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