fft                   package:base                   R Documentation

Fast Discrete Fourier Transform

Description:

     Performs the Fast Fourier Transform of an array.

Usage:

     fft(z, inverse = FALSE)
     mvfft(z, inverse = FALSE)

Arguments:

       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)').

Value:

     When `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, inverse = 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.

References:

     Singleton, R. C. (1979) Mixed Radix Fast Fourier Transforms, in
     Programs for Digital Signal Processing, IEEE Digital Signal
     Processing Committee eds. IEEE Press.

See Also:

     `convolve', `nextn'.

Examples:

     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))[2:(n%/%2 + 1)]^2 / (2*pi*n) # drop I(0)
     }
     data(sunspots)
     plot(10*log10(periodogram(sunspots)), type = "b", col = "blue")

