## Contents

## # OVERVIEW OF THE SECOND MAIN CHAPTER #

In this chapter periodic signals are considered and described mathematically in the time and frequency domain.

This chapter contains in detail:

- Some basic terms like "period duration", "basic frequency" and "circular frequency",
- the properties of a "DC signal" as a boundary case of a periodic signal,
- the definition and interpretation of the "Dirac function",
- the spectral representation of a DC signal or a DC signal component,
- the time and frequency representation of "harmonic oscillations", and finally
- the application of "Fourier series" for spectral analysis of periodic signals.

## Features and applications

Periodic signals are of great importance for communications engineering:

- They belong to the class of deterministic signals, whose time function can be specified in analytical form.
- Their signal path is thus known for all times $t$ and can be clearly predicted for the future.
- They are therefore never information-carrying signals.

Nevertheless, periodic signals are often also required in communications engineering, for example

- for modulation and demodulation in carrier frequency systems,
- for synchronization and clock regeneration in digital systems,
- as test and verification signals during system implementation.

$\text{Example 1:}$ The oscilloscope image shows two typical representatives of periodic signals:

- above a cosine signal,
- below a triangular signal.

As can be seen from the displayed settings, the period duration of both signals is one millisecond and the amplitude one volt.

## Definition and parameters

Before we turn to the signal parameters of a periodic signal, the term "periodicity" shall be clearly defined:

$\text{Definition:}$ A $\text{periodic signal}$ $x(t)$ is present if for all arbitrary values of $t$ and all integer values of $i$ with an appropriate $T_{0}$ applies:

- $$x(t+i\cdot T_{0}) = x(t).$$

This results in the following parameters:

- The $\text{period duration}$ $T_{0}$ indicates the smallest possible value, which satisfies the above equation.
- The $\text{basic frequency}$ $f_{0} = 1/T_{0}$ describes the number of periods per time unit (mostly per second).
- The unit "1/s" is also called "Hz", named after the German physicist Heinrich Hertz.
- The $\text{basic circular frequency}$ $\omega_{0}$ represents the angular rotation per second, usually given in radians.
- In contrast to the basic frequency, the unit "Hz" is not common here, but "1/s". The following equation applies:

- $$\omega_{0}=2\pi f_{0} = {2\pi}/{T_{0}}.$$

$\text{Example 2:}$ Here, a periodic time signal is shown:

- The period duration is $T_{0} = 2.5 \ \rm ms$.
- From this the basic frequency $f_0 = 400 \ \rm Hz$ is calculated.
- The basic circular frequency results to $\omega_{0}=2513 \ \rm 1/s.$

## Resulting period duration

If a signal $x(t)$ consists of the sum of two periodic signals $x_{1}(t)$ and $x_{2}(t)$ with the period durations $T_{1}$ or $T_{2}$, the resulting period duration of the sum signal is the smallest common multiple of $T_{1}$ and $T_{2}$.

- This statement applies independently of the amplitude and phase relations.
- On the other hand, if $T_{1}$ and $T_{2}$ don't have a rational common multiple $($Example: $T_{2} = \pi \cdot T_{1})$,

then the sum signal $x(t)$ is in contrast to its two components $x_{1}(t)$ and $x_{2}(t)$ not periodic.

$\text{Example 3:}$ Here, a cosinusoidal signal $x_{1}(t)$ with period duration $T_{1} = 2\; {\rm ms}$ (blue signal course) is added with a sinusoidal signal $x_{2}(t)$ with period duration $T_{2} = 5\; {\rm ms}$ and twice the amplitude (green curve).

- The (red) sum signal $x(t) = x_{1}(t) + x_{2}(t)$ then shows the resulting period duration $T_{0} = 10\; {\rm ms}$ ⇒ basic frequency $f_{0} = 100\; {\rm Hz}$.
- The frequency $f_{0}$ itself is not contained in $x(t)$ only integer multiples of it, namely

- $f_{1} = 500\; {\rm Hz}$ and $f_{2} = 200\; {\rm Hz}$.

With the interactive applet Period Duration of Periodic Signals the resulting period of two harmonic oscillations can be determined.

## Exercises for the chapter