This simple notebook lets you interactively explore the intermodulation effects of two superimposed sinusoids.
# standard bookkeeping
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from IPython.display import Audio, display
# interactivity library:
from ipywidgets import interact, interactive, fixed, interact_manual
import ipywidgets as widgets
plt.rcParams["figure.figsize"] = (14,4)
1. Introduction¶
Modulation is a technique used in telecommunication system in order to shift the spectral occupancy of a signal and adapt it to the transmission medium. In AM radio, for instance, a voice signal $s(t)$ whose maximum frequency content is below 10KHz, is modulated into the MHz range for wireless transmission:
$$ x(t) = s(t)\cos(2\pi f_0 t); $$
at the receiver, the $x(t)$ is demodulated to recover $s(t)$. The modulated sinusoid is usually called the carrier while $s(t)$ is the modulating signal.
Consider now the well-known trigonometric formula:
$$ \cos a + \cos b = 2\cos\left(\frac{a + b}{2}\right)\cos\left(\frac{a-b}{2}\right); $$
using two sinusoidal signals, we have
$$ \cos(\omega_a n) + \cos(\omega_b n) = 2\cos\left(\omega_0 n\right)\cos\left(\omega_c n\right) $$
with $\omega_0 = (\omega_a - \omega_b)/2$ and $\omega_c = (\omega_a + \omega_b)/2$. While the left-hand side describes the sum of two cosines, which is what we obtain if we play two sinusoids together, the right-hand side describes the same signal as a product, that is, as a modulation.
If the frequencies are very close to each other, $\omega_a \approx \omega_b$, then their average will be approximately equal to either of them, that is, $\omega_c m\approx \omega_a \approx \omega_b$ while their difference will be very small, that is, the modulating signal at frequency $\omega_0$ will have a very low frequency.
When these sinusoids are mapped to real-world frequencies in the audible range, the perceived effect will be that of a single sinusoid whose volume fluctuates in time at a rate inversely proportional to $\omega_0$. Acoustically, this volume fluctuation is called a frequency beat.
1.1 Tuning an instrument¶
Interestingly, frequency beats are useful to tune a stringed instrument when a reference tone (such as that of a tuning fork) is available. The idea is to play the reference note and the string at the same time. If the string is playing a frequency in the vicinity of the reference, the composite sound will be perceived as a modulated sinusoid. The number of volume fluctuations per second provide an estimate of the difference in frequency between the two notes.
2. Let's play¶
This simple fuction generates, plots and plays two sinusoids at the given frequencies:
def beat_freq(f1=220.0, f2=224.0):
# the clock of the system
LEN = 4 # seconds
Fs = 8000.0
n = np.arange(0, int(LEN * Fs))
s = np.cos(2*np.pi * f1/Fs * n) + np.cos(2*np.pi * f2/Fs * n)
# start from the first null of the beating frequency
if f2 != f1:
K = int(Fs / (2 * abs(f2-f1)))
s = s[K:]
# play the sound
display(Audio(data=s, rate=Fs))
# display one second of audio
plt.plot(s[0:int(Fs)])
The interactivity module in the notebooks allows us to easily change one of the frequencies and observe the results
interact(beat_freq, f1=(200.0,300.0), f2=(200.0,300.0));
