代写ELEN 4810 Midterm Exam 2022代写C/C++编程

ELEN 4810 Midterm Exam

1. Systems in Time and Frequency. Consider the causal linear, time invariant system corresponding to the following block diagram:

Here, D1 denotes an ideal delay by one sample, D2 denotes an ideal delay by two samples, and the triangular blocks denote multiplication by scalars −1 and 4/3, respectively.

(a) Is the system stable? Why or why not?

(b) What is the frequency response H(e jω) of the system?

(c) What is the output of the system when the input is x[n] = (−1)n? What about when x[n] = 1 for all n?

(d) Suppose that the output of the system is y[n] = 16(−1)n. What are the possible values of x[n]? Please make your answer as broad as possible for full credit.

2. Sampling and Downsampling. A bandlimited continuous-time signal xc(t), with bandlimit ΩM is sampled with rate Ωs to produce a discrete-time signal x[n]. The below graph shows the DTFT X(e jω).

Please answer the following questions:

Part (a). Using the graph of X(e jω), please determine the ratio Ωs/ΩM of the sampling rate Ωs to the bandlimit ΩM.

Part (b). Consider the following block diagram:

Here, Hu(e jω) is the ideal upsampling filter

What is the largest integer M for which y[n] = x[n]?

Part (c). Now consider the following modified block diagram:

Can you increase M by choosing ω0 appropriately? Please specify the largest possible M and corresponding best choice of ω0.

3. Correlation and Convolution. Consider discrete-time signals

and

The signals x and w are plotted below:

Let ˜w[n] = w[−n], and let y be the convolution of ˜w and x: y = ˜w ∗ x.

Part a: Where are the nonzeros? The nonzero entries of y[n] satisfy N1 ≤ n ≤ N2. Please determine N1 and N2 – make N1 as large as possible and N2 as small as possible.

Part b: Where are the maximum / minimum values? For which value/values of n is y[n] maximized? For which value/values of n is y[n] minimized?