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Non-sinusoidal Periodic Waveforms

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• Author: Diego Inácio
• GitHub: github.com/diegoinacio
• Notebook: waveform_non-sinusoidal.ipynb

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Overview about non-sinusoidal periodic waveforms.

%matplotlib inline
import matplotlib.pyplot as plt
from IPython.display import Audio
import numpy as np

from _utils import *

plt.rcParams['figure.figsize'] = (16, 4)

fs = 44100                      # Sampling rate
dur = 3                         # Duration in seconds
t = np.linspace(0, dur, fs*dur) # Time vector

Sine wave model

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$$ \large x(t)=A sin(2\pi f t + \phi)=A sin(\omega t + \phi) $$

where,

• $A$ = The aplitude;
• $f$ = The ordinary frequency;
• $\omega$ = The angular frequency;
• $\phi$ = The phase.

A = 0.25
f = 110
omega = 2*np.pi*f
phi = 0

1. Square wave

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$$ \large x_\textrm{sqr}(t)=A \textrm{sgn}(sin(\omega t + \phi)) \quad ; \quad sgn(x)=\begin{cases}\begin{aligned} -1, x<0 \\\ 0, x=0 \\\ 1, x>0 \end{aligned}\end{cases} $$

x_sqr = A*np.sign(np.sin(omega*t + phi))
x_sqr[-1] = 1 # fix normalized output

Audio(x_sqr, rate=fs)

audiovis(x_sqr[:-1], tlim=[0, 0.04])

audio mono

1.1. Harmonic form

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$$ \large x_\textrm{sqrh}(t)=A\frac{4}{\pi}\sum_{k=1}^{\infty}\frac{sin((2k-1)(\omega t + \phi))}{2k-1} $$

N = 32
K = np.array([range(N)]).T + 1.0

_x_sqrh = np.sin((2*K - 1)*(omega*t + phi))/(2*K - 1)
x_sqrh = A*4/np.pi*np.sum(_x_sqrh, axis=0)
x_sqrh[-1] = 1 # fix normalized output

for h in _x_sqrh:
    plt.plot(t, h)
plt.xlim([0, 1/f])
plt.title('Harmonics')
plt.show()

Audio(x_sqrh, rate=fs)

audiovis(x_sqrh[:-1], tlim=[0, 0.04])

audio mono

spectrogram(x_sqrh)

audio mono

2. Triangular wave

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$$ \large x_\textrm{tri}(t)=A\frac{2}{a}\left(t+\frac{\phi}{\omega}-a\left\lfloor\frac{t}{a}+\frac{\phi}{\omega a}+\frac{1}{2}\right\rfloor\right)(-1)^{\left\lfloor\frac{t}{a}+\frac{\phi}{\omega a}+\frac{1}{2}\right\rfloor} \quad ; \quad a=\frac{1}{2f} $$

A = 0.5
a = 1/(2*f)
floor = np.vectorize(np.floor)
tpw = t + phi/omega # only for saving space
x_tri = A*2/a*(tpw - a*floor(tpw/a + 1/2))*(-1)**floor(tpw/a + 1/2)
x_tri[-1] = 1 # fix normalized output

Audio(x_tri, rate=fs)