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ELEC 3004 / 7312 – Digital Linear Systems: Signals & Controls
2015
Problem Set 3: Filters (Digital & Analog) Total points:
50
Due Date (extended):
May 5, 2015 (at 11:59pm, AEST) 2
Note: This assignment is worth
12% of the final course mark. Please submit answers via
Platypus
. It is requested that solutions, including equations, should be typed please. The final grade is the median of the marks from the peer reviews and the staff (with provisions for review). Finally, the tutors will
not assist you further unless there is real evidence you have attempted the questions. Thank you. :) Questions Explain your solutions as if you are trying to
teach a peer
. Demonstrate your insight and understanding. For all questions,
please justify your solutions
. Answers alone are not sufficient. At all times please (note: remember to justify your solution). Answering an entire question with bare equations, lone numbers or without any verbal explanation is not acceptable. Although a rubric will be provided to guide peer marking, peers may reduce marks if they believe an answer is of poor quality, demonstrates little effort or significant misunderstanding. Furthermore, if a solution demonstrates understanding in a way not anticipated by the rubric, peers may also award higher marks than would be awarded by strict adherence to the rubric. In other words, you are free to use your judgment when marking your peers. Please explain and show reasoning/work
. (as in any University course, answers alone are not sufficient). Q1. Dancing Around Poles and Filters
[10 points] Consider the filter functions: −1)
H a(z) = (1−a)(1+z
2(1−az−1) −1
)
1
H b(z) = (1+b)(1−z
(1+b)(1z
)
term in
H
b]
2(1−bz−1) [note the
where the filters are assumed stable (ie, |z|>|a| and |z|>|b| respectively) a. Please draw a Pole/Zero Plot for H
and H
. a
b
b. Which one is a lowpass filter (LPF) and highpass filter (HPF)? (please explain) c. What is the order of these filter functions? (please explain) d. Is this filter FIR? or IIR? 1
⇒ [ Correction (24/4/2015): H
had a typo in the numerator. It should be (1+b)(1z
) ] ⇐ b
⇒ [Update (30/4/2015]: The |z|>|a| and |z|>|b| just means the filter is stable] ⇐
Q2. Towards Digital Filters
[10 points] Consider the simple firstorder FIR filter H (z) = 1 + z−1 a. Show that this is a discrete time lowpass filter [
hint:
what is H (ejω) ?] b. Plot the magnitude and phase response for this filter [
hint:
plot this on the ω axis] c. What is the magnitude of this filter at DC (0 Hz) and where is the magnitude equal to 1? d. Is the response periodic? Please explain. [
hint:
Consider Problem Set 2, Question 2] ⇒ [ Correction (28/4/2015): hint should reference PS2, Question 2 not PS2Q3 ] ⇐ 1 Q3. A Simple FIR Filter
[10 points] A lowpass filter is desired with the following specifications: ● Passband cutoff: 150 Hz ● Stopband: > 250 Hz ● Sampling rate: 1 kHz ● Stopband attenuation: >50 dB a. What window function should be used for designing the filter? (Please explain) b. Given your selection in (a), what order is required? c. Is this the lowest possible order required to achieve the required specification? (Please explain) Q4. A PoorMan's FIR Jacket
[10 points] A poor, but common, technique for smoothing a noisy data sequence is to average the sequence over the M+1 adjacent data samples, such that: M
1
y[n] = M+1
∑ x[n − k] k=0
Thus the output
y[n]
is the mean of the input
x[n]
and the preceding M samples (
x[n1], x[n2], …, x[nM]
). Hence the name, a moving simple average filter. It has the following impulse response: M
1
1
h[n] = M+1
∑ δ[n − k] = M+1
(u[n] − u[n − M − 1]) k=0
a. What is the system function in the
z
domain, H(z)? b. What are the zeros and poles of the system function? (
hint:
use the solution from part (a)). c. For
M=[The last digit of your student number]+1
, please plot an impulse and polezero plot for the simple average filter (
hint/note:
please be sure to explicitly state the value of M picked, but
not
your student number). d. Plot and comment on the magnitude and phase response for this filter (relative to normalized frequency, π on the frequency axis (i.e., for ω = [π, π])) (
hint:
set z = ejω , remember to label your plots and include units) 2 Q5. More FIRmly Filtered
[10 points] One of the advantages of the a nonrecursive filter is that it can represent a diverse range of filter types. Let’s consider the design of a multiband FIR filter. 4
For a sampling rate of 2 kHz (i.e. T=5×10
s, or a Nyquist range of 0 to 1000 Hz),
design

an th
n order nonrecursive (FIR) filter to
keep
frequencies from:
200300 Hz
,
500600 Hz
and 800900 Hz
where the
nth
order is a value from the set of 20 to 200. As part of this design please consider: ● What value of
n
should you use? In your analysis please consider at least two significantly different values of
n (e.g.,
n1

=20 and
n2

=100) ● Please clearly and fully justify/discuss the value of
n
you chose and the filter designed. In particular, please consider the tradeoffs of
rolloff, passband gain
and latency
. Hint /Reading Guide: Before solving the question, please review the following material: ● Lathi §12.8 (pp. 761777) ● Lathi 12.8.2 ● Matlab’s
Mutliband FIR Filter Documentation . 3