Given an audio signal sampled at 441 kHz a What is the frequ

Given an audio signal sampled at 44.1 kHz.

a. What is the frequency resolution for the Discrete Fourier Transform of the signal sample if 8192 samples are collected?

b. What is the frequency resolution for the Discrete Fourier Transform of the signal sample if 32,768 samples are collected?

c. How many operations will it take to compute the DFT directly as compared to computing the DFT using the FFT algorithm for cases a and b?

Solution

The Frequency Resolution=Fs/N

Fs-sampling Frequency,N=no of samples

a)44.1K/8192=5.3833Hz/bin

b)44.1K/32768=1.3458Hz/bin

c)

The No of Multiplication req for DFT=N^2; for FFT=(N/2)log₂(N) ;

The No of Addition req for DFT=N²-N;for FFT=Nlog₂(N)

for case a) using DFT we need 8192²=67108864 multiplication and 67108864-8192=67100672 addition process

   using FFTwe need (8192/2)log₂(8192)= 53248 multiplication and 8192*log2(8192)= 106496 addition process