Question

Show details of calculation for full credit in solving the following DFT related problems.

(a) Compute the N-point DFT of a length- N signal: x[n]= δ[n] + sin(2πn/N), 0≤n≤N−1.

Answer 1

To solve the problem, apply the definition of the DFT to the given signal, considering the contribution of the Kronecker delta function at n=0 and the periodic sinusoidal function separately. The component calculations show the frequency content of the original signal.

Step-by-step explanation:

To compute the N-point Discrete Fourier Transform (DFT) of the given length-N signal x[n] = δ[n] + sin(2πn/N), 0 ≤ n ≤ N−1, we apply the definition of the DFT:

X[k] = Σ_{n=0}^{N-1} x[n] e^{-j(2π/N)nk}, where 0 ≤ k ≤ N-1

Here, x[n] has two components: the Kronecker delta function δ[n], which is 1 at n=0 and 0 everywhere else; and the sinusoidal function sin(2πn/N), which is periodic over n with period N.

Using the DFT definition:

• X[k] = e^{0} for the Kronecker delta part, as only the term at n=0 is non-zero (which equals 1).
• For the sinusoidal part, the calculation involves summing complex exponentials, which map the sinusoid onto a circle in the complex plane.

The resulting X[k] will have distinct values for each k, representing the frequency content of the original signal x[n]. The zero-frequency component (k=0) corresponds to any DC component in x[n], while other values of k correspond to sinusoidal components at different frequencies.