Fast Fourier Transforms

# Title: Fast Fourier Transforms

## 1 GSL::FFT::Complex::PackedArray class

In GSL, the complex FFT routines are provided to apply to the type `gsl_complex_packed_array` defined as

    typedef double *gsl_complex_packed_array ;

In Ruby/GSL, the `GSL::FFT::Complex::PackedArray` class is defined as a derived class of the GSL::Vector class, and the complex FFT routines are provided as methods for the `PackedArray` class.

`GSL::FFT::Complex::PackedArray.new(size)`
`GSL::FFT::Complex::PackedArray.alloc(size)`
  Constructors. In the case of FFT for a real vector of length n, the size must be size = 2*n.
`GSL::FFT::Complex::PackedArray#get(i)`
`GSL::FFT::Complex::PackedArray#[]`
  This returns the i-th complex number in the array.
`GSL::FFT::Complex::PackedArray#real(i)`
`GSL::FFT::Complex::PackedArray#imag(i)`
  Return the real or the imaginary part of the i-th complex number.
`GSL::FFT::Complex::PackedArray#set(i, arg)`
`GSL::FFT::Complex::PackedArray#[] =`

  Set the i-th complex number.

  ex) Set 2-th elements to 1 + 2i.

      ex1:
            pary.set(2, 1, 2)
      ex2:
            pary.set(2, [1, 2])
      ex3:
            z = GSL::Complex.new(1, 2)
            pary.set(2, z)

`GSL::FFT::Complex::PackedArray#set_real(i, val)`
`GSL::FFT::Complex::PackedArray#set_imag(i, val)`
  Set the real or imaginary part of the i-th complex number.

Here is a table which shows the layout of the array data, and the correspondence between the time-domain data z, and the frequency-domain data x.

    index    z               x = FFT(z)

    0        z(t = 0)        x(f = 0)
    1        z(t = 1)        x(f = 1/(N Delta))
    2        z(t = 2)        x(f = 2/(N Delta))
    .        ........        ..................
    N/2      z(t = N/2)      x(f = +1/(2 Delta),
                                   -1/(2 Delta))
    .        ........        ..................
    N-3      z(t = N-3)      x(f = -3/(N Delta))
    N-2      z(t = N-2)      x(f = -2/(N Delta))
    N-1      z(t = N-1)      x(f = -1/(N Delta))

When N is even the location N/2 contains the most positive and negative frequencies +1/(2 Delta), -1/(2 Delta)) which are equivalent. If N is odd then general structure of the table above still applies, but N/2 does not appear.

## 2 Radix-2 FFT routines for complex data

The radix-2 algorithms are simple and compact, although not necessarily the most efficient. They use the Cooley-Tukey algorithm to compute in-place complex FFTs for lengths which are a power of 2 -- no additional storage is required. The corresponding self-sorting mixed-radix routines offer better performance at the expense of requiring additional working space.

`GSL::FFT::Complex::PackedArray#radix2_forward(stride = 1, n = half length of the array)`
`GSL::FFT::Complex::PackedArray#radix2_backward(stride = 1, n = half length of the array)`
`GSL::FFT::Complex::PackedArray#radix2_inverse(stride = 1, n = half length of the array)`
`GSL::FFT::Complex.radix2_forward(packedarray, ....)`
`GSL::FFT::Complex.radix2_backward(packedarray, ....)`
`GSL::FFT::Complex.radix2_inverse(packedarray, ....)`
`GSL::FFT::Complex::Radix2.forward(packedarray, ....)`
`GSL::FFT::Complex::Radix2.backward(packedarray, ....)`
`GSL::FFT::Complex::Radix2.inverse(packedarray, ....)`

  These functions compute forward, backward and inverse FFTs of length n with stride stride, on the packed complex array using an in-place radix-2 decimation-in-time algorithm. The length of the transform n is restricted to powers of two.

  The methods return a value of `GSL::SUCCESS` if no errors were detected, or `GSL::EDOM` if the length of the data n is not a power of two.

`GSL::FFT::Complex::PackedArray#radix2_transform(stride = 1, n = half length of the array, sign)`
`GSL::FFT::Complex.radix2_transform(packedarray, ....)`
`GSL::FFT::Complex::Radix2.transform(packedarray, ....)`
  The sign argument can be either `GSL::FFT::Forward` or `GSL::FFT::Backward`.
`GSL::FFT::Complex::PackedArray#radix2_dif_forward(stride = 1, n = half length of the array)`
`GSL::FFT::Complex::PackedArray#radix2_dif_backward(stride = 1, n = half length of the array)`
`GSL::FFT::Complex::PackedArray#radix2_dif_inverse(stride = 1, n = half length of the array)`
`GSL::FFT::Complex::PackedArray#radix2_dif_transform(stride = 1, n = half length of the array, sign)`
`GSL::FFT::Complex.radix2_dif_forward(packedarray, ...)`
`GSL::FFT::Complex.radix2_dif_backward(packedarray, ...)`
`GSL::FFT::Complex.radix2_dif_inverse(packedarray, ...)`
`GSL::FFT::Complex::Radix2.dif_forward(packedarray, ...)`
`GSL::FFT::Complex::Radix2.dif_backward(packedarray, ...)`
`GSL::FFT::Complex::Radix2.dif_inverse(packedarray, ...)`
  These are decimation-in-frequency versions of the radix-2 FFT functions.

### 2.1 Example of complex FFT

Here is an example program which computes the FFT of a short pulse in a sample of length 128. To make the resulting Fourier transform real the pulse is defined for equal positive and negative times (-10 ... 10), where the negative times wrap around the end of the array.

    require("gsl")
    n = 128
    data = FFT::Complex::PackedArray.new(2*n)

    data.set_real(0, 1.0)
    for i in 1..10 do
      data[i] = [1.0, 0.0]
      data[n-i] = [1.0, 0.0]
    end

    #for i in 0...n do
    #  printf("%d %e %e\n", i, data.real(i), data.imag(i))
    #end

    # You can choose whichever you like
    #data.radix2_forward()
    #data.radix2_transform(1, n, FFT::Forward)
    data.radix2_transform(n, FFT::Forward)
    #data.radix2_transform(FFT::Forward)
    #FFT::Complex::Radix2.forward(data)
    #FFT::Complex.radix2_transform(data, FFT::Forward)
    sqrtn = Math::sqrt(n)
    for i in 0...n do
      printf("%d %e %e\n", i, data.real(i)/sqrtn, data.imag(i)/sqrtn)
    end

## 3 Mixed-radix FFT routines for complex data

### 3.1 GSL::FFT::Complex::Wavetable class

`GSL::FFT::Complex::Wavetable.new(n)`
`GSL::FFT::Complex::Wavetable.alloc(n)`

  These methods prepare a trigonometric lookup table for a complex FFT of length n. The length n is factorized into a product of subtransforms, and the factors and their trigonometric coefficients are stored in the wavetable. The trigonometric coefficients are computed using direct calls to sin and cos, for accuracy. Recursion relations could be used to compute the lookup table faster, but if an application performs many FFTs of the same length then this computation is a one-off overhead which does not affect the final throughput.

  The `Wavetable` object can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The same wavetable can be used for both forward and backward (or inverse) transforms of a given length.

`GSL::FFT::Complex::Wavetable#n`
`GSL::FFT::Complex::Wavetable#nf`
`GSL::FFT::Complex::Wavetable#factor`

### 3.2 GSL::FFT::Complex::Workspace class

`GSL::FFT::Complex::Workspace.new(n)`
`GSL::FFT::Complex::Workspace.alloc(n)`
  Create a workspace for a complex transform of length n.

### 3.3 Methods to compute transform

`GSL::FFT::Complex::PackedArray#forward(stride = 1, n = array half length, table, work)`
`GSL::FFT::Complex::PackedArray#forward(n, table, work)`
`GSL::FFT::Complex::PackedArray#forward(table, work)`
`GSL::FFT::Complex::PackedArray#forward(n)`
`GSL::FFT::Complex::PackedArray#forward(table)`
`GSL::FFT::Complex::PackedArray#forward(work)`
`GSL::FFT::Complex::PackedArray#forward()`
`GSL::FFT::Complex::PackedArray#backward(arguments same as forward)`
`GSL::FFT::Complex::PackedArray#inverse(arguments same as forward)`
`GSL::FFT::Complex::PackedArray#transform(arguments same as forward, sign)`
`GSL::FFT::Complex.forward(packedarray, ....)`
`GSL::FFT::Complex.backward(packedarray, ....)`
`GSL::FFT::Complex.inverse(packedarray, ....)`
`GSL::FFT::Complex.transform(packedarray, ....)`
  These methods compute forward, backward and inverse FFTs of length n with stride stride, on the packed complex array self, using a mixed radix decimation-in-frequency algorithm. There is no restriction on the length n. Efficient modules are provided for subtransforms of length 2, 3, 4, 5, 6 and 7. Any remaining factors are computed with a slow, O(n^2), general-n module. The caller can supply a table containing the trigonometric lookup tables and a workspace work (they are optional). The sign argument for the method `transform` can be either `GSL::FFT::Forward` or `GSL::FFT::Backward`. These methods return a value.
  - `GSL::SUCCESS`: if no errors were detected.
  - `GSL::EDOM`: the length of the data n is not a positive integer (i.e. n is zero),
  - `GSL::EINVAL`: the length of the data n and the length used to compute the given wavetable do not match.

### 3.4 Example to use the mixed-radix FFT algorithm

    require 'gsl'

    n = 630
    data = FFT::Complex::PackedArray.new(2*n)

    table = FFT::Complex::Wavetable.new(n)
    space = FFT::Complex::Workspace.new(n)

    data.set_real(0, 1.0)
    for i in 1..10 do
      data[i] = [1.0, 0.0]
      data[n-i] = [1.0, 0.0]
    end

    data.forward(table, space)
    #data.forward()
    #data.transform(FFT:Forward)
    #FFT::Complex.forward(data, table, space)
    #FFT::Complex.forward(data)
    #FFT::Complex.transform(data, table, space, FFT::Forward)
    #FFT::Complex.transform(data, FFT::Forward)

    sqrtn = Math::sqrt(n)
    for i in 0...n do
      printf("%d %e %e\n", i, data.real(i)/sqrtn, data.imag(i)/sqrtn)
    end

end

## 4 Radix-2 FFT routines for real data

The routines for real FFTs are provided as methods of the GSL::Vector class.

`GSL::Vector#real_radix2_transform(stride = 1, n = vector length)`
`GSL::Vector#radix2_transform(stride = 1, n = vector length)`
`GSL::Vector#real_radix2_forward(stride = 1, n = vector length)`
`GSL::Vector#radix2_forward(stride = 1, n = vector length)`
`GSL::FFT::Real.radix2_transform(vector, ...)`
`GSL::FFT::Real.radix2_forward(vector, ...)`
`GSL::FFT::Real::Radix2.transform(vector, ...)`
`GSL::FFT::Real::Radix2.forward(vector, ...)`

  These methods compute an in-place radix-2 FFT of length n and stride stride on the real array self. The output is a half-complex sequence, which is stored in-place. The arrangement of the half-complex terms uses the following scheme: for k < N/2 the real part of the k-th term is stored in location k, and the corresponding imaginary part is stored in location N-k. Terms with k > N/2 can be reconstructed using the symmetry z_k = z^*_{N-k}. The terms for k=0 and k=N/2 are both purely real, and count as a special case. Their real parts are stored in locations 0 and N/2 respectively, while their imaginary parts which are zero are not stored.

  The following table shows the correspondence between the output self and the equivalent results obtained by considering the input data as a complex sequence with zero imaginary part,

      complex[0].real    =    self[0]
      complex[0].imag    =    0
      complex[1].real    =    self[1]
      complex[1].imag    =    self[N-1]
          ...............         ................
      complex[k].real    =    self[k]
      complex[k].imag    =    self[N-k]
          ...............         ................
      complex[N/2].real  =    self[N/2]
      complex[N/2].real  =    0
          ...............         ................
      complex[k'].real   =    self[k]        k' = N - k
      complex[k'].imag   =   -self[N-k]
          ...............         ................
      complex[N-1].real  =    self[1]
      complex[N-1].imag  =   -self[N-1]

`GSL::Vector#halfcomplex_radix2_inverse(stride = 1, n = vector length)`
`GSL::Vector#radix2_inverse(stride = 1, n = vector length)`
`GSL::Vector#halfcomplex_radix2_backward(stride = 1, n = vector length)`
`GSL::Vector#radix2_backward(stride = 1, n = vector length)`
`GSL::FFT::HalfComplex.radix2_inverse(vector, ...)`
`GSL::FFT::HalfComplex.radix2_backward(vector, ...)`
`GSL::FFT::HalfComplex::Radix2.inverse(vector, ...)`
`GSL::FFT::HalfComplex::Radix2.backward(vector, ...)`
  These methods compute the inverse or backwards in-place radix-2 FFT of length n and stride stride on the half-complex sequence data stored according the output scheme used by gsl_fft_real_radix2. The result is a real array stored in natural order.

## 5 Mixed-radix FFT routines for real data

### 5.1 Data storage scheme

The table below shows the output for an odd-length sequence, n=5. The two columns give the correspondence between the 5 values in the half-complex sequence computed `real_transform`, halfcomplex[] and the values complex[] that would be returned if the same real input sequence were passed to `complex_backward` as a complex sequence (with imaginary parts set to 0),

    complex[0].real  =  halfcomplex[0]
    complex[0].imag  =  0
    complex[1].real  =  halfcomplex[1]
    complex[1].imag  =  halfcomplex[2]
    complex[2].real  =  halfcomplex[3]
    complex[2].imag  =  halfcomplex[4]
    complex[3].real  =  halfcomplex[3]
    complex[3].imag  = -halfcomplex[4]
    complex[4].real  =  halfcomplex[1]
    complex[4].imag  = -halfcomplex[2]

The upper elements of the complex array, `complex[3]` and `complex[4]` are filled in using the symmetry condition. The imaginary part of the zero-frequency term `complex[0].imag` is known to be zero by the symmetry.

The next table shows the output for an even-length sequence, n=5 In the even case there are two values which are purely real,

    complex[0].real  =  halfcomplex[0]
    complex[0].imag  =  0
    complex[1].real  =  halfcomplex[1]
    complex[1].imag  =  halfcomplex[2]
    complex[2].real  =  halfcomplex[3]
    complex[2].imag  =  halfcomplex[4]
    complex[3].real  =  halfcomplex[5]
    complex[3].imag  =  0
    complex[4].real  =  halfcomplex[3]
    complex[4].imag  = -halfcomplex[4]
    complex[5].real  =  halfcomplex[1]
    complex[5].imag  = -halfcomplex[2]

The upper elements of the complex array, `complex[4]` and `complex[5]` are filled in using the symmetry condition. Both `complex[0].imag` and `complex[3].imag` are known to be zero.

### 5.2 Wavetable and Workspace classes

`GSL::FFT::Real::Wavetable.new(n)`
`GSL::FFT::Real::Wavetable.alloc(n)`
`GSL::FFT::HalfComplex::Wavetable.new(n)`
`GSL::FFT::HalfComplex::Wavetable.alloc(n)`

  These methods create trigonometric lookup tables for an FFT of size n real elements. The length n is factorized into a product of subtransforms, and the factors and their trigonometric coefficients are stored in the wavetable. The trigonometric coefficients are computed using direct calls to sin and cos, for accuracy. Recursion relations could be used to compute the lookup table faster, but if an application performs many FFTs of the same length then computing the wavetable is a one-off overhead which does not affect the final throughput.

  The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The appropriate type of wavetable must be used for forward real or inverse half-complex transforms.

`GSL::FFT::Real::Workspace.new(n)`
`GSL::FFT::Real::Workspace.alloc(n)`
  This creates a workspace object for a real transform of length n. The same workspace can be used for both forward real and inverse halfcomplex transforms.

### 5.3 Methods for real FFTs

`GSL::Vector#real_transform(stride = 1, n = vector length, table, work)`
`GSL::Vector#halfcomplex_transform(stride = 1, n = vector length, table, work)`
`GSL::FFT::Real.transform(vector, ...)`
`GSL::FFT::Halfcomplex.transform(vector, ...)`
  These methods compute the FFT of self, a real or half-complex array of length n, using a mixed radix decimation-in-frequency algorithm. For `real_transform` self is an array of time-ordered real data. For `halfcomplex_transform` self contains Fourier coefficients in the half-complex ordering described above. There is no restriction on the length n. Efficient modules are provided for subtransforms of length 2, 3, 4 and 5. Any remaining factors are computed with a slow, O(n^2), general-n module. The caller can supply a table containing trigonometric lookup tables and a workspace work (optional).
`GSL::Vector#halfcomplex_inverse(stride = 1, n = vector length, table, work)`
`GSL::Vector#halfcomplex_backward(stride = 1, n = vector length, table, work)`
`GSL::FFT::HalfComplex.inverse(vector, ...)`
`GSL::FFT::HalfComplex.backward(vector, ...)`

### 5.4 Example

    require("gsl")

    n = 100
    data = Vector.new(n)

    for i in (n/3)...(2*n/3) do
      data[i] = 1.0
    end

    rtable = FFT::Real::Wavetable.new(n)
    rwork = FFT::Real::Workspace.new(n)

    # you can choose...
    #data.real_transform(1, n, rtable, rwork)
    #data.real_transform(n, rtable, rwork)
    data.real_transform(rtable, rwork)
    #data.real_transform(n, rtable)
    #data.real_transform(n, rwork)
    #data.real_transform(n)
    #data.real_transform(rtable)
    #data.real_transform(rwork)
    #data.real_transform()
    #FFT::Real.transform(data)

    for i in 11...n do
      data[i] = 0.0
    end

    hctable = FFT::HalfComplex::Wavetable.new(n)

    #data.halfcomplex_inverse(hctable, rwork)
    data.halfcomplex_inverse()
    #FFT::HalfComplex.inverse(data)

    for i in 0...n do
      printf("%d %e\n", i, data[i])
    end

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