

In [1]:

    
# Graphing helper function
def setup_graph(title='', x_label='', y_label='', fig_size=None):
    fig = plt.figure()
    if fig_size != None:
        fig.set_size_inches(fig_size[0], fig_size[1])
    ax = fig.add_subplot(111)
    ax.set_title(title)
    ax.set_xlabel(x_label)
    ax.set_ylabel(y_label)

Fourier Transform

The Fourier Transform is like a prism (not the NSA one)

Fourier Transform Definition

$$G(f) = \int_{-\infty}^\infty g(t) e^{-i 2 \pi f t} dt$$

For our purposes, we will just be using the discrete version...

Discrete Fourier Transform (DFT) Definition

$$G(\frac{n}{N}) = \sum_{k=0}^{N-1} g(k) e^{-i 2 \pi k \frac{n}{N} }$$

Meaning:

$N$ is the total number of samples
$g(k)$ is the kth sample for the time-domain function (i.e. the DFT input)
$G(\frac{n}{N})$ is the output of the DFT for the frequency that is $\frac{n}{N}$ cycles per sample; so to get the frequency, you have to multiply $n/N$ by the sample rate.

How to represent waves



In [3]:

    
freq = 1 #hz - cycles per second
amplitude = 3
time_to_plot = 2 # second
sample_rate = 100 # samples per second
num_samples = sample_rate * time_to_plot

t = linspace(0, time_to_plot, num_samples)
signal = [amplitude * sin(freq * i * 2*pi) for i in t] # Explain the 2*pi

Why the 2*pi?

If we want a wave which completes 1 cycle per second, so sine must come back to the same position on a circle as the starting point
So one full rotation about a circle - $2 \pi$ (in radians)

Plot the wave



In [4]:

    
setup_graph(x_label='time (in seconds)', y_label='amplitude', title='time domain')
plot(t, signal)









    Out[4]:





[<matplotlib.lines.Line2D at 0x104d15290>]

Convert to the Frequency Domain



In [5]:

    
fft_output = numpy.fft.rfft(signal)
magnitude_only = [sqrt(i.real**2 + i.imag**2)/len(fft_output) for i in fft_output]
frequencies = [(i*1.0/num_samples)*sample_rate for i in range(num_samples/2+1)]



In [6]:

    
setup_graph(x_label='frequency (in Hz)', y_label='amplitude', title='frequency domain')
plot(frequencies, magnitude_only, 'r')









    Out[6]:





[<matplotlib.lines.Line2D at 0x104ddf190>]

Question: So what does the Fourier Transform give us?

The amplitudes of simple sine waves
Their starting position - phase (we won't get into this part much)

Question: what sine wave frequencies are used?

Answer: This is determined by how many samples are provided to the Fourier Transform
Frequencies range from 0 to (number of samples) / 2
Example: If your sample rate is 100Hz, and you give the FFT 100 samples, the FFT will return the amplitude of the components with frequencies 0 to 50Hz.



In [6]: