Question

By means of the DFT and IDFT, determine the response of the FIR filter with impulse response h(n)={1,2,3} to the input sequence x(n)={1,2,2,1}?

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

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

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

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

The question was posed to me in examination.

Origin of the question is Linear Filtering Methods Based on DFT in section Discrete Fourier Transform – Properties and Applications of Digital Signal Processing

Answer 1

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

To explain I would say: The input sequence has a length N=4 and impulse response has a length M=3. So, the response must have a length of 6(4+3-1).

We know that, Y(k)=X(k).H(k)

Thus we obtain Y(k)={36,-14.07-j17.48,j4,0.07+j0.515,0,0.07-j0.515,-j4,-14.07+j17.48}

By applying IDFT to the above sequence, we get y(n)={1,4,9,11,8,3,0,0}

Thus the output of the system is {1,4,9,11,8,3}.