Fourier DFT FFT reconstruction from important periodogram contributions

fourier-fft-periodogram-pick-components.R

## This is just a quick exploration, answering the question:
## How do I reconstruct a signal by picking the highest-power frequency from a periodogram?
## Assumes I keep the constant (DC) term.
## -----------------------------------------------------------------------------
## Fourier decomposition exploration
## -----------------------------------------------------------------------------

x <- 10 * sin(2 * pi * (1:75) / 25) + 300 + rnorm(1:75, 0, 2.5)
x

fourier_coeff <- fft(x)
fourier_coeff

iftt <- fft(fourier_coeff, inverse = TRUE) / length(fourier_coeff)


## Get the series
x <- dat$X1

## get the periodogram
xp <- TSA::periodogram(dat$X1, plot = FALSE)
#which.max(xp$spec)

## Get the fourier transform of x
xf <- fft(x)

## Make a filtered fft, picking out only the component with the max power from periodogram
## Since if periodogram has N entries, the fft has 2N + 1 entries.  If I want to keep the constant
## term and the biggest periodogram frequency (call it entry K in periodogram), I need to pick out from the fft:
## 1 (constant term)
## K + 1 (corresponding to K in periodogram)
## 2N + 1 - K (corresponding to K in periodogram, which is the equivalent of the c_{-K} fft coefficient)

# xffilt <- ifelse(1:length(xf) %in% which.max(xp$spec), xf, 0)
# xffilt <- ifelse(Mod(xf) > 100, xf, 0)
xffilt <- ifelse(1:length(xf) %in% c(1, which.max(xp$spec) + 1, length(xf) - which.max(xp$spec)), xf, 0)



## reconstruct x
xr <- Re(fft(xffilt, inverse=TRUE) / length(xffilt))

## plot it
plot(x, type="p")
par(new=TRUE)
plot(xr, type="l")