Question

What is the sequence y(n) that results from the use of four point DFTs if the impulse response is h(n)={1,2,3} and  the input sequence x(n)={1,2,2,1}?

(a) {9,9,7,11}

(b) {1,4,9,11,8,3}

(c) {7,9,7,11}

(d) {9,7,9,11}

I got this question in an online interview.

This key question is from Linear Filtering Methods Based on DFT topic in chapter Discrete Fourier Transform – Properties and Applications of Digital Signal Processing

Answer 1

The correct choice is (d) {9,7,9,11}

The explanation: The four point DFT of h(n) is H(k)=1+2e^-jkπ/2+3 e^-jkπ (k=0,1,2,3)

Hence H(0)=6, H(1)=-2-j2, H(3)=2, H(4)=-2+j2

The four point DFT of x(n) is X(k)= 1+2e^-jkπ/2+2 e^-jkπ+3e^-3jkπ/2(k=0,1,2,3)

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

The product of these two four point DFTs is

Ŷ(0)=36, Ŷ(1)=j4, Ŷ(2)=0, Ŷ(3)=-j4

The four point IDFT yields ŷ(n)={9,7,9,11}

We can verify as follows

We know that from the previous question y(n)={1,4,9,11,8,3}

ŷ(0)=y(0)+y(4)=9

ŷ(1)=y(1)+y(5)=7

ŷ(2)=y(2)=9

ŷ(3)=y(3)=11.