fft(z, inverse=F)
TRUE
, the inverse of the
transform is computed.
z,
or the inverse transform if
inverse=TRUE.
This is of mode
"complex".
Because the unnormalized
transform is computed, the commands
fft(fft(z),
inverse=T)
and
fft(fft(z,
inverse=T))
return
nz
for an
n-dimensional vector
z
. If you require
Fourier coefficients, you should divide the value that
fft
returns by the length of the input
vector.
The fast Fourier transform algorithm is used.
No padding of the input data is done;
length(z)
is factored,
if possible, and the factorization is used in the algorithm.
Therefore, if
length(z)
is prime, there will be no advantage
in using
fft
over computing the transform explicitly.
If
z
is an array,
fft
will return the multi-dimensional
unnormalized discrete Fourier transform of
z
-a complex array with the
same shape as
z
.
Therefore, using
fft
on a multivariate time series will not compute the
time transform.
The discrete Fourier transform is used to compute an approximation to the
continuous Fourier transform of a periodic function F. In the usual
definition, n points are sampled from F symmetrically around 0; that is,
the domain of the sampled points is [-N/2,N/2], where N is the period of
F. However, S-PLUS assumes the n points are sampled from the interval
[0,N]. When this convention is followed, the resulting frequencies are
shifted. For example, let j be the index of the sampled points and
suppose n is even. In S-PLUS, the zero frequency corresponds to j=1, the
positive frequencies correspond to 2 <= j <= n/2, the negative
frequencies correspond to n/2+2 <= j <= n, and the Nyquist critical
frequency corresponds to j = n/2+1. The definitions are analagous if n is
odd. For more details, see Press et al. (1996).
Bloomfield, P. (1976).
Fourier Analysis of Time Series: An Introduction.
Wiley, New York.
The chapter "Analyzing Time Series" in the S-PLUS Guide to Statistical and Mathematical Analysis.
Press, W.H., Teukolsky, S.A., Vetterling, W.T., & Flannery, B.P. (1996).
Numerical Recipes in Fortran 77: The Art of Scientific Computing
(2nd ed.). New York: Cambridge University Press.