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**Unformatted text preview: **The Cooper Union Department of Electrical Engineering
Prof. Fred L. Fontaine ECE211 Signal Processing & Systems Analysis
Problem Set III: Discrete-Time Systems
February 21, 2017 1. Given the following transfer function: (2 — 2) (32 — 1) 11(2): 10(2z — 3)2 (42 + 1) (a) Specify the poles and zeros, with multiplicity, including any at 00. (b) Sketch the poles and zeros. Then indicate all possible ROCS (on the sketch, and
also specify them mathematically). (c) Identify the ROC, if any, associated with the following: 1. H (2:) represents the transfer function of a direct form 11 transposed digital
ﬁlter structure. 2. H (2) represents a stable system. 3. H (z) is associated with a well—deﬁned frequency response.
(d) Find the conditions on L such that zLH (2) does not have an ROC associated
with a causal system. 2. An FIR ﬁlter has impulse response h [n] with support 0 _<_ n S 3 given by:
h = {3,4, —1, 5} (a) Specify the length and order of the ﬁlter. (b) Express h [n] as a superposition of impulses. (c) Write explicit formulas for H (z) and H (w). ((1) Write a difference equation with input an, output 3). (e) Sketch a transversal ﬁlter realization. 3. Given the following discrete-time transfer function: 522 + 3;: — 1
H = K
(2) 10::2 + z + 6
(a) Write a difference equation with input 3:, output y.
(b) Sketch a direct form II realization. (c) Sketch a direct form II transposed realization. 4. Consider the system given in the previous problem. Suppose it is operated at a
sampling rate of 10M H z. The input signal is :c(t) = 64/7 cos(27rf0t)u(t)
7‘ = 0.2,usec and f0 = 2MHz. (a) Use impz to compute the impulse response h [n] for 0 g n S 49. Draw a stern
plot (discrete—time, do not label the time axis in seconds). (b) Use tf2zp to compute the poles and zeros, and zplane to sketch them. (c) Use freqz to compute (not sketch) the frequency response at 1000 equally spaced
points from DC to the Nyquist bandwidth. Then plot the frequency responsck
magnitude response in dB, phase response unwrapped in degrees, frequency axis
in Hz (use subplot to put the magnitude and phase responses in the same ﬁgure). (d) Create a block of 50 samples of a: (t). Use filter to compute the ﬁrst 50 samples
of the output signal y (t) Use subplot to show stem plots for Mn] and y [n]
for 0 S n S 49 (i.e., discrete—time graphs, with discrete—time indexing, i.e., not
continuous-time) . 5. Refer to Figure 5, which shows a DTFT X (to). (Actually it is just the magnitude
spectrum, we are not concerned with the phase here). Sketch each of the following (by
hand, on separate axes) in the range —27r S w s 271'. (Yes, —27r to 27r). (21 )X (w )
(b) X (2w) Note: This 18 the spectrum of y— — (T 2):c.
( (C) X 00/2) (d )X (W/2 + 7T) (e) % [X (cu/2) + X (LU/2 + 7r)] . Note: This is spectrum of y = (l 2) :1: (f) Neither (0) nor (d) can possibly represent the DTFT of any discrete—time signal.
Why not? 6. Let y = (T 2) 2:. Show that Y (z) = X (22) by using the z-transform formula. From
this, show directly that Y (w) = X (2w), i..e., do not use the DTFT formula, instead
go from the z—transform to the DTFT. Q) Hp F Ska an \rnpuksg; ms? ‘32 M vﬁhfmppw} 0:511:35 Bum l=§3.kr1.53
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