1 chapter 6 random processes and spectral analysis
TRANSCRIPT
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 stationar
y to the order N if , for any t1,t2,…,tN, :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
15a)-(6 andconstant = )( 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=>)(<=
6c)-(6 = )(][=])([
6b)-(6 ])([1
lim=)]([
6a)-(6 ][][
222
∞
∞-
T/2
T/2-
∫
∫
xxrms
xx
T→→
xdc
mtxX
mdxxfxtx
dttxT
tx
=mx(t)=x(t)=x
6
Introduction
• Definition : the autocorrelation function of a real process x(t) is:
13)-(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.
14)-(6 )()(=)τ( 21 txtxRx
• Properties of the autocorrelation function of a real wide-sense stationary process are as follows:
2
2
22
σ=)∞(-)0( )5(
power 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
powerveragexEtxR
7
Introduction
• Definition : the cross-correlation function for two real process x(t) and y(t) is:
19)-(6 ),()()(),( ∫ ∫∞∞-
∞∞- 212121 dxdyyxxyftytxttR 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 :
27)-(6 )]([)]([)( yxxy mmtytxR • For all value of τ, similarly, two random processes x(t)
and y(t) are said to be orthogonal if28)-(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:
29)-(6 )]()][([)]()][([)( tytxtytxRxy
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:
33)-(6 )()(),( 21*
21 tgtgttRg
Where the asterisk denotes the complex conjugate. the autocorrelation for a wide-sense stationary complex random process has the Hermitian symmetry property:
34)-(6 )()( * gg RR
10
Introduction
• For a Gaussian process, the one-dimension PDF can be represented by:
]σ2
)m-(-exp[
σπ2
1=)( 2
2xx
xf
• 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=)(=)( and 1=)( ∫∫∫∞
m
m
∞-
∞
∞- x
x dxxfdxxfdxxf
11
Introduction• The cumulative distribution function (CDF) for the Gaussian
distribution is:
)σ2
-()2/1(=)
σ
-(=]
σ2
)-(-exp[
σπ2
1=)( ∫ ∞- 2
2xxx x mx
erfcmx
Qdzmz
xF
• Where the Q function is defined by:
λ)2
λ-(exp
π2
1=)(
2∞
∫ dzQz
• And the error function (erf) defined as:
λ)λ-(expπ2
1=)( 2∞
∫ dzerfcz
• And the complementary error function (erfc) defined as:
λ)λ-(expπ2
1=)( 2z
0∫ dzerf
• And
1-22=)(or 222=)(
1=)(
z)Q(zerfz)Q(-zerfc
-erf(z)zerfc
12
6.2 Power Spectral Density(definition)
• The definition of the PSD for the case of deterministic waveform is Eq.(2-66):
66)-(2 )(
lim=)(
2
∞→ω T
fWf
TP
• Definition: The power spectral density (PSD) for a random process x(t) is given by:
42)-(6 )])([
(lim=)(
2
∞→ T
fXf
T
TxP
• where43)-(6 )()( ∫
T/2
T/2π2
--= dtetxfX ftj
T
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:
44)-(6 τ)τ(=)]τ([=)( ∫∞
∞-τπ2-
ω deRRf fjxxFP
• Conversely,45)-(6 )(=)]([=)τ( ∫
∞
∞-τπ21 dfeffR fj
xxx PP-F• Provided that R(τ) becomes sufficiently small for large values of τ, so that
46)-(6 ∞<τ|)τ(τ|∫∞
∞-dRx
• 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 )])([
(lim=)( 1
2
∞→ T
fXf
T
TxP methoddirect
• 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,
54)-(6 (0) xP)( 2∞∞-∫ xx Rdff =P
55)-(6 )()0( ∫ ∞∞- dRxx 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 )()(
nsn 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:
68)-(6 mnknn aaaaR(k) • By truncating x(t) we get:
)()(
N
NnsnT nTtfatx
• Where T/2=(N+1/2)Ts, its Fourier transform is:57)-(6 )()]([)}({)(
N
Nn
nTjn
N
NnsnTT
seafFnTtfatxfX FF
17
6.2 Power Spectral Density
• According to the definition of PSD, we get:
58)-(6 )()12(
12lim
|)(|
)()12(
1lim|)(|
1lim|)(|
|)(|1
lim )(1
lim)(
∞→
∞→
2
)(
∞→
2
22
∞→
2
∞→
nN
nNk
Tjk
sNs
N
Nn
nN
nNk
Tjk
sN
N
Nn
N
Nm
Tnmjmn
T
N
Nn
nTjn
TT
Tx
s
s
s
s
ekRTN
N
T
fF
ekRTN
fF
eaaT
fF
eafFT
fXT
f
P
• Thus:70b)-(6 )(
|)(|)(
k
Tjk
sx
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
sx
ss
ss
s
ekRekRRT
fF
ekRekRRT
fF
ekRT
fFf
P
70a)-(6 )2cos()(2)0( |)(|
)(1
ks
sx kfTkRR
T
fFf P
• Thus an equivalent expression of PSD is:
• Where the autocorrelation of the data is:
70c)-(6 )( 1
I
iiiknnknn PaaaaR(k)
• In which Pi 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 frequen
cy 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
222n
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 )(|)(|)(|)(|
)( |)(|
)|(|
)(
spectrum
22
spectrum
22
222
222
discrete
na
continuous
a
naa
s
k
Tjkaa
sx
nDfnDFDmfFD
nDfDmT
fF
emT
fFf 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
70e)-(6 )(|)(|)()(|)(|)(
spectrum discrete
22
spectrum continuous
22
naax nDfnDFDmffFDf WP
70f)-(6 )()( 2
k
kfTj sekf W
• is a spectral weight function obtained form the Fourier transform of the normalized autocorrelation impulse train
)()(
kskTk
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:
71)-(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 N
Rx
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
2var;0 mean22 2/
2
1)(
tetf
)(2
)()(
)(2
)(
02
20
NR
fN
f
n
n
P
24
6.2 Power Spectral Density• Bandpass White Gaussian Noise : n(t) is a (narrow) bandpass r
andom process (random signal) of 2BHz, while the baseband signal is BHz)
00122 2)()0(var;0 BNNffRmean n
22 2/
2
1)(
nenf
)2
2cos(2
2sin2)(
,0
,2/)(
120
210
ff
B
BBNR
otherwise
fffNf
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.
42)-(6 )])([
(lim)(
2
∞→ T
fXf
T
Tx P
)()()(or ])([
1)(
2
TRFffX
Tf 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:
82b)-(6 )(*)(*)()(
or
82a)-(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:
154)-(6 )(
)(20
20
tn
ts
N
S
out
• 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)/<n0
2(t)>, and that has a transfer function given by:
155)-(6 )(
)()( 0
*tj
n
ef
fSKfH 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()( 20
20 0
P• Then:
156)-(6 )(|)(|
|)()(|
)(
)(2
2
20
20
0
dfffH
dfefSfH
tn
ts
N
S
n
tj
out P
• With the aid of Schwarz inequality:157)-(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:
)(
)()( and )()()(
0
f
efSfBffHfA
n
tj
nP
P
• then:
159)-(6 )(
)(
)(|)(|
)(
)()()(
-
2
2
22
dff
fS
dfffH
dff
fSdfffH
N
S
n
n
nn
out
P
P
PP
• 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 FiltersResults for White Noise
• For white noise, Pn(f)=N0/2, thus we get:
0)(2
)( *
0
tjefSN
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 :
161)-(6 2
)(2
2/
)(
0-
2
0-
0
2
N
Edtts
Ndf
N
fS
N
S s
out
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 2TW )(
)/( 2
0 in
s
out N
S
WN
TETW
N
S
34
6.8 Matched Filters
• Example 6-11 Integrate-and-Dump (Matched) filter
t
t
t
t
( )S t
( )S t
0 ( )S tT
T
0.75T
0.5T
0.25T
1 t 2t
1- t2-t
0t0t T
1
1
2T
1
T
t1
a) Input signal
b) “backwards” signal
c) matched-Filter impulse response
d) Signal output of matched filter
t0==t2h(t)=s(t0-t)
35
6.8 Matched Filters
• Fig.6-17
Waveform at A(input signal
and noise)
Waveform at B
Waveform at C
Waveform at D
Integrator reset to zero initial condition at
clocking time
Integrator Reset
Sample and Hold
DD
Clocking signal(bit sync)
BD CD AD
r(t)=s(t)+n(t) r0(t) output
36
6.8 Matched Filterscorrelation 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) (Known signal Reference input)
0
0)()(
t
Ttdttstr r(t)=s(t)+n(t) r0(t0)
167)-(6 )()()( 0
000 t
Ttdttstrtr
Where 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
000000 t
Ttdthrthtrtr
Because of h(t)=Cs(t0-t) (6-160)
otherwhere
Tt0
0
)()( 0 tts
th
so
0
0
0
0
0
0
))()())()(
))(()()( 0000
t
Tt
t
Tt
t
Tt
dttstrdsr
dttsrtr
This is over
38
6.8 Matched Filters
• Example 6-12 Matched Filter for Detection 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
Delay
T Delay
T Delay
T Delay
T
a11 a21 a
31 aN1
r0(t)=s0(t)+n0(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:
168)-(6 ))1(()(
))1(()2()()()(
1000
003020100
N
kk
N
Tktsatsor
TNtsaTtsaTtsatsats
• Similarly, the output noise at time t=t0 is:
169)-(6 ))1(()(1
0
N
kk Tktnatn
• The average noise power is:
170)-(6 )(
))1(())1(()(
1 1
1 1
20
N
k
N
llk
N
k
N
lklk
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
10
)(
)(20
020
N
k
N
llk
N
kk
tn
ts
lTkTRaa
Tktsa
• Using Lagrange’s method of maximizing the numerator while constraining the denominator to be a constant, we can get:
177)-(6 or
174)-(6 )())1((1
0
Ras
N
kk 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
181)-(6 1sRa
43
6.8 Matched Filters(Transversal Matched Filter)
44
6.8 Matched Filters(Transversal Matched Filter)
45
Homework