# Communication Systems: Mathematical Fundamentals

I’m using my website as platform for taking and storing notes from my classes. This is part of my collection of notes for 525.416 Communication Systems Engineering at JHU. Content is derived from JHU EP and the course instructor, Rob Nichols. These notes primarily follow the format and topics of the course.

# Classes of Signals

## Continuous and Discrete Time

Continuous signals have values at all points in time.

Discrete signals have values at specific times only. This a result of the sampling stage of digitization.

## Analog and Digital

Analog signals can take on any value.

Digital signals have a finite number of levels.

## Periodic and Aperiodic, Power and Energy

Periodic signals repeat a waveform over and over forever. Energy is infinite so the signal strength is characterized by power (energy over a period of time, the waveform period).

The signal power is defined as

Aperiodic signals have a definite beginning and a definite ending. However there is no period over which to average so we use energy to define signal strength.

The signal energy is defined as

# Important Signals

Unit Step Function - Transitions from 0 to 1 at time t=0.

Rect Function - A rectancular pulse.

Trangle Function - Produces a triangle pulse.

Sinc function - Fourier transform of a rectangular pulse. $\frac{sin(t)}{t}$

Delta Function

Also called the unit impulse signal

Shown as a line at t=0. See image below.

Shifting Property

$\delta (t - t_0)$ shifts the function to the right.

$\delta (t + t_0)$ shifts the function to the left.

This works because the delta function turns on when the argument, $t - t_0$ or $t + t_0$, is equal to zero.

Unit Area Property

Sampling Property

Grabing the value of the function at a given time.

Consider the function

We get the value of $x(t)$ at $t_0$ by the following:

Delta Function Expansion

# Signal Operations

## Time Shifting

If we have a signal s(t)

• s(t - T) shifts the signal to the right by T seconds
• s(t + T) shifts the signal to the left by T seconds

In other words, the shifting process shifts the time axis by T.

## Time Scaling

If we have a signal s(t)

• s(t/2) increases the period of the signal by a factor of two.
• s(2t) reduces the period of the signal by a factor of two.

## Time Inversion

Making s(t) into s(-t). Flips the time axis.

# The Trigonometric Fourier Series

Orthogonality - When two vectors are oriented 90 degrees to one another.

For orthogonal vectors, the dot product is zero: i.e. $\vec{x_1} \cdot \vec{x_2} = 0$

Signal Orthogonality

Signal dot product is defined as

Trigonometric Fourier Series

Compact Fourier Series

# References

[1] Nichols, R., Module 02. 525.416 Communication Systems Engineering. Summer 2017.