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# Fourier Transform

Fourier Transform (FT) is one of the most prominent transformations/algorithms in signal processing (and really in computer science and mathematics in general), which enables us to express and manipulate a signal (such as a sound or picture) in terms of frequencies it is composed of (rather than in terms of individual samples). It is so important because frequencies (basically sine waves) are actually THE important essence in signals, they allow us to detect things (voices, visual objects, chemical elements, ...), compress signals, modify them in useful ways (e.g. filter out noise in specific frequency band, enhance specific frequency bands, change audio speed without changing its pitch, pitch-shift a song without changing its speed and so on). There also exists a related algorithm called Fast Fourier Transform (FFT) which is able to compute one specific version of FT very quickly and so is often used in practice.

FT is typically not easy to understand right away, it takes time to wrap one's head around it. Confusing terminology doesn't help in this regard either; there exist slightly different kinds of Fourier Transform bearing similar names and sometimes they're all just called "Fourier Transform" -- what programmers usually mean by FT is DFT or FFT. There also exist Fourier Transforms in higher dimensions (2D, 3D, ...) -- the base case is called one dimensional (because our input signal has one coordinate). All this mess will be explained below.

What FT does in essence: it transforms an input signal (which can also be though of as a function) from time (also space) domain, i.e. the usual representation that for each x says the sample value f(x), to frequency domain, another function that for each frequency f says "how much of the frequency is present" (amplitude and phase). For example an FT of a simple sine wave will be a function with a single spike at the frequency of the sine wave. There is also an inverse Fourier Transform that does the opposite (transforms the signal from frequencies back to time samples). The time and frequency representations are EQUIVALENT in that either one can be used to represent the signal -- it turns out that even "weird" looking functions can be decomposed into just sums of many differently shifted and scaled sine waves. In the frequency domain we can usually do two important things we cannot do in time domain: firstly analyze what frequencies are present (which can help e.g. in voice recognition, spectral analysis, earthquake detection, music etc.) and also MODIFY them (typicaly example is e.g. music equalizer or compression that removes or quantizes some frequencies); if we modify the frequencies, we may use the inverse FT to get back the "normal" (time representation) signal back. Some things are also easier to do in the frequency domain, for example convolution becomes mere multiplication.

FT is actually just one of many so called integral transforms that are all quite similar -- they always transform the signal to some other domain and back, they use similar equation but usually use a different kind of function. Other integral transforms are for example discrete cosine transformation (DCT) or wavelet transform. DCT is actually a bit simpler than FT, so if you are having hard time with FT, go check out DCT.

If you know linear algebra, this may help you understand what (D)FT really does: Picture the signal we work with as a POINT (we can also say a vector) in many dimensional space; if for instance we have a recorded sound of 1000 samples, it is really a 1000 dimensional vector, a point in 1000 dimensional space, expressed as an "array" of 1000 numbers (vector components). A short note: since we assume a finite number of discrete samples here, we are actually dealing with what's called DISCRETE FT here, not the "vanilla" FT, but for now let's not digress. (D)FT does nothing more than transforming from one vector basis ("system of coordinates", "frame of reference") to another basis; i.e. by default the signal is expressed in time domain (our usual vector basis), the numbers in the sound "array" are such because we are viewing them from the time "frame of reference" -- (D)FT will NOT do anything with to the signal itself (it is a vector/point in space, which will stay where it is, the recorded sound itself will not change), it will merely express this same point/vector from a different "point of view"/"frame of reference" (set of basis vectors) -- that of frequencies. That's basically how all the integral transforms work, they just have to ensure the basis they are transforming to is orthogonal (i.e. kind of legit, "usable") of course. In addition the FT equation is nothing complex, it literally just uses a dot product of the whole input signal with each checked frequency wave to find out how similar the signal is to that particular frequency, as dot product simply says "how similar two vectors are" -- really, think about the equation and you will see it's really doing just that.

TODO: alternatives (like performing FIR filtering without actually doing FT etc.)

## Details

First and foremost let's clarify the whole terminology around FT a bit:

- Fourier Series (FS): Transforms a PERIODIC (repeating) signal into a DISCRETE (non-continuous) spectrum. We can see this spectrum also as an infinite SERIES of coefficients c0, c1, c2, etc. The input signal can generally be complex, in which case the output spectrum also has negative part (c-1, c-2, c-3 etc.) and shows us COMPLEX EXPONENTIALS (i.e. not mere sine waves) of the input signal; however if the input signal is real (probably most signals we practically deal with), the spectrum's negative part is symmetric to the positive part and the corresponding positive and negative complex exponentials always together give a sine wave, so for "normal" signals we can see the spectrum only being in the non-negative part and showing us the sine waves of the signal.
- Fourier Transform (FT): Generalization of FS to work on any signal, not just periodic ones, i.e. FT takes a NON-PERIODIC signal and transforms it into a CONTINUOUS spectrum. This is achieved simply by considering the period of the signal to be infinite -- the spectrum now becomes continuous exactly because the input is non-periodic (this is a relationship that generally holds); since the output is continuous, we now rather see it as a function rather than a series. Same as with FS the input can be complex (in which case the same implications apply), but we usually work with real signals.
- Inverse Fourier Transform (IFT): Does the opposite of FT, i.e. transforms the signal back from frequency domain to time domain.
- Discrete Time Fourier Transform (DTFT) (not to be confused with DFT!): Fourier Transform for DISCRETE (non-continuous) input signals (e.g. sound pressure captured only at specific points in time) -- since the input is discrete, the spectrum will be PERIODIC (this is another relationship that generally holds).
- Discrete Fourier Series (DFS): Version of FS for discrete (non-continuous) signals, transforms the input DISCRETE and PERIODIC signal to a spectrum (series of coefficients) of which there are infinitely many and are also PERIODIC (with the same period as the input signal).
- Discrete Fourier Transform (DFT) (not to be confused with DTFT!): Uses DFS to transform a FINITE DISCRETE signal to a FINITE DISCRETE spectrum (with the same period as the input) by simply "pretending" the finite input signal is actually repeating over and over and then, after the transform, only leaving in the first period of the result (since the rest is just repeating). This is actually what programmers usually mean by Fourier Transform because in computers we practically always only deal with finite discrete signals (i.e. arrays of data).
- Fast Fourier Transform (FFT): Computes DFT (NOT FT!) that's faster than the naive implementation, i.e. computing the equation that defines DFT as it's written has time complexity O(n^2) while FFT improves this to O(n * log(n)).

From now on we will implicitly be talking about DFT of a real function (we'll ignore the possibility of complex input), the most noteworthy transform here.

The input to DFT is a real function, i.e. the time domain representation of the signal. The output is a complex valued function of frequency, i.e. the spectrum -- for each frequency it says a complex number whose magnitude and phase say the magnitude and phase of that frequency (a sine wave) in the signal (many programs will visualize just the magnitude part as that's usually the important thing, however keep in mind there is always also the phase part as well).

The general equations defining DFT and IDFT, for signal with N samples, are following

              ___ N - 1
              \
    DFT[k] =  /__    x[n] * e^(-2 * i * pi * k * n / N)
                  n = 0
                    ___ N - 1
                    \
    IDFT[k] = 1/N * /__   x[n] * e^(2 * i * pi * k * n / N)
                        n = 0

OK, this is usually where every noob ragequits if he hasn't already because of all the pis and es and just generally ununderstable mess of weird symbols etc. What the heck does this all mean? As said above, it's doing nothing else than dot product or vectors really: one vector is the input signal and the other vectors are the individual frequencies (sine waves) we are trying to discover in the signal -- this looks so complicated because here we are actually viewing the general version for a possible complex input signal, the e to something part is actually the above mentioned complex exponential, it is the exponential way of writing a complex number (see e.g. Euler's identity). Anyway, considering only real input signal, we can simplify this to a more programmer friendly form:

    DFT:
      init DFT_real and DFT_imag to 0s

      for k = 0 to N - 1
        for n = 0 to N - 1
          angle = -2 * i * pi * k * n / N
          DFT_real[k] += x[n] * cos(angle)
          DFT_imag[k] += x[n] * sin(angle)

    IDFT:
      init data to 0s

      for k = 0 to N - 1
        for n = 0 to N - 1
          angle = 2 * i * pi * k * n / N
          data[k] += DFT_real[n] * cos(angle) - DFT_imag[n] * sin(angle)

        data[k] /= N

Example: take a look at the following array of 8 kind of arbitrary values and what their DFT looks like:

                                #      #
                  #             #      #
                  #      #      #      #
                  #      #      #      #
                  #      #      #      #             #
                  #      #      #      #      #      #
    data:        5.00   4.71   6.00   6.54   1.00   2.29   0.00  -0.54

    DFT:
                  #
                  #
                  #
                  #      #                                         #
                  #      #                                         #
                  #      #             #             #             #
      magn.:    25.00  12.74   1.00   7.33   1.00   7.33   1.00  12.74
      phase:     0.00  -1.52  -1.57  -0.10  -3.14   0.10   1.57   1.52
      -----
      real:     25.00   0.70   0.00   7.30  -1.00   7.30  -0.00   0.70
      imag.:     0.00 -12.72  -1.00  -0.72  -0.00   0.72   1.00  12.72

    restored:
    data:        5.00   4.71   6.00   6.54   1.00   2.29   0.00  -0.54

At the top we have the input data: notice the data kind of looks similar to a low-frequency sine wave, so the frequencies in the spectrum below are mostly low, but there's also some high frequency noise that's deforming the wave. For convenience here we show the spectrum values in both formats (magnitude/phase and real/imaginary part), but keep in mind it's just different formats of the same complex number values; for analysis we are mostly interested in the magnitude of the complex numbers as that shows as the amplitude of the frequency, i.e. the "amount" of the frequency in the signal. Here we notice the greatest peak is at frequency 0 -- this is a "constant" component, the lowest possible frequency that just represents a constant vertical offset of the signal (a constant number added to all samples); this component here is so big because our input signal doesn't really oscillate around the value 0 as it doesn't even go to negative values -- DFT sees this as our signal being shifted quite a lot "up". Frequencies 1 and 7 are the second biggest here: DFT is telling us the signal looks mostly like an addition of a sine wave with very low frequency and very high frequency (which it does), it doesn't see many middle value frequencies here. At the end we also see the original values computed back using IDFT, just to check everything is working as expected.

Here is the C code that generates the above, you may use it as a snippet and/or to play around with different inputs to see what their spectra look like (for "readability" we commit the sin of using floating point numbers here, implementation of DFT without floats is left as an exercise :]):

    #include <stdio.h>
    #include <math.h>

    #define PI 3.141592
    #define N (sizeof(data) / sizeof(double))   // size of input data
    #define NUM_FORMAT "%6.2lf"
    #define STR_FORMAT "%-10s"
    #define DRAW_HEIGHT 6

    double data[] =                             // enter input data here
      {5.00, 4.71, 6.00, 6.54, 1.00, 2.29, 0.00, -0.54};
                                                // output DFT:
    double dftR[N];                             // real part of DFT
    double dftI[N];                             // imaginary part of DFT
                                                // just for printing:
    double dftM[N];                             // magnitude of DFT
    double dftA[N];                             // argument (angle/phase) of DFT

    void printArray(double *array)
    {
      for (int i = 0; i < N; ++i)
        printf(" " NUM_FORMAT,array[i]);

      putchar('\n');
    }

    void drawArray(double *array, double scale)
    {
      for (int y = 0; y < DRAW_HEIGHT; ++y)
      {
        printf("        ");

        for (int x = 0; x < N; ++x)
        {
          printf("      ");
          putchar(((int) array[x] * scale) >= (DRAW_HEIGHT - y) ? '#' : ' ');
        }

        putchar('\n');
      }
    }

    void printDft(void)
    {
      printf(STR_FORMAT,"  magn.:"); printArray(dftM);
      printf(STR_FORMAT,"  phase:"); printArray(dftA);
      puts("  -----");
      printf(STR_FORMAT,"  real:"); printArray(dftR);
      printf(STR_FORMAT,"  imag.:"); printArray(dftI);
    }

    void dft(void)
    {
      for (int i = 0; i < N; ++i)
      {
        dftR[i] = 0;
        dftI[i] = 0;
      }

      for (int k = 0; k < N; ++k)
      {
        for (int n = 0; n < N; ++n)
        {
          double angle = (-2 * PI * k * n) / N;
          dftR[k] += data[n] * cos(angle);
          dftI[k] += data[n] * sin(angle);
        }

        // just for printing also precompute magnitudes and phases
        dftM[k] = sqrt(dftR[k] * dftR[k] + dftI[k] * dftI[k]);
        dftA[k] = atan2(dftI[k],dftR[k]);
      }
    }

    void idft(void)
    {
      for (int i = 0; i < N; ++i)
        data[i] = 0;

      for (int k = 0; k < N; ++k)
      {
        for (int n = 0; n < N; ++n)
        {
          double angle = (2 * PI * k * n) / N;
          data[k] += dftR[n] * cos(angle) - dftI[n] * sin(angle);
        }

        data[k] /= N;
      }
    }

    int main(void)
    {
      drawArray(data,1);
      printf(STR_FORMAT,"data:"); printArray(data);

      puts("\nDFT:");
      dft();
      drawArray(dftM,0.25);
      printDft();
      idft();

      puts("\nrestored:");
      printf(STR_FORMAT,"data:"); printArray(data);

      return 0;
    }

TODO: pictures, 2D version

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