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I'm trying to get to grips with the relationship between a signal $x(t)$, its Fourier transform $X(F)$ and the graph representation of $t$ plotted against $|X(F)|$. I've been using Matlab to perform the calculations and plot the graphs and see if I can predict where the spike will be on the X axis for a particular signal.

Firstly I generate a vector called xaxis which contains values from 0 to 300 in steps of 1. If I generate another vector called signal by using $signal = sin(3*(xaxis/100)*2*pi)$ and plot it, it comes out as expected - the wave oscillates 9 times between 0 and 300. The Fourier transform for this wave is also straightforward - the spike on the graph of $|X(F)|$ occurs at 9, which makes sense.

If, however, I use the xaxis range $0$ to $2\pi$ in steps of $\frac{\pi}{100}$ and define signal with $signal = sin(3*xaxis)$, the signal wave clearly oscillates three times over the interval but the frequency spike does not occur at 3 - it's roughly 0.95 from what I can see. I have a feeling that I'm interpreting the X axis value of the spike incorrectly but I can't get my head around exactly how the value of 0.95 is related to the correct answer. Could anyone enlighten me?

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The first signal $y(t) = \sin (2\pi \cdot \frac{3}{100} t)$ has a frequency of $\frac{3}{100}$, not $9$.

When using python to plot the two signals it works out just fine. I suspect you made some mistake in your Matlab code.

Two signals next to their Fourier transform

Code:

 1 # Illustrating the relationship between a signal 2 # and its Fourier transform. 3 4 from __future__ import division 5 from matplotlib import pyplot as plt 6 import numpy as np 7 8 # Sine function. 9 def f(freq): 10 return np.sin(2*np.pi*freq*t) 11 12 fig = plt.figure("A signal and its Fourier transform") 13 plt.suptitle("A signal and its Fourier transform") 14 15 # Proper spacing between subplots 16 fig.subplots_adjust(hspace=.5) 17 18 # Use of TeX 19 plt.rc('text',usetex=True) 20 plt.rc('font',family='serif') 21 22 23 # For the first signal we have steps of dt = 1 and 24 # 300 samples. 25 num_samples = 300 26 dt = 1 27 t = dt * np.arange(num_samples) 28 29 # Plot the signal. 30 fig.add_subplot(221) 31 plt.title("Frequency = $3/100$ and $dt = 1$") 32 plt.xlabel("t") 33 plt.ylabel("y(t)") 34 y = f(3./100) 35 36 plt.plot(t,y) 37 38 # Plot the Fourier transform. 39 fig.add_subplot(222) 40 plt.title("Fourier transform") 41 ft = np.fft.rfft(y) 42 frq = np.fft.rfftfreq(num_samples) 43 plt.xlabel("Frequency") 44 plt.ylabel("Amplitude") 45 46 plt.plot(frq,abs(ft)) 47 48 # For the second signal we have steps of dt = pi/100 49 # and 100 samples, to get an interval from 0 to 2 pi. 50 num_samples = 100 51 dt = np.pi/100 52 t = dt * np.arange(num_samples) 53 54 # Plot the signal. 55 fig.add_subplot(223) 56 plt.title(r"Frequency =$3$ and $dt=\pi/100$") 57 plt.xlabel("t") 58 plt.ylabel("y(t)") 59 y = f(3) 60 plt.plot(t,y) 61 62 # Plot the Fourier transform. 63 fig.add_subplot(224) 64 plt.title("Fourier transform") 65 ft = np.fft.rfft(y) 66 frq = np.fft.rfftfreq(num_samples,d=np.pi/100) 67 plt.xlabel("Frequency") 68 plt.ylabel("Amplitude") 69 plt.plot(frq,abs(ft)) 70 71 plt.show() 72
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  • $\begingroup$ Perhaps I'm getting mixed up with the addition of time - the module Fourier transforms are being taught in is to do with signal processing, so a lot of the time things like period are defined relative to time. $\endgroup$ Commented Jun 9, 2014 at 13:09

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