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  • EE4601Communication Systems

    Week 4Ergodic Random Processes, Power Spectrum

    Linear Systems

    0 c2011, Georgia Institute of Technology (lect4 1)

  • Ergodic Random Processes

    An ergodic random process is one where time averages are equal to ensembleaverages. Hence, for all g(X) and X

    E[g(X)] =

    g(X)pX(t)(x)dx

    = limT

    1

    2T

    T

    Tg[X(t)]dt

    = < g[X(t)] >

    For a random process to be ergodic, it must be strictly stationary. However, notall strictly stationary random processes are ergodic.

    A random process is ergodic in the mean if

    < X(t) >= X

    and ergodic in the autocorrelation if

    < X(t)X(t+ ) >= XX()

    0 c2011, Georgia Institute of Technology (lect4 2)

  • Example (contd)

    Recall the random process

    X(t) = A cos(2fct+)

    where A and fc are constants, and is a uniformly distributed random phase.

    p() =

    1/(2) , 0 2

    0 , elsewhere

    The time average mean of X(t) is

    < X(t) >= limT

    1

    2T

    T

    TA cos(2fct+ )dt = 0

    In this example X(t) =< X(t) >, so the random process X(t) is ergodic in themean.

    N.B. Make sure you understand the difference between the time average andensemble average.

    0 c2011, Georgia Institute of Technology (lect4 3)

  • Example (contd)

    The time average autocorrelation of X(t) is

    < X(t)X(t+ ) > = limT

    1

    2T

    T

    TA2 cos(2fct+ 2fc + ) cos(2fct+ )dt

    = limT

    A2

    4T

    T

    T[cos(2fc) + cos(4fct+ 2fc + 2)] dt

    =A2

    2cos(2fc)

    The random process X(t) is ergodic in the autocorrelation.

    It follows that the random process X(t) in this example is ergodic in the meanand autocorrelation.

    0 c2011, Georgia Institute of Technology (lect4 4)

  • Example

    Consider the random process shown below.

    X (t) = a P = 1/4

    P = 1/4

    X (t) = 0 P = 1/2

    X (t) = -a

    1 1

    22

    3 3

    0 c2011, Georgia Institute of Technology (lect4 5)

  • Example (contd)

    For this example, the ensemble and time average means are, respectively,

    X = E[X(t)] = 0

    X(t) =

    a with probability 1/40 with probability 1/2

    a with probability 1/4

    Hence, X(t) is not ergodic in the mean.

    The ensemble and time average autocorrelations are

    XX() = E[X(t)X(t+ )] = a2(1/4) + 0(1/2) + (a)2(1/4) = a2/2

    X(t)X(t + ) =

    a2 with probability 1/20 with probability 1/2

    Hence, X(t) is not ergodic in the autocorrelation.

    0 c2011, Georgia Institute of Technology (lect4 6)

  • Example (contd)

    Note that

    E[X(t)] = X

    E[X(t)X(t+ )] = XX()

    Because of this property X(t) and X(t)X(t+ ) are said to provide unbiasedestimates of X and XX(), respectively.

    0 c2011, Georgia Institute of Technology (lect4 7)

  • Power Spectral Density

    The power spectral density (psd) of a random process X(t) is the Fourier trans-form of its autocorrelation function, i.e.,

    XX(f) = =

    XX()e

    j2fd

    XX() =

    XX(f)e

    j2fdf .

    We have seen that XX() is real and even. Therefore, XX(f) = XX(f)

    meaning that XX(f) is also real and even.

    The total power (ac + dc), P , in a random process X(t) is

    P = E[X2(t)] = XX(0) =

    XX(f)df

    a famous result known as Parsevals theorem.

    0 c2011, Georgia Institute of Technology (lect4 8)

  • Example

    X(t) = A cos(2fct+)

    where A and fv are constants and

    p() =

    12

    ,

    0 , elsewhere

    We have seen before that

    XX() =A2

    2cos(2fc)

    Hence,

    XX(f) =A2

    2F [cos(2fc)]

    =A2

    4((f fc) + (f + fc))

    0 c2011, Georgia Institute of Technology (lect4 9)

  • Properties of XX(f )

    1. XX(0) = XX()d

    2. 0+0 XX(f)df = dc power

    3. XX(0) =XX(f)df = total power

    4. XX(f) 0 for all f . Power is never negative.

    5. XX(f) = XX(f) (even function) if X(t) is a real random process.

    6. XX(f) is always real.

    0 c2011, Georgia Institute of Technology (lect4 10)

  • Discrete-time Random Processes

    Consider a discrete-time real random process Xn, that consists of an ensembleof sample sequences {xn}.The ensemble mean of Xn is defined as

    Xn = E[Xn] =

    xnfXn(xn)dxn

    The ensemble autocorrelation of Xn is

    XX(n, k) = E[XnXk] =

    XnXkfXn,Xk(xn, xk)dxndxk

    For a wide-sense stationary discrete-time real random process, we have

    Xn = X , n

    XX(n, k) = XX(n k)

    From Parsevals theorem, the total power in the process Xn is

    P = E[X2n] = XX(0)

    0 c2011, Georgia Institute of Technology (lect4 11)

  • Power Spectrum of Discrete-time RP

    The power spectrum of the real wide-sense stationary discrete-time random pro-cess Xn is the discrete-time Fourier transform of the autocorrelation function,

    i.e.,

    XX(f) =

    n=XX(n)e

    j2fn

    XX(n) = 1/2

    1/2XX(f)e

    j2fndf

    Observe that the power spectrum XX(f) is periodic in f with a period of unity.In other words XX(f) = XX(f + k), for k = 1,2, . . . This is a character-

    istic of any discrete-time sequence. For example, one obtained by sampling acontinuous-time random process Xn = x(nTs), where Ts is the sample period.

    0 c2011, Georgia Institute of Technology (lect4 12)

  • Linear Systems

    Yt t

    XX

    XX

    YY

    YY

    f

    t( )h H( )fX( ) ( )

    ( )

    ( )

    ( )f( )

    0 c2011, Georgia Institute of Technology (lect4 13)

  • Linear Systems

    Suppose that the input to the linear system h(t) is a wide sense stationary ran-dom process X(t), with mean X and autocorrelation XX().

    The input and output waveforms are related by the convolution integral

    Y (t) =

    h()X(t )d .

    Hence,Y (f) = H(f)X(f) .

    The output mean is

    Y =

    h()E[X(t )]d = X

    h()d = XH(0) .

    This is just the mean (dc component) of the input signal multiplied by the dcgain of the filter.

    0 c2011, Georgia Institute of Technology (lect4 14)

  • Linear Systems

    The output autocorrelation is

    Y Y () = E[Y (t)Y (t+ )]

    = E[

    h()X(t )d

    h()X(t+ )d

    ]

    =

    h()h()E [X(t )X(t+ )] dd

    =

    h()h()XX( + )dd

    =

    h()

    h()XX( + )dd

    ={

    h()XX( + )d

    }

    h()

    = h() XX() h() .

    Taking transforms, the output psd is

    Y Y (f) = H(f)XX(f)H(f)

    = |H(f)|2XX(f) .

    0 c2011, Georgia Institute of Technology (lect4 15)

  • Cross-correlation and Cross-covariance

    If X(t) and Y (t) are each wide sense stationary and jointly wide sense stationary,then

    XY (t, t+ ) = E[X(t)Y (t+ )] = XY ()

    XY (t, t+ ) = XY () = XY () xy

    The crosscorrelation function XY () has the following properties.

    1. XY () = Y X()

    2. |XY ()| 12[XX(0) + Y Y (0)]

    3. |XY ()|2 XX(0)Y Y (0) if X(t) and Y (t) have zero mean.

    0 c2010, Georgia Institute of Technology (lect4 17)

  • Example

    Consider the linear system shown in the previous example. The crosscorrelationbetween the input process X(t) and the output process Y (t) is

    Y X() = E[Y (t)X(t+ )]

    = E[

    h()X(t )dX(t+ )

    ]

    =

    h()E [X(t )X(t+ )] d

    =

    h()XX( + )d

    = h() XX()

    The cross power spectral density is

    Y X(f) = H(f)XX(f)

    Note also that

    Y X() = XY ()

    0 c2011, Georgia Institute of Technology (lect4 17)

  • Example

    R

    CX(t) Y(t)

    0 c2011, Georgia Institute of Technology (lect4 18)

  • Example

    The transfer function of the filter is

    H(f) =1

    1 + j2fRC

    Suppose that XX() = e| |. What is Y Y ()?

    We haveY Y (f) = |H(f)|

    2XX(f)

    where

    |H(f)|2 =1

    1 + (2fRC)2

    XX(f) =2

    2 + (2f)2

    Hence,

    Y Y (f) =1

    1 + (2fRC)2

    2

    2 + (2f)2

    0 c2011, Georgia Institute of Technology (lect4 19)

  • Example

    Do you remember partial fractions? Now you need them!We write

    Y Y (f) =A

    2 + (2f)2+

    B

    1 + (2fRC)2

    and solve for A and B. We have

    A(1 + (2fRC)2) + B(2 + (2f)2) = 2

    Clearly,

    A+ B2 = 2

    A(2fRC)2 +B(2f)2 = 0

    From the second equation

    A = B

    (RC)2= B2

    where = 1/(RC).

    0 c2011, Georgia Institute of Technology (lect4 20)

  • Example

    Then using the first equation

    B =2

    2 2

    Also,

    A = B2 = 22

    2 2

    Finally,

    Y Y (f) =2

    2 2

    2

    2 + (2f)2+

    2 2

    2

    2 + (2f)2

    Now take inverse Fourier transforms to get

    Y Y () =2

    2 2 e| | +

    2 2 e| |

    0 c2011, Georgia Institute of Technology (lect4 21)

  • Discrete-time Random Processes

    Consider a wide-sense stationary discrete-time random process Xn that is inputto a discrete-time linear time invariant filter with impulse response hn.The transfer function of the filter is

    H(f) =

    n=hne

    j2fn

    The output of the filter is the convolution sum

    Yk =

    n=hnXkn

    It follows that the output mean is

    Y = E[Yk] =

    n=hnE[Xkn]

    = X

    n=hn

    = XH(0)

    0 c2011, Georgia Institute of Technology (lect4 22)

  • Discrete-time Random Processes

    The autocorrelation function of the output process is

    Y Y (k) = E[YnYn+k]

    =

    i=

    j=

    hihjE[XnihjXn+kj]

    =

    i=

    j=

    hihjXX(k j + i)]

    By taking the discrete-time Fourier transform of Y Y (k) and using the above

    relationship, we can obtain

    Y Y (f) = XX(f)|H(f)|2

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