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Fast Fourier Transform

Preamble

In [1]:
# used to create block diagrams
%reload_ext xdiag_magic
%xdiag_output_format svg
    
import numpy as np                   # for multi-dimensional containers
import pandas as pd                  # for DataFrames
import plotly.graph_objects as go    # for data visualisation
import plotly.io as pio              # to set shahin plot layout
from plotly.subplots import make_subplots
import scipy.fftpack                 # discrete Fourier transforms

pio.templates['shahin'] = pio.to_templated(go.Figure().update_layout(margin=dict(t=0,r=0,b=40,l=40))).layout.template
pio.templates.default = 'shahin'
pio.renderers.default = "notebook_connected"

In a previous section we looked at how to create a single Sine Wave and visualise it in the time domain.

In [2]:
sample_rate = 1000
start_time = 0
end_time = 10

time = np.arange(start_time, end_time, 1/sample_rate)

frequency = 3
amplitude = 1
theta = 0

sinewave = amplitude * np.sin(2 * np.pi * frequency * time + theta)

fig = go.Figure(layout=dict(xaxis=dict(title='Time (sec)'),yaxis=dict(title='Amplitude')))
fig.add_scatter(x=time, y=sinewave)
fig.show()

Fourier Transform

In [3]:
freq = scipy.fftpack.fft(sinewave)/len(time)
In [4]:
hz = np.linspace(0, sample_rate/2, int(np.floor(len(time)/2)+1))

fig = go.Figure(
    layout=dict(
        xaxis=dict(title='Frequency (Hz)', range=[0, np.max(frequency) * 1.2]),
        yaxis=dict(title='Amplitude'))
)

fig.add_scatter(x=hz, y=2 * np.abs(freq))
fig.show()
In [ ]:
 

Summing Sine Waves

In [5]:
sample_rate = 1000
start_time = 0
end_time = 10
theta = 0

time = np.arange(start_time, end_time, 1/sample_rate)

frequency = [3, 5, 2, 1, 10]
amplitude = [1, 2, 7, 3, 0.1]

fig = go.Figure(layout=dict(xaxis=dict(title='Time (sec)'),yaxis=dict(title='Amplitude')))
fig.add_scatter(x=time, y=sinewave)
fig.show()
In [6]:
fig = go.Figure(layout=dict(xaxis=dict(title='Time (sec)'),yaxis=dict(title='Amplitude')))
fig.add_scatter(x=time, y=sinewave)
fig.show()