EEL5701: Exam 1

1.

(25 points)

Consider the following system:

$$y[n]= (1-\alpha)y[n-1] + \alpha x[n]$$

(a)

(5 points) Compute the impulse response (h[n]) of this system.

(b)

(5 points) For what values of $\alpha$ is this a BIBO stable system?

Consider the following system:

y[n]= (1-x[n])y[n-1] + (x[n])²

(c)

(5 points) Compute the impulse response of this system.

(d)

(5 points) Compute the response of the system to $x[n]=\delta[n]+\delta[n-2]$

(e)

(5 points) Is this system BIBO stable? Justify why or why not.

2.

(25 points)

You are given the following N-point digital sequence x[n]: $1,-1,1,-1,1,-1,1,-1 \ldots$ where N is even and $0 \le n \le N-1$Answer the following questions

(a)

(5 points) Compute X[0] (the first point of the DFT of x[n])

(b)

(5 points) Compute $\sum_{k=0}^{N-1} \vert X[k]\vert^2$

(c)

(15 points) Compute the DFT X[k] for all values of k where $0 \le k \le N$.
Show all of your work!

3.

(25 points)

Consider

$$X(z)=\frac{z^2-4z+3}{z^2-6z+8}$$

(a)

(5 points) Draw a pole-zero diagram for X(z).

(b)

(10 points) Compute the partial fraction expansion of X(z).

(c)

(5 points) Compute the inverse z-transform of X(z)as a right-sided sequence x[n].

(d)

(5 points) Compute the inverse z-transform of X(z) as a double-sided sequence x[n].

4.

(25 points) Short Answer

(a)

(5 points) Compute the following convolution:

x[n]=aⁿu[n] * bⁿu[n]

Assume $a \neq b$. Express your answer in simplest form, e.g., you shouldn't require any summations in your answer.

(b)

(5 points) Derive the 4-point DFT (X(k)) of the following length-4 sequence

x[n]=+1,+1,-1,-1.

(c)

(5 points) Sketch the magnitude of the DTFT ( $\vert X(e^{j \omega}\vert$) of

$$x[n]=\delta[n-1] - 2\delta[n] + \delta[n+1]$$

(d)

(5 points) Assuming $Y(e^{j\omega})=\vert X(e^{j\omega})\vert^2$, express y[n] as a function of x[n].

(e)

(5 points) Compute the DTFT ( $X(e^{j \omega}$) for the infinite sequence given by

x[n]=...,+1,+1,-1,-1,+1,+1,-1,-1,...

Show all of your work.