Random Process

1
Random Process

• Linear Filtering of Random Processes
• Lowpass and Bandpass Processes

2
LTI System and Random Process

• The relation between input and output ???
• Transfer function of LTI system
• Output random process Y(t)
• Note
• If X(t) is Gaussian, Y(t) is also Gaussian with
  change of mean and covariance

3
Mean and covariance of Y(t)

• Mean
• Covariance

Interchange E and ?? h(.) is deterministic mx
EX(t) Replace t-? to ? H(0) H(f)f0
4
Power Spectrum of Y(t)

• Power spectrum
• Illustrative Problem 2.6
• X(t) is white random process
• Flat Power spectrum Sx(f)
• Impulse autocorrelation Rx(?)
• The output Y(t)
• See figure 2.15 and 2.16 at page 65 and 66

5
Discrete time case

• Convolution sum
• Input X(n), output Y(n) and impulse response h(n)
• Transfer Function

6
Mean and covariance of Y(n)

• Mean
• Covariance

7
Spectral density of Y(n)

• Spectral density of Y(n)
• Where

8
Homework

• Illustrative Problem 2.8
• Find autocorrelation function using inverse
  FFTSy(f), and compare it with the result of
  Illustrative problem 2.3 (Figure 2.8)
• Gauss-Markov process is output process of the LTI
  system with White process input

9
Lowpass Processes

• Definition of Lowpass
• A random process is called lowpass if most of its
  power concentrated at low frequencies
• Definition of Bandlimited
• A lowpass random process X(t) is bandlimited if
  the power spectrum Sx(f)0 for fgtB
• B is called Bandwidth of the random process

10
Illustrative Problem 2.9

• Generating lowpass random process
• Lowpass Filter TF and difference equation
• Compare Autocorrelation and Power spectrum of
  input and output process
• Figure 2.18, 2.19 at page 71 and 72

11
Definition of bandpass

• A random process is called bandpass if its
  spectrum is large in a band of frequencies
  centered in the neighborhood of central frequency
  ?f0 and relatively small outside of this band
  frequencies
• A random process is called narrowband if its
  bandwidth B ltlt f0

12
Modulation and Bandpass

• Information bearing signal
• Usually lowpass random process
• Modulation translate lowpass random process to
  bandpass random process

13
Using In-phase and Quadrature

• See Lowpass equivalent of Bandpass signals in
  chapter 1
• Theorem
• If X(t) is zero-mean stationary random process,
  the process Xc(t), Xs(t) are also zero-mean,
  jointly stationary process

14
Autocorrelation of Xc(t), Xs(t)

• Rc(?) Rs(?)
• Where
• Cross correlation
• Relations between Rx(?), Rc(?), Rcs(?)

15
Homework

• Illustrative Problem 2.10
• This form is widely used in communication system
• Modulation / Demodulation
• Quiz