### 22.2 Functions and Variables for fft

Function: polartorect (r, t)

  Translates complex values of the form `r %e^(%i t)` to the form `a + b %i`, where r is the magnitude and t is the phase. r and t are 1-dimensional arrays of the same size. The array size need not be a power of 2.

  The original values of the input arrays are replaced by the real and imaginary parts, `a` and `b`, on return. The outputs are calculated as

      a = r cos(t)
      b = r sin(t)

  `polartorect` is the inverse function of `recttopolar`.

  `load("fft")` loads this function. See also `fft`.

  Categories:Package fft ·Complex variables ·

Function: recttopolar (a, b)

  Translates complex values of the form `a + b %i` to the form `r %e^(%i t)`, where a is the real part and b is the imaginary part. a and b are 1-dimensional arrays of the same size. The array size need not be a power of 2.

  The original values of the input arrays are replaced by the magnitude and angle, `r` and `t`, on return. The outputs are calculated as

      r = sqrt(a^2 + b^2)
      t = atan2(b, a)

  The computed angle is in the range `-%pi` to `%pi`.

  `recttopolar` is the inverse function of `polartorect`.

  `load("fft")` loads this function. See also `fft`.

  Categories:Package fft ·Complex variables ·

Function: inverse_fft (y)

  Computes the inverse complex fast Fourier transform. y is a list or array (named or unnamed) which contains the data to transform. The number of elements must be a power of 2. The elements must be literal numbers (integers, rationals, floats, or bigfloats) or symbolic constants, or expressions `a + b*%i` where `a` and `b` are literal numbers or symbolic constants.

  `inverse_fft` returns a new object of the same type as y, which is not modified. Results are always computed as floats or expressions `a + b*%i` where `a` and `b` are floats. If bigfloat precision is needed the function `bf_inverse_fft` can be used instead as a drop-in replacement of `inverse_fft` that is slower, but supports bfloats.

  The inverse discrete Fourier transform is defined as follows. Let `x` be the output of the inverse transform. Then for `j` from 0 through `n - 1`,

      x[j] = sum(y[k] exp(-2 %i %pi j k / n), k, 0, n - 1)

  As there are various sign and normalization conventions possible, this definition of the transform may differ from that used by other mathematical software.

  `load("fft")` loads this function.

  See also `fft` (forward transform), `recttopolar`, and `polartorect`.

  Examples:

  Real data.

      (%i1) load ("fft") $
      (%i2) fpprintprec : 4 $
      (%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
      (%i4) L1 : inverse_fft (L);
      (%o4) [0.0, 14.49 %i - .8284, 0.0, 2.485 %i + 4.828, 0.0,
                             4.828 - 2.485 %i, 0.0, - 14.49 %i - .8284]
      (%i5) L2 : fft (L1);
      (%o5) [1.0, 2.0 - 2.168L-19 %i, 3.0 - 7.525L-20 %i,
      4.0 - 4.256L-19 %i, - 1.0, 2.168L-19 %i - 2.0,
      7.525L-20 %i - 3.0, 4.256L-19 %i - 4.0]
      (%i6) lmax (abs (L2 - L));
      (%o6)                       3.545L-16

  Complex data.

      (%i1) load ("fft") $
      (%i2) fpprintprec : 4 $
      (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
      (%i4) L1 : inverse_fft (L);
      (%o4) [4.0, 2.711L-19 %i + 4.0, 2.0 %i - 2.0,
      - 2.828 %i - 2.828, 0.0, 5.421L-20 %i + 4.0, - 2.0 %i - 2.0,
      2.828 %i + 2.828]
      (%i5) L2 : fft (L1);
      (%o5) [4.066E-20 %i + 1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i,
      1.55L-19 %i - 1.0, - 4.066E-20 %i - 1.0, 1.0 - 1.0 %i,
      1.0 %i + 1.0, 1.0 - 7.368L-20 %i]
      (%i6) lmax (abs (L2 - L));
      (%o6)                       6.841L-17

  Categories:Package fft ·

Function: fft (x)

  Computes the complex fast Fourier transform. x is a list or array (named or unnamed) which contains the data to transform. The number of elements must be a power of 2. The elements must be literal numbers (integers, rationals, floats, or bigfloats) or symbolic constants, or expressions `a + b*%i` where `a` and `b` are literal numbers or symbolic constants.

  `fft` returns a new object of the same type as x, which is not modified. Results are always computed as floats or expressions `a + b*%i` where `a` and `b` are floats. If bigfloat precision is needed the function `bf_fft` can be used instead as a drop-in replacement of `fft` that is slower, but supports bfloats. In addition if it is known that the input consists of only real values (no imaginary parts), `real_fft` can be used which is potentially faster.

  The discrete Fourier transform is defined as follows. Let `y` be the output of the transform. Then for `k` from 0 through `n - 1`,

      y[k] = (1/n) sum(x[j] exp(+2 %i %pi j k / n), j, 0, n - 1)

  As there are various sign and normalization conventions possible, this definition of the transform may differ from that used by other mathematical software.

  When the data x are real, real coefficients `a` and `b` can be computed such that

      x[j] = sum(a[k]*cos(2*%pi*j*k/n)+b[k]*sin(2*%pi*j*k/n), k, 0, n/2)

  with

      a[0] = realpart (y[0])
      b[0] = 0

  and, for k from 1 through n/2 - 1,

      a[k] = realpart (y[k] + y[n - k])
      b[k] = imagpart (y[n - k] - y[k])

  and

      a[n/2] = realpart (y[n/2])
      b[n/2] = 0

  `load("fft")` loads this function.

  See also `inverse_fft` (inverse transform), `recttopolar`, and `polartorect`.. See `real_fft` for FFTs of a real-valued input, and `bf_fft` and `bf_real_fft` for operations on bigfloat values. Finally, for transforms of any size (but limited to float values), see `fftpack5_fft` and `fftpack5_real_fft`.

  Examples:

  Real data.

      (%i1) load ("fft") $
      (%i2) fpprintprec : 4 $
      (%i3) L : [1, 2, 3, 4, -1, -2, -3, -4] $
      (%i4) L1 : fft (L);
      (%o4) [0.0, 1.811 %i - .1036, 0.0, 0.3107 %i + .6036, 0.0,
                               0.6036 - 0.3107 %i, 0.0, (- 1.811 %i) - 0.1036]
      (%i5) L2 : inverse_fft (L1);
      (%o5) [1.0, 2.168L-19 %i + 2.0, 7.525L-20 %i + 3.0,
      4.256L-19 %i + 4.0, - 1.0, - 2.168L-19 %i - 2.0,
      - 7.525L-20 %i - 3.0, - 4.256L-19 %i - 4.0]
      (%i6) lmax (abs (L2 - L));
      (%o6)                       3.545L-16

  Complex data.

      (%i1) load ("fft") $
      (%i2) fpprintprec : 4 $
      (%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
      (%i4) L1 : fft (L);
      (%o4) [0.5, 0.5, 0.25 %i - 0.25, (- 0.3536 %i) - 0.3536, 0.0, 0.5,
                                              (- 0.25 %i) - 0.25, 0.3536 %i + 0.3536]
      (%i5) L2 : inverse_fft (L1);
      (%o5) [1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i, - 1.0, - 1.0, 1.0 - 1.0 %i,
                                                                   1.0 %i + 1.0, 1.0]
      (%i6) lmax (abs (L2 - L));
      (%o6)                       0.0

  Computation of sine and cosine coefficients.

      (%i1) load ("fft") $
      (%i2) fpprintprec : 4 $
      (%i3) L : [1, 2, 3, 4, 5, 6, 7, 8] $
      (%i4) n : length (L) $
      (%i5) x : make_array (any, n) $
      (%i6) fillarray (x, L) $
      (%i7) y : fft (x) $
      (%i8) a : make_array (any, n/2 + 1) $
      (%i9) b : make_array (any, n/2 + 1) $
      (%i10) a[0] : realpart (y[0]) $
      (%i11) b[0] : 0 $
      (%i12) for k : 1 thru n/2 - 1 do
         (a[k] : realpart (y[k] + y[n - k]),
          b[k] : imagpart (y[n - k] - y[k]));
      (%o12)                        done
      (%i13) a[n/2] : y[n/2] $
      (%i14) b[n/2] : 0 $
      (%i15) listarray (a);
      (%o15)          [4.5, - 1.0, - 1.0, - 1.0, - 0.5]
      (%i16) listarray (b);
      (%o16)           [0, - 2.414, - 1.0, - .4142, 0]
      (%i17) f(j) := sum (a[k]*cos(2*%pi*j*k/n) + b[k]*sin(2*%pi*j*k/n),
                          k, 0, n/2) $
      (%i18) makelist (float (f (j)), j, 0, n - 1);
      (%o18)      [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0]

  Categories:Package fft ·

Function: real_fft (x)

  Computes the fast Fourier transform of a real-valued sequence x. This is equivalent to performing `fft(x)`, except that only the first `N/2+1` results are returned, where `N` is the length of x. `N` must be power of two.

  No check is made that x contains only real values.

  The symmetry properties of the Fourier transform of real sequences to reduce he complexity. In particular the first and last output values of `real_fft` are purely real. For larger sequences, `real_fft` may be computed more quickly than `fft`.

  Since the output length is short, the normal `inverse_fft` cannot be directly used. Use `inverse_real_fft` to compute the inverse.

  Categories:Package fft ·

Function: inverse_real_fft (y)

  Computes the inverse Fourier transform of y, which must have a length of `N/2+1` where `N` is a power of two. That is, the input x is expected to be the output of `real_fft`.

  No check is made to ensure that the input has the correct format. (The first and last elements must be purely real.)

  Categories:Package fft ·

Function: bf_inverse_fft (y)

  Computes the inverse complex fast Fourier transform. This is the bigfloat version of `inverse_fft` that converts the input to bigfloats and returns a bigfloat result.

  Categories:Package fft ·

Function: bf_fft (y)

  Computes the forward complex fast Fourier transform. This is the bigfloat version of `fft` that converts the input to bigfloats and returns a bigfloat result.

  Categories:Package fft ·

Function: bf_real_fft (x)

  Computes the forward fast Fourier transform of a real-valued input returning a bigfloat result. This is the bigfloat version of `real_fft`.

  Categories:Package fft ·

Function: bf_inverse_real_fft (y)

  Computes the inverse fast Fourier transform with a real-valued bigfloat output. This is the bigfloat version of `inverse_real_fft`.

  Categories:Package fft ·