stationary random process and digital signal processing ...uet.vnu.edu.vn/~thle/stationary random...

Download Stationary random process and Digital signal processing ...uet.vnu.edu.vn/~thle/Stationary random process

Post on 29-Mar-2020

2 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

  • 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

    )( fQxy : Quadratic 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

Recommended

View more >