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• 1 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing

NOTES ON

STATIONARY RANDOM PROCESS AND DIGITAL SIGNAL PROCESSING

Prepared by Le Thai Hoa

2004

• 2 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing

1. STATIONARY RANDOM PROCESS

1.1. Basic concepts (1) Continuous random process:

  )}()}...(),(),({)( 321 txtxtxtxtx Kk  ,  t Where: { }: Ensemble of sample functions xk(t)

k: Index of sample function (k=1,2,3…K)

t: Time variable

Random process {xk(t)} = Ensemble of sample function xk(t)

(2) The random process is called as the K-variate random process (multi-variate

random process)

Ensemble (sample records) of random signal

(3) For discrete sample function, discrete values of any sample random function are

measured at certain time points t1, t2, t3, … tN (N: number of sampling values of

sample function)

t

t

t t+

x1(t)

kth sample function

1st sample function

xk(t) Time shift 

• 3 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing

)}()}...(),(),()( 321 Nkkkkk txtxtxtxtx  : Discrete sample function

1.2. Classification of random process (1) Classification of random process can be widely expressed as follows

Classification of random processes

1.3. Representation of random process: (1) Time-domain representation (as raw formats and sources)

(2) Frequency-domain representation (due to Fourier Transform)

(3) Time-frequency representation (due to Wavelet Transform)

1.4. Characteristics of random process

Basic statistical characteristics of two arbitrary random processes  )(txk and  )(tyk (1) Mean value (Expectation): First-order statistical moment

 

  dttxN

LimtxEt kNkx )( 1)]([)(

Random process or Stochastic field

Stationary process

Non-stationary processes

Ergodic process

Non-ergodic signals

Gaussian process

Non-Gaussian process

• 4 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing

 

  dttyN

LimtyEt kNky )( 1)]([)(

(2) Variance and covariance: Second-order moment

)()]([]))()([()( 2222 ttxEttxEt kkxkx   : Variance

)()]([]))()([()( 2222 ttyEttyEt ykyky   : Variance

)(*)()](*)([ ))]()())(()([()(

tttytxE ttyttxEtC

yxkk

ykxkxy







 : Covariance

))]()())(()([()(   ttxttxEC xkxkxx : Covariance

))]()())(()([()(   ttyttyEC ykykyy : Covariance

Note: Zero mean value process xk(t): 0)( tx

]))([()( 22 txEt kx  : Variance

]))([()( 22 tyEt ky  : Variance

)()0( 2 tC xxx 

)()0( 2 tC yyy 

(3) Mean square and root mean square

)]([)( 2 txEtC kxx  : Mean square

)]([)( 2 tyEtC kyy  : Mean square

Note: Zero mean value process xk(t): 0)( tx

• 5 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing

Variance )(2 tx = Mean Square Cxx(t)

(4) Correlations: Second-order moment

)](*)([)(   txtxER kkxx : Autocorrelation

)](*)([)(   tytyER kkyy : Autocorrelation

)](*)([)(   tytxER kkxy : Cross-correlation

 : arbitrary time (time shift or time lag)

Note: xxxxx RC 2)()(  

yyyyy RC 2)()(  

yxxyxy RC   )()(

Zero mean random process: 0)( tx , 0)( ty

)()(  xxxx RC 

)()(  yyyy RC 

)()(  xyxy RC 

(5) Correlation coefficients

)( )(

)( )()(

2

 

 

xx

xxx

xx

xx xx R

R R C 

 : Auto-correlation coefficient

)( )(

)( )(

)( 2

 

 

 yy

yyy

yy

yy yy R

R R C 

 : Auto-correlation coefficient

)( )(

)( )(

)( 

  

 xy

yxxy

xy

xy xy R

R R C 

 : Cross-correlation coefficient

• 6 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing

x

xx

xx

xx xx

C R C

2 )(

)( )()(

 

  

y

yy

yy

yy yy

C R C

2

)( )( )(

)( 

  

 

yx

xy

xy

xy xy

C R C

 

 

 )(

)( )(

)( 

Note 1:

i) )()(  xxxx RC 

ii) )0()( xxxx CC 

iii) )0()( xxxx RR 

iv) 2)0( xxxC  and

2)0( yyyC 

v) )0(*)0(|)(| 2 yyxxxy CCC 

vi) 222|)(| yxxyC  

vii) )0(*)0(|)(| 2 yyxxxy RRR 

viii) )()(  xxxx RR  and )()(  yyyy RR 

)()(  yxxy RR 

Note 2: 0

i) 2)0( xxxC  : Variance

• 7 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing

ii) 2)0( yyyC  : Variance

iii) 22 )]([)0( xxx txER  : Mean square

iv) 22 )]([)0( yyy tyER  : Mean square

(6) Power spectral density (PSD) function in frequency-domain

1.5. Power spectral density (PSD) PSD function can be computed by following methods: i) Via correlation function (by

definition), ii) Via Fourier transform and iii) Via filter-squaring-averaging computation

(1) Spectra via correlation (by Fourier Transform of correlation)

By definition of spectral density through the Fourier Transform:

  deRfS fjxxxx 2*)()( 

 : Auto-spectral density function

  deRfS fjyyyy 2*)()( 

 : Auto-spectral density function

  deRfS fjxyxy 2*)()( 

 : Cross-spectral density function

Inverse Fourier Transform:

dfefSR fjxxxx  2*)()( 

  : Auto-correlation

dfefSR fjyyyy  2*)()( 

  : Auto-correlation

dfefSR fjxyxy  2*)()( 

  : Cross-correlation

• 8 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing

Sxx(f), Syy(f), Sxy(f): Two-sided spectra, f[-,]

One-sided spectral densities

 dfRdfRfS xxxxxx 2cos*)(22cos*)()( 0 

 

 

 0

2cos*)(2)(  dffSR xxxx

Changing the two-sided spectral density Sxx(f) with f[-,] to the one-sided spectral

density Gxx(f) with f[0,]

)(2)( fSfG xxxx 

)(2)( fSfG yyyy 

)(2)( fSfG xyxy 

Thus,

 dfRfG xxxx 2cos*)(4)( 0 

 ; f[0,]

dfffGR xxxx  2cos*)()( 0 

 ; f[0,]

Real part and imaginary part of one-sided cross-spectral density:

f(Hz)

Spectra

[-,0] [0,]

Gxx(f)=2Sxx(f): One-sided

Sxx(f): Two-sided

• 9 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing

)()(*)(2)( 2 fiQfCdeRfG xyxy fj

xyxy  

  

)( fCxy : Co-spectrum

Writing in standard form: )(|)(|)( fjxyxy xyefGfG



Where:

)()(|)(| 22 fQfCfG xyxyxy 

)( )(

tan)( 1 fC fQ

f xy

xy xy



)(cos|)(|)( ffGfC xyxyxy 

)(sin|)(|)( ffGfQ xyxyxy 

dfffiQffCR xyxyxy ]2sin)(2cos)([)( 0    

dffCR xyxy  

 0

)()0(

(2) Spectra via Fourier transform

Fourier Transform (Kinchint-Weiner’s pair):

dtetxfX ftjkk   

0

2)()( 

dfefXfx ftjkk  

 0

2)()( 

• 10 | Le Thai Hoa - Notes on Stationary Random Process and Digital Signal Processing

Spectral density function:

]|)([|12)( 2fXE T

LimfG kTxx 

]|)([|12)( 2fYE T

LimfG kTyy 

)](*)([12)( fYfXE T

LimfG kkTxy 

1.5. Coherence Coherence plays the same role as the correlation coefficient. The correlation coefficient

is expressed in time domain, whereas the coherence in frequenc

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