Theory
Sinusoidal signals are the most basic signals in the theory of signals and
systems. This tool tries to help the student in getting more familiar with
operations that need to be performed on Sinusoids. Manipulating sinusoids can be
done by representing them as Phasors and Complex exponentials. In this section
we give a brief overview of the different kinds of representation of sinusoids
and some basic operations that can be performed on them.
Sinusoids include both 'cosine' and 'sine' signals. The mathematical formula
representing a sinusoidal signal is

Analysis and manipulation of sinusoidal signals in the above form will be
difficult to achieve. So, often they are converted to related Complex
exponential signals which are easier to manipulate and analyze. These complex
exponentials depend on the polar form of representation of complex numbers.
In general for a complex number represented in cartesian form

its modulus and phase are represented as

Using the above relations any complex number can be represented in polar form,

A better polar form can be obtained using Euler's formula for complex
exponentials, given below.

Making use of the above relation any complex number can be represented as,

Now we will look at the relation between sinusoidal signal and its
representation as a complex exponential signal. The complex exponential signal
is defined in the next equation and represnted using Euler's formula as shown,

From this the representation cosine part can be obtained as

Having been exposed to different ways of representing sinusoidal signals, next
we look at ways to manipulate such signals using the above representations. One
useful operation would be to add 2 or more sinusoids. This can be achieved by
first representing the sinusoids as complex exponentials and then converting to
the cartesian or rectangular forms. Addition of 2 or more components can be
easily done in rectangular form. After this they can be converted back to
sinusoid form.
The Examples section provides examples for doing several forms
of this manipulation.i.e., between cosine, complex exponentials, phasors and
rectangular forms of complex numbers.
Sampling
The continuous time sinusoid given as

can be represented in its discrete-time form for a given sampling period
Ts
as shown below,

Here we do not deal with issues related to sampling of signals, please refer to
other related tools to know more about sampling of signals.
Pole-Zeros from Transfer function
The general form of representing system transfer function is

Let us consider the second-order case, which is represented as

The numerator and denominator polynomials each have 2 roots. The numerator roots
are called the 'Zeros', since for these roots H(z) becomes zero. The denominator
roots are called 'Poles' of the system and for these roots, H(z) becomes
undefined (infinite). The poles and zeros are plotted in the complex z-plane and
can be determined as shown in the Example section below.
The next section deals with specific examples related to the various types of
tests in the Phasor Races GUI.
Examples
In this section, the arithmetic involved in solving the different types of
Questions in the tool are illustrated with examples.
Complex
In this type of test, the sum of given 2 complex numbers needs to be found. By
default, the numbers are in exponential form. So, in order to find the sum
Convert each complex number to rectangular form
The real and imaginary parts can be added separately to obtain the required
complex number (sum) in rectangular form.
The Rectangular form can than be converted to Polar form, by finding the
magnitude and phase.
These steps are shown below.

Sinusoid
This type of test deals with adding sinusoids that have the same frequency, but
different (or same) amplitude and phase. One way to do this is
Represent the sinusoids as Phasors
Convert the Phasors to their Rectangular form
Perform addition of the complex numbers in rectangular form
Convert the result obtained in previous step to the polar form to obtain the
phasor.
Express the phasor in its cosine form.
These steps are shown below.

Real Part
The goal in this type of test is, given a complex number in mixed form, we need
to determine the Real part of the number. One way to do this is,
Convert the rectangular form of the complex number to its exponential form,
by finding its magnitude and phase.
Once the exponential form is available, Euler's formula can be used to
express it as sum of cosine and sine.
Then the required real part is the cosine term.
These steps are shown below.

Spectrum
In this type of test, there are two kinds of question. They are
- Converting exponentials to cosine form
- Converting cosine to exponential form
The Inverse form of Euler's formula can be used to obtain the required
form. This can be done as shown below,
Exponential to Cosine form

Cosine to Exponential form

Sampling
In this type of test, we are to determine the Discrete-time representation of a
signal from its continuous-time representation and given sampling period. This can
be done as shown in below,

Z-Transform
In this type of test, we are required to determine the location of Poles and
Zeros of a given transfer function. The way to determine the poles and zeros is
as shown below,

In the Pole-Zero plot shown below, 'O'
represent a Zero and 'x' represent a Pole.

All
In this mode, one of the above types of tests is presented in a random order.
Does it all make sense to you?  If you are not sure go over it one more
time.
If you still do not get it, let me know what is confusing
you.  Send
me by clicking on my name in the Overview section.  I want to make this
tutorial understandable and any feedback is appreciated!