Question

Can you explain the difference between the following two methods:

1)zero padding

2)periodic extension

when computing DFT of finite extend signals.

Answer 1

Zero padding is a simple concept; it simply refers to adding zeros to end of a time-domain signal to increase its length. The example 1 MHz and 1.05 MHz real-valued sinusoid waveforms we will be using throughout this article is shown in the following plotZero padding is a simple concept; it simply refers to adding zeros to end of a time-domain signal to increase its length. The example 1 MHz and 1.05 MHz real-valued sinusoid waveforms we will be using throughout this article is shown in the following plot:

There are two aspects of FFT resolution. I’ll call the first one “waveform frequency resolution” and the second one “FFT resolution”. These are not technical names, but I find them helpful for the sake of this discussion. The two can often be confused because when the signal is not zero padded, the two resolutions are equivalent.

The “waveform frequency resolution” is the minimum spacing between two frequencies that can be resolved. The “FFT resolution” is the number of points in the spectrum, which is directly proportional to the number points used in the FFT.

It is possible to have extremely fine FFT resolution, yet not be able to resolve two coarsely separated frequencies.

It is also possible to have fine waveform frequency resolution, but have the peak energy of the sinusoid spread throughout the entire spectrum (this is called FFT spectral leakage).

The waveform frequency resolution is defined by the following equation:

where T is the time length of the signal with data. It’s important to note here that you should not include any zero padding in this time! Only consider the actual data samples.

It’s important to make the connection here that the discrete time Fourier transform (DTFT) or FFT operates on the data as if it were an infinite sequence with zeros on either side of the waveform. This is why the FFT has the distinctive sinc function shape at each frequency bin.

You should recognize the waveform resolution equation 1/T is the same as the space between nulls of a sinc function.

The FFT resolution is defined by the following equation: