If x(n) and X(k) are an N-point DFT pair, then X(k+N)=?Select one:a. X(k)b. X(-k)c. None of the mentionedd. -X(k)

Break Down the Problem

1. Identify the properties of the Discrete Fourier Transform (DFT).
2. Understand the periodicity characteristics of DFT.

Relevant Concepts

• The DFT is periodic with a period of N. This implies that the DFT coefficients wrap around after N samples.

Analysis and Detail

1. For a given N-point DFT pair $x(n)$ and $X(k)$, we know: $X(k) = \sum_{n=0}^{N-1} x(n) e^{-j(2\pi/N)kn}$
2. The periodicity property of the DFT states: $X(k + N) = X(k)$ This means that adding N to the index of the DFT output does not change the output value.

Verify and Summarize

• Given the periodic nature of the DFT, we have established that $X(k + N)$ is equal to $X(k)$.

Final Answer

$X(k + N) = X(k)$ (Select option a).