In [1]:
%matplotlib inline
import seaborn
import numpy, scipy, matplotlib.pyplot as plt, librosa, IPython.display as ipd
Fourier Transform¶
Let's download an audio file:
In [2]:
import urllib
filename = 'c_strum.wav'
urllib.urlretrieve('http://audio.musicinformationretrieval.com/c_strum.wav', filename=filename)
x, sr = librosa.load(filename)
In [3]:
print x.shape
print sr
Listen to the audio file:
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ipd.Audio(x, rate=sr)
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Fourier Transform¶
The Fourier Transform (Wikipedia) is one of the most fundamental operations in applied mathematics and signal processing.
It transforms our time-domain signal into the frequency domain. Whereas the time domain expresses our signal as a sequence of samples, the frequency domain expresses our signal as a superposition of sinusoids of varying magnitudes, frequencies, and phase offsets.
To compute a Fourier transform in NumPy or SciPy, use scipy.fft:
In [5]:
X = scipy.fft(x)
X_mag = numpy.absolute(X)
f = numpy.linspace(0, sr, len(X_mag)) # frequency variable
Plot the spectrum:
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plt.figure(figsize=(13, 5))
plt.plot(f, X_mag) # magnitude spectrum
plt.xlabel('Frequency (Hz)')
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Zoom in:
In [7]:
plt.figure(figsize=(13, 5))
plt.plot(f[:5000], X_mag[:5000])
plt.xlabel('Frequency (Hz)')
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