Consider the six-point sequence x[n] = 6δ[n] + 5δ[n − 1] + 4δ[n − 2] + 3δ[n − 3] +...

Consider the six-point sequence

x[n] = 6δ[n] + 5δ[n − 1] + 4δ[n − 2] + 3δ[n − 3] + 2δ[n − 4] + δ[n − 5]

shown in Figure P8.28.

(a) Determine X[k], the six-point DFT of x[n]. Express your answer in terms of W₆ = e^−j2π/6.

(b) Plot the sequence w[n], n = 0,1,...,5, that is obtained by computing the inverse six-point DFT of W[k] = W₆^−2kX[k].

(c) Use any convenient method to evaluate the six-point circular convolution of x[n] with the sequence h[n] = δ[n] + δ[n − 1] + δ[n − 2]. Sketch the result.

(d) If we convolve the given x[n] with the given h[n] by N-point circular convolution, how should N be chosen so that the result of the circular convolution is identical to the result of linear convolution? That is, choose N so that

(e) In certain applications, such as multicarrier communication systems (see Starr et al, 1999), the linear convolution of a finite-length signal x[n] of length L samples with a shorter finite-length impulse response h[n] is required to be identical (over 0 ≤ n ≤ L−1) to what would have been obtained by L-point circular convolution of x[n] with h[n]. This can be achieved by augmenting the sequence x[n] appropriately. Starting with the graph of Figure P8.28, where L = 6, add samples to the given sequence x[n] to produce a new sequence x₁[n] such that with the sequence h[n] given in part (c), the ordinary convolution y₁[n] = x₁[n] ∗ h[n] satisfies the equation

( f ) Generalize the result of part (e) for the case where h[n] is nonzero for 0 ≤ n ≤ M and x[n] is nonzero for 0 ≤ n ≤ L − 1, where M < L; i.e., show how to construct a sequence x₁[n] from x[n] such that the linear convolution x₁[n] ∗ h[n] is equal to the circular convolution x[n] h[n] for 0 ≤ n ≤ L − 1.