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# Digital Signal Processing Homework 10

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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
(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
π
( )
π

ω
ω
– π
d Η
2
3
π π π
π
3
π
3 2
π
2 –
3

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
(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
 Digital Signal Processing Homework 10
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