代做ELEN 4810 Final Exam 2022调试SPSS

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