22.3 Functions and Variables for FFTPACK5

FFTPACK5 provides several routines to compute Fourier transforms for both real and complex sequences and their inverses. The forward transform is defined the same as for fft. The major difference is the length of the sequence is not constrained to be a power of two. In fact, any length is supported, but it is most efficient when the length has the form \(2^r*3^s*5^t\).

load("fftpack5") loads this function.

Function: fftpack5_fft (x) ¶

Like fft (fft), this computes the fast Fourier transform of a complex sequence. However, the length of x is not limited to a power of 2.

load("fftpack5") loads this function.

Examples:

Real data.

(%i1) load("fftpack5") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 2, 3, 4, -1, -2 ,-3, -4] $
(%i4) L1 : fftpack5_fft(L);
(%o4) [0.0, 1.811 %i - 0.1036, 0.0, 0.3107 %i + 0.6036, 0.0, 
                                0.6036 - 0.3107 %i, 0.0, (- 1.811 %i) - 0.1036]
(%i5) L2 : fftpack5_inverse_fft(L1);
(%o5) [1.0, 4.441e-16 %i + 2.0, 1.837e-16 %i + 3.0, 4.0 - 4.441e-16 %i, 
     - 1.0, (- 4.441e-16 %i) - 2.0, (- 1.837e-16 %i) - 3.0, 4.441e-16
       %i - 4.0]
(%i6) lmax (abs (L2-L));
(%o6)                       4.441e-16
(%i7) L : [1, 2, 3, 4, 5, 6]$
(%i8) L1 : fftpack5_fft(L);
(%o8) [3.5, (- 0.866 %i) - 0.5, (- 0.2887 %i) - 0.5, (- 1.48e-16 %i) - 0.5, 
                                               0.2887 %i - 0.5, 0.866
                                                      %%i - 0.5]
(%i9) L2 : fftpack5_inverse_fft (L1);
(%o9) [1.0 - 1.48e-16 %i, 3.701e-17 %i + 2.0, 3.0 - 1.48e-16 %i, 
                     4.0 - 1.811e-16 %i, 5.0 - 1.48e-16 %i, 5.881e-16
                           %i + 6.0]
(%i10) lmax (abs (L2-L));
(%o10)                             9.064e-16

Complex data.

(%i1) load("fftpack5") $
(%i2) fpprintprec : 4 $
(%i3) L : [1, 1 + %i, 1 - %i, -1, -1, 1 - %i, 1 + %i, 1] $
(%i4) L1 : fftpack5_inverse_fft (L);
(%o4) [4.0, 2.828 %i + 2.828, (- 2.0 %i) - 2.0, 4.0, 0.0, 
                                       (- 2.828 %i) - 2.828, 2.0 %i - 2.0, 4.0]
(%i5) L2 : fftpack5_fft(L1);
(%o5) [1.0, 1.0 %i + 1.0, 1.0 - 1.0 %i, (- 2.776e-17 %i) - 1.0, - 1.0, 
                                1.0 - 1.0 %i, 1.0 %i + 1.0, 1.0 -
                                          %2.776e-17 %i]
(%i6) lmax(abs(L2-L));
(%o6)                              1.11e-16

Categories: Package fftpack5 ·

Function: fftpack5_inverse_fft (y) ¶

Computes the inverse complex Fourier transform, like inverse_fft, but is not constrained to be a power of two.

Categories: Package fftpack5 ·

Function: fftpack5_real_fft (x) ¶

Computes the fast Fourier transform of a real-valued sequence x, just like real_fft, except the length is not constrained to be a power of two.

Examples:

(%i1) fpprintprec : 4 $
(%i2) L : [1, 2, 3, 4, 5, 6] $
(%i3) L1 : fftpack5_real_fft(L);
(%o3)       [3.5, (- 0.866 %i) - 0.5, (- 0.2887 %i) - 0.5, - 0.5]
(%i4) L2 : fftpack5_inverse_real_fft(L1, 6);
(%o4)                  [1.0, 2.0, 3.0, 4.0, 5.0, 6.0]
(%i5) lmax(abs(L2-L));
(%o5)                            1.332e-15
(%i6) fftpack5_inverse_real_fft(L1, 7);
(%o6)            [0.5, 2.083, 2.562, 3.7, 4.3, 5.438, 5.917] 

The last example shows how important it to set the length correctly for fftpack5_inverse_real_fft.

Categories: Package fftpack5 ·

Function: fftpack5_inverse_real_fft (y, n) ¶

Computes the inverse Fourier transform of y, which must have a length of floor(n/2) + 1. The length of sequence produced by the inverse transform must be specified by n. This is required because the length of y does not uniquely determine n. The last element of y is always real if n is even, but it can be complex when n is odd.

Categories: Package fftpack5 ·