# Why does Fourier zero-fill interpolation pad the middle frequencies?

In this: http://dspguru.com/dsp/howtos/how-to-interpolate-in-time-domain-by-zero-padding-in-frequency-domain the author says that we can interpolate a signal by zero-padding the middle frequencies in the DFT. But then the high frequency components are pushed even higher, so shouldn't the result look more wiggly? Why is that not so?

## 1 Answer

Usually a fast Fourier transform (FFT) returns a vector that starts with the zero-frequency bin. This means that the Nyquist frequency (sampling frequency / 2) bin is in the middle of the vector. So you are not pushing apart the low and high frequencies by zero padding at the middle. The zeros go between the highest positive frequency bin and the highest-absolute-frequency negative frequency bin.

Rick omits the details of how to deal with the Nyquist frequency bin (which is there with even-length DFT): If you have a non-zero Nyquist frequency bin then you need to split it into a half-amplitude positive Nyquist frequency bin and a half-amplitude negative Nyquist frequency bin, and insert the zero padding between those. ...I think.

• It's confusing because Figure 3 is in kHz. That suggests that the frequencies are constantly going up. – ogogmad Apr 17 '16 at 6:54
• If the FFT were zero-centred then would the padding go half to the left and half to the right? – ogogmad Apr 17 '16 at 6:55
• 1. Yeah that is kind of an error, but if the sampling frequency is for example 10 kHz then 9 kHz and -1 kHz complex exponential functions have identical discrete representations, so from the discrete perspective they are equivalent. 2. Yes. – Olli Niemitalo Apr 17 '16 at 8:14