ELEN 4810 Final Exam
1. Discrete Fourier Transform. and Fast Fourier Transform. Consider two discrete time signals u[n] and v[n], which satisfy
u[n] ≠ 0, n = 0, 4, 8, 12, . . . , 60, u[n] = 0 else,
v[n] ≠ 0, n = 0, 8, 16, 24, . . . , 56, v[n] = 0 else.
Set y = u ∗ v. Please answer the following questions:
Part A. Suppose we compute y via the Discrete Fourier Transform, via
For what choices of N does this operation correctly compute y?
Part B. In this part, we use the structure of u and v to compute y[n] more efficiently (similar to the Fast Fourier Transform). Let ¯u and ¯v be downsampled versions of u and v:
u¯ = u ↓ 4,
v¯ = v ↓ 8.
Let
U¯[k] = DFT32n u¯
o [k],
V¯ [k] = DFT16n v¯
o [k],
Y [k] = DFT128n y
o [k].
Please give an expression for Y [k] in terms of U¯ and V¯ .
2. Z Transform. Consider the following rational transfer function H(z):
Part a. What are the poles and zeros of H?
Part b. Assuming the system is causal, please specify the region of convergence (ROC) and the impulse response h[n].
Part c. Assuming the system is stable, please specify the region of convergence (ROC) and the impulse response h[n].
Part d. Which of the following best describes the system?
LOW PASS BAND PASS HIGH PASS ALL PASS
3. Spectrograms. The following question has two parts.
Part (a). A signal x[n] has the form.
for some scalars α, β, γ, τ .
Which of the four figures above is the spectrogram of the signal? For full credit, please justify your answer.
Part (b). A linear chirp signal
is passed through a canonical generalized linear phase system whose impulse response has length 5,and satisfies h[0] = 1.
Above are the spectrograms for z[n] (left) and yn] = h*x[n] (right). Both spectrograms are generated with a Discrete Fourier Transform. (DFT) of length N = 512.
Please answer the following questions as accurately as possible, given the available information:
(b.i) What type of canonical generalized linear phase system is this?
(b.ii) What is the group delay grd[H(ew)]?
(b.iii) Please sketch the pole-zero diagram of H(z), using the axes on the next page. Please labelany repeated poles and zeros with their multiplicity.
4. Filter Design by Windowing. In this problem, we design a low-pass filter by windowing. We set
The corresponding time-domain target is
We use a rectangular window
and set h[n] = w[n] htarget[n]. The impulse response h[n] is plotted below, for L = 80:
Part A. Does the filter h[n] have generalized linear phase? Why or why not?
Part B. In lecture, we discussed Kaiser windowing, which uses a different choice of w[n]. What is the main advantage of Kaiser windowing compared to the rectangular window used in part A?
Part C. What are the two main advantages of design by L∞ optimization, compared to design by windowing?
Part D. Let h[n] be our designed impulse response, H(z) its Z-transform, and let ζ1, . . . , ζM denote the zeros of H(z).
Suppose we generate a new filter by setting hnew[n] = (−1)nh[n]. Please give an expression for the zeros ζ1
′
, . . . , ζM
′ of Hnew(z) in terms of the zeros ζ1, . . . , ζM.
Part E. Which of the following best characterizes the filter hnew[n]? Why?
LOW PASS BAND PASS HIGH PASS ALL PASS