Given a signal x[n] = δ[n] + 0.9 δ [n − 6]. The Discrete Time Fourier Transform for 8 points is __________Select one:1. 1 + 0.9 e−j2π8k62. 1 – 0.9 ej2π8k63. 1 + 0.9 ej2π8k64. 1 – 0.9 e−j2π8k6

To find the Discrete Time Fourier Transform (DTFT) for the given signal x[n] = δ[n] + 0.9 δ [n − 6] with 8 points, we can use the formula:

X[k] = Σ x[n] * e^(-j2πnk/N)

where X[k] is the DTFT of x[n], x[n] is the input signal, k is the frequency index, and N is the number of points.

In this case, we have N = 8. Let's calculate the DTFT step by step:

For k = 0: X[0] = Σ x[n] * e^(-j2π0n/8) = Σ x[n] = x[0] + x[1] + x[2] + x[3] + x[4] + x[5] + x[6] + x[7] = δ[0] + 0.9 δ[0-6] = 1 + 0.9 * 0 = 1

For k = 1: X[1] = Σ x[n] * e^(-j2π1n/8) = Σ x[n] * e^(-jπn/4) = x[0] * e^(-jπ0/4) + x[1] * e^(-jπ1/4) + x[2] * e^(-jπ2/4) + x[3] * e^(-jπ3/4) + x[4] * e^(-jπ4/4) + x[5] * e^(-jπ5/4) + x[6] * e^(-jπ6/4) + x[7] * e^(-jπ7/4) = δ[0] * e^(-jπ0/4) + 0.9 δ[1-6] * e^(-jπ1/4) = 1 * e^0 + 0.9 * 0 * e^(-jπ1/4) = 1

Similarly, for k = 2, 3, 4, 5, 6, and 7, we will get the same result as k = 1, which is 1.

Therefore, the Discrete Time Fourier Transform for 8 points is 1 + 0.9 ej2π8k6.