fft(z, inverse = FALSE) mvfft(z, inverse = FALSE)
z
| a real or complex array containing the values to be transformed |
inverse
|
if TRUE, the unnormalized inverse transform is
computed (the inverse has a + in the exponent of e,
but here, we do not divide by 1/length(x)).
|
z is a vector, the value computed and returned by
fft is the unnormalized univariate Fourier transform of the
sequence of values in z.
When z contains an array, fft computes and returns the
multivariate (spatial) transform. If inverse is TRUE,
the (unnormalized) inverse Fourier transform is returned, i.e.,
if y <- fft(z), then z is fft(y, inv=TRUE) / length(y).
By contrast, mvfft takes a real or complex matrix as argument,
and returns a similar shaped matrix, but with each column replaced by
its discrete Fourier transform. This is useful for analyzing
vector-valued series.
The FFT is fastest when the length of of the series being transformed is highly composite (i.e. has many factors). If this is not the case, the transform may take a long time to compute and will use a large amount of memory.
convolve, nextn.
x <- 1:4
fft(x)
all(fft(fft(x), inverse = TRUE)/(x*length(x)) == 1+0i)
eps <- 1e-11 ## In general, not exactly, but still:
for(N in 1:130) {
cat("N=",formatC(N,wid=3),": ")
x <- rnorm(N)
if(N %% 5 == 0) {
m5 <- matrix(x,ncol=5)
cat("mvfft:",all(apply(m5,2,fft) == mvfft(m5)),"")
}
dd <- Mod(1 - (f2 <- fft(fft(x), inverse=TRUE)/(x*length(x))))
cat(if(all(dd < eps))paste(" all < ", formatC(eps)) else
paste("NO: range=",paste(formatC(range(dd)),collapse=",")),"\n")
}
plot(fft(c(9:0,0:13, numeric(301))), type = "l")
periodogram <- function(x, mean.x = mean(x)) { # simple periodogram
n <- length(x)
x <- unclass(x) - mean.x
Mod(fft(x))[1:(n%/%2 + 1)]^2 / (2*pi*n)
}
data(sunspots)
plot(10*log10(periodogram(sunspots)), type = "b", col = "blue")