$30
ECE 4250
Assignment 2
General Instructions
The assignment submissions are via Canvas. There are two parts: a problem set and a coding
(programming) component. Your answers to the problems can be written up any way you want
(e.g., using Latex or scanning your handwritten answers). As long as we can read it, we will
grade it. For the programming part, you will code in Python 3, using Jupyter Notebook. Make
sure that all cell outputs are clearly visible before saving the notebook. Thus, we recommend
that you re-run your code after you are done editing. You should then print this file into a pdf.
For the coding component, please submit both the “.pdf” and “.ipynb” files. Please zip up all
your work into a single file and submit that.
Any acknowledgments of collaboration, answers to questions, comments to code or other notes
should all be included in your problem set answer sheet and/or Jupyter Notebook. General
guidelines regarding assignments apply. In other words, you can collaborate at the ideation/brainstorming stage, but whatever you submit should represent your own individual work. You are
not allowed to copy/paste others’ code/answers or work on same code with someone else. You
are not allowed to use functions outside of the Python Standard Library, unless specified otherwise or you have received permission to do so. Your code will be graded on readability,
executability, and accuracy. You are strongly advised to use detailed commenting that explains the algorithmic steps in your code. Explain every function and loop. Use piazza to ask
questions. Take advantage of TA and Instructor Office Hours to seek help.
Problem Set
Question 1. Derive the analytic form for the causal signal that has the following z-transform:
X(z) = 2 −
1
2
z
−1 +
3
5
z
−2
1 −
1
2
z−1 +
3
5
z−2
Note that you are supposed to get an expression that is as compact as possible.
Question 2. A linear interpolator constructs a continuous signal by connecting successive
samples. For example:
xˆ(t) = x(nTs − Ts) + x(nTs) − x(nTs − Ts)
Ts
(t − nTs), nTs ≤ t ≤ (n + 1)Ts
This interpolator has a delay of Ts seconds. I.e. ˆx(nTs) = x(nTs − Ts).
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1. Derive the impulse response of the linear filter that yields the interpolation function described above when applied to an impulse train P∞
k=−∞ x(kTs)δ(t − kTs).
2. Is this a causal filter?
3. Derive the corresponding frequency response. How does this correspond to the ideal
interpolator’s frequency response?
Question 3. Consider a finite duration sequence x(n), with non-zero values for 0 ≤ n ≤ 7. Its
z-transform is X(z). Consider following points on the z-plane:
zk = 0.8e
j[2πk/8+π/8]
, 0 ≤ k ≤ 7
1. Sketch these points in the complex z-plane.
2. Determine a sequence s(n) such that its DFT will give the samples of X(z) at these points.
Note we are looking for an expression for s(n) in terms of x(n).
Question 4. Consider a band-limited analog signal xa(t), where Xa(ω) = 0, for all |ω| > 2πB.
Derive the Nyquist rate for:
1. dxa(t)
dt
2. xa(t) cos 6πBt
3. xa(3t)
Question 5. If X(k) is the N-point DFT of x(n), what is the N-point DFT of:
xc(n) = x(n) cos 2πkon
N
, 0 ≤ n ≤ N − 1
in terms of X(k)?
Question 6. Consider x(n) such that x(−2) = 1, x(−1) = 2, x(0) = 3, x(1) = 2, x(2) =
1, x(3) = 0 and all else is 0. Determine its Fourier transform X(ω). Now, compute the 6-point
DFT Y (k) of y(n) = [3, 2, 1, 0, 1, 2]. How is X(ω) and Y (k) related?
Programming Questions
1 Convolutions
Define the following signals
x =
3, 4, 1, 2, 5, 6, 7, 8, 2, 4
h =
1
3
,
1
3
,
1
3
Calculate the convolution between these two signals, x ∗ h, via the frequency domain. Confirm
your answer by computing the convolution directly in the time-domain (you did this in assignment 1). You can use numpy’s fft function. Note that fft is a fast algorithm that computes
the Discrete Fourier Transform and stands for Fast Fourier Transform.
Hint: remember to pad the signals so that they are the same size, so their fft’s can be multiplied.
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