linear filter that
maximizes (S/N)out=s02(t0)/<n02(t)>,and that has a transfer
function given by,
1 5 5 )-(6 )( )()( 0
*
tj
n
effSKfH ??? P
Where s(f)=F[s(t)] is the Fourier transform of the known
input signal s(t) of duration T sec,Pn(f) is the PSD of the
input noise,t0 is the sampling time when (S/N)out is
evaluated,and K is an arbitrary real nonzero constant,
Matched filter
h(t)
H(f)
r(t)=s(t)+n(t)
Fig.6-15 matched filter
r0(t)=s0(t)+n0(t)
30
6.8 Matched Filters
? Proof,the output signal at time t0 is,
????? dfefSfHts tj 0)()()( 00 ?? The average power of the output noise is,
?????? dfffHRtn nn )(|)(|)0()( 2020 0 P? Then,
156)-(6
)(|)(|
|)()(|
)(
)(
2
2
2
0
2
0
0
?
?
?
??
?
??????
?
?
???
?
dfffH
dfefSfH
tn
ts
N
S
n
tj
o u t P
?
? With the aid of Schwarz inequality,
1 5 7 )-(6 )()( |)()(| 222 ??? ????????? ? dffBdffAdffBfA
? Where A(f) and B(f) may be complex function of the real
variable f,equality is obtained only when,
158)-(6 )()( * fKBfA ?
31
6.8 Matched Filters
? Leting,
)(
)()( a n d )()()( 0
f
efSfBffHfA
n
tj
n PP
?
??? then,
1 5 9 )-(6
)(
)(
)(|)(|
)(
)(
)()(
-
2
2
2
2
?
?
? ?
?
?
?
??
?
??
?
??
?
???
?
?
??
?
?
df
f
fS
dfffH
df
f
fS
dfffH
N
S
n
n
n
n
o u t
P
P
P
P
? The maximum (S/N)out is obtained when H(f) is chosen
such that equality is attained,This occurs when
A(f)=KB*(f),Or,
)(
)()(
)(
)( )()( 00 **
f
efKSfH
f
efKSffH
n
tj
n
tj
n PPP
?? ??
???
32
6.8 Matched Filters
Results for White Noise
? For white noise,Pn(f)=N0/2,thus we get,
0)(2)( *
0
tjefS
N
KfH ???
? Theorem when the input noise is white,the impulse
response of the matched filter becomes,
? h(t)=Cs(t0-t) (6-160)
Where C is an arbitrary real positive constant,t0 is the time
of the peak signal output,and s(t) is the known input
signal waveshape,
? The impulse response of the matched filter (white–noise
case) is simply the known signal waveshape that is
“played backward” and translated by an amount to,
? Thus,the filter is said to be,matched” to the signal,
33
6.8 Matched Filters
? An important property,
the actual value of (S/N)out that is obtained form the
matched filter is,
1 6 1 )-(6 2 )(22/ )(
0-
2
0- 0
2
N
Edtts
NdfN
fS
N
S s
o u t
?????
?
?
???
? ?? ?
?
?
?
The result states that (S/N)out depends on the signal energy
and PSD level of the noise,and not on the particular signal
waveshape that is used,It can also be written in another
terms,Assume that the input noise power is measured in a
band that is W hertz wide,The signal has a duration of T
seconds,Then,
162)-(6 2 T W )( )/( 2
0 in
s
o u t N
S
WN
TETW
N
S
??????????????????
34
6.8 Matched Filters
? Example 6-11 Integrate-and-Dump (Matched) filter
t
t
t
t
()St
()St?
0()StT
T
0.75T
0.5T
0.25T
1 t 2t
1- t2-t
0t
0tT?
1
1
2T
1
T
t1
a) Input signal
b),backwards” signal
c) matched-Filter
impulse response
d) Signal output of
matched filter
t0==t2 h(t)=s(t
0-t)
35
6.8 Matched Filters
? Fig.6-17
W a ve f or m a t
A ( i np ut s i gn a l
a nd no i s e )
W a ve f or m
a t B
W a ve f or m
a t C
W a ve f or m
a t D
I nt e gr a t or r e s e t t o z e r o
i ni t i a l c on di t i on a t
c l oc ki ng t i m e
Int e g ra t or
Re s e t
S a m pl e a n d
H ol d
D D
C l oc ki ng s i gn a l ( bi t s yn c )
B D C D A D
r ( t ) = s ( t ) + n( t ) r 0 ( t ) ou t pu t
36
6.8 Matched Filters
correlation processing
Theorem,
The matched filter may be realized by correlating the input
with s(t) for the case of white noise,
that is,
s ( t ) ( K n o w n s i g n a l
R e f e r e n c e i n p u t )
? ?
0
0
)()(
t
Tt
dttstr r ( t ) = s ( t ) + n ( t ) r 0 (t 0 )
1 6 7 )-(6 )()()( 0
000 ? ?
? t Tt dttstrtrWhere s(t) is the known signal waveshape and r(t) is the
processor input,as illustrated in Fig.6-18
Fig.6-18 Matched-filter realization by correlation processing
37
6.8 Matched Filters
? Proof,the output of the matched filter at time t0 is,
)()()(*)()( 0
0 00000 ? ?
??? t Tt dthrthtrtr ???
Because of h(t)=Cs(t0-t) (6-160)
??
? ????
o t h e rw h e re
Tt0
0
)()( 0 ttsth
so
??
?
??
?
??
???
0
0
0
0
0
0
))()())()(
))(()()( 0000
t
Tt
t
Tt
t
Tt
dttstrdsr
dttsrtr
???
???
This is over
38
6.8 Matched Filters
? Exampl
e 6-12
Match
ed
Filter
for
Detect
ion of
a
BPSK
signal
39
6.8 Matched Filters
(Transversal Matched Filter)
? we wish to find the set of transversal filter coefficients
{ai;i=1,2,…..,N} such that signal -to-average–noise–power
ratio is maximized
D e l a y
T
D e l a y
T
D e l a y
T
D e l a y
T
a 1
1
a 2
1
a 3
1
a N
1
r 0 ( t ) = s 0 ( t ) + n 0 ( t )
r ( t ) = s ( t ) + n( t )
? Fig,6-20 Transversal matched filter
40
6.8 Matched Filters
(Transversal Matched Filter)
? The output signal at time t=t0 is,
1 6 8 )-(6 ))1(()(
))1(()2()()()(
1
000
003020100
?
?
???
?????????
N
k
k
N
Tktsatsor
TNtsaTtsaTtsatsats ?
? Similarly,the output noise at time t=t0 is,
169)-(6 ))1(()(
10
?
?
??? N
k k
Tktnatn
? The average noise power is,
170)-(6 )(
))1(())1(()(
1 1
1 1
2
0
? ?
? ?
? ?
? ?
??
?????
N
k
N
l
lk
N
k
N
l
klk
lTkTRaa
TltnaTktnaatn
41
6.8 Matched Filters
(Transversal Matched Filter)
? Thus the output-peak-signal to average-noise-power ratio
is,
171)-(6
)(
))1((
1 1
2
1
0
)(
)(
2
0
0
2
0
? ?
?
? ?
?
?
?
?
?
?
?
?
??
?
N
k
N
l
lk
N
k
k
tn
ts
lTkTRaa
Tktsa
? Using Lagrange’s method of maximizing the numerator
while constraining the denominator to be a constant,we
can get,
1 7 7 )-(6 o r
1 7 4 )-(6 )())1((
1
0
Ras ?
???? ?
?
N
k
k iTkTRaTits
? Is the matrix notation of (6-174)
42
6.8 Matched Filters
(Transversal Matched Filter)
? Where the known signal vector and the known
autocorrelation matrix for the input noise and the unknown
transversal matched filter coefficient vector are given by
,,
2
1
21
22221
11211
2
1
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
NNNNN
N
N
N
a
a
a
rrr
rrr
rrr
s
s
s
?
?
???
?
?
?
aRs
? the transversal matched filter coefficient vector are given by
1 8 1 )-(6 1 sRa ??
43
6.8 Matched Filters
(Transversal Matched Filter)
44
6.8 Matched Filters
(Transversal Matched Filter)
45
Homework
Chapter 6
Random Processes and
Spectral Analysis
2
Introduction
(chapter objectives)
? Power spectral density
? Matched filters
Recall former Chapter that random signals are
used to convey information,Noise is also
described in terms of statistics,Thus,
knowledge of random signals and noise is
fundamental to an understanding of
communication systems,
3
Introduction
? Signals with random parameter are random singals ;
? All noise that can not be predictable are called random
noise or noise ;
? Random signals and noise are called random process ;
? Random process (stochastic process) is an indexed set of
function of some parameter( usually time) that has
certain statistical properties,
? A random process may be described by an indexed set of
random variables,
? A random variable maps events into constants,whereas
a random process maps events into functions of the
parameter t,
4
Introduction
? Random process can be classified as strictly stationary
or wide-sense stationary;
? Definition,A random process x(t) is said to be
stationary to the order N if,for any t1,t2,…,t N,,
3)-(6 ))+(),.,,,+(),+((=))(),.,,,(),(( 0020121 ttxttxttxftxtxtxf NxNx
? Where t0 si any arbitrary real constant,Furthermore,
the process is said to be strictly stationary if it is
stationary to the order N→infinite
? Definition,A random process is said to be wide-sense
stationary if
15b)-(6 )τ(=),( 2
1 5 a)-(6 an dco n s t an t = )( 1
21 xx RttR
tx
? Where τ=t2-t1,
5
Introduction
? Definition,A random process is said to be ergodic if all
time averages of any sample function are equal to the
corresponding ensemble averages(expectations)
? Note,if a process is ergodic,all time and ensemble
averages are interchangeable,Because time average
cannot be a function of time,the ergodic process must
be stationary,otherwise the ensemble averages would be
a function of time,But not all stationary processes are
ergodic,
7)-(6 +σ=>)(<=
6 c )-(6 = )(][=])([
6b)-(6 ])([
1
l i m=)]([
6 a )-(6 ][][
222
∞
∞-
T / 2
T / 2-
∫
∫
xxr m s
xx
T →→
xdc
mtxX
mdxxfxtx
dttx
T
tx
=mx ( t )=x ( t )=x
6
Introduction
? Definition, the autocorrelation function of a real process
x(t) is,
1 3 )-(6 ),()()(),( ∫ ∫∞ ∞- 21∞ ∞- 21212121 dxdxxxfxxtxtxttR xx ??
? Where x1=x(t1),and x2=x(t2),if the process is a second-
order stationary,the autocorrelation function is a
function only of the time difference τ=t2-t1,
1 4 )-(6 )()(=)τ( 21 txtxR x? Properties of the autocorrelation function of a real wide-
sense stationary process are as follows,
2
2
22
σ=)∞(-)0( )5(
p o w e r dc=)]([=)∞( )4(
18)-(6 )0(≤|)τ(| )3(
17)-(6 )τ(=)τ-( )2(
16)-(6 a=( t ) }{=)(=)0( )1(
xx
x
xx
xx
x
RR
txER
RR
RR
p o w erv e r a g exEtxR
7
Introduction
? Definition, the cross-correlation function for two real
process x(t) and y(t) is,
1 9 )-(6 ),()()(),( ∫ ∫∞ ∞- ∞ ∞- 212121 d x d yyxx y ftytxttR xxy ??
? if x=x(t1),and y=x(t2) are jointly stationary,the cross-
correlation function is a function only of the time
difference τ=t2-t1,
)(),( 21 ?xyxy RttR ?? Properties of the cross-correlation function of two real
jointly stationary process are as follows,
22)-(6 )]0()0([
2
1
|)(| )3(
21)-(6 )0()0(|)(| )2(
20)-(6 )()( )1(
yxx
yxxy
yxxy
RRR
RRR
RR
??
?
??
?
?
??
8
? Two random processes x(t) and y(t) are said to be
uncorrelated if,
2 7 )-(6 )]([)]([)( yxxy mmtytxR ???
? For all value of τ,similarly,two random processes x(t)
and y(t) are said to be orthogonal if
2 8 )-(6 0)( ??xyR
? For all value of τ,If the random processes x(t) and y(t)
are jointly ergodic,the time average may be used to
replace the ensemble average,For correlation
functions,this becomes,
2 9 )-(6 )]()][([)]()][([)( tytxtytxR xy ???
Introduction
9
Introduction
? Definition,a complex random process is,
31)-(6 )()()( tjytxtg ??Where x(t) and y(t) are real random processes,
? Definition,the autocorrelation for complex random
process is,
3 3 )-(6 )()(),( 21*21 tgtgttR g ?
Where the asterisk denotes the complex conjugate,the
autocorrelation for a wide-sense stationary complex
random process has the Hermitian symmetry property,
3 4 )-(6 )()( * ?? gg RR ??
10
Introduction
? For a Gaussian process,the one-dimension PDF can
be represented by,
]σ2 )m-(-ex p [σπ21=)( 2
2
xxxf
? some properties of f(x) are,
? (1) f(x) is a symmetry function about x=mx;
? (2) f(x) is a monotony increasing function at(-
infinite,mx) and a monotony decreasing funciton at
(mx,),the maximum value at mx is 1/[(2π)(1/2)σ];
5.0=)(=)( a n d 1=)( ∫∫∫ ∞mm∞-∞ ∞-
x
x dxxfdxxfdxxf
11
Introduction
? The cumulative distribution function (CDF) for the Gaussian
distribution is,
)σ2-()2/1(=)σ-(=]σ2 )-(-ex p [σπ2 1=)( ∫ ∞- 2 2 xxx x mxer f cmxQdzmzxF
? Where the Q function is defined by,
λ)2λ-(e x pπ21=)( 2∞∫ dzQ z
? And the error function (erf) defined as,
λ)λ-(e x pπ21=)( 2∞∫ dze r f c z
? And the complementary error function (erfc) defined as,
λ)λ-(e x pπ21=)( 2z0∫ dze r f
? And
1-22=)(o r 222=)(
1=)(
z)Q(ze r fz)Q(-ze r f c
- e r f ( z )ze r f c
12
6.2 Power Spectral Density
(definition)
? The definition of the PSD for the case of deterministic
waveform is Eq.(2-66),
6 6 )-(2 )(l i m=)(
2
∞→ω T
fWf
TP
? Definition,The power spectral density (PSD) for a
random process x(t) is given by,
42)-(6 )])([(l i m=)(
2
∞→ T
fXf T
Tx
P
? where
43)-(6 )()( ∫ T / 2T / 2 π2－ －＝ dtetxfX ftjT
13
6.2 Power Spectral Density
(Wiener-Khintchine Theorem)
? When x(t) is a wide-sense stationary process,the PSD
can be obtained from the Fourier transform of the
autocorrelation function,
4 4 )-(6 τ)τ(=)]τ([=)( ∫ ∞ ∞- τπ2-ω deRRf fjxxFP? Conversely,
4 5 )-(6 )(=)]([=)τ( ∫ ∞ ∞- τπ21 dfeffR fjxxx PP－F? Provided that R(τ) becomes sufficiently small for large
values of τ,so that
4 6 )-(6 ∞<τ|)τ(τ|∫ ∞ ∞- dR x? This theorem is also valid for a nonstationary process,
provided that we replace R(τ) by < R(t,t+τ) >,
? Proof,(notebook p)
14
6.2 Power Spectral Density
(Wiener-Khintchine Theorem)
? There are two different methods that may be used to
evaluate the PSD of a random process,
42)-(6 )])([(l i m=)( 1
2
∞→ T
fXf T
TxP m e t h o dd i r e c t ? 2 using the indirect method by evaluating the Fourier
transform of Rx(τ),where Rx(τ) has to obtained first
? Properties of the PSD,
? (1) Px(f) is always real;
? (2) Px(f)>=0;
? (3) When x(t) is real,Px(-f)= Px(f);
? (4) When x(t) is wide-sense stationary,
5 4 )-(6 ( 0 ) xP)( 2∞ ∞-∫ xx Rdff ??＝P
55)-(6 )()0( ∫ ∞ ∞- ?? dR xx ?P ( 5 )
15
6.2 Power Spectral Density
? Example 6-3,(notebook p)
16
6.2 Power Spectral Density
? summary,the general expression for the PSD of a digital
signal can obtained by starting from,
56)-(6 )()( ?
?
???
??
n
sn nTtfatx
? Where f(t) is the signaling pulse shape,and Ts is the
duration of one symbol,{an} is a set of random variables
that represent the data,The autocorrelation of data is,
6 8 )-(6 mnknn aaaaR ( k ) ?? ?? By truncating x(t) we get,
)()( ?
??
??
N
Nn
snT nTtfatx? Where T/2=(N+1/2)T
s,its Fourier transform is,
5 7 )-(6 )()]([)}({)( ??
??
?
??
????
N
Nn
nTj
n
N
Nn
snTT seafFnTtfatxfX
?FF
17
6.2 Power Spectral Density
? According to the definition of PSD,we get,
58)-(6 )(
)12(
12
l i m
|)(|
)(
)12(
1
l i m|)(|
1
l i m|)(|
|)(|
1
l i m )(
1
l i m)(
∞→
∞→
2
)(
∞→
2
2
2
∞→
2
∞→
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
? ?
?
?
???
??
?
???
?? ??
?
??
?
nN
nNk
Tjk
s
N
s
N
Nn
nN
nNk
Tjk
s
N
N
Nn
N
Nm
Tnmj
mn
T
N
Nn
nTj
n
T
T
T
x
s
s
s
s
ekR
TN
N
T
fF
ekR
TN
fF
eaa
T
fF
eafF
T
fX
T
f
?
?
?
?
P
? Thus,
7 0 b )-(6 )( |)(|)( ??
?
?
???
?? ??
???k
Tjk
s
x sekRT
fFf ?P
18
6.2 Power Spectral Density
? furthermore
70b)-(6 )()()0(
|)(|
)()()0(
|)(|
)(
|)(|
)(
11
1
1
?
?
?
?
?
?
?
?
????
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
?
?
?
??
??
?
?
?
?
?
?
?
?
?
???
?
???
k
Tjk
k
Tjk
s
k
Tjk
k
Tjk
s
k
Tjk
s
x
ss
ss
s
ekRekRR
T
fF
ekRekRR
T
fF
ekR
T
fF
f
??
??
?
P
7 0 a )-(6 )2c o s ()(2)0( |)(| )(
1 ?
?
?
?
???
? ?? ??
?k
s
s
x k f TkRRT
fFf ?P
? Thus an equivalent expression of PSD is,
? Where the autocorrelation of the data is,
7 0 c )-(6 )(
1
?
?
?? ??
I
i
iiknnknn PaaaaR ( k )? In which P
i is the probability of getting the product
(anan+k),of which there are I possible value
19
6.2 Power Spectral Density
? Note that the quantity in brackets in Eq.(6.70b) is
similar to the discrete Fourier transform of the data
autocorrelation function R(k),except that the frequency
variable ω is continuous; that the PSD of the baseband
digtial signal is influenced by both the,spectrum” of the
data and the spectrum of the pulse shape used for the
line code; that spectrum may contain delta functions if
the mean value of data,an,is nonzero,that is,
0
0
0
0
2
222
n
??
?
?
?
?
???
??
?
?
?
?
???
?
? k
k
m
m
k
k
aa
aaaR( k )
a
aa
knn
knn
?
? this is the case that the data symbols are uncorrelated,
20
6.2 Power Spectral Density
70d)-(6 )(|)(|)(|)(|
)(
|)(|
)|(|
)(
s p e c t r u m
22
s p e c t r u m
22
22
2
22
2
????? ?????? ??
?? ??? ??
d i s cr e t e
n
a
c o n t i n u o u s
a
n
aa
s
k
Tjk
aa
s
x
nDfnDFDmfFD
nDfDm
T
fF
em
T
fF
f
s
?
?
?
?
???
?
???
?
???
???
?
?
?
?
?
?
?
?
???
?
?
?
?
?
?
?
?
??
?
??
?
?
P
? Where D=1/Ts,And the Poisson sum formula is used,For
the general case where there is correlation between the
data,let the data autocorrelation function R(k) be
expressed in terms of the normalized–data autocorrelation
function ρ(k),the PSD of the digital signal is
? thus
21
6.2 Power Spectral Density
? where
7 0 e )-(6 )(|)(|)()(|)(|)(
s p ec t r u m d i s c r et e
22
s p ec t r u m co n t i n u o u s
22
????? ?????? ????? ???? ??
?
?
???
???
n
aax nDfnDFDmffFDf ?? WP
7 0 f )-(6 )()( 2??
???
??
k
k f Tj sekf ?
? ?W
? is a spectral weight function obtained form the Fourier
transform of the normalized autocorrelation impulse train
)()(?
?
???
?
k
skTk ???
22
6.2 Power Spectral Density
? White noise processes,
? Definition,A random process x(t) is said to be a white-
noise process if the PSD is constant over all frequencies;
that is,
7 1 )-(6 2)( 0Nfx ?P
? Where N0 is a positive constant,
? The autocorrelation function for the white-noise process is
obtained by taking the inverse Fourier transform of eq,
Above,The result is,
)(2)( 0 ??? NR x ?
23
6.2 Power Spectral Density
? White Guassian Noise,n(t) is a random process (random
signal)
? Gaussian – Gaussian PDF(probability-density-function)
? White -- a flat PSD (Power-Spectrum-Density) or a
impulse-like auto-correlation
2v a r;0 ???m e a n
22 2/
2
1)( ?
??
tetf ??
)(
2
)()(
)(
2
)(
02
20
??????
?
N
R
f
N
f
n
n
??
????????P
24
6.2 Power Spectral Density
? Bandpass White Gaussian Noise,n(t) is a (narrow)
bandpass random process (random signal) of 2BHz,
while the baseband signal is BHz)
00122 2)()0(v a r;0 BNNffRm e a n n ?????? ?
22 2/
2
1)( ?
?
nenf ??
)
2
2c o s (
2
2s i n
2)(
,0
,2/
)(
12
0
210
??
??
??
?
ff
B
B
BNR
o t h er w i s e
fffN
f
n
n
?
?
?
?
?
?
??
??
?
?
?
?P
?Gaussian – Gaussian PDF (probability-density-function)
?White -- a flat PSD (Power-Spectrum-Density) in a band
of BHz or a sinc-like auto-correlation
25
6.2 Power Spectral Density
? Measurement of PSD
? Analog techniques
? Numerical computation of the PSD
? Note,in either case the measurement can only
approximate the true PSD,because the measurement is
carried out over a finite time interval instead of the infinite
interval,
4 2 )-(6 )
])([
(l i m)(
2
∞→ T
fX
f T
Tx
?P
??
?
??
??? )()()(o r ])([1)( 2
TRFffXTf xTTT
???PP
26
Input-Output Relationships for
Linear System
? Theorem,if a wide-sense stationary random process x(t) is
applied to the input of a time-invariant linear network
with impulse response h(t) the output autocorrelation is,
8 2 b )-(6 )(*)(*)()(
o r
8 2 a )-(6 )()()()( 211221
????
????????
xy
xy
RhhR
ddRhhR
??
??? ? ??
??
?
??
? The output PSD is,
83)-(6 )( |)(| )( 2 ffHf xy PP ?
? Where H(f)=F{h(t)},Linear network h(t)
H(f)
Input x(t)
output y(t)
X(f)
Rx(τ)
Px(f)
Y(f)
Ry(τ)
Py(f) Fig.6-6 Linear system
27
6.8 Matched Filters
? Matched filtering is a technique for designing a linear
filter to minimize the effect of noise while maximize the
signal,
? A general representation for a matched filter is illustrated
as follows,
Matched filter
h(t)
H(f)
r(t)=s(t)+n(t)
Fig.6-15 matched filter
r0(t)=s0(t)+n0(t)
The input signal is denoted by s(t) and the output signal by
s0(t),Similar notation is used for the noise,The signal is
assumed to be (absolutely) time limited to the interval (0,T)
and is zero otherwise,The PSD,Pn(f),of the additive input
noise n(t) is known,if signal is present,its waveform is also
known,
28
6.8 Matched Filters
? The matched-filter design criterion,
? Finding a h(t) or,equivalently H(f),so that the
instantaneous output signal power is maximized at a
sampling time t0,that is,
1 5 4 )-(6 )( )(2
0
2
0
tn
ts
N
S
o u t
???
?
?
???
?
? Is a maximum at t=t0,
? Note,the matched filter does not preserve the input signal
waveshape,Its objective is to distort the input signal
waveshape and filter the noise so that at the sampling time
t0,the output signal level will be as large as possible with
respect to the rms output noise level,
29
6.8 Matched Filters
? Theorem,the matched filter is the linear filter that
maximizes (S/N)out=s02(t0)/<n02(t)>,and that has a transfer
function given by,
1 5 5 )-(6 )( )()( 0
*
tj
n
effSKfH ??? P
Where s(f)=F[s(t)] is the Fourier transform of the known
input signal s(t) of duration T sec,Pn(f) is the PSD of the
input noise,t0 is the sampling time when (S/N)out is
evaluated,and K is an arbitrary real nonzero constant,
Matched filter
h(t)
H(f)
r(t)=s(t)+n(t)
Fig.6-15 matched filter
r0(t)=s0(t)+n0(t)
30
6.8 Matched Filters
? Proof,the output signal at time t0 is,
????? dfefSfHts tj 0)()()( 00 ?? The average power of the output noise is,
?????? dfffHRtn nn )(|)(|)0()( 2020 0 P? Then,
156)-(6
)(|)(|
|)()(|
)(
)(
2
2
2
0
2
0
0
?
?
?
??
?
??????
?
?
???
?
dfffH
dfefSfH
tn
ts
N
S
n
tj
o u t P
?
? With the aid of Schwarz inequality,
1 5 7 )-(6 )()( |)()(| 222 ??? ????????? ? dffBdffAdffBfA
? Where A(f) and B(f) may be complex function of the real
variable f,equality is obtained only when,
158)-(6 )()( * fKBfA ?
31
6.8 Matched Filters
? Leting,
)(
)()( a n d )()()( 0
f
efSfBffHfA
n
tj
n PP
?
??? then,
1 5 9 )-(6
)(
)(
)(|)(|
)(
)(
)()(
-
2
2
2
2
?
?
? ?
?
?
?
??
?
??
?
??
?
???
?
?
??
?
?
df
f
fS
dfffH
df
f
fS
dfffH
N
S
n
n
n
n
o u t
P
P
P
P
? The maximum (S/N)out is obtained when H(f) is chosen
such that equality is attained,This occurs when
A(f)=KB*(f),Or,
)(
)()(
)(
)( )()( 00 **
f
efKSfH
f
efKSffH
n
tj
n
tj
n PPP
?? ??
???
32
6.8 Matched Filters
Results for White Noise
? For white noise,Pn(f)=N0/2,thus we get,
0)(2)( *
0
tjefS
N
KfH ???
? Theorem when the input noise is white,the impulse
response of the matched filter becomes,
? h(t)=Cs(t0-t) (6-160)
Where C is an arbitrary real positive constant,t0 is the time
of the peak signal output,and s(t) is the known input
signal waveshape,
? The impulse response of the matched filter (white–noise
case) is simply the known signal waveshape that is
“played backward” and translated by an amount to,
? Thus,the filter is said to be,matched” to the signal,
33
6.8 Matched Filters
? An important property,
the actual value of (S/N)out that is obtained form the
matched filter is,
1 6 1 )-(6 2 )(22/ )(
0-
2
0- 0
2
N
Edtts
NdfN
fS
N
S s
o u t
?????
?
?
???
? ?? ?
?
?
?
The result states that (S/N)out depends on the signal energy
and PSD level of the noise,and not on the particular signal
waveshape that is used,It can also be written in another
terms,Assume that the input noise power is measured in a
band that is W hertz wide,The signal has a duration of T
seconds,Then,
162)-(6 2 T W )( )/( 2
0 in
s
o u t N
S
WN
TETW
N
S
??????????????????
34
6.8 Matched Filters
? Example 6-11 Integrate-and-Dump (Matched) filter
t
t
t
t
()St
()St?
0()StT
T
0.75T
0.5T
0.25T
1 t 2t
1- t2-t
0t
0tT?
1
1
2T
1
T
t1
a) Input signal
b),backwards” signal
c) matched-Filter
impulse response
d) Signal output of
matched filter
t0==t2 h(t)=s(t
0-t)
35
6.8 Matched Filters
? Fig.6-17
W a ve f or m a t
A ( i np ut s i gn a l
a nd no i s e )
W a ve f or m
a t B
W a ve f or m
a t C
W a ve f or m
a t D
I nt e gr a t or r e s e t t o z e r o
i ni t i a l c on di t i on a t
c l oc ki ng t i m e
Int e g ra t or
Re s e t
S a m pl e a n d
H ol d
D D
C l oc ki ng s i gn a l ( bi t s yn c )
B D C D A D
r ( t ) = s ( t ) + n( t ) r 0 ( t ) ou t pu t
36
6.8 Matched Filters
correlation processing
Theorem,
The matched filter may be realized by correlating the input
with s(t) for the case of white noise,
that is,
s ( t ) ( K n o w n s i g n a l
R e f e r e n c e i n p u t )
? ?
0
0
)()(
t
Tt
dttstr r ( t ) = s ( t ) + n ( t ) r 0 (t 0 )
1 6 7 )-(6 )()()( 0
000 ? ?
? t Tt dttstrtrWhere s(t) is the known signal waveshape and r(t) is the
processor input,as illustrated in Fig.6-18
Fig.6-18 Matched-filter realization by correlation processing
37
6.8 Matched Filters
? Proof,the output of the matched filter at time t0 is,
)()()(*)()( 0
0 00000 ? ?
??? t Tt dthrthtrtr ???
Because of h(t)=Cs(t0-t) (6-160)
??
? ????
o t h e rw h e re
Tt0
0
)()( 0 ttsth
so
??
?
??
?
??
???
0
0
0
0
0
0
))()())()(
))(()()( 0000
t
Tt
t
Tt
t
Tt
dttstrdsr
dttsrtr
???
???
This is over
38
6.8 Matched Filters
? Exampl
e 6-12
Match
ed
Filter
for
Detect
ion of
a
BPSK
signal
39
6.8 Matched Filters
(Transversal Matched Filter)
? we wish to find the set of transversal filter coefficients
{ai;i=1,2,…..,N} such that signal -to-average–noise–power
ratio is maximized
D e l a y
T
D e l a y
T
D e l a y
T
D e l a y
T
a 1
1
a 2
1
a 3
1
a N
1
r 0 ( t ) = s 0 ( t ) + n 0 ( t )
r ( t ) = s ( t ) + n( t )
? Fig,6-20 Transversal matched filter
40
6.8 Matched Filters
(Transversal Matched Filter)
? The output signal at time t=t0 is,
1 6 8 )-(6 ))1(()(
))1(()2()()()(
1
000
003020100
?
?
???
?????????
N
k
k
N
Tktsatsor
TNtsaTtsaTtsatsats ?
? Similarly,the output noise at time t=t0 is,
169)-(6 ))1(()(
10
?
?
??? N
k k
Tktnatn
? The average noise power is,
170)-(6 )(
))1(())1(()(
1 1
1 1
2
0
? ?
? ?
? ?
? ?
??
?????
N
k
N
l
lk
N
k
N
l
klk
lTkTRaa
TltnaTktnaatn
41
6.8 Matched Filters
(Transversal Matched Filter)
? Thus the output-peak-signal to average-noise-power ratio
is,
171)-(6
)(
))1((
1 1
2
1
0
)(
)(
2
0
0
2
0
? ?
?
? ?
?
?
?
?
?
?
?
?
??
?
N
k
N
l
lk
N
k
k
tn
ts
lTkTRaa
Tktsa
? Using Lagrange’s method of maximizing the numerator
while constraining the denominator to be a constant,we
can get,
1 7 7 )-(6 o r
1 7 4 )-(6 )())1((
1
0
Ras ?
???? ?
?
N
k
k iTkTRaTits
? Is the matrix notation of (6-174)
42
6.8 Matched Filters
(Transversal Matched Filter)
? Where the known signal vector and the known
autocorrelation matrix for the input noise and the unknown
transversal matched filter coefficient vector are given by
,,
2
1
21
22221
11211
2
1
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
NNNNN
N
N
N
a
a
a
rrr
rrr
rrr
s
s
s
?
?
???
?
?
?
aRs
? the transversal matched filter coefficient vector are given by
1 8 1 )-(6 1 sRa ??
43
6.8 Matched Filters
(Transversal Matched Filter)
44
6.8 Matched Filters
(Transversal Matched Filter)
45
Homework
