# The DC bin of an FFT should be the mean of all the other bins in the spectrum, why does the FFT produce 0?

I am experimenting a bit with FFT's. From what I have read the first bin of an FFT is called the DC bin -- that is it is the mean of the other components in the FFT. I have also read that it should be close to 0.

When I actually perform the FFT and take the absolute value, what I find is that the first element of this is always 0. This makes sense mathematically if you consider the definition of the DFT itself.

Is there a best practice to reconstructing this first DC bin?

So it seems that because I was making my test FFT with a sine wave as the transient signal, this is what caused the 0 valued first FFT bin. I changed it to a cosine and got a non-zero first element...And the same with any random data. I should have thought a bit first.

• fft([1,1,1,1]) = [4,0,0,0] - far from being zero! The first bin is the DC which is the sum/average of the samples in time domain. It's zero if your signal is zero mean. Otherwise it's not. What do you mean by "reconstruct"? – Florian Jul 12 at 13:12
• There is absolutely no reason for the DC bin to be close to 0. Can you cite your source ? – Hilmar Jul 12 at 15:46

FFT computes the DFT which is

$$X[k] = \sum_{n=0}^{N-1} x[n] e^{-j \frac{2\pi}{N} k n }$$

where $$x[n]$$ is a sequence of length $$N$$ defined in $$0 \leq n \leq N-1$$.

The DC bin is $$X[0]$$; for $$k=0$$

$$X[0] = \sum_{n=0}^{N-1} x[n]$$

is the sum of all samples of the signal $$x[n]$$. It's not the average, but the average is obtained simply by dividing it by $$N$$.

• Thanks for the answer. I peform my FFT via mathematica, and the very first element of the output of this is always 0 once I take the absolute value. Does this imply my FFT algorithm is making some strange normalisation? – Q.P. Jul 12 at 13:05
• is the first element ( X[0] ) still zero even for the input signal x = [1,1,1,1,1] ? – Fat32 Jul 12 at 16:17

Depending on your normalization factor the DC bin is either the sum of the signal values in your frame (1), the average (1/N), a "representation" (1/sqr(N)),

In the DFT formula the exponential value is $$e^0$$, or 1, for each term.