a. Why do we use Discrete Fourier Transform (DFT) instead of Discrete-Time Fourier Transform (DTFT) in real-world signal processing applications? b. Given the 4-point discrete-time sequence x[0], x[1], x[2] and x[3], write a separate formula for each of its 4-point Discrete Fourier Transform X[0], X[1], X[2] and X[3].(Hint: X[.] is a function of x[0],x[1],x[2], and x[3]) c. Consider the "time-reversed" sequence y(n) = x[3 - n] for n = 0,1,2,3. Write its 4- point Discrete Fourier Transform Y[0], Y[1], Y[2] and Y[3] in terms of X[0], X[1], X[2] and X[3], where they are the Discrete Fourier Transform of x[n].

Suppose a function f(x) = x between 0 and 1, and f(x) = 0 elsewhere. Solve for the fourier transform g(k) of the original function f(x).

4. Find the characteristic polynomial of A = w53

1 Practice with Fourier Transforms (**) For the listed functions A) through D), do the following: • Plot the function f(x) and its magnitude |ƒ(x)|. • Compute the Fourier transform of f(x): A) f(x) = e-|x| 1, B) f(x) = { 1 g(k) |x| ≤ 1 | • Compute and plot the magnitude |g(k)| as a function of k. • For each function and its transform, think about (but do not compute!) Parseval's Theorem: lg(k)|²dk = |f(x)|²da 0, otherwise = -∞ 1 √2π Lx f(x)e-iker dar (1) Based on your drawings of the function and its Fourier transform, which side of Parseval's theorem would be easier to compute? (2)