Reconstruct Time Series from FFT frequency and strength data using R

Question

After applying a Fourier Transform to an EEG measurement, I want to compare the approximation by FFT with the original signal in the form of a plot. I have to convert the data (frequency and strength) from the FFT back to a time series. To transform the original time series I use the eegfft method of the eegkit package. I get a list of frequencies and amplitudes to approximate the original signal.

Here the two results of the FFT are shown as shortened examples:

# Frequency in Hz
freq <- c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)  

# Strength in uV
ampl <- c(4.1135352, 5.1272713, 3.2069741, 1.5336438, 2.4301334, 1.0974758, 1.8238327, 0.9637886, 1.1401306, 0.2224472)

Is there a package or method that I can use to reconstruct a time series from the frequency and amplitude data that has been approximated by FFT?

EDIT:

For the reconstruction of the original signal, do I also need the phase information that the eegfft method returns in the result?

# Phase shift in range -pi to pi
phase <- c(0.0000000, 1.1469542, -2.1930702, 2.7361738,1.1597980, 2.6118647, -0.6609641, -2.1508755,1.6584852, -1.2906986)

There are 11 frequencies but only 10 amplitudes and phases. I presume you didn't mean to include frequency of zero? — Jon Spring
@JonSpring Yeah good point, a frequency of zero Hertz doesn´t make sense. I will correct that — JohnDizzle

Jon Spring Jon Spring · Accepted Answer · 2019-04-15T15:54:24

I expect something like this should work.

Edit: I have set phases to default to zero if missing and not passed into data_from_fft.

freq <- c(1, 2, 3, 4, 5, 6, 7, 8, 9, 10)  
ampl <- c(4.1135352, 5.1272713, 3.2069741, 1.5336438, 2.4301334, 1.0974758, 1.8238327, 0.9637886, 1.1401306, 0.2224472)
phase <- c(0.0000000, 1.1469542, -2.1930702, 2.7361738,1.1597980, 2.6118647, -0.6609641, -2.1508755,1.6584852, -1.2906986)
sampl_freq = 1000

data_from_fft <- function(xmin, xmax, sample_freq, 
                          frequencies, amplitudes, phases = 0) {
  x_vals <- seq(xmin, xmax, length.out = sample_freq * (xmax-xmin))
  y_vals <- x_vals * 0
  for (i in seq_along(x_vals)) {
    # Note, I don't understand why the pi/2 phase adjustment is needed here,
    #   but I couldn't get the right answers out eegfft without it... :-(
    y_vals[i] <- sum(amplitudes * sin(2*pi*frequencies * x_vals[i] + phase + pi/2))
  }
  data.frame(x_vals, y_vals)
}

library(tidyverse)

plot_from_FFT <- data_from_fft(0, 1, sampl_freq, freq, ampl, phase)
ggplot(plot_from_FFT, aes(x_vals, y_vals)) +
  geom_line()

Now, let's see if we can use that output to reconstruct the inputs:

eegkit::eegfft(plot_from_FFT$y_vals, lower = 1, upper = 20, Fs = sampl_freq) %>% 
  filter(abs(strength) > 0.1)

   frequency  strength  phase.shift
1          1 4.1158607  0.004451123
2          2 5.1177070  1.154553861
3          3 3.2155744 -2.185185998
4          4 1.5319350  2.739953054
5          5 2.4283426  1.173258629
6          6 1.0813858  2.645126993
7          7 1.8323207 -0.644216053
8          8 0.9598727 -2.138381646
9          9 1.1427380  1.685081744
10        10 0.2312619 -1.265466418

Yes! These are pretty close to the inputs.

eegkit::eegfft(plot_from_FFT$y_vals, lower = 1, upper = 20, Fs = sampl_freq) %>% 
      filter(abs(strength) > 0.1) %>%
      left_join(
        tibble(frequency = freq,
               strength_orig = ampl,
               phase_orig   = phase)
      ) %>%
      gather(stat, value, -frequency) %>%
      mutate(category = if_else(stat %>% str_detect("str"), "strength", "phase"),
             version  = if_else(stat %>% str_detect("orig"), "plot inputs", "reconstructed inputs"),) %>%
      ggplot(aes(frequency, value, shape = version, size = version)) + 
      geom_point() +
      scale_x_continuous(breaks = 1:10, minor_breaks = NULL) +
      scale_shape_manual(values = c(16, 21)) +
      scale_size_manual(values = c(1,5)) +
      facet_wrap(~category)

Reconstruct Time Series from FFT frequency and strength data using R

1 Answers