Imaginary Dust

Numerical error can sometimes cause what should be real output to be complex. Below we take the inverse fast Fourier transform (FFT) of the FFT of a real sequence. The result should be real (we should get back what we started with), but the result is a complex sequence, which is not what we started with!

x <- fft(fft(1:10),inverse = TRUE)/10
x

##  [1]  1+0i  2+0i  3-0i  4+0i  5-0i  6+0i  7-0i  8+0i  9-0i 10+0i

The complex part is small but shouldn’t be there. We could use the Re function to remove the imaginary part, but getting a large imaginary part when we were expecting a real result indicates an error. If we use the Re function, the error is hidden.

The function removeDust removes the small imaginary part but returns an error if the imaginary part is larger than expected. How small is up to you, but the default is 1e-10.

removeDust <- function(x, imTol=1e-10){
  if (max(abs(Im(x))) > imTol) {
    stop("The imaginary part was > ", imTol)
  }
  Re(x)
}

Below, we use the removeDust function to remove the imaginary dust.

x

##  [1]  1+0i  2+0i  3-0i  4+0i  5-0i  6+0i  7-0i  8+0i  9-0i 10+0i

removeDust(x)

##  [1]  1  2  3  4  5  6  7  8  9 10