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.

[Back to Top]

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
    1. Converting exponentials to cosine form
    2. 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!

    [Back to Top]