Units of frequency for Fourier transforms

Question

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?

Answer 1

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.

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