7/27/2019 DSP_Test1_2005

1/2

1

Discrete-Time Signal Processing Test 1 Fall 2005 Close Book

2005 10 17 120

1. (10%) Consider an LTI system with the frequency response H(ejw). Show thatwhen a sinusoidal input x[n] = Asin[w0n+] is applied to this system, the output is

also a sinusoidal sequence with the same frequency w0. {Hint: You may need to

represent the output in terms of the magnitude response |H(ejw)| and phase

response H(ejw), and use the symmetry property that the Fourier transform of

any real sequence is conjugate symmetric, i.e., H(ejw) = H*(ejw).}

2. (a) (5%) What is the initially-at-rest condition of a constant-coefficient differenceequation?

(b) [Fibonacci number] Consider the difference equation

y[n] = y[n1] + y[n2] + x[n1].

Assume that it is initially at rest.

(b1) (10%) Write down the frequency response of this system.

(b2) (10%) Is this system stable? Explain your answer.

(b3) (10%) A Fibonacci sequence f[n] is recursively computed by

[ ] [ ] [ ]

=

=+

=

00

11221

n

nnnfnf

nf

Show that the impulse response of the above system is the Fibonacci

sequence.

3. Consider a moving-difference system as follows:[ ] ( ) [ ]

=

=0

3

14

1

k

kknxny

(a) (5%) Is this system FIR or IIR?(b)(5%) Is this system causal? Explain your answer.(c) (10%) Write down the system function (in z-transform) of this system. List

all the poles and zeros of the system function.

(d)(5%) Show that the ROC of the system function is the entire z-plane exceptat z = 0 orz = .

7/27/2019 DSP_Test1_2005

2/2

2

(e) (5%) Write down the magnitude response of this system. Evaluate themagnitude response at w = 0, (1/4), (1/2), (3/4), , and draw a

rough sketch of the magnitude response within the range [, ].

( 41412 . )

(f) (5%) According to the magnitude response of this system, is this systemapproximate to a high-pass or a low-pass filter? Explain your answer.

4. (a) (5%) Consider a real-valued signal x[n] with frequency response X(ejw). Showthat the frequency response of the time-reversed signal x[n] is X*(ejw).

(b) (10%) Let a FIR system with system function H(z) be given as follows:

Assume that the impulse response h[n] is real-valued. Consider a cascading

system having the real-valued input x[n] and the output y[n]. The output is

obtained by setting s[n] = v[n] andy[n] = g[n].

Question: Find the magnitude and phase responses of the cascading system.

5. (5%) Determine the inverse z-transform of the following:

( ) 1

1

311

3

11

+

=

z

z

zX

, x[n] is a right-sided sequence.

H(z)s[n] g[n]H(z)x[n] v[n]