Question

What is the DFT of the four point sequence x(n)={0,1,2,3}?

(a) {6,-2+2j-2,-2-2j}

(b) {6,-2-2j,2,-2+2j}

(c) {6,-2-2j,-2,-2+2j}

(d) {6,-2+2j,-2,-2-2j}

The question was posed to me in quiz.

This intriguing question originated from Sampling Rate Conversion by a Rational Factor I/D topic in portion Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

The correct option is (d) {6,-2+2j,-2,-2-2j}

Explanation: Given x(n)={0,1,2,3}

We know that the 4-point DFT of the above given sequence is given by the expression

X(k)=\(\sum_{n=0}^{N-1} x(n)e^{-j2πkn/N} \)

In this case N=4

=>X(0)=6, X(1)=-2+2j, X(2)=-2, X(3)=-2-2j.