Why does my NAudio FFT result differ from MATLAB by a factor of 4?

Question

The following C# NAudio code produces a different result to MATLAB by a factor of 4. Why does this occur and is one of them incorrect?

Complex[] tmp = new Complex[4];
tmp[0].X = 1.0f;
tmp[1].X = 0.5f;
tmp[2].X = 1.0f;
tmp[3].X = 0.25f;
tmp[0].Y = 0.0f;
tmp[1].Y = 0.0f;
tmp[2].Y = 0.0f;
tmp[3].Y = 0.0f;
FastFourierTransform.FFT(true, 2, tmp);

NAUDIO OUTPUT:

0.6875 + 0.0000i
0.0000 - 0.0625i
0.3125 + 0.0000i
0.0000 + 0.0625i

MATLAB OUTPUT:

2.7500 + 0.0000i
0.0000 - 0.2500i
1.2500 + 0.0000i
0.0000 + 0.2500i

Answer 1

The Discrete Fourier transform and its inverse require a certain normalization so that ifft(fft(x))==x . How this normalization is done changes from implementation to implementation.

It seems that, in this case, NAudio has chosen a different normalization than MATLAB.

MATLAB uses the most common normalization, where fft(x) at k=0 is equal to sum(x) , and the inverse transform does the same thing but divides by n (number of samples). This is also the equation as described in the Wikipedia page for the DFT . In this case, the inverse transform matches the equation for the Fourier series.

NAudio seems to do the division by n in the forward transform, such that at k=0 you have mean(x) .

Given the above, you can use the first frequency bin (the DC component) to verify what normalization is used (assuming there is a DC component, if the signal has a zero mean this will not work): If the DC component is equal to the sum of all the sample values, then the "common" normalization is used. It can also be equal to the sum divided by sqrt(n) , in the case of a symmetric definition, where the forward and inverse transform carry the same normalization. In the case of NAudio it will be equal to the sum divided by n (ie the mean of the sample values). In general, take the DC component and divide it by the sum of the sample values. The result q is the normalization term used. The inverse transform should have a normalization term 1/qn .