Consider the signal x[n] = δ[n − 4] + 2δ[n − 5] + δ[n − 6]. (a) Find X(ejω) the dis...

Consider the signal x[n] = δ[n − 4] + 2δ[n − 5] + δ[n − 6].

(a) Find X(e^jω) the discrete-time Fourier transform of x[n]. Write expressions for the magnitude and phase of X(e^jω), 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:

where X[k] is the three-point DFT of x[n]?