Linear Shift Invariant System with Random Inputs



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Response of Linear time-invariant system to WSS input

In many applications, physical systems are modeled as linear time invariant (LTI) systems. The dynamic behavior of an LTI system to deterministic inputs is described by linear differential equations. We are familiar with time and transform domain (such as Laplace transform and Fourier transform) techniques to solve these equations. In this lecture, we develop the technique to analyze the response of an LTI system to WSS random process.


The purpose of this study is two-folds:


  1. Analysis of the response of a system

  2. Finding an LTI system that can optionally estimate an unobserved random process from an observed process. The observed random process is statistically related to the unobserved random process. For example, we may have to find LTI system (also called a filter) to estimate the signal from the noisy observations.


Basics of Linear Time Invariant Systems:
A system is modeled by a transformation T that maps an input signal to an output signal y(t). We can thus write,



Linear system

The system is called linear if superposition applies: the weighted sum of inputs results in the weighted sum of the corresponding outputs. Thus for a linear system




Example : Consider the output of a linear differentiator, given by

Then,



Hence the linear differentiator is a linear system.


Linear time-invariant system
Consider a linear system with y(t) = T x(t). The system is called time-invariant if

It is easy to check that that the differentiator in the above example is a linear time-invariant system.


Causal system

The system is called causal if the output of the system at depends only on the present and past values of input. Thus for a causal system




Response of a linear time-invariant system to deterministic input

A


LTI

system



x(n)



linear system can be characterised by its impulse response where is the Dirac delta function.
Recall that any function x(t) can be represented in terms of the Dirac delta function as follows

If x(t) is input to the linear system y(t) = T x(t), then




Where is the response at time t due to the shifted impulse .

If the system is time invariant,



Therefore for a linear time invariant system,



where * denotes the convolution operation.


Thus for a LTI System,







Taking the Fourier transform, we get


Response of an LTI System to WSS input
Consider an LTI system with impulse response h(t). Suppose is a WSS process input to the system. The output of the system is given by

where we have assumed that the integrals exist in the mean square (m.s.) sense.


Mean and autocorrelation of the output process
The mean of the output process is given by,

where is the frequency response at 0 frequency () given by


Therefore, the mean of the output process is a constant
The Cross correlation of the input X(t) and the out put Y(t) is given by











Thus we see that is a function of lag only. Therefore, and are jointly wide-sense stationary.

The autocorrelation function of the output process Y(t) is given by,



Thus the autocorrelation of the output process depends on the time-lag , i.e., .

Thus



The above analysis indicates that for an LTI system with WSS input

  1. the output is WSS and

  2. the input and output are jointly WSS.

The average power of the output process is given by




Power spectrum of the output process

Using the property of Fourier transform, we get the power spectral density of the output process given by



Also note that

Taking the Fourier transform of we get the cross power spectral density given by





Example:
(a) White noise process with power spectral density is input to an ideal low pass filter of band-width B. Find the PSD and autocorrelation function of the output process.


The input process is white noise with power spectral density .


The output power spectral density is given by,



The output PSD and the output autocorrelation function are illustrated in Fig. below.





Example 2:
A random voltage modeled by a white noise process with power spectral density is input to an RC network shown in the fig.
Find (a) output PSD

(b) output auto correlation function

(c) average output power

The frequency response of the system is given by






Therefore,

(a)


(b) Taking inverse Fourier transform



(c) Average output power




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