Lab 5: Tremolo/Beats Effects 1#
Last updated 8/12/24
00. Content #
Mathematics#
Laws of sines/cosines
Programming Skills#
Type here
Embedded Systems#
N/A
0. Required Hardware #
N/A
Write your name and email below:
Name: me
Email: me @purdue.edu
import numpy as np
1. Introduction to Amplitude Modulation #
Amplitude modulation is the process of multiplying a CT signal by a another signal \(c(t)\) called a “carrier.”
If the signal is denoted by \(x(t)\), then the modulated signal \(y(t)\) is given by
For example, we can modulate a signal \(x(t)\) using a cosine carrier \(c(t)=\cos (2 \pi f_0 t) \) with frequency \(f_0\). Then the modulated signal \(y(t)\) is given by
As you may have learned in your companion course (e.g. ECE301), periodic carriers play a special role in signal processing as they can be used to shift the frequencies of a signal. This is because of the multiplication property of the Fourier transform/series. We will come back to the idea of shifting frequencies at the end of this lab. But first, let us explore a fun and intuitive effect that can be created with amplitude modulation: the tremolo effect (also called “beat effect”).
2. The Tremolo Effect #
When the frequency of the carrier \(c(t)\) is low, more specifically when it is below the audible range (the audible range is about 20Hz to 20KHz), then modulation creates a “tremolo” effect” on the original signal. This is something you can hear if \(x(t)\) is an audio signal. It sounds like one is turning the volume up and down, periodically. Let’s try it out!
Modulation with Sine Wave Exercises#
Exercise 1 #
Let \(x(t)= \sin 2 \pi 440 t\) (this is a middle A). Plot and play \(x(t)\). Now consider the carrier \(c(t)=\cos (2 \pi f_0 t )\) with \(f_0=2 Hz\). Plot and play the modulated sound \(y(t)= x(t) c(t).\)
Exercise 2 #
Change the frequency \(f_0\). Describe what happens to the modulated sound when the frequency is very low (<20Hz) versus when it is very high (20-20kHZ).
Exercise 3 #
Create a sine wave of frequency 220Hz, and modulate it with \(c(t)=\cos (2 \pi f_0 t )\)
Modulation with Triangular Wave Exercises#
Exercise 4 #
Create a periodic triangular wave with frequency of 3 Hz. Plot and play the triangular wave. Change the triagular wave frequency and play the resulting sound.
Exercise 5 #
Modulate a middle A with the triangular wave you created in the previous exercise. Play the modulated sound. Change the triagular wave frequency and play the resulting modulated sound. Describe what happens when the frequency is very low (<20Hz) versus when it is very high (20-20kHZ).
Modulation with Square Wave Exercises#
Exercise 6 #
Create a periodic square wave with frequency of 3 Hz. Plot and play the square wave.
Exercise 7 #
Modulate a middle A with the square wave you created in the previous exercise. Play the modulated sound.
3. The Math Behind Multiplication by a Periodic Carrier #
Because of the properties of complex exponentials, multiplying a signal by a periodic carrier signal results in changing the frequencies of the components of the signal.
For example, multiplying two sine wave together is the same as adding a wave at their average frequency and a wave at their frequency difference.
Indeed, we have
In other words, the frequency \(f_1\) in the signal \(\sin \left( 2\pi f_1 t\right)\) is divided into two components which are “shifted” to new frequencies on either sides of \(f_1\); the extent of the shift is given by the carrier frequency \(f_2\).
Exercise 8 #
Prove the above formula. Hint: Transform the sines into complex exponentials.
Reflection #
Do not skip this section! Lab will be graded only on completion of this section.
1. What parts of the lab, if any, do you feel you did well?
2. What are some things you learned today?
3. Are there any topics that could use more clarification?
4. Do you have any suggestions on parts of the lab to improve?