**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*.

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:

- Add
- Subtract
- Multiply
- Divide
- Inverse
- Conjugate

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:

- Add
- Subtract
- Multiply
- Divide
- Inverse
- Conjugate

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]
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]