ECE 310 Digital Signal Processing

Homework 10

1. The transfer functions of three LSI systems are given below. For each system, determine if it is an

FIR or an IIR filter (justify your answer).

(a) H(z)=1+ z−1 + 7z−6

(b) H(z) = z2 + 3z + 2

z + 1

(c) H(z) = z + 1

z2 + 3z + 2

2. Derive the transfer function and the corresponding difference equation for the following block diagram

Z

-1

1/2

-3

-1/4

1/3

x(n) y(n)

-1 Z

3. Draw a Direct Form I block diagram for the system in Problem 2.

4. Draw a block diagram implementation (in direct form I and II, respectively) of the system described

by

y[n + 1] − 2y[n] − 5y[n − 1] = x[n]+8x[n + 1] − 2x[n − 1]

5. The frequency response of a GLP filter can be expressed as Hd(ω) = R(ω)ej(α−Mω) where R(ω) is

a real function. For each of the following filters, determine whether it is a GLP filter. If it is, find

R(ω), M, and α, and indicate whether it is also a linear phase filter.

(a) {hn}2

n=0 = {2, 1, 2}

(b) {hn}2

n=0 = {1, 2, 3}

(c) {hn}2

n=0 = {−1, 3, 1}

(d) {hn}4

n=0 = {1, 1, 1, −1, −1}

(e) {hn}2

n=0 = {1, 0, −1}

(f) {hn}3

n=0 = {2, 1, 1, 2}

In each case, the remaining terms of the unit pulse response of the filter are zero.

6. Given the following phase response ∠Hd(ω) of a generalized linear-phase FIR filter, answer the

following questions. Explain you answers.

π

( )

π

–

ω

ω

– π

d Η

2

3

π π π

π

3

π

3 2

π

2 –

3

2π

π

2

π

2 –

(a) Is the filter (i) type-1 GLP, (ii) type-2 GLP, or (iii) neither type-1 GLP nor type-2 GLP?

(b) Determine the filter length from the phase plot.

(c) Can you characterize the filter as (i) possibly low-pass, (ii) possibly high-pass, (iii) neither highpass nor low-pass, or (iv) the given information is insufficient to make any of the preceding

statements? (Specify all correct answers).

(d) Determine Hd(π

2 ).

7. The frequency response of a length-N symmetric or antisymmetric FIR filter with unit pulse response

h[n] can be expressed as

Hd(ω) = R(ω)ej(α−( N−1

2 )ω).

For ONE of the following, show that

(a) for symmetric h[n] with N even,

R(ω)=2

N

2 −1

n=0

h[n] cos

ω

N − 1

2 − n

(b) for symmetric h[n] with N odd,

R(ω) = h

N − 1

2

+ 2

N−3

2

n=0

h[n] cos

ω

N − 1

2 − n

(c) for antisymmetric h[n] with N even,

R(ω)=2

N

2 −1

n=0

h[n] sin

ω

N − 1

2 − n

(d) for antisymmetric h[n] with N odd,

R(ω)=2

N−3

2

n=0

h[n] sin

ω

N − 1

2 − n