A causal and linear time-invariant filter has the output...

A causal and linear time-invariant filter has the output $Y(z) = \frac{1 - z^{-1}}{(1 - 3z^{-1})(1 - 2z^{-1})}$ if the input is given by $x[n] = -\frac{1}{3}(\frac{1}{4})^n u[n] - \frac{5}{3}(2)^n u[-n + 1]$. (a) Find the z-transform of x[n]. (b) What are the possible choices for the region of convergence of Y(z)? (c) What are the possible system transfer functions H(z) and regions of convergence? (d) What are the possible impulse responses of the system?

Transcript

00:01 So, here in this question we are considering about a casual system which is represented by the following differential equation that is y of n plus 0 .25 of y of n minus 1 is equals to x of n plus 0 .5 of x of n minus 1.

00:16 So, this is the equation we are given here.

00:18 In the first part, we have to find out the system function of hg giving the corresponding result of the correspond.

00:24 So, here we can apply the z transform here.

00:29 So, that from here is equals to y of z plus 0 .25 of z inverse y of z is equals to x of z plus 0 .5 of z inverse which is multiplied by the x of z bar.

00:44 So, the value of y of z multiplied by the 1 plus 0 .25 of z inverse become equals to x of z which is multiplied by the 1 plus 0 .5 of z inverse.

00:55 So, we can say that h of g become equals to y of z which is divided by x of z that become equals to 1 plus 0 .5 of z inverse that is divided by 1 plus 0 .25 of z inverse.

01:09 So, the value of hg from here is equals to z plus 0 .5 which is divided by z plus 0 .25.

01:17 This is the value of the hg from here and we are considering about we have to find the reason of the convergence.

01:23 So, let's say we are considering z from here is greater than 0 .25 is the reason of convergence.

01:29 So, this is the answer to the part a of the question.

01:31 Now, we are considering about the part b where we are considering about the unit step response.

01:37 So, unit step response is what we are considering here.

01:40 So, here we can say that x of z is equals to z which is divided by z minus 1...

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