1. Consider the complex sequence
ejω0n, 0 ≤ n ≤ N - 1.
x[n] =
0, otherwise.
(a) Find the Fourier transform X(eiω) of x[n].
(b) Find the N-point DFT X [k] of the finite-length sequence x[n).
(c) Find the DFT of x[n] for the case ω0 = 2πk0/N, where k0 is an integer.
2. Suppose we have two four-point sequences x[n] and h[n] as follows:
x[n] = cos (πn/2), n = 0, 1, 2, 3.
h[n] = 2n, n =0, 1, 2, 3.
(a) Calculate the four-point DFT X [k].
(b) Calculate the four-point DFT H[k].
(c) Calculate y[n] = x[n]4h[n] by doing the circular convolution directly.
(d) Calculate y[n] of pan (c) by multiplying the DFTs of x[n] and h[n) and performing an inverse DFT.
3. Figure P8.18-1 shows a sequence x[n] for which the value of x[3] is an unknown constant c. The sample with amplitude c is not necessarily drawn to scale. Let X1[k] = X [k]ej2π3k/5, where X [k] is the five-point DFT of x[n]. The sequence x1[n] plotted in Figure P8.18-2 is the inverse DFT of X1[k]. What is the value of c?
4. Consider the signal x[n] = δ[n - 4] + 2δ[n - 5] + δ[n - 6].
(a) Find X(ejω) the discrete-time Fourier transform of x[n]. Write expressions for the magnitude and phase of X (ejω), and sketch these functions.
(b) Find all values of N for which the N-point DFT is a set of real numbers.
(c) Can you find a three-point causal signal x1[n] (i.e., x1[n] = 0 for n < 0 and n > 2) for which the three-point DFT of x1[n] is:
X1[k] = |X[k]| k = 0, 1, 2
where X [k] is the three-point DFT of x[n]?
5. An FIR filter has a 10-point impulse response, i.e.,
H[n] = 0 for n < 0 and for n > 9.
Given that the 10-point DFT of h[n] is given by
H[k] = 1/5δ[k - 1] + 1/3δ[k - 7], find H(eJω), the DTFT of h[n].
6. Suppose that x1[n] and x2[n] are two finite-length sequences of length N, i.e.. x1[n] = x2[n] = 0 outside 0 ≤ n ≤ N -1. Denote the z-transform of x1[n] by X1(z), and denote the N-point DFT of x2[n] by X2[k]. The two transforms X1(z) and X2[k] are related by:
X2[k] = X1(z)|z = 1/2e-j(2πk/N). , k = 0, 1, . . . , N - 1
Determine the relationship between x1[n] and x2(n].