Reconstructing a signal from its discrete fourier transform in R - r

Question

I am trying to replicate the following figure in R: (adapted from http://link.springer.com/article/10.1007/PL00011669)
The basic concept of the figure is to show the first few components of a DFT, plotted in the time domain, and then show a reconstructed wave in the time domain using only these components (X') relative to the original data (X). I would like to slightly modify the above figure such that all of the lines shown are overlaid on a single plot.
I have been trying to adapt the figure with some real data sampled at 60 Hz. For example:
## 3 second sample where: time is in seconds and var is the variable of interest
temp = data.frame(time=seq(from=0,to=3,by=1/60),
var = c(0.054,0.054,0.054,0.072,0.072,0.072,0.072,0.09,0.09,0.108,0.126,0.126,
0.126,0.126,0.126,0.144,0.144,0.144,0.144,0.144,0.162,0.162,0.144,0.126,
0.126,0.108,0.144,0.162,0.18,0.162,0.126,0.126,0.108,0.108,0.126,0.144,
0.162,0.144,0.144,0.144,0.144,0.162,0.162,0.126,0.108,0.09,0.09,0.072,
0.054,0.054,0.054,0.036,0.036,0.018,0.018,0.018,0.018,0,0.018,0,
0,0,-0.018,0,0,0,-0.018,0,-0.018,-0.018,0,-0.018,
-0.018,-0.018,-0.018,-0.036,-0.036,-0.054,-0.054,-0.072,-0.072,-0.072,-0.072,-0.072,
-0.09,-0.09,-0.108,-0.126,-0.126,-0.126,-0.144,-0.144,-0.144,-0.162,-0.162,-0.18,
-0.162,-0.162,-0.162,-0.162,-0.144,-0.144,-0.144,-0.126,-0.126,-0.108,-0.108,-0.09,
-0.072,-0.054,-0.036,-0.018,0,0,0,0,0.018,0.018,0.036,0.054,
0.054,0.054,0.054,0.054,0.054,0.054,0.054,0.054,0.054,0.072,0.054,0.072,
0.072,0.072,0.072,0.072,0.072,0.054,0.054,0.054,0.036,0.036,0.036,0.036,
0.036,0.054,0.054,0.072,0.09,0.072,0.036,0.036,0.018,0.018,0.018,0.018,
0.036,0.036,0.036,0.036,0.018,0,-0.018,-0.018,-0.018,-0.018,-0.018,0,
-0.018,-0.036,-0.036,-0.018,-0.018,-0.018,-0.036,0,0,-0.018,-0.018,-0.018,-0.018))
##plot the original data
ggplot(temp, aes(x=time, y=var))+geom_line()
I believe that I can use fft() to eventually accomplish this goal however the leap from the output of fft() to my goal is a bit unclear.
I realize that this question is somewhat similar to: How do I calculate amplitude and phase angle of fft() output from real-valued input? but I am more specifically interested in the actual code for the specific data above.
Please note that I am relatively new to time series analysis so any clarity you could provide w.r.t. putting the output of fft() in context, or any package you could recommend that would accomplish this task efficiently would be appreciated.
Thank you

Answer 1

Matlab is your best tool, and the specific function is just fft(). To use it, first determine several basic parameters of your time domain data:
1, time duration (T), which equals to 3s.
2, Sampling interval T_s, which equals to 1/60 s.
3, Frequency domain revolution f_s, which equals to the frequency difference between two adjacent Fourier basis. You may define f_s according to your needs. However, the smallest possible f_s equals to 1/T=0.333 Hz. As a result, if you want better frequency domain revolution (smaller f_s), you need longer time domain data.
4, Maximum frequency f_M, which equals to 1/(2T_s)=30 according to Shannon sampling theory.
5, DFT length N, which equals to 2*f_M/f_s.
Then find out the specific frequencies of four Fourier basis that you want to use to approximate the data. For example, 3,6,9 and 12 Hz. So f_s = 3 Hz. Then N=2*f_M/f_s=20.
Your Matlab code looks like this:
var=[0.054,0.054,0.054 ...]; % input all your data points here
f_full=fft(var,20); % Do 20-point fft
f_useful=f_full(2:5); % You are interested with the lowest four frequencies except DC
Here f_useful contains the four complex coefficients of four Fourier basis. To reconstruct var, do the following:
% Generate basis functions
dt=0:1/60:3;
df=[3:3:12];
basis1=exp(1j*2*pi*df(1)*dt);
basis2=exp(1j*2*pi*df(2)*dt);
basis3=exp(1j*2*pi*df(3)*dt);
basis4=exp(1j*2*pi*df(4)*dt);
% Reconstruct var
var_recon=basis1*f_useful(1)+...
basis2*f_useful(2)+...
basis3*f_useful(3)+...
basis4*f_useful(4);
var_recon=real(var_recon);
% Plot both curves
figure;
plot(var);
hold on;
plot(var_recon);
Adapt this code to your paper :)

Answer 2

Adapting my own post from Signal Processing. I think it's still relevant for those in Python.
I am no expert in this topic, but have some useful examples to share.
The more Fourier components you keep, the closer you'll mimic the original signal.
This example shows what happens when you keep 10, 20, ...up to n components. Assuming x and y are your data vectors.
import numpy
from matplotlib import pyplot as plt
n = len(y)
COMPONENTS = [10, 20, n]
for c in COMPONENTS:
colors = numpy.linspace(start=100, stop=255, num=c)
for i in range(c):
Y = numpy.fft.fft(y)
numpy.put(Y, range(i+1, n), 0.0)
ifft = numpy.fft.ifft(Y)
plt.plot(x, ifft, color=plt.cm.Reds(int(colors[i])), alpha=.70)
plt.title("First {c} fourier components".format(c=c))
plt.plot(x,y, label="Original dataset", linewidth=2.0)
plt.grid(linestyle='dashed')
plt.legend()
plt.show()
For the book's dataset, keeping up to 4, 10, and n components:
For your dataset, keeping up to 4, 10, and n components: