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					ELEC 303 – Random Signals

 Lecture 20 – Random processes
       Dr. Farinaz Koushanfar
      ECE Dept., Rice University
            Nov 11, 2010
               Lecture outline
•   Basic concepts
•   Random processes and linear systems
•   Power spectral density of stationary processes
•   Power spectra in LTI systems
•   Power spectral density of a sum process
•   Gaussian processes
           RP and linear systems
• When a RP passes a linear time-invariant system the
  output is also a RP
• Assuming a stationary process X(t) is input, the linear
  time-invariant system with the impulse response h(t),
  output process Y(t)
• Under what condition the output process would be
  stationary?
• Under what conditions will the input/output jointly
  stationary?
• Find the output mean, autocorrelation, and
  crosscorrelation

             X(t)        h(t)         Y(t)
     Linear time invariant systems
• If a stationary RP with mean mX and autocorrelation
  function RX()
• Linear time invariant (LTI) system with response h(t)
• Then, the input and output process X(t) and Y(t) will be
  jointly stationary with




               X(t)         h(t)         Y(t)
            The response mean
• Using the convolution integral to relate the
  output Y(t) to the input X(t), Y(t)=X()h(t-)d




          This proves that mY is independent of t

             X(t)         h(t)          Y(t)
               Cross correlation
• The cross correlation function between output
  and the input is




       This shows that RXY(t1,t2) depends only on =t1-t2
              Output autocorrelation
   • The autocorrelation function of the output is




This shows that RY and RXY depend only on =t1-t2,
 Output process is stationary, and input/output are jointly stationary
Power spectral density of a stationary
              process
• If the signals in the RP are slowly varying, then
  the RP would mainly contain the low
  frequencies in its power concentration
• If the signal changes very fast, most of the
  power will be concentrated at high frequency
• The power spectral density of a RP X(t) is
  denoted by SX(f) showing the strength of the
  power in RP as a function of frequency
• The unit for SX(f) is Watts/Hz
      Wiener-Khinchin theorem
• For a stationary RP X(t), the power spectral
  density is the Fourier transform of the
  autocorrelation function, i.e.,
                 Example 2
• Randomly choose a phase  ~ U[0,2]
• Generate a sinusoid with fixed amplitude (A)
  and fixed freq (f0) but a random phase 
• The RP is X(t)= A cos(2f0t + )
• From the previous lecture, we know
                   Example 3
• X(t)=X
• Random variable X~U[-1,1]
• In this case

• Thus,


• For each realization of the RP, we have a different
  power spectrum
           Power spectral density
• The power content of a RP is the sum of the powers at
  all frequencies in that RP
• To find the total power, need to integrate the power
  spectral density across all frequencies


• Since SX(f) is the Fourier transform of RX(), then RX()
  will be the inverse Fourier transform of SX(f), Thus


• Substituting =0, we get
                 Example 4
• Find the power in the process of example 2
  Translation to frequency domain
• For the LTI system and stationary input, find the translation
  of the relationships between the input/output in frequency
  domain


• Compute the Fourier transform of both sides to obtain



• Which says the mean of a RP is its DC value. Also, phase is
  irrelevant for power. Only the magnitude affects the power
  spectrum, i.e., power dependent on amplitude, not phase
                   Example 5
•   If a RP passes through a differentiator
•   H(f)=j2f
•   Then, mY=mX H(0) = 0
•   Also, SY(f) = 42 f2 SX(f)
Cross correlation in frequency domain
• Let us define the cross spectral density SXY(f)


• Since RYX() = RXY(-), we have


• Although SX(f) and SY(f) are real nonnegative
  functions, SXY(f) and SYX(f) can generally be
  complex functions
                 Example 6
• Randomly choose a phase  ~ U[0,2]
• Generate a sinusoid with fixed amplitude (A)
  and fixed freq (f0) but a random phase 
• The RP is X(t)= A cos(2f0t + )
• The X(t) goes thru a differentiator H(f)=j2f
                  Example 7
• X(t)=X
• Random variable X~U[-1,1]
• If this goes through differentiation, then

           SY(f) = 42 f2 ((f)/3) = 0
           SXY(f) = -j2f ((f)/3) = 0
     Power spectral density of a sum
                process
• Z(t) = X(t)+Y(t)
• X(t) and Y(t) are jointly stationary RPs
• Z(t) is a stationary process with
        RZ() = RX() + RY() + RXY() + RYX()
• Taking the Fourier transform from both sides:
        SZ(f) = SX(F) + SY(f) + 2 Re[SXY(f)]
• The power spectral density of the sum process is the sum
  of the power spectral of the individual processes plus a
  term, that depends on the cross correlation
• If X(t) and Y(t) are uncorrelated, then RXY()=mXmY
• If at least one of the processes is zero mean, RXY()=0, and
  we get: SZ(f) = SX(F) + SY(f)
                    Example 8
•   X(t)=X
•   Random variable X~U[-1,1]
•   Z(t) = X(t) + d/dt X(t), then
•   SXY(f) = jA2f0 /2 [(f+f0) - (f-f0)]
•   Thus,
•   Re[SXY(f)] = 0
•   SZ(f)= SX(f)+SY(f) = A2(1/4+2f02)[(f+f0)+(f-f0)]
              Gaussian processes
• Widely used in communication
• Because thermal noise in electronics is produced
  by the random movement of electrons closely
  modeled by a Gaussian RP
• In a Gaussian RP, if we look at different instances
  of time, the resulting RVs will be jointly Gaussian:
  Definition 1: A random process X(t) is a Gaussian process if
  for all n and all (t1,t2,…,tn), the RVs {X(ti)}, i=1,…,n have a
  jointly Gaussian density function.
      Gaussian processes (Cont’d)
Definition 2: The random processes X(t) and Y(t) are jointly
Gaussian if for all n and all (t1,t2,…,tn), and (1,2,…,m)the
random vector {X(ti)}, i=1,…,n, {Y(j}, j=1,…,m have an n+m
dimensional jointly Gaussian density function.

• It is obvious that if X(t) and Y(t) are jointly
  Gaussian, then each of them is individually
  Gaussian
• The reverse is not always true
• The Gaussian processes have important and
  unique properties
   Important properties of Gaussian
              processes
• Property 1: If the Gaussian process X(t) is
  passed through an LTI system, then the output
  process Y(t) will also be a Gaussian process.
  Y(t) and X(t) will be jointly Gaussian processes
• Property 2: For jointly Gaussian processes,
  uncorrelatedness and independence are
  equivalent

				
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posted:4/18/2013
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