Discrete-Time Linear Shift Invariant System with WSS Random Inputs

Discrete-time Linear Shift Invariant System with WSS Random Inputs

Discrete-time Linear Shift Invariant System with Deterministic Inputs

We have seen that the Dirac delta function plays a very important role in the analysis of the response of the continuous-time LTI systems to deterministic and random inputs. Similar role in the case of the discrete-time LTI system is played by the unit sample sequence defined by

Any discrete-time signal can be expressed in terms of as follows:

A discrete-time linear shift-invariant system is characterized by the unit sample response which is the output of the system to the unit sample sequence

The DTFT of the unit sample response is the transfer function of the system and given by

The transfer function in terms of is given by

where is a function of the complex variable It is defined on a region of convergence (ROC) on the

An analysis similar to that for the continuous-time LTI system can be applied to the discrete-timeLTI system. Such an analysis shows that the response of a the linear time-invariant system with impulse response to a deterministic input is

By taking the DTFT of both sides, we get

More generally, we can take the z-transform of the input and the response and show that

Remark

• If the LTI system is causal, then

In this case, the ROC of is a region in the given by For example, suppose

Then,

• Similarly, if the LTI system is anti-causal, then

In this case, the ROC of is a region in the given by

• The contour is called the unit circle. Thus represents evaluated on the unit circle.
• can be expressed as the ratio of two polynomials in

The polynomials and helps us in analyzing the properties of a linear system in terms of the zeros and poles of defined by

Zero- the point in the where at such a point.

Pole- the point in the where at such a point. The ROC of does not contain any pole.

• For the stability of the LTI system, the unit-sample response should decay to zero as A necessary and sufficient condition for the stabilityof a discrete-time LTI system is that all its poles lie strictly inside the unit circle.
• A discrete-time LTI system is called a minimum- phase system if all its poles and zeros lie inside the unit circle. A minimum-phase system is always stable as its poles lie inside the unit circle. Because the zeros of the system lie inside the unit circle, the inverse system with a transfer function will have all its poles inside the unit circle and be stable.
• A discrete-time LTI system is called a maximum- phase system if all its poles and zeros lie outside the unit circle.

Response of a discrete-time LTI system to WSS input

Consider a discrete-time linear time-invariant system with impulse response and input as shown in Fig. Assume to be a WSS processwith meanand autocorrelation function

The output random process is given by

Given the WSS input the output process is also WSS. We establish this result in the following section.

Mean and Autocorrelation of the output

The mean of the output is given by

where is the dc gain of the system given by

Thus the mean of the output process is constant. We write

The cross-correlation between the output and the input random processes is given by

.

Thus does not depend on , but on lag and we can write

The autocorrelation function of the out put is

.

is a function of lag only and we write

The mean-square value of the output process is

Thus if is WSS then is also WSS.

Taking the DTFT of we get

In terms of we get

Notice that if is causal, then is anti-causal and vice versa.

Similarly if is minimum-phase then is maximum-phase.

Remark

Finding the probability density function of the output process is a difficult task. However, if is a WSS Gaussian random process, then the output process is also Gaussian with the probability density function determined by its mean and the autocorrelation function.

Example

Suppose and is a zero-mean unity-variance white-noise sequence. Then

By partial fraction expansion and inverse transform, we get

Remark

• Though the input is an uncorrelated process, the output is a correlated process.
• For the same white noise input, we can generate random processes with different autocorrelation functions or power spectral densities.

Spectral factorization theorem

Consider a discrete-time LTI system with the transfer function and the white noise sequence as the input random process as shown in the Fig. below.

Then

We have seen that is the product of a constant and two transfer functions This result is of fundamental importance in modeling a WSS process because of the spectral factorization theorem stated below:

If is an analytic function of and, then

where

is the causal minimum phase transfer function

is the anti-causal maximum phase transfer function

and a constant and interpreted as the variance of a white-noise sequence.

Thus a WSS random signal with continuous spectrum that satisfies the Paley Wiener condition can be considered as an output of a linear filter fed by a white noise sequence.

Innovation sequence

Figure Innovation Filter

Minimum phase filter => the corresponding inverse filter exists.

Proof of the spectral factorization theorem

Since is analytic in an annular region that includes the unit circle,

where is the order cepstral coefficient.

For a real signal

and

and are both analytic

Therefore, is a minimum phase filter.

Similarly let

where

Remarks

• Note that of a real process is a function of. Therefore, is a function of Consider rational spectrum so that where and are polynomials in If is a root of so is Thus the roots of are symmetrical about the unit circle groups the poles and zeros inside the unit circle and groups the poles and zeros outside the unit circle.
• can be factorized into a minimum-phase and a maximum-phase factors i.e. and
• In general spectral factorization is difficult, however for a signal with rational power spectrum, spectral factorization can be easily done.
• Since is a minimum phase filter, the inverse filter exists and stable. Therefore we can have a filter to filter the given signal to get the innovation sequence.
• and are related through an invertible transform; so they contain the same information.

Example

Suppose the power spectral density of a discrete random sequence is given by

Then

Wold’s Decomposition

Any WSS signal can be decomposed as a sum of two mutually orthogonal processes

• a regular process and a predictable process ,
• can be expressed as the output of linear filter using a white noise sequence as input.
• is a predictable process, that is, the process can be predicted from its own past with zero prediction error.