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					ELEC 303 – Random Signals

 Lecture 21 – Random processes
       Dr. Farinaz Koushanfar
      ECE Dept., Rice University
            Nov 19, 2009
               Lecture outline
•   Basic concepts
•   Gaussian processes
•   White processes
•   Filtered noise processes
•   Noise equivalent bandwidth
           Things to remember
• Stationary
  – A random process is stationary if time shift does
    not affect its properties
  – For all T, and for all sets of sample times, (t0,…,tn),
    P(X(t0)x0,…,X(tn)xn) = P(X(T+t0)x0,…,X(T+tn)xn)
  – Stationary random processes have constant mean,
    defined as E[X(t)] = mX
  – For stationary RPs, autocorrelation depends on
    the time difference between the samples
  – RX(t1,t2)=E[X(t1)X(t2)] = RX(=t1-t2)
            Exact definition WSS
• A process is wide sense stationary if its expected
  power is finite |E[X2(t)|<, its mean is constant,
  and its autocorrelation depends only on the time
  difference between samples
• WSS processes: stationary in 1st and 2nd moment
• Stationary processes are WSS, but not vice versa
• Power spectral density (PSD)
   – Defined only for WSS processes
   – The Fourier transform of the autocorrelation function
   – Expected power is the integral of the PSD
              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
                  White processes
  Definition 3: A random process X(t) is a called a white
  process if it has a flat spectral density, i.e., SX(f) is constant
  for all f

• White processes are those where all frequency
  components appear with equal power
• Thermal noise can be modeled as a white noise
  over a wide range of frequencies
• A wide range of information sources can be
  modeled as the output of LTI systems driven by a
  white process
        Power of a white process
• SX(f)=C, (C is a constant), then


• Obviously, no real physical process can have an
  infinite power
• Thus, the white process is not a meaningful
  physical process.
• Quantum mechanical analysis of natural noise
  shows it has a power spectral density given by
             White processes
• Quantum mechanical analysis of natural noise
  shows it has a power spectral density given by
                   White noise

• Thermal noise, though not precisely white, can be
  modeled as a white process for all practical purposes
• PSD is Sn(f) = kT/2 (denoted by N0) = N0/2
• Autocorrelation Rn() = -1[N0/2]=N0/2 (t)
• For all 0, we have RX()=0
• Thus, two samples of noise at t1 and t2 will be
  uncorrelated
• If the RP is white and Gaussian, any pair of RVs X(t1)
  and X(t2) are independent for t1t2
                        Example 1
• A stationary RP passes through a quadrature filter defined
  by h(t)=1/t
• What are the mean and autocorrelation functions of the
  output?
• What is the cross correlation between input and output?




  Using the fact that     and that RX() has no DC component.
       Properties of thermal noise
•   Thermal noise is a stationary process
•   Thermal noise has a zero mean process
•   Thermal noise is a Gaussian process
•   Thermal noise is a white process with a power
    spectral density Sn(f) = kT/2
    – Thermal noise increases with increasing ambient
      temperature, cooling circuits lowers the noise
            Filtered noise process
• In many cases, the noise in one stage of the process gets
  filtered by a bandpass filter
• Frequency of bandpass is fc, away from zero
• The bandpass filters can be expressed in terms of the
  inphase and quadrature components:
   – E.g., single frequency signal is an extreme case:
   – x(t) = A Cos(2fct + ) = A Cos()Cos(2fct)–A Sin() Sin(2fct)
         = xc Cos(2fct) - xs Sin(2fct)       {Phasor: Aej = xc + j xs}
   – More generally: x(t) = xc(t) Cos(2fct) - xs(t) Sin(2fct)
   – In phase component:         xc(t) = A(t) Cos ((t))
   – Quadrature component: xs(t) = A(t) Sin ((t))
              Bandpass Filter
• X(t) is the output of an ideal bandpass filter of
  bandwidth W centered at fc
• Examples:
                   Filtered noise
• Filtered thermal noise is Gaussian but not
  white                                               2
                              For ideal filter:|H(f)| =H(f)


• Power spectral density:
• For the examples on the last slides,
      Filtered noise components
• All filtered noise signals have in-phase and
  quadrature components that are lowpass, i.e.,
      X(t) = Xc(t) Cos(2fct) - Xs(t) Sin(2fct)
• In-phase and quadrature components:
  – Xc(t) and Xs(t) are zero-mean, low pass, jointly
    stationary, and jointly Gaussian random processes
  – If the power in process X(t) is PX, then the power
    in each of the processes Xc(t) and Xs(t) is also PX
         Properties of Xc and Xs
• Both have a common amplitude
  – Shifting the positive frequencies to the left by fc
  – Shifting the negative frequencies to the right by fc
  – If H1(f) and H2(f) are used, then
  – P1=4WN0/2=2N0W,              P2=2WN0/2=N0W
     Noise equivalent bandwidth
• A white Gaussian noise passing through a
  filter would be Gaussian but not white
• We have        SY(f) = SX(f)|H(f)|2=.5 N0|H(f)|2
• We have to integrate SY(f) to get the power



• Define Bneq, the noise equivalent bandwidth
                            Hmax is the maximum of |H(f)| in the
                            Filter’s passband
      Noise equivalent bandwidth
                             Hmax is the maximum of |H(f)| in the
                             Filter’s passband




• Thus, given Bneq, finding the output noise becomes a
  simple task
• The of filters and amplifiers are usually given by the
  manufacturers
                  Example
• Find the noise equivalent bandwidth of a low
  pass filter
                                   =RC
    Summary: Gaussian processes
• X(t) is a Gaussian process if Yg=0T g(t) X(t) dt is
  Gaussian for any T and function g
• Linear filtering of a Gaussian process results in a
  Gaussian process
• Samples of a Gaussian process are jointly
  Gaussian random variables
• Uncorrelated samples of a Gaussian process are
  independent
         Summary: white noise
• White noise is defined as a WSS random
  processes with a flat PSD: Sn(f) = N0/2
• The autocorrelation of white noise is N0/2 (t)
• White noise is the most random form of noise
  since it decorrelates randomly!




               http://www.stanford.edu/class/ee179/multi/lecture16-multi.pdf
        Summary: filtered noise
• Filtered thermal noise is Gaussian but not white
• The bandpass filters can be expressed in terms of
  the inphase and quadrature components
      x(t) = xc(t) Cos(2fct) - xs(t) Sin(2fct)
  – In phase component: xc(t) = A(t) Cos ((t))
  – Quadrature component: xs(t) = A(t) Sin ((t))
• Define Bneq, the noise equivalent bandwidth

				
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