Taking the FFT of a sinusoidal signal and going back

Question

I am a computer science student and want to do some stuff with audio data. I want to use the DFT to analyze and synthesize some sounds. Before going to the more complex stuff I experimented with basic tones and found some issues I do not understand - therefore I want to ask for help in this physics forum.

I created a single sinus wave function with an audioprogram of $440\textrm{ Hz}$. From this I take the DFT and want to recreate the original signal from this. While doing so, first I emit the negative frequencies, thus take only the first $n/2$ frequencies. $N$ is my blocksize (I divide the signal into blocks of length $n$). The sampling rate of my audio signal is $44100\textrm{ Hz}$. So when I do this, and take the IFT from the resulting data, I can perfectly reconstruct my signal.

Now to my issues and questions:

1. When I set the phase to $0$ in the Fourier space (the imaginary part), with different $n$, I recover a sinus signal with some periodic noise.
   • I assume this is the phase difference which can be heard between the blocks?
2. When I set $n$ to $44100$ (the sampling frequency), I get complete noise.
   • Why is that?
3. Now I want to take only the strongest frequency amplitude wise (which in my opinion should perfectly work) - thus I set all other frequencies in the Fourier space to $0$ and then do the IFT. For different $n$ this kind of works, I get a sinus signal with some periodic noise.
   • Why the noise?
   • Moreover, the frequency of the resulting tone changes with $n$. Why is that?
   • With $n = 44100$ I get complete silence. Why?
4. When I set the phases to $0$ again I get the same results.

I hope it's somewhat interesting for you too. Could you explain to me these "phenomena"?

Answer 1

Part of the problem is that you are using sine waves. The imaginary components of an FFT result contains all the information about pure sine waves (e.g. that are equivalent to a sin() function that starts with phase of zero at the start of the FFT aperture, and that are exactly integer periodic in aperture). If you use only the real part of an FFT result, you can only see cosine waves (not sine waves, periodic in aperture), thus your silent result.

For shorter FFTs, the real component of the result might contain windowing artifacts if the frequency of your sine wave is between FFT result bins for the given FFT length (e.g. not integer periodic in aperture). So, in synthesis, you are hearing an artifact, quantized in frequency.

Answer 2

Mathematically, an FFT is a specific form of Fourier transform. First you only sample it at a fixed grid of points, leading to a periodic frequency representation (you cannot distinguish sine waves whose frequency differs by whole multiples of the sampling frequencies). This is something you already need to deal with at the sampling stage by prepending appropriate low-pass filtering.

Then you take the sample window of your data and repeat it an infinite number of times. As a result of that operation, the resulting Fourier transform will only contain components with a periodicity of whole multiples of your sample window length. As another result of that operation, sine signals with a period not an exact multiple of the sampling window will have discontinuities at the points they are "pasted together".

One uses "windowing" in order to get a good compromise between the resulting frequency bleed over frequency bins and a reasonable symbolic accuracy of signal representation.

Note that as long as you do nothing with your transformed data but transform it back unchanged, it will be reconstructed perfectly: the FFT is an information-preserving transform.

It's just not a Fourier transform in the manner that is so nice for calculating frequency responses of passive circuitry because it works on time-discretized forced-periodic data with resulting discrete frequency bins.