Time series analysis - extract and sustract the periodic components - r

Question

The spectrum of my data set shows 3 periodic components in the time serie. I would like to substract the periodic components to keep the data without these periodicities.
It points out periodic events with the periodicity of (1/144 = daily), (1/72 = 1/2daily), and (1/6 = hourly).
My idea was to find out the Fourier components (mag and phase) of my dataset and to extract the Fourier components for these 3 specific frequencies and to create a new signal with is :
Data - PeriodicSignal_1h - PeriodicSignal_1/2day - PeriodicSignal_1day
I try with fft but I do not know how to extract the signal at these specific frequencies.
My dataset is complicated but I'm working on an example to understand the process. Here is the example :
samplingFrequency = 1000;
timeInterval = 1/samplingFrequency;
signalIndex = seq(0, 1, by=timeInterval);
N = 1000
a1 = 2;
a2 = 3;
f1 = 10;
f2 = 20;
signal1 = a1 * sin(2 * pi * f1 * signalIndex);
signal2 = a2 * sin(2 * pi * f2 * signalIndex);
inputSignal = signal1 + signal2;
Y <- fft(inputSignal)
mag <-sqrt(Re(Y)^2+Im(Y)^2)*2/length(inputSignal)
phase <-atan(Im(Y)/Re(Y)) Yr <- Re(Y) Yi <- Im(Y)
I'm trying to extract the mag and the phase of the signal with frequency f1. And I would like to generate a new signal with is :
ImputSignal - Signal_f1

Answer 1

I believe the following does what you are looking for ... I changed some of the variable names. At the bottom is the frequency selection that you asked about.
Set up the time and frequency parameters
samplingFrequency = 1000;
f_Hz = samplingFrequency
N = 1000
df_Hz = f_Hz / N
T = 1 / df_Hz
dt=T/N
t = dt*(seq(1,N)-1)
Generate a fake signal, no noise
a1 = 2;
a2 = 3;
f1 = 10;
f2 = 20;
signal1 = a1 * sin(2 * pi * f1 * t);
signal2 = a2 * sin(2 * pi * f2 * t);
inputSignal = signal1 + signal2;
Plot the fake signal
plot(t, signal1,type='l',col='green',ylim=c(-6,6))
lines(t, signal2,col='red')
lines(t, inputSignal,col='black')
Get the fft, and plot positive frequency portion
Y <- fft(inputSignal)
m <- floor(N/2)-1
posFreqIndices <- 2:(m+1)
negFreqIndices <- N:(m+3)
mag <-sqrt(Re(Y)^2+Im(Y)^2)*2/length(inputSignal)
phase <-atan(Im(Y)/Re(Y))
Yr <- Re(Y)
Yi <- Im(Y)
freq <- seq(df_Hz,f_Hz/2-df_Hz,df_Hz)
plot(freq,mag[posFreqIndices],type='l',xlab='Freq (Hz)', ylab='Magnitude',xlim=c(0,30))
# plot(freq,10*log10(mag[posFreqIndices]),type='l',xlab='Freq (Hz)', ylab='Magnitude (db)',xlim=c(0,30))
# plot(freq,phase[posFreqIndices]*180/pi,type='l',xlab='Freq (Hz)', ylab='Phase (deg)',xlim=c(0,30))
Identify the frequencies for the filtered signal based on amplitude
ampSelectIndices <- which(mag>1.9 & mag < 2.1)
Generate the filtered fft for the selected frequencies
YAmpSelect <- Y*0
YAmpSelect[ampSelectIndices] = Y[ampSelectIndices]
Calculate the inverse fft
yAmpSelect = Re(fft(YAmpSelect, inverse = TRUE))/length(YAmpSelect)
Plot the filtered signal
plot(t,yAmpSelect,t='l',xlab='t (sec)',ylab='Filtered for mag ~ 2')
Plot the original signal minus the filtered signal
plot(t,inputSignal-yAmpSelect,type='l')
The fft is calculated with frequencies folded. The following checks the unfolding process, this check works for real valued signals (not complex value time signals). The process is correct for complex valued time signals.
checkFreqWrapping = all.equal(mag[posFreqIndices], mag[negFreqIndices])
stopifnot(checkFreqWrapping)
Select fft values by frequency
freqSelectIndices_a <- which(9.95 < freq & freq < 10.05)
freqSelectIndices = union(posFreqIndices[freqSelectIndices_a],negFreqIndices[freqSelectIndices_a])
Create the fft for selected frequencies
YFreqSelect <- Y*0
YFreqSelect[freqSelectIndices ] = Y[freqSelectIndices ]
Calculate the time signal, plot it.
yFreqSelect = Re(fft(YFreqSelect, inverse = TRUE))/length(YFreqSelect)
plot(t,yFreqSelect,t='l',xlab='t (sec)',ylab='Filtered for mag ~ 2')
plot(t,inputSignal-yFreqSelect,type='l')
OK, I think that explains how to select fft values based on frequencies... Good luck...