Question

Please explain and give an example for each of the following:

What is the relationship between the FFT of a signal and the length of the sinewave, i.e. the number of samples?

What is the relationship between the FFT of a signal and the frequency of the sine wave?

What the region of main spectral energy is

Answer 1

According to Nyquist sampling thereom, we can reconstruct the message signal from its sampled version only if sampling frequency is greater than two times frequency of message signal. While taking FFT of Sine wave signal we are doing the same thing. First we are sampling the signal and then we are reconstructing it by taking FFT.

The total no of samples are depend on the sampling frequency. We will consider the Sine wave with 1Khz frequency. and it is sampled with 2Khz Frequency. So in one second we will get 2000 samples i.e. sampling rate will be 2000 samples/Sec.

If we take FFT of the signal we will get 2000 frequency bins. The FFT will give us the double sided spectrum with complex conjugate spectrum. The inforamation is repeated in each side. So we will neglect negative frequencies. So we will get 2000/2 i.e.1000 bins.

So we can here say that the no of frequency bins are equal to no of samples.

The frequency resolution in the FFT is sampling Freq / No of samples so in our case 2000/2000 i.e. So on frequency axis you will get resolution of 1Hz.

There is always trade off between FFT resolution and computing power.

The main spectral energy is located at the frequency content in the original signal. You will always find a peaks at the contecnt frequency,