## 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 limiting case of a periodic signal,
- the definition and interpretation of the »Dirac delta 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 »**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 »
**period duration**« $T_{0}$ indicates the smallest possible value, which satisfies the above equation.

- The »
**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 $\text{Heinrich Hertz}$.

- The »
**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 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