What Makes a Fourier Transform Fast?

If there were ever an algorithm to radically change the landscape of computer science and engineering by making seemingly impossible problems possible, it would be the Fast Fourier Transform (FFT). On the surface, the algorithm seems like a simple application of recursion, and in principle, that is exactly what it is; however, the Fourier Transform is no ordinary transform -- it allows researchers and engineers to easily bounce back and forth between real space and frequency space and is the heart of many physics and engineering applications. From calculating superfluid vortex positions to super-resolution imaging, Fourier Transforms lay at the heart of many scientific disciplines and are essential to many algorithms we will cover later in this book.

Simply put, the Fourier Transform is a beautiful application of complex number systems; however, it would rarely be used today if not for the ability to quickly perform the operation with Fast Fourier Transform, first introduced by the great Frederick Gauss in 1805 and later independently discovered by James Cooley and John Tukey in 1965 [1]. Gauss (of course) already had too many things named after him and Cooley and Tukey both had cooler names, so the most common algorithm for FFT's today is known as the Cooley-Tukey algorithm.

What is a Fourier Transform?

To an outsider, the Fourier Transform looks like a mathematical mess -- certainly a far cry from the heroic portal between two domains I have depicted it to be; however, like most things, it's not as bad as it initially appears to be. So, here it is in all it's glory!

and

Where represents a function in frequency space, represents a value in frequency space, represents a function in real space, and represents a value in the real space. Note here that the only difference between the two exponential terms is a minus sign in the transformation to frequency space. As I mentioned, this is not intuitive syntax, so please allow me to explain a bit.

Firstly, what does the Fourier Transform do?

If we take a sum sinusoidal functions (like or ), we might find a complicated mess of waves between . Each constituent wave can be described by only one value: . So, instead of representing these curves as seen above, we could instead describe them as peaks in frequency space, as shown below.

This is what the Fourier Transform does! After performing the transform, it is now much, much easier to understand precisely which frequencies are in our waveform, which is essential to most areas of signal processing.

Now, how does this relate to the transformations above? Well, the easiest way is to substitute in the Euler's formula:

This clearly turns our function in frequency space into:

and our function in real space into:

Here, the and functions are clearly written in the formulas, so it looks much friendlier, right? This means that a point in real space is defined by the integral over all space of it's corresponding frequency function multiplied by sinusoidal oscillations.

Truth be told, even after seeing this math, I still didn't understand Fourier Transforms. Truth be told, I didn't understand it fully until I discretized real and frequency space to create the Discrete Fourier Transform (DFT), which is the only way to implement Fourier Transforms in code.

What is a Discrete Fourier Transform?

In principle, the Discrete Fourier Transform (DFT) is simply the Fourier transform with summations instead of integrals:

and

Where and are sequences of numbers in frequency and real space, respectively. In principle, this is no easier to understand than the previous case! For some reason, though, putting code to this transformation really helped me figure out what was actually going on.

function DFT(x)
    N = length(x)

    # We want two vectors here for real space (n) and frequency space (k)
    n = 0:N-1
    k = n'
    transform_matrix = exp.(-2im*pi*n*k/N)
    return transform_matrix*x

end
void dft(double complex *X, const size_t N) {
    double complex tmp[N];
    for (size_t i = 0; i < N; ++i) {
        tmp[i] = 0;
        for (size_t j = 0; j < N; ++j) {
            tmp[i] += X[j] * cexp(-2.0 * M_PI * I * j * i / N);
        }
    }

    memcpy(X, tmp, N * sizeof(*X));
}
(defn dft
  "take a vector of real numbers and return a vector of frequency
  space"
  [vx]
  (let [len (count vx)]
     (matrix-mult
      (partition len
                 (for [n (range len)
                       k (range len)]
                   ;; expresion below is
                   ;; e^(n*k*2*pi*(1/len)*(-i))
                   (c/exp (c/* n k
                               2 Math/PI
                               (/ len)
                               (c/complex 0 -1)))))
      vx)))
template <typename Iter>
void dft(Iter X, Iter last) {
  const auto N = last - X;
  std::vector<complex> tmp(N);
  for (auto i = 0; i < N; ++i) {
    for (auto j = 0; j < N; ++j) {
      tmp[i] += X[j] * exp(complex(0, -2.0 * M_PI * i * j / N));
    }
  }
  std::copy(std::begin(tmp), std::end(tmp), X);
}
dft :: [Complex Double] -> [Complex Double]
dft x = matMult dftMat x
  where
    n = length x
    w = exp $ (-2) * pi * (0 :+ 1) / fromIntegral n
    dftMat = [[w ^ (j * k) | j <- [0 .. n - 1]] | k <- [0 .. n - 1]]
    matMult m x = map (sum . zipWith (*) x) m
def dft(X):
    N = len(X)
    temp = [0]*N
    for i in range(N):
        for k in range(N):
            temp[i] += X[k] * exp(-2.0j*pi*i*k/N)
    return temp
function DFT(x)
    N = length(x)

    # We want two vectors here for real space (n) and frequency space (k)
    n = 0:N-1
    k = n'
    transform_matrix = exp.(-2im*pi*n*k/N)
    return transform_matrix*x

end
# rdi - array ptr
# rsi - array size
dft:
  push   rbx
  push   r12
  push   r13
  push   r14
  push   r15
  mov    r12, rdi                    # Save parameters
  mov    r13, rsi
  sub    rsp, r13                    # Make a double complex array
  xor    r14, r14                    # Set index to 0
dft_loop_i:
  cmp    r14, r13                    # Check if index is equal to array size
  je     dft_end_i
  lea    rax, [rsp + r14]            # Set tmp array to zero at r14
  mov    QWORD PTR [rax], 0
  mov    QWORD PTR [rax + 8], 0
  xor    r15, r15                    # Set second index to 0
dft_loop_j:
  cmp    r15, r13                    # Check if the index is equal to array size
  je     dft_end_j
  movsd  xmm1, two_pi                # Calculate xmm1 = -2pi * i * j / N
  mov    rax, r14
  imul   rax, r15
  shr    rax, 4
  cvtsi2sdq xmm2, rax
  mulsd  xmm1, xmm2
  cvtsi2sdq xmm2, r13
  divsd  xmm1, xmm2
  pxor   xmm0, xmm0                  # Set xmm0 to 0
  call   cexp
  lea    rax, [r12 + r15]            # Calculate X[i] * cexp(-2pi * i * j / N)
  movsd  xmm2, QWORD PTR [rax]
  movsd  xmm3, QWORD PTR [rax + 8]
  call   __muldc3
  lea    rax, [rsp + r14]
  movsd  xmm6, QWORD PTR [rax]       # Sum to tmp array
  movsd  xmm7, QWORD PTR [rax + 8]
  addsd  xmm6, xmm0
  addsd  xmm7, xmm1
  movsd  QWORD PTR [rax], xmm6       # Save to tmp array
  movsd  QWORD PTR [rax + 8], xmm7
  add    r15, 16
  jmp    dft_loop_j
dft_end_j:
  add    r14, 16
  jmp    dft_loop_i
dft_end_i:
  mov    rdi, r12                    # Move tmp array to array ptr
  mov    rsi, rsp
  mov    rdx, r13
  call   memcpy
  add    rsp, r13
  pop    r15
  pop    r14
  pop    r13
  pop    r12
  pop    rbx
  ret
function dft(x) {
  const N = x.length;

  // Initialize an array with N elements, filled with 0s
  return Array(N)
    .fill(new Complex(0, 0))
    .map((temp, i) => {
      // Reduce x into the sum of x_k * exp(-2*sqrt(-1)*pi*i*k/N)
      return x.reduce((a, b, k) => {
        return a.add(b.mul(new Complex(0, (-2 * Math.PI * i * k) / N).exp()));
      }, new Complex(0, 0)); // Start accumulating from 0
    });
}

In this function, we define n to be a set of integers from and arrange them to be a column. We then set k to be the same thing, but in a row. This means that when we multiply them together, we get a matrix, but not just any matrix! This matrix is the heart to the transformation itself!

M = [1.0+0.0im  1.0+0.0im           1.0+0.0im          1.0+0.0im;
     1.0+0.0im  6.12323e-17-1.0im  -1.0-1.22465e-16im -1.83697e-16+1.0im;
     1.0+0.0im -1.0-1.22465e-16im   1.0+2.44929e-16im -1.0-3.67394e-16im;
     1.0+0.0im -1.83697e-16+1.0im  -1.0-3.67394e-16im  5.51091e-16-1.0im]

It was amazing to me when I saw the transform for what it truly was: an actual transformation matrix! That said, the Discrete Fourier Transform is slow -- primarily because matrix multiplication is slow, and as mentioned before, slow code is not particularly useful. So what was the trick that everyone used to go from a Discrete Fourier Transform to a Fast Fourier Transform?

Recursion!

The Cooley-Tukey Algorithm

The problem with using a standard DFT is that it requires a large matrix multiplications and sums over all elements, which are prohibitively complex operations. The Cooley-Tukey algorithm calculates the DFT directly with fewer summations and without matrix multiplications. If necessary, DFT's can still be calculated directly at the early stages of the FFT calculation. The trick to the Cooley-Tukey algorithm is recursion. In particular, we split the matrix we wish to perform the FFT on into two parts: one for all elements with even indices and another for all odd indices. We then proceed to split the array again and again until we have a manageable array size to perform a DFT (or similar FFT) on. We can also perform a similar re-ordering by using a bit reversal scheme, where we output each array index's integer value in binary and flip it to find the new location of that element. With recursion, we can reduce the complexity to , which is a feasible operation.

In the end, the code looks like:

function cooley_tukey(x)
    N = length(x)

    if (N > 2)
        x_odd = cooley_tukey(x[1:2:N])
        x_even = cooley_tukey(x[2:2:N])
    else
        x_odd = x[1]
        x_even = x[2]
    end
    n = 0:N-1
    half = div(N,2)
    factor = exp.(-2im*pi*n/N)
    return vcat(x_odd .+ x_even .* factor[1:half],
                x_odd .- x_even .* factor[1:half])

end
void cooley_tukey(double complex *X, const size_t N) {
    if (N >= 2) {
        double complex tmp [N / 2];
        for (size_t i = 0; i < N / 2; ++i) {
            tmp[i] = X[2*i + 1];
            X[i] = X[2*i];
        }
        for (size_t i = 0; i < N / 2; ++i) {
            X[i + N / 2] = tmp[i];
        }

        cooley_tukey(X, N / 2);
        cooley_tukey(X + N / 2, N / 2);

        for (size_t i = 0; i < N / 2; ++i) {
            X[i + N / 2] = X[i] - cexp(-2.0 * I * M_PI * i / N) * X[i + N / 2];
            X[i] -= (X[i + N / 2]-X[i]);
        }
    }
}
(defn fft [vx]
  (let [len (count vx)]
    (if (= len 1)
      vx
      ;;else
      (let [;; take values of vx in the even indices
            even-indices (keep-indexed #(if (even? %1) %2) vx)
            ;; take values in the odd indices
            odd-indices (keep-indexed #(if (odd? %1) %2) vx)
            ;; recursion
            even-fft (fft even-indices)
            odd-fft (fft odd-indices)
            ;; make a sequence of e^(-2pi*i*k/N) where N is the length
            ;; vx and k range from 0 to N/2
            omegas-half (map
                         (comp c/exp
                               (partial c/*
                                        (/ len)
                                        2 Math/PI
                                        (c/complex 0 -1)))
                         (range 0 (quot len 2)))
            ;; take the negative of the first sequence because
            ;; e^(-2pi*i*(k+N/2)/N=-e^(-2pi*i*k/N) where k ranges from
            ;; 0 to N/2 
            omegas-2half (map c/- omegas-half)
            mult-add (partial map #(c/+ %3 (c/* %1 %2)))]
        (concat (mult-add omegas-half odd-fft even-fft)
                (mult-add omegas-2half odd-fft even-fft))))))
template <typename Iter>
void cooley_tukey(Iter first, Iter last) {
  auto size = last - first;
  if (size >= 2) {
    // split the range, with even indices going in the first half,
    // and odd indices going in the last half.
    auto temp = std::vector<complex>(size / 2);
    for (int i = 0; i < size / 2; ++i) {
      temp[i] = first[i * 2 + 1];
      first[i] = first[i * 2];
    }
    for (int i = 0; i < size / 2; ++i) {
      first[i + size / 2] = temp[i];
    }

    // recurse the splits and butterflies in each half of the range
    auto split = first + size / 2;
    cooley_tukey(first, split);
    cooley_tukey(split, last);

    // now combine each of those halves with the butterflies
    for (int k = 0; k < size / 2; ++k) {
      auto w = std::exp(complex(0, -2.0 * pi * k / size));

      auto& bottom = first[k];
      auto& top = first[k + size / 2];
      top = bottom - w * top;
      bottom -= top - bottom;
    }
  }
}
fft :: [Complex Double] -> [Complex Double]
fft x = fft' x
  where
    n = length x
    w0 = exp ((-2) * pi * (0 :+ 1) / fromIntegral n)
    w = M.fromList [(k % n, w0 ^ k) | k <- [0 .. n - 1]]
    fft' [x] = [x]
    fft' x =
      let (evens, odds) = partition (even . fst) $ zip [0 ..] x
          e = fft' $ map snd evens
          o = fft' $ map snd odds
          x1 = zipWith3 (\e o k -> e + o * w ! (k %n)) e o [0 ..]
          x2 = zipWith3 (\e o k -> e - o * w ! (k %n)) e o [0 ..]
       in x1 ++ x2
def cooley_tukey(X):
    N = len(X)
    if N <= 1:
        return X
    even = cooley_tukey(X[0::2])
    odd =  cooley_tukey(X[1::2])

    temp = [i for i in range(N)]
    for k in range(N//2):
        temp[k] = even[k] + exp(-2j*pi*k/N) * odd[k]
        temp[k+N//2] = even[k] - exp(-2j*pi*k/N)*odd[k]
    return temp
function cooley_tukey(x)
    N = length(x)

    if (N > 2)
        x_odd = cooley_tukey(x[1:2:N])
        x_even = cooley_tukey(x[2:2:N])
    else
        x_odd = x[1]
        x_even = x[2]
    end
    n = 0:N-1
    half = div(N,2)
    factor = exp.(-2im*pi*n/N)
    return vcat(x_odd .+ x_even .* factor[1:half],
                x_odd .- x_even .* factor[1:half])

end
# rdi - array ptr
# rsi - array size
cooley_tukey:
  cmp    rsi, 16                     # Check if size if greater then 1
  jle    cooley_tukey_return
  push   rbx
  push   r12
  push   r13
  push   r14
  push   r15
  mov    r12, rdi                    # Save parameters
  mov    r13, rsi
  mov    r14, rsi                    # Save N / 2
  shr    r14, 1
  sub    rsp, r14                    # Make a tmp array
  xor    r15, r15
  mov    rbx, r12
cooley_tukey_spliting:
  cmp    r15, r14
  je     cooley_tukey_split
  lea    rax, [r12 + 2 * r15]        # Moving all odd entries to the front of the array
  movaps xmm0, XMMWORD PTR [rax + 16]
  movaps xmm1, XMMWORD PTR [rax]
  movaps XMMWORD PTR [rsp + r15], xmm0
  movaps XMMWORD PTR [rbx], xmm1
  add    rbx, 16
  add    r15, 16
  jmp    cooley_tukey_spliting
cooley_tukey_split:
  mov    rax, rsp
  lea    rdi, [r12 + r13]
cooley_tukey_mov_data:
  cmp    rbx, rdi
  je     cooley_tukey_moved
  movaps xmm0, XMMWORD PTR [rax]
  movaps XMMWORD PTR [rbx], xmm0
  add    rbx, 16
  add    rax, 16
  jmp    cooley_tukey_mov_data
cooley_tukey_moved:
  add    rsp, r14
  mov    rdi, r12                   # Makking a recursive call
  mov    rsi, r14
  call   cooley_tukey
  lea    rdi, [r12 + r14]           # Makking a recursive call
  mov    rsi, r14
  call   cooley_tukey
  lea    rbx, [r12 + r14]
  mov    r14, rbx
  mov    r15, r12
cooley_tukey_loop:
  cmp    r15, rbx
  je     cooley_tukey_end
  pxor   xmm0, xmm0                 # Calculate cexp(-2.0 * I * M_PI * i / N)
  movsd  xmm1, two_pi
  mov    rax, r14
  sub    rax, rbx
  cvtsi2sdq xmm2, rax
  cvtsi2sdq xmm3, r13
  divsd  xmm2, xmm3
  mulsd  xmm1, xmm2
  call   cexp
  movq   xmm2, QWORD PTR [r14]      # Calculating X[i] - cexp() * X[i + N / 2]
  movq   xmm3, QWORD PTR [r14 + 8]
  call   __muldc3
  movq   xmm2, QWORD PTR [r15]
  movq   xmm3, QWORD PTR [r15 + 8]
  subsd  xmm2, xmm0
  subsd  xmm3, xmm1
  movq   QWORD PTR [r14], xmm2      # Save value in X[i + N / 2]
  movq   QWORD PTR [r14 + 8], xmm3
  movq   xmm0, QWORD PTR [r15]      # Calculating X[i] -= X[i + N / 2] - X[i]
  movq   xmm1, QWORD PTR [r15 + 8]
  subsd  xmm2, xmm0
  subsd  xmm3, xmm1
  subsd  xmm0, xmm2
  subsd  xmm1, xmm3
  movq   QWORD PTR [r15], xmm0
  movq   QWORD PTR [r15 + 8], xmm1
  add    r14, 16
  add    r15, 16
  jmp    cooley_tukey_loop
cooley_tukey_end:
  pop    r15
  pop    r14
  pop    r13
  pop    r12
  pop    rbx
cooley_tukey_return:
  ret
function cooley_tukey(x) {
  const N = x.length;
  const half = Math.floor(N / 2);
  if (N <= 1) {
    return x;
  }

  // Extract even and odd indexed elements with remainder mod 2
  const evens = cooley_tukey(x.filter((_, idx) => !(idx % 2)));
  const odds = cooley_tukey(x.filter((_, idx) => idx % 2));

  // Fill an array with null values
  let temp = Array(N).fill(null);

  for (let i = 0; i < half; i++) {
    const arg = odds[i].mul(new Complex(0, (-2 * Math.PI * i) / N).exp());

    temp[i] = evens[i].add(arg);
    temp[i + half] = evens[i].sub(arg);
  }

  return temp;
}

As a side note, we are enforcing that the array must be a power of 2 for the operation to work. This is a limitation of the fact that we are using recursion and dividing the array in 2 every time; however, if your array is not a power of 2, you can simply pad the leftover space with 0's until your array is a power of 2.

The above method is a perfectly valid FFT; however, it is missing the pictorial heart and soul of the Cooley-Tukey algorithm: Butterfly Diagrams.

Butterfly Diagrams

Butterfly Diagrams show where each element in the array goes before, during, and after the FFT. As mentioned, the FFT must perform a DFT. This means that even though we need to be careful about how we add elements together, we are still ultimately performing the following operation:

However, after shuffling the initial array (by bit reversing or recursive subdivision), we perform the matrix multiplication of the terms in pieces. Basically, we split the array into a series of omega values:

And at each step, we use the appropriate term. For example, imagine we need to perform an FFT of an array of only 2 elements. We can represent this addition with the following (radix-2) butterfly:

Here, the diagram means the following:

However, it turns out that the second half of our array of values is always the negative of the first half, so , so we can use the following butterfly diagram:

With the following equations:

By swapping out the second value in this way, we can save a good amount of space. Now imagine we need to combine more elements. In this case, we start with simple butterflies, as shown above, and then sum butterflies of butterflies. For example, if we have 8 elements, this might look like this:

Note that we can perform a DFT directly before using any butterflies, if we so desire, but we need to be careful with how we shuffle our array if that's the case. In the code snippet provided in the previous section, the subdivision was performed in the same function as the concatenation, so the ordering was always correct; however, if we were to re-order with bit-reversal, this might not be the case.

For example, take a look at the ordering of FFT (found on wikipedia) that performs the DFT shortcut:

Here, the ordering of the array was simply divided into even and odd elements once, but they did not recursively divide the arrays of even and odd elements again because they knew they would perform a DFT soon thereafter.

Ultimately, that's all I want to say about Fourier Transforms for now, but this chapter still needs a good amount of work! I'll definitely come back to this at some point, so let me know what you liked and didn't like and we can go from there!

Bibliography

1Cooley, James W and Tukey, John W, An algorithm for the machine calculation of complex Fourier series, JSTOR, 1965.

Example Code

To be clear, the example code this time will be complicated and requires the following functions:

  • An FFT library (either in-built or something like FFTW)
  • An approximation function to tell if two arrays are similar

As mentioned in the text, the Cooley-Tukey algorithm may be implemented either recursively or non-recursively, with the recursive method being much easier to implement. I would ask that you implement either the recursive or non-recursive methods (or both, if you feel so inclined). If the language you want to write your implementation in is already used, please append your code to the already existing codebase. As before, pull requests are favoured.

Note: I implemented this in Julia because the code seems more straightforward in Julia; however, if you wish to write better Julia code or better code in your own language, please feel free to do so! I do not claim that this is the most efficient way to implement the Cooley-Tukey method, so if you have a better way to do it, feel free to implement it that way!

using FFTW

#simple DFT function
function DFT(x)
    N = length(x)

    # We want two vectors here for real space (n) and frequency space (k)
    n = 0:N-1
    k = n'
    transform_matrix = exp.(-2im*pi*n*k/N)
    return transform_matrix*x

end

# Implementing the Cooley-Tukey Algorithm
function cooley_tukey(x)
    N = length(x)

    if (N > 2)
        x_odd = cooley_tukey(x[1:2:N])
        x_even = cooley_tukey(x[2:2:N])
    else
        x_odd = x[1]
        x_even = x[2]
    end
    n = 0:N-1
    half = div(N,2)
    factor = exp.(-2im*pi*n/N)
    return vcat(x_odd .+ x_even .* factor[1:half],
                x_odd .- x_even .* factor[1:half])

end

function bitreverse(a::Array)
    # First, we need to find the necessary number of bits
    digits = convert(Int,ceil(log2(length(a))))

    indices = [i for i = 0:length(a)-1]

    bit_indices = []
    for i = 1:length(indices)
        push!(bit_indices, bitstring(indices[i]))
    end

    # Now stripping the unnecessary numbers
    for i = 1:length(bit_indices)
        bit_indices[i] = bit_indices[i][end-digits:end]
    end

    # Flipping the bits
    for i =1:length(bit_indices)
        bit_indices[i] = reverse(bit_indices[i])
    end

    # Replacing indices
    for i = 1:length(indices)
        indices[i] = 0
        for j = 1:digits
            indices[i] += 2^(j-1) * parse(Int, string(bit_indices[i][end-j]))
        end
       indices[i] += 1
    end

    b = [float(i) for i = 1:length(a)]
    for i = 1:length(indices)
        b[i] = a[indices[i]]
    end

    return b
end

function iterative_cooley_tukey(x)
    N = length(x)
    logN = convert(Int,ceil(log2(length(x))))
    bnum = div(N,2)
    stride = 0;

    x = bitreverse(x)

    z = [Complex(x[i]) for i = 1:length(x)]
    for i = 1:logN
       stride = div(N, bnum)
       for j = 0:bnum-1
           start_index = j*stride + 1
           y = butterfly(z[start_index:start_index + stride - 1])
           for k = 1:length(y)
               z[start_index+k-1] = y[k]
           end
       end
       bnum = div(bnum,2)
    end

    return z
end

function butterfly(x)
    N = length(x)
    half = div(N,2)
    n = [i for i = 0:N-1]
    half = div(N,2)
    factor = exp.(-2im*pi*n/N)

    y = [0 + 0.0im for i = 1:length(x)]

    for i = 1:half
        y[i] = x[i] + x[half+i]*factor[i]
        y[half+i] = x[i] - x[half+i]*factor[i]
    end

    return y
end

function main()
    x = rand(128)
    y = cooley_tukey(x)
    z = iterative_cooley_tukey(x)
    w = fft(x)
    if(isapprox(y, w))
        println("Recursive Cooley Tukey matches fft() from FFTW package.")
    end
    if(isapprox(z, w))
        println("Iterative Cooley Tukey matches fft() from FFTW package.")
    end
end

main()
#include <complex.h>
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <time.h>
#include <fftw3.h>

void fft(double complex *x, int n) {
    double complex y[n];
    memset(y, 0, sizeof(y));
    fftw_plan p;

    p = fftw_plan_dft_1d(n, (fftw_complex*)x, (fftw_complex*)y,
                         FFTW_FORWARD, FFTW_ESTIMATE);

    fftw_execute(p);
    fftw_destroy_plan(p);

    for (size_t i = 0; i < n; ++i) {
        x[i] = y[i] / sqrt((double)n);
    }
}

void dft(double complex *X, const size_t N) {
    double complex tmp[N];
    for (size_t i = 0; i < N; ++i) {
        tmp[i] = 0;
        for (size_t j = 0; j < N; ++j) {
            tmp[i] += X[j] * cexp(-2.0 * M_PI * I * j * i / N);
        }
    }

    memcpy(X, tmp, N * sizeof(*X));
}

void cooley_tukey(double complex *X, const size_t N) {
    if (N >= 2) {
        double complex tmp [N / 2];
        for (size_t i = 0; i < N / 2; ++i) {
            tmp[i] = X[2*i + 1];
            X[i] = X[2*i];
        }
        for (size_t i = 0; i < N / 2; ++i) {
            X[i + N / 2] = tmp[i];
        }

        cooley_tukey(X, N / 2);
        cooley_tukey(X + N / 2, N / 2);

        for (size_t i = 0; i < N / 2; ++i) {
            X[i + N / 2] = X[i] - cexp(-2.0 * I * M_PI * i / N) * X[i + N / 2];
            X[i] -= (X[i + N / 2]-X[i]);
        }
    }
}

void bit_reverse(double complex *X, size_t N) {
    for (int i = 0; i < N; ++i) {
        int n = i;
        int a = i;
        int count = (int)log2((double)N) - 1;

        n >>= 1;
        while (n > 0) {
            a = (a << 1) | (n & 1);
            count--;
            n >>= 1;
        }
        n = (a << count) & ((1 << (int)log2((double)N)) - 1);

        if (n > i) {
            double complex tmp = X[i];
            X[i] = X[n];
            X[n] = tmp;
        }
    }
}

void iterative_cooley_tukey(double complex *X, size_t N) {
    bit_reverse(X, N);

    for (int i = 1; i <= log2((double)N); ++i) {
        int stride = pow(2, i);
        double complex w = cexp(-2.0 * I * M_PI / stride);
        for (size_t j = 0; j < N; j += stride) {
            double complex v = 1.0;
            for (size_t k = 0; k < stride / 2; ++k) {
                X[k + j + stride / 2] = X[k + j] - v * X[k + j + stride / 2];
                X[k + j] -= (X[k + j + stride / 2] - X[k + j]);
                v *= w;
            }
        }
    }
}

void approx(double complex *X, double complex *Y, size_t N) {
    for (size_t i = 0; i < N; ++i) {
        if (cabs(X[i]) - cabs(Y[i]) > 1E-5) {
            printf("This is not approximate.\n");
            return;
        }
    }
    printf("This is approximate.\n");
}

int main() {
    srand(time(NULL));
    double complex x[64], y[64], z[64];
    for (size_t i = 0; i < 64; ++i) {
        x[i] = rand() / (double) RAND_MAX;
        y[i] = x[i];
        z[i] = x[i];
    }

    fft(x, 64);
    cooley_tukey(y, 64);
    iterative_cooley_tukey(z, 64);

    approx(x, y, 64);
    approx(x, z, 64);

    return 0;
}
(ns fft.core
  (:require [complex.core :as c]))
;; complex is a jar for complex numbers
;; https://github.com/alanforr/complex
;; add [complex "0.1.11"] to :dependencies in your project.clj
;; and run lein repl or lein deps in the terminal
(defn matrix-mult
  "take a matrix m and a vector v which length is number of columns
  ,return a vector of applying dot-product between v and each row of
  m. the returned vector's length is the number of rows of m"
  [m v]
  (mapv (comp (partial apply c/+)
              (partial map c/* v))
        m))
(defn dft
  "take a vector of real numbers and return a vector of frequency
  space"
  [vx]
  (let [len (count vx)]
     (matrix-mult
      (partition len
                 (for [n (range len)
                       k (range len)]
                   ;; expresion below is
                   ;; e^(n*k*2*pi*(1/len)*(-i))
                   (c/exp (c/* n k
                               2 Math/PI
                               (/ len)
                               (c/complex 0 -1)))))
      vx)))
(defn fft [vx]
  (let [len (count vx)]
    (if (= len 1)
      vx
      ;;else
      (let [;; take values of vx in the even indices
            even-indices (keep-indexed #(if (even? %1) %2) vx)
            ;; take values in the odd indices
            odd-indices (keep-indexed #(if (odd? %1) %2) vx)
            ;; recursion
            even-fft (fft even-indices)
            odd-fft (fft odd-indices)
            ;; make a sequence of e^(-2pi*i*k/N) where N is the length
            ;; vx and k range from 0 to N/2
            omegas-half (map
                         (comp c/exp
                               (partial c/*
                                        (/ len)
                                        2 Math/PI
                                        (c/complex 0 -1)))
                         (range 0 (quot len 2)))
            ;; take the negative of the first sequence because
            ;; e^(-2pi*i*(k+N/2)/N=-e^(-2pi*i*k/N) where k ranges from
            ;; 0 to N/2 
            omegas-2half (map c/- omegas-half)
            mult-add (partial map #(c/+ %3 (c/* %1 %2)))]
        (concat (mult-add omegas-half odd-fft even-fft)
                (mult-add omegas-2half odd-fft even-fft))))))
(defn -main [& args]
    (let [vx [0 1 2 3]
          len (count vx)
          ;; calculate the next power of 2 after len
          ;; the reason behind this is to fill them with zeros for fft
          next-len (->>
                    [len 2]
                    (map #(Math/log %))
                    (apply /)
                    Math/ceil
                    (Math/pow 2)
                    int)
          ;; add zeros at the end of vx
          complete-vx (into vx (repeat (- next-len len) 0))
          fft-cvx (fft complete-vx)
          dft-cvx (dft complete-vx)
          diffv (mapv c/- fft-cvx dft-cvx)]
    (println "vx:" vx)
    (println "complete-vx:" complete-vx)
    (println "result from fft:" (map c/stringify fft-cvx))
    (println "result from dft:" (map c/stringify dft-cvx))
    (println "difference: " (map c/stringify diffv))))
// written by Gathros, modernized by Nicole Mazzuca.

#include <algorithm>
#include <array>
#include <complex>
#include <cstdint>
#include <vector>

// These headers are for presentation not for the algorithm.
#include <iomanip>
#include <iostream>
#include <random>

using std::begin;
using std::end;
using std::swap;

using std::size_t;

using complex = std::complex<double>;
static const double pi = 3.14159265358979323846264338327950288419716;

template <typename Iter>
void dft(Iter X, Iter last) {
  const auto N = last - X;
  std::vector<complex> tmp(N);
  for (auto i = 0; i < N; ++i) {
    for (auto j = 0; j < N; ++j) {
      tmp[i] += X[j] * exp(complex(0, -2.0 * M_PI * i * j / N));
    }
  }
  std::copy(std::begin(tmp), std::end(tmp), X);
}

// `cooley_tukey` does the cooley-tukey algorithm, recursively
template <typename Iter>
void cooley_tukey(Iter first, Iter last) {
  auto size = last - first;
  if (size >= 2) {
    // split the range, with even indices going in the first half,
    // and odd indices going in the last half.
    auto temp = std::vector<complex>(size / 2);
    for (int i = 0; i < size / 2; ++i) {
      temp[i] = first[i * 2 + 1];
      first[i] = first[i * 2];
    }
    for (int i = 0; i < size / 2; ++i) {
      first[i + size / 2] = temp[i];
    }

    // recurse the splits and butterflies in each half of the range
    auto split = first + size / 2;
    cooley_tukey(first, split);
    cooley_tukey(split, last);

    // now combine each of those halves with the butterflies
    for (int k = 0; k < size / 2; ++k) {
      auto w = std::exp(complex(0, -2.0 * pi * k / size));

      auto& bottom = first[k];
      auto& top = first[k + size / 2];
      top = bottom - w * top;
      bottom -= top - bottom;
    }
  }
}

// note: (last - first) must be less than 2**32 - 1
template <typename Iter>
void sort_by_bit_reverse(Iter first, Iter last) {
  // sorts the range [first, last) in bit-reversed order,
  // by the method suggested by the FFT
  auto size = last - first;

  for (std::uint32_t i = 0; i < size; ++i) {
    auto b = i;
    b = (((b & 0xaaaaaaaa) >> 1) | ((b & 0x55555555) << 1));
    b = (((b & 0xcccccccc) >> 2) | ((b & 0x33333333) << 2));
    b = (((b & 0xf0f0f0f0) >> 4) | ((b & 0x0f0f0f0f) << 4));
    b = (((b & 0xff00ff00) >> 8) | ((b & 0x00ff00ff) << 8));
    b = ((b >> 16) | (b << 16)) >> (32 - std::uint32_t(log2(size)));
    if (b > i) {
      swap(first[b], first[i]);
    }
  }
}

// `iterative_cooley_tukey` does the cooley-tukey algorithm iteratively
template <typename Iter>
void iterative_cooley_tukey(Iter first, Iter last) {
  sort_by_bit_reverse(first, last);

  // perform the butterfly on the range
  auto size = last - first;
  for (int stride = 2; stride <= size; stride *= 2) {
    auto w = exp(complex(0, -2.0 * pi / stride));
    for (int j = 0; j < size; j += stride) {
      auto v = complex(1.0);
      for (int k = 0; k < stride / 2; k++) {
        first[k + j + stride / 2] =
            first[k + j] - v * first[k + j + stride / 2];
        first[k + j] -= (first[k + j + stride / 2] - first[k + j]);
        v *= w;
      }
    }
  }
}

int main() {
  // initalize the FFT inputs
  std::random_device random_device;
  std::mt19937 rng(random_device());
  std::uniform_real_distribution<double> distribution(0.0, 1.0);

  std::array<complex, 64> initial;
  std::generate(
      begin(initial), end(initial), [&] { return distribution(rng); });

  auto recursive = initial;
  auto iterative = initial;

  // Preform an FFT on the arrays.
  cooley_tukey(begin(recursive), end(recursive));
  iterative_cooley_tukey(begin(iterative), end(iterative));

  // Check if the arrays are approximately equivalent
  std::cout << std::right << std::setw(16) << "idx" << std::setw(16) << "rec"
            << std::setw(16) << "it" << std::setw(16) << "subtracted" << '\n';
  for (size_t i = 0; i < initial.size(); ++i) {
    auto rec = recursive[i];
    auto it = iterative[i];
    std::cout << std::setw(16) << i << std::setw(16) << std::abs(rec)
              << std::setw(16) << std::abs(it) << std::setw(16)
              << (std::abs(rec) - std::abs(it)) << '\n';
  }
}
import Data.Complex
import Data.List (partition)
import Data.Map ((!))
import qualified Data.Map as M
import Data.Ratio

dft :: [Complex Double] -> [Complex Double]
dft x = matMult dftMat x
  where
    n = length x
    w = exp $ (-2) * pi * (0 :+ 1) / fromIntegral n
    dftMat = [[w ^ (j * k) | j <- [0 .. n - 1]] | k <- [0 .. n - 1]]
    matMult m x = map (sum . zipWith (*) x) m

fft :: [Complex Double] -> [Complex Double]
fft x = fft' x
  where
    n = length x
    w0 = exp ((-2) * pi * (0 :+ 1) / fromIntegral n)
    w = M.fromList [(k % n, w0 ^ k) | k <- [0 .. n - 1]]
    fft' [x] = [x]
    fft' x =
      let (evens, odds) = partition (even . fst) $ zip [0 ..] x
          e = fft' $ map snd evens
          o = fft' $ map snd odds
          x1 = zipWith3 (\e o k -> e + o * w ! (k %n)) e o [0 ..]
          x2 = zipWith3 (\e o k -> e - o * w ! (k %n)) e o [0 ..]
       in x1 ++ x2

main = do
  print $ dft [0, 1, 2, 3]
  print $ fft [0, 1, 2, 3]
from random import random
from cmath import exp, pi
from math import log2

def dft(X):
    N = len(X)
    temp = [0]*N
    for i in range(N):
        for k in range(N):
            temp[i] += X[k] * exp(-2.0j*pi*i*k/N)
    return temp

def cooley_tukey(X):
    N = len(X)
    if N <= 1:
        return X
    even = cooley_tukey(X[0::2])
    odd =  cooley_tukey(X[1::2])

    temp = [i for i in range(N)]
    for k in range(N//2):
        temp[k] = even[k] + exp(-2j*pi*k/N) * odd[k]
        temp[k+N//2] = even[k] - exp(-2j*pi*k/N)*odd[k]
    return temp

def bitReverse(X):
    N = len(X)
    temp = [i for i in range(N)]
    for k in range(N):
        b =  sum(1<<(int(log2(N))-1-i) for i in range(int(log2(N))) if k>>i&1)
        temp[k] = X[b]
        temp[b] = X[k]
    return temp

def iterative_cooley_tukey(X):
    N = len(X)

    X = bitReverse(X)

    for i in range(1, int(log2(N)) + 1):
        stride = 2**i
        w = exp(-2j*pi/stride)
        for j in range(0, N, stride):
            v = 1
            for k in range(stride//2):
                X[k + j + stride//2] = X[k + j] - v*X[k + j + stride//2];
                X[k + j] -= (X[k + j + stride//2] - X[k + j]);
                v *= w;
    return X

X = []

for i in range(64):
    X.append(random())

Y = cooley_tukey(X)
Z = iterative_cooley_tukey(X)
T = dft(X)

print(all(abs([Y[i] - Z[i] for i in range(64)][j]) < 1 for j in range(64)))
print(all(abs([Y[i] - T[i] for i in range(64)][j]) < 1 for j in range(64)))

Some rather impressive scratch code was submitted by Jie and can be found here: https://scratch.mit.edu/projects/37759604/#editor

.intel_syntax noprefix

.section .rodata
  two:           .double 2.0
  one:           .double 1.0
  two_pi:        .double -6.28318530718
  rand_max:      .long 4290772992
                 .long 1105199103
  fmt:           .string "%g\n"

.section .text
  .global main
  .extern printf, memset, memcpy, srand, rand, time, cexp, __muldc3, cabs, log2

# rdi - array ptr
# rsi - array size
dft:
  push   rbx
  push   r12
  push   r13
  push   r14
  push   r15
  mov    r12, rdi                    # Save parameters
  mov    r13, rsi
  sub    rsp, r13                    # Make a double complex array
  xor    r14, r14                    # Set index to 0
dft_loop_i:
  cmp    r14, r13                    # Check if index is equal to array size
  je     dft_end_i
  lea    rax, [rsp + r14]            # Set tmp array to zero at r14
  mov    QWORD PTR [rax], 0
  mov    QWORD PTR [rax + 8], 0
  xor    r15, r15                    # Set second index to 0
dft_loop_j:
  cmp    r15, r13                    # Check if the index is equal to array size
  je     dft_end_j
  movsd  xmm1, two_pi                # Calculate xmm1 = -2pi * i * j / N
  mov    rax, r14
  imul   rax, r15
  shr    rax, 4
  cvtsi2sdq xmm2, rax
  mulsd  xmm1, xmm2
  cvtsi2sdq xmm2, r13
  divsd  xmm1, xmm2
  pxor   xmm0, xmm0                  # Set xmm0 to 0
  call   cexp
  lea    rax, [r12 + r15]            # Calculate X[i] * cexp(-2pi * i * j / N)
  movsd  xmm2, QWORD PTR [rax]
  movsd  xmm3, QWORD PTR [rax + 8]
  call   __muldc3
  lea    rax, [rsp + r14]
  movsd  xmm6, QWORD PTR [rax]       # Sum to tmp array
  movsd  xmm7, QWORD PTR [rax + 8]
  addsd  xmm6, xmm0
  addsd  xmm7, xmm1
  movsd  QWORD PTR [rax], xmm6       # Save to tmp array
  movsd  QWORD PTR [rax + 8], xmm7
  add    r15, 16
  jmp    dft_loop_j
dft_end_j:
  add    r14, 16
  jmp    dft_loop_i
dft_end_i:
  mov    rdi, r12                    # Move tmp array to array ptr
  mov    rsi, rsp
  mov    rdx, r13
  call   memcpy
  add    rsp, r13
  pop    r15
  pop    r14
  pop    r13
  pop    r12
  pop    rbx
  ret

# rdi - array ptr
# rsi - array size
cooley_tukey:
  cmp    rsi, 16                     # Check if size if greater then 1
  jle    cooley_tukey_return
  push   rbx
  push   r12
  push   r13
  push   r14
  push   r15
  mov    r12, rdi                    # Save parameters
  mov    r13, rsi
  mov    r14, rsi                    # Save N / 2
  shr    r14, 1
  sub    rsp, r14                    # Make a tmp array
  xor    r15, r15
  mov    rbx, r12
cooley_tukey_spliting:
  cmp    r15, r14
  je     cooley_tukey_split
  lea    rax, [r12 + 2 * r15]        # Moving all odd entries to the front of the array
  movaps xmm0, XMMWORD PTR [rax + 16]
  movaps xmm1, XMMWORD PTR [rax]
  movaps XMMWORD PTR [rsp + r15], xmm0
  movaps XMMWORD PTR [rbx], xmm1
  add    rbx, 16
  add    r15, 16
  jmp    cooley_tukey_spliting
cooley_tukey_split:
  mov    rax, rsp
  lea    rdi, [r12 + r13]
cooley_tukey_mov_data:
  cmp    rbx, rdi
  je     cooley_tukey_moved
  movaps xmm0, XMMWORD PTR [rax]
  movaps XMMWORD PTR [rbx], xmm0
  add    rbx, 16
  add    rax, 16
  jmp    cooley_tukey_mov_data
cooley_tukey_moved:
  add    rsp, r14
  mov    rdi, r12                   # Makking a recursive call
  mov    rsi, r14
  call   cooley_tukey
  lea    rdi, [r12 + r14]           # Makking a recursive call
  mov    rsi, r14
  call   cooley_tukey
  lea    rbx, [r12 + r14]
  mov    r14, rbx
  mov    r15, r12
cooley_tukey_loop:
  cmp    r15, rbx
  je     cooley_tukey_end
  pxor   xmm0, xmm0                 # Calculate cexp(-2.0 * I * M_PI * i / N)
  movsd  xmm1, two_pi
  mov    rax, r14
  sub    rax, rbx
  cvtsi2sdq xmm2, rax
  cvtsi2sdq xmm3, r13
  divsd  xmm2, xmm3
  mulsd  xmm1, xmm2
  call   cexp
  movq   xmm2, QWORD PTR [r14]      # Calculating X[i] - cexp() * X[i + N / 2]
  movq   xmm3, QWORD PTR [r14 + 8]
  call   __muldc3
  movq   xmm2, QWORD PTR [r15]
  movq   xmm3, QWORD PTR [r15 + 8]
  subsd  xmm2, xmm0
  subsd  xmm3, xmm1
  movq   QWORD PTR [r14], xmm2      # Save value in X[i + N / 2]
  movq   QWORD PTR [r14 + 8], xmm3
  movq   xmm0, QWORD PTR [r15]      # Calculating X[i] -= X[i + N / 2] - X[i]
  movq   xmm1, QWORD PTR [r15 + 8]
  subsd  xmm2, xmm0
  subsd  xmm3, xmm1
  subsd  xmm0, xmm2
  subsd  xmm1, xmm3
  movq   QWORD PTR [r15], xmm0
  movq   QWORD PTR [r15 + 8], xmm1
  add    r14, 16
  add    r15, 16
  jmp    cooley_tukey_loop
cooley_tukey_end:
  pop    r15
  pop    r14
  pop    r13
  pop    r12
  pop    rbx
cooley_tukey_return:
  ret

# rdi - array ptr
# rsi - array size
bit_reverse:
  push   rbx
  push   r12
  push   r13
  push   r14
  push   r15
  mov    r12, rdi                  # Save parameters
  mov    r13, rsi
  shr    r13, 4
  xor    r14, r14                  # Loop through all entries
bit_reverse_entries:
  cmp    r14, r13
  je     bit_reverse_return
  cvtsi2sdq xmm0, r13              # Calculating the number of bit in N
  call   log2
  cvttsd2si rcx, xmm0
  mov    rdi, 1                    # Calculating (1 << log2(N)) - 1
  sal    edi, cl
  sub    edi, 1
  sub    ecx, 1
  mov    rax, r14
  mov    r15, r14
bit_reverse_loop:
  sar    r15                       # Check if r15 is 0
  je     bit_reverse_reversed
  sal    rax, 1                    # Calculating (rax << 1) | (r15 & 1)
  mov    rsi, r15
  and    rsi, 1
  or     rax, rsi
  sub    ecx, 1                    # Decrement bit count
  jmp    bit_reverse_loop
bit_reverse_reversed:
  sal    eax, cl                   # Calculate (rax << rcx) & (1 << bit count)
  and    rax, rdi
  cmp    rax, r14                  # Check if rax is greater then r14
  jle    bit_reverse_no_swap       # If so then swap entries
  shl    rax, 4                    # Times index by 16 to get bytes to entry
  shl    r14, 4
  movaps xmm0, XMMWORD PTR [r12 + rax]
  movaps xmm1, XMMWORD PTR [r12 + r14]
  movaps XMMWORD PTR [r12 + rax], xmm1
  movaps XMMWORD PTR [r12 + r14], xmm0
  shr    r14, 4
bit_reverse_no_swap:
  add    r14, 1
  jmp    bit_reverse_entries
bit_reverse_return:
  pop    r15
  pop    r14
  pop    r13
  pop    r12
  pop    rbx
  ret

# rdi - array ptr
# rsi - array size
iterative_cooley_tukey:
  push   r12
  push   r13
  push   r14
  push   r15
  push   rbx
  sub    rsp, 48
  mov    r12, rdi
  mov    r13, rsi
  call   bit_reverse              # Bit reversing array
  sar    r13, 4                   # Calculate log2(N)
  cvtsi2sdq xmm0, r13
  call   log2
  cvttsd2si rax, xmm0
  mov    QWORD PTR [rsp], rax     # Save it to the stack
  mov    r14, 1
iter_ct_loop_i:
  cmp    r14, rax                 # Check if r14 is greater then log2(N)
  jg     iter_ct_end_i
  movsd  xmm0, two                # Calculate stride = 2^(r14)
  cvtsi2sdq xmm1, r14
  call   pow
  cvttsd2si r10, xmm0
  mov    QWORD PTR [rsp + 40], r10# move stride to stack
  movsd  xmm1, two_pi             # Calculating cexp(-2pi * I / stride)
  divsd  xmm1, xmm0
  pxor   xmm0, xmm0
  call   cexp
  movq   QWORD PTR [rsp + 8], xmm0  # Save it to stack
  movq   QWORD PTR [rsp + 16], xmm1
  xor    r15, r15
iter_ct_loop_j:
  cmp    r15, r13                 # Check if r15 is less then array size
  je     iter_ct_end_j
  movsd  xmm4, one                # Save 1 + 0i to stack
  pxor   xmm5, xmm5
  movsd  QWORD PTR [rsp + 24], xmm4
  movsd  QWORD PTR [rsp + 32], xmm5
  xor    rbx, rbx
  mov    rax, QWORD PTR [rsp + 40]# Calculate stride / 2
  sar    rax, 1
iter_ct_loop_k:
  cmp    rbx, rax                 # Check if rbx is less then stride / 2
  je     iter_ct_end_k
  mov    r8, r15                  # Saving pointers to X[k + j + stride / 2] and X[k + j]
  add    r8, rbx
  sal    r8, 4
  mov    r9, QWORD PTR [rsp + 40]
  sal    r9, 3
  add    r9, r8
  lea    r9, [r12 + r9]
  lea    r8, [r12 + r8]
  movsd  xmm0, QWORD PTR [r9]     # Calculate X[k + j] - v * X[k + j + stride / 2]
  movsd  xmm1, QWORD PTR [r9 + 8]
  movsd  xmm2, QWORD PTR [rsp + 24]
  movsd  xmm3, QWORD PTR [rsp + 32]
  call   __muldc3
  movsd  xmm2, QWORD PTR [r8]
  movsd  xmm3, QWORD PTR [r8 + 8]
  subsd  xmm2, xmm0
  subsd  xmm3, xmm1
  movsd  QWORD PTR [r9], xmm2     # Saving answer
  movsd  QWORD PTR [r9 + 8], xmm3
  movsd  xmm0, QWORD PTR [r8]     # Calculating X[k + j] - (X[k + j + stride / 2] - X[k + j])
  movsd  xmm1, QWORD PTR [r8 + 8]
  subsd  xmm2, xmm0
  subsd  xmm3, xmm1
  subsd  xmm0, xmm2
  subsd  xmm1, xmm3
  movsd  QWORD PTR [r8], xmm0     # Saving answer
  movsd  QWORD PTR [r8 + 8], xmm1
  movsd  xmm0, QWORD PTR [rsp + 24] # Calculating v * w
  movsd  xmm1, QWORD PTR [rsp + 32]
  movsd  xmm2, QWORD PTR [rsp + 8]
  movsd  xmm3, QWORD PTR [rsp + 16]
  call   __muldc3
  movsd  QWORD PTR [rsp + 24], xmm0 # Saving answer
  movsd  QWORD PTR [rsp + 32], xmm1
  add    rbx, 1
  mov    rax, QWORD PTR [rsp + 40]
  sar    rax, 1
  jmp    iter_ct_loop_k
iter_ct_end_k:
  add    r15, QWORD PTR [rsp + 40]
  jmp    iter_ct_loop_j
iter_ct_end_j:
  add    r14, 1
  mov    rax, QWORD PTR [rsp]
  jmp    iter_ct_loop_i
iter_ct_end_i:
  add    rsp, 48
  pop    rbx
  pop    r15
  pop    r14
  pop    r13
  pop    r12
  ret

# rdi - array a ptr
# rsi - array b ptr
# rdx - array size
approx:
  push   r12
  push   r13
  push   r14
  push   r15
  mov    r12, rdi
  mov    r13, rsi
  mov    r14, rdx
  lea    r15, [rdi + rdx]
  sub    rsp, 8
approx_loop:
  cmp    r12, r15
  je     approx_return
  movsd  xmm0, QWORD PTR[r13]
  movsd  xmm1, QWORD PTR[r13 + 8]
  call   cabs
  movsd  QWORD PTR [rsp], xmm0
  movsd  xmm0, QWORD PTR[r12]
  movsd  xmm1, QWORD PTR[r12 + 8]
  call   cabs
  movsd  xmm1, QWORD PTR [rsp]
  subsd  xmm0, xmm1
  mov    rdi, OFFSET fmt
  mov    rax, 1
  call   printf
  add    r12, 16
  add    r13, 16
  jmp    approx_loop
approx_return:
  add    rsp, 8
  pop    r15
  pop    r14
  pop    r13
  pop    r12
  ret

main:
  push   r12
  sub    rsp, 2048
  mov    rdi, 0
  call   time
  mov    edi, eax
  call   srand
  lea    r12, [rsp + 1024]
loop:
  cmp    r12, rsp
  je     end_loop
  sub    r12, 16
  call   rand
  cvtsi2sd xmm0, rax
  divsd  xmm0, rand_max
  lea    rax, [r12 + 1024]
  movsd  QWORD PTR [r12], xmm0
  movsd  QWORD PTR [rax], xmm0
  mov    QWORD PTR [r12 + 8], 0
  mov    QWORD PTR [rax + 8], 0
  jmp    loop
end_loop:
  mov    rdi, rsp
  mov    rsi, 1024
  call   iterative_cooley_tukey
  lea    rdi, [rsp + 1024]
  mov    rsi, 1024
  call   cooley_tukey
  mov    rdi, rsp
  lea    rsi, [rsp + 1024]
  mov    rdx, 1024
  call   approx
  xor    rax, rax
  add    rsp, 2048
  pop    r12
  ret
const Complex = require("complex.js");

function dft(x) {
  const N = x.length;

  // Initialize an array with N elements, filled with 0s
  return Array(N)
    .fill(new Complex(0, 0))
    .map((temp, i) => {
      // Reduce x into the sum of x_k * exp(-2*sqrt(-1)*pi*i*k/N)
      return x.reduce((a, b, k) => {
        return a.add(b.mul(new Complex(0, (-2 * Math.PI * i * k) / N).exp()));
      }, new Complex(0, 0)); // Start accumulating from 0
    });
}

function cooley_tukey(x) {
  const N = x.length;
  const half = Math.floor(N / 2);
  if (N <= 1) {
    return x;
  }

  // Extract even and odd indexed elements with remainder mod 2
  const evens = cooley_tukey(x.filter((_, idx) => !(idx % 2)));
  const odds = cooley_tukey(x.filter((_, idx) => idx % 2));

  // Fill an array with null values
  let temp = Array(N).fill(null);

  for (let i = 0; i < half; i++) {
    const arg = odds[i].mul(new Complex(0, (-2 * Math.PI * i) / N).exp());

    temp[i] = evens[i].add(arg);
    temp[i + half] = evens[i].sub(arg);
  }

  return temp;
}

function bit_reverse_idxs(n) {
  if (!n) {
    return [0];
  } else {
    const twice = bit_reverse_idxs(n - 1).map(x => 2 * x);
    return twice.concat(twice.map(x => x + 1));
  }
}

function bit_reverse(x) {
  const N = x.length;
  const indexes = bit_reverse_idxs(Math.log2(N));
  return x.map((_, i) => x[indexes[i]]);
}

// Assumes log_2(N) is an integer
function iterative_cooley_tukey(x) {
  const N = x.length;

  x = bit_reverse(x);

  for (let i = 1; i <= Math.log2(N); i++) {
    const stride = 2 ** i;
    const half = stride / 2;
    const w = new Complex(0, (-2 * Math.PI) / stride).exp();
    for (let j = 0; j < N; j += stride) {
      let v = new Complex(1, 0);
      for (let k = 0; k < half; k++) {
        // perform butterfly multiplication
        x[k + j + half] = x[k + j].sub(v.mul(x[k + j + half]));
        x[k + j] = x[k + j].sub(x[k + j + half].sub(x[k + j]));
        // accumulate v as powers of w
        v = v.mul(w);
      }
    }
  }

  return x;
}

// Check if two arrays of complex numbers are approximately equal
function approx(x, y, tol = 1e-12) {
  let diff = 0;
  for (let i = 0; i < x.length; i++) {
    diff += x[i].sub(y[i]).abs();
  }
  return diff < tol;
}

const X = Array.from(Array(8), () => new Complex(Math.random(), 0));
const Y = cooley_tukey(X);
const Z = iterative_cooley_tukey(X);
const T = dft(X);

// Check if the calculations are correct within a small tolerance
console.log("Cooley tukey approximation is accurate: ", approx(Y, T));
console.log("Iterative cooley tukey approximation is accurate: ", approx(Z, T));

License

Code Examples

The code examples are licensed under the MIT license (found in LICENSE.md).

Text

The text of this chapter was written by James Schloss and is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Images/Graphics
Pull Requests

After initial licensing (#560), the following pull requests have modified the text or graphics of this chapter:

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