# General Description

## # 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:

1. Some basic terms like  »period duration«,  »basic frequency«  and  »circular frequency«,
2. the properties of a  »DC signal«  as a limiting case of a periodic signal,
3. the definition and interpretation of the  »Dirac delta function«,
4. the  »spectral representation«  of a DC signal or a DC signal component,
5. the time and frequency representation of  »harmonic oscillations«,  and finally
6. the application of  »Fourier series«  for spectral analysis of periodic signals.

## Features and applications

Periodic signals are of great importance for Communications Engineering:

• 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.

Oscilloscope image of cosine and triangular signals

$\text{Example 1:}$  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  »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:
$$\omega_{0}=2\pi f_{0} = {2\pi}/{T_{0}}.$$

Given signal and period duration

$\text{Example 2:}$  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).

Resulting period duration of the sum of cosine and sine signal
• 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.