[R] FFT Normalization Documentation

[Eike Petersen]

Hello everyone,

the docpage for the fft function states:

“Description: Performs the Fast Fourier Transform of an array.”

and

“Arguments – 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)’).”

Judging from this, I would expect ‘fft(x)’ to yield the correct FFT of x, and
‘fft(X, inverse = TRUE) / length(X)’ to yield the correct inverse FFT of X.

However, it seems to me that actually the result of ‘fft(x)’ should be scaled by
‘1/length(x)’, while ‘fft(X, inverse=TRUE)’ seems to yield a correct result by
default:

t <- seq(0.001, 1, 0.001)
y <- 1 + sin(2*pi*20*t) + 1.5 * sin(2*pi*30*t)
Y <- fft(y)
dev.new()
plot(abs(Y)) ## Shows peaks at amplitudes 1000, 500 and 750, while they should
be at amplitude 1, 0.5 and 0.75.
y2 <- Re(fft(Y / length(Y), inverse = TRUE))
max(abs(y-y2)) ## The IFFT yields a correctly scaled result by default, if
applied to a correctly scaled FFT.

Did I get something wrong? If not, having spent quite some time figuring this
out, I would like to see the documentation clearly pointing this out. I find the
current text rather confusing.

On another note: I have spent some time working on demo files that showcase some
of the properties of the FFT and their implementation in R. I have done this
primarily for myself, as I keep forgetting how these things work, but I thought
that it might be helpful to others as well. Any hints on where/how I should
publish such a thing?

Kind regards and many thanks in advance,

Eike