Question

What is the convolution of the sequences of x1(n)=x2(n)={1,1,1}?

(a) {1,2,3,2,1}

(b) {1,2,3,2,1}

(c) {1,1,1,1,1}

(d) {1,1,1,1,1}

This question was posed to me by my school teacher while I was bunking the class.

My question is based upon Properties of Fourier Transform  for Discrete Time Signals topic in division Frequency Analysis of Signals and Systems of Digital Signal Processing

Answer 1

Correct answer is (a) {1,2,3,2,1}

Best explanation: Given x1(n)=x2(n)={1,1,1}

By calculating the Fourier transform of the above two signals, we get

X1(ω)= X2(ω)=1+ e^jω + e^-jω = 1+2cosω

From the convolution property of Fourier transform we have,

X(ω)= X1(ω). X2(ω)=(1+2cosω)^2=3+4cosω+2cos2ω

By applying the inverse Fourier transform of the above signal, we get

x1(n)*x2(n)={1,2,3,2,1}