1 chapter 6 random processes and spectral analysis

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1 Chapter 6 Random Processes and Spectral Analysis

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Page 1: 1 Chapter 6 Random Processes and Spectral Analysis

1

Chapter 6

Random Processes and Spectral Analysis

Page 2: 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.

Page 3: 1 Chapter 6 Random Processes and Spectral Analysis

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.

Page 4: 1 Chapter 6 Random Processes and Spectral Analysis

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.

Page 5: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 6: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 7: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 8: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 9: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 10: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 11: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 12: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 13: 1 Chapter 6 Random Processes and Spectral Analysis

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)

Page 14: 1 Chapter 6 Random Processes and Spectral Analysis

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)

Page 15: 1 Chapter 6 Random Processes and Spectral Analysis

15

6.2 Power Spectral Density

• Example 6-3: (notebook p)

Page 16: 1 Chapter 6 Random Processes and Spectral Analysis

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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

Page 17: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 18: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 19: 1 Chapter 6 Random Processes and Spectral Analysis

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.

Page 20: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 21: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 22: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 23: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 24: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 25: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 26: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 27: 1 Chapter 6 Random Processes and Spectral Analysis

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.

Page 28: 1 Chapter 6 Random Processes and Spectral Analysis

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.

Page 29: 1 Chapter 6 Random Processes and Spectral Analysis

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)

Page 30: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 31: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 32: 1 Chapter 6 Random Processes and Spectral Analysis

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.

Page 33: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 34: 1 Chapter 6 Random Processes and Spectral Analysis

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)

Page 35: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 36: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 37: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 38: 1 Chapter 6 Random Processes and Spectral Analysis

38

6.8 Matched Filters

• Example 6-12 Matched Filter for Detection of a BPSK signal

Page 39: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 40: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 41: 1 Chapter 6 Random Processes and Spectral Analysis

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)

Page 42: 1 Chapter 6 Random Processes and Spectral Analysis

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

Page 43: 1 Chapter 6 Random Processes and Spectral Analysis

43

6.8 Matched Filters(Transversal Matched Filter)

Page 44: 1 Chapter 6 Random Processes and Spectral Analysis

44

6.8 Matched Filters(Transversal Matched Filter)

Page 45: 1 Chapter 6 Random Processes and Spectral Analysis

45

Homework