2. Sinusoids

Overview: In chapter two, the most basic waveform in signal processing, the cosine wave, is presented. The mathematical formula for the cosine wave, in its most general form is given below: $$x(t)=A \cos(2 \pi f_0 t + \varphi)$$ Where $$x(t)$$ is a function of the time variable $$t$$. The amplitude of the cosine is given by the real number $$A$$. The frequency of the of the cosine wave is $$f_0$$, and in the audio experiments that follow, it is the frequency that determines what we hear. Finally, the phase of the sinusoid is given by the parameter $$\varphi$$. A plot of a cosine is given in the figure below:

Also in chapter two, the phasor representation of sinusoids is presented. A new signal is introduced called the complex exponential:

$$x(t)= A e^{j(2 \pi f_0 t + \varphi)}$$

The generalization to complex exponentials is important for later work in Fourier analysis, so we are laying a foundation for the future. The real part of the complex exponential is a cosine, and its imaginary part is the sine function, so a plot of the complex exponential is a rotating vector with a constant length $$A$$. This signal is called a rotating phasor.
Databases:

Demos - MATLAB6

Here's a demo that shows the nearly sinusoidal waveforms produced by two clay whistles.
Here are four movies showing rotating phasors and how the real part of the phasor traces out a sinusoid versus time. Two of the movies show how rotating phasors of different frequencies interact to produce complicated waveforms such as beat signals.
Sine Drill (sindrill) is a program that tests the users ability to determine basic parameters of a sinusoid. After a plot of a sinusoid is displayed, the user must correctly guess its amplitude, frequency, and phase.
This is an introduction to plotting sinusoids (both sine and cosine waves) from equations. The tutorial also reviews how to write the equation of the sinusoid given a plot of the waveform.
This demo shows how the size and stiffness of a tuning fork affect the tone produced by three different tuning forks.
ZDrill is a program that tests the users ability to calculate the result of simple operations on complex numbers. The program emphasizes the vectorial view of a complex number. The following six operations are supported:
• Subtract
• Multiply
• Divide
• Inverse
• Conjugate

Demos - LabVIEW3

Shows how the real part of the rotating phasors traces out a sinusoid versus time.
Tests the users' ability to determine basic parameters of a sinusoid.
Tests users ability to calculate the result of simple operations on complex numbers. The program emphasizes the vectorial view of a complex number. The following six operations are supported:
• Subtract
• Multiply
• Divide
• Inverse
• Conjugate

Labs - MATLAB4

In this lab we introduce the fundamentals of Matlab. Matlab is a programming environment that you will find helpful for many of the exercises in this text.
Manipulating sinusoid functions using complex exponentials turns trigonometric problems into simple arithmetic and algebra. In this lab, we first review the complex exponential signal and the phasor addition property needed for adding cosine waves. Then we will use Matlab to make plots of phasor diagrams that show the vector addition needed when combining sinusoids.
Manipulating sinusoid functions using complex exponentials turns trigonometric problems into simple arithmetic and algebra. In this lab, we first review the complex exponential signal and the phasor addition property needed for adding cosine waves. Then we will use Matlab to make plots of phasor diagrams that show the vector addition needed when combining sinusoids. [Files]
The objective of this lab is to learn how the outputs from two spatially separated sensors that receive signals from the same source can be used to estimate the direction to the source of the signal. The key to this processing is phase difference or time difference of arrival (TDOA) at the two receivers. [Files]

Labs - LabVIEW2

In this lab we introduce the fundamentals of LabVIEW. LabVIEW is a programming environment that you will find helpful for many of the exercises in this text.
Manipulating sinusoid functions using complex exponentials turns trigonometric problems into simple arithmetic and algebra. In this lab, we first review the complex exponential signal and the phasor addition property needed for adding cosine waves. Then we will use Matlab to make plots of phasor diagrams that show the vector addition needed when combining sinusoids. [Files]