The discrete cosine transform (DCT) is a transform similar to the Fourier transform, but it tries to represent a signal by a sum of cosine terms only (refer to equation 6.11). The DCT is used for signal compression and in the calculation of the mel frequency spectrum, which I mentioned in the Analyzing the frequency spectrum of audio recipe. We can convert normal frequencies to the mel frequency (a frequency more appropriate for the analysis of speech and music) with the following equation:

The steps to create the mel frequency spectrum are not complicated, but there are quite a few of them. The relevant Wikipedia page is available at https://en.wikipedia.org/wiki/Mel-frequency_cepstrum (retrieved September 2015). If you do a quick web search, you can find a couple of Python libraries that implement the algorithm. I implemented a very simple version of the computation in this recipe.

How to do it...

1. The imports are as follows:

   import dautil as dl
   from scipy.fftpack import dct
   import matplotlib.pyplot as plt
   import ch6util
   import seaborn as sns
   import numpy as np
   from IPython.display import HTML

2. Load the data and transform it as follows:

   rate, audio = ch6util.read_wav()
   transformed = dct(audio)

3. Plot the amplitude spectrum using DCT:

   sp = dl.plotting.Subplotter(2, 2, context)
   freqs = np.fft.fftfreq(audio.size, 1./rate)
   indices = np.where(freqs > 0)[0]
   sp.ax.semilogy(np.abs(transformed)[indices])
   sp.label()

4. Plot the distribution of the amplitude:

   sns.distplot(np.log(np.abs(transformed)), ax=sp.next_ax())
   sp.label()

5. Plot the distribution of the phase:

   sns.distplot(np.angle(transformed), ax=sp.next_ax())
   sp.label()

6. Plot the mel amplitude spectrum as follows:

   magnitude = ch6util.amplitude(audio)
   cepstrum = dl.ts.power(np.fft.ifft(np.log(magnitude ** 2)))
   mel = 1127 * np.log(1 + freqs[indices]/700)
   sp.next_ax().plot(mel, ch6util.amplitude(dct(np.log(magnitude[indices] ** 2))))
   sp.label()
   HTML(sp.exit())

Refer to the following screenshot for the end result:

The code for this recipe is in the analyzing_dct.ipynb file in this book's code bundle.