ii - random signals/ .ii - random signals/ processes ... random processes are characterized by joint

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  • 11/10/12 EC3500.FallFY13/MPF - Section II 1

    II - Random Signals/ Processes [p. 3] Random signal/sequence definition [p. 4] Signal mean, variance, autocorrelation & autocovariance, normalized cross-correlation [p. 15] Statistical characterization of random signals

    I.I.D. Random process Stationarity

    Wide sense stationarity (wss)

    Jointly wide sense stationarity (jointly wss)

    Correlation & cross-correlation for stationary RPs

    Signal average

    Ergodicity

    [p. 34] Specific examples of RPs:

    Binary signal, white noise, colored noise, Bernoulli process, Random walk, counting process, Poisson process, MA process

    [p. 75] Periodic Random process Definition [p. 82] Uncorrelated Random process Definition [p. 83] Cyclostationary Random process Definition

    [p. 85] Multiple Random Processes Joint Properties [p. 90] Application to data analysis

    How to assess signal stationarity [p. 94] Application to data analysis -

    How to check IID assumption--

    Autocorrelation --

    Lag plot [p. 98] Application to data analysis -

    How to extract an IID sequence out of an non IID sequence [p. 105] Application: target range detection [p. 107] Introduction to the spectrogram [p. 111] Application: Gas furnace reaction time [p. 114] Application: Detection of the periodicity of stationary signals in noisy environments [p. 116] How to estimate correlation lags; biased/unbiased estimator issues [p. 125] Frequency domain description for a stationary process

    Power spectral density (PSD), definition & properties

    Cross Power spectral density , definition & properties [p. 146] Appendices [p. 170] References

  • 11/10/12 EC3500.FallFY13/MPF - Section II 2

    Examples

    [p. 5] Example 1

    [p. 7] Example 2

    [p. 9] Example 3

    [p. 24] Example 4

    [p. 29] Example 5

    [p. 32] Example 6

    [p. 34] Example 7

    [p. 43] Example 8

    [p. 50] Example 9

    [p. 64] Example 10

    [p. 67] Example 11

    [p. 70] Example 12

    [p. 76] Example 13

    [p. 78] Example 14

    [p. 80] Example 15

    [p. 87] Example 16

    [p. 93] Example 17

    [p. 97] Example 18

    [p. 104] Example 19

    [p. 110] Example 20

    [p. 114] Example 21

    [p. 115] Example 22

    [p. 121] Example 23

    [p. 128] Example 24

    [p. 129] Example 25

    [p. 130] Example 26

    [p. 132] Example 27

  • 11/10/12 EC3500.FallFY13/MPF - Section II 3

    Consider sequence x(t) =x(t,)

    for a fixed t, x(t) is a Random Variable (RV)

    x(t): random signal ( can be infinite dimensional) x(t,) for fixed RV : called realization/trial of the random process

    if x(t)is

    discrete for a fixed , the RP x(n)= x(nT, )is called a discrete

    random process

    Random Process Signal/Sequence:

    A RP is a mapping function that attributes a function x(t) = x(t,) to each outcome of the random experiment

    x(t, 1

    )

    x(t, 3

    )

    x(t, 2

    )

    1

    2

    3

    t

    t

    t

  • 11/10/12 EC3500.FallFY13/MPF - Section II 4

    4 Types of Random Processes

    Discrete time / discrete valued

    Discrete time/ continuous valued

    Continuous time/ discrete valued

    Continuous time / continuous valued

  • 11/10/12 EC3500.FallFY13/MPF - Section II 5

    Example 1: x(t)=x(t,) = cos(t/10), where = U[0,1].

    x(n)=x(nT, ) = cos(t/10+), where = U[0,2].

  • 11/10/12 EC3500.FallFY13/MPF - Section II 6

    Signal mean value (ensemble average):

    Signal variance: ( ) ( ){ }( ){ } ( )

    22

    2 2

    ( )x x

    x

    t E x t m t

    E x t m t

    =

    =

    n1n

    x[n]

    n2

    2 1n n= Lag

    (dimensionless)

    ( ) ( ){ }xm t E x t=

    Discrete process case Continuous process case

    t1t

    x(t) 2 1t t = Time lag (sec)

    t2

  • 11/10/12 EC3500.FallFY13/MPF - Section II 7

    x(t,) = cos(t/10), where = U[0,1].Example 2

    Compute process mean and variance

  • 11/10/12 EC3500.FallFY13/MPF - Section II 8

    Signal autocorrelation function:

    ( ) ( ) ( ) ( ){ }*1 2 1 2 1 2, ,xx xR t t R t t E x t x t= =measures the dependency between 2 values of the process at two different times.

    Allows to evaluate: 1) How quickly a random signal changes with respect to time.2) The amount of memory

    a signal may have.

    2) Whether the process has a periodic component and what the expected frequency might be, etc

    t1t

    x(t) 2 1t t = lag

    t2

  • 11/10/12 EC3500.FallFY13/MPF - Section II 9

    Let x(t) be a real valued process defined as

    Compute mx (t)Compute: Rx (t1 ,t2 )

    Example 3

    x(n, 1

    )

    x(n, 3

    )

    x(n, 2

    )

    1

    2

    3

    x(t,) = cos(t/10+), where = U[0,2].

  • 11/10/12 EC3500.FallFY13/MPF - Section II 10

  • 11/10/12 EC3500.FallFY13/MPF - Section II 11

  • 11/10/12 EC3500.FallFY13/MPF - Section II 12

    ( ) ( )( ) ( ) ( )( ){ }( ) ( )

    *

    1 2 1 1 2 2

    *

    1 2 1 2

    ( , )

    ( , )

    xx x x

    x x x

    C t t E x m x m

    R t t m m

    t t t t

    t t

    =

    =

    Signal autocovariance function (remove impact due to process mean):

    Signal normalized correlation function (remove impact due to process mean and normalizes max value to 1):

    ( ) ( )( ) ( )1 2

    1 21 2

    ,, xx

    x x

    C t tt t

    t t

    = ( )1 2| , | 1 !!x t t

  • 11/10/12 EC3500.FallFY13/MPF - Section II 13

    Signal cross-correlation function:

    Signal cross-covariance function:

    ( ) ( ) ( ) ( )*1 2 1 2 1 2, ,xy xy x yC t t R t t m t m t=

    ( )1 2,xyR t t =

    Measures the dependency between values of two processes at two different times.

    Allows to evaluate whether/how two processes are related

    Similar to cross-correlation function: measures the dependency between values of two processes at two different times,

    but also

    Removes

    impact of mean values.

    Note: unless there is a good reason to keep the signal

    means, remove or use covariance based

    expressions!

  • 11/10/12 EC3500.FallFY13/MPF - Section II 14

    Normalized cross-correlation function:

    ( ) ( )( ) ( )1 2

    1 21 2

    ,, xyxy

    x y

    C t tt t

    t t

    = ( )1 2| , | 1 !!xy t t

  • 11/10/12 EC3500.FallFY13/MPF - Section II 15

    Random processes are characterized by joint distribution (or density) of samples

    Fx (x1

    , x2

    , , xk , t1

    ,, tk ) = Pr [x(t1

    )

    x1

    ,

    x(tk )

    xk ]

    F(.) is highly complex to compute -

    difficult or impossible to obtain in practice

    Statistical Characterization of Random Processes:

  • 11/10/12 EC3500.FallFY13/MPF - Section II 16

    Independent, Identically Distributed (I.I.D.) Random Process:

    A Random Process is said to be:

    An independent process (i.e., independent of itself at earlier and/or later times) if for any ti :

    Fx (x1

    , x2

    ,,xk ;t1 ,,tk ) = F1 (x1 ;t1 )Fk (x2 ;tk )

    An I.I.D. process if all RVs have the same pdf fx (x)Note: I.I.D. processes have no memory, where a future value would depend on past values they can be viewed as building blocks for more realistic random processes

    Mean of I.I.D. Process:

    mx (t) = E{x(t)} =

  • 11/10/12 EC3500.FallFY13/MPF - Section II 17

    Independent, Identically Distributed (I.I.D.) RP, cont

    Autocovariance of IID RP:

    { }{ } { }

    { }

    *1 2 1 1 2 2

    *1 1 2 2 1 2

    21 1 1 2

    ( , ) ( ( ) ( ))( ( ) ( ))

    ( ( ) ( )) ( ( ) ( )) ,

    | ( ( ) ( )) | ,

    =

    x x x

    x x

    x

    C t t E x t m t x t m t

    E x t m t E x t m t t t

    E x t m t t t

    =

    = =

    Autocorrelation of IID RP:

    Rx (t1

    , t2

    ) =

  • 11/10/12 EC3500.FallFY13/MPF - Section II 18

    ( , ) 0.05* , ~ (0,1)x t t

    N

    = +

    I.I.D. process ?

    [ ( , )]E x t

  • 11/10/12 EC3500.FallFY13/MPF - Section II 19

    Stationarity Concept:

    If x(t) is stationary for all orders N = 1, 2, x(t) is said to be strict-sense stationary.

    If x(t) is stationary for order N = 1,

    Stationary up to order 2 wide-sense stationary (WSS).

    ( ; ) ( ; ) x xf x t f x t T = +

    Pdf is identical for all times instants t

    Definition: a RP is said to be stationary if any joint density or distribution function depends only on the spacing between time instants, not

    where on the timeline the time instants occur.

    fx (x1

    , , xN ; t1

    , , tN ) = fx (x1

    , , xN ; t1

    +T ,, tN +T) for any ti , T & any joint pdf

  • 11/10/12 EC3500.FallFY13/MPF - Section II 20

    Stationarity of order N=1 - Physical interpretation for a discrete process

    ...

    ...

    ...

    n

    n

    n

    x(n, 1

    )

    x(n, 3

    )

    x(n, 2

    )

    ...

    ...

    n

    n

    n

    x(n, 4

    )

    ... x(n, P

    )

    x(n, 5

    )

    Experiment is performed P timesleads to P time sequences

    How to computeFx (x1

    ; n1

    ) = Pr [x(n1

    )

    x1

    ][Probability that the functions x(n,) do not exceed x1 at time n1 ]

    Select values for x1 and n1

    Count the number of trials K for which x(n1 ) x1

    Fx (x1 ; n1 ) = Pr [x(n1 ) x1 ]= K/P

    x1

    x1

    x1

    x1

    x1

    x1

    n1[11]

  • 11/10/12 EC3500.FallFY13/MPF - Section II 21

    Stationarity of order N=2 - Physical interpretati

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