## Contents

## # OVERVIEW OF THE SECOND MAIN CHAPTER #

$\text{Definition:}$
In general, **»distortion«** is understood to be undesirable deterministic changes in a message signal caused by a transmission system.

In addition to stochastic interferences $($noise, crosstalk, etc.$)$, such deterministic distortions are a critical limitation on the transmission quality and rate for many transmission systems.

This chapter presents these distortions in a summarizing way, in particular:

- The quantitative description of such signal falsifications via the »distortion power«,
- the distinguishing features between »nonlinear and linear distortions«,
- the meaning and computation of the »distortion factor in nonlinear systems«, and
- the effects of »linear attenuation and phase distortions«.

## Prerequisites for the second main chapter

In the following, we consider always a »system«

- whose input is the signal $x(t)$ with the corresponding spectrum $X(f)$, and

- the output signal is denoted by $y(t)$ and its spectrum by $Y(f).$

The block labelled »**system**« can be a part of an »electrical circuit« or a complete transmission system« consisting of

- »transmitter«,
- »channel«, and
- »receiver«.

For the whole main chapter »Signal Distortions and Equalization« the following shall apply:

- The system be »
**time-invariant**«. If the input signal $x(t)$ results in the output signal $y(t)$, then a later input signal of the same form – in particular $x(t - t_0)$ – will result in the signal $y(t - t_0)$.

- In the following, »
**no noise**« is considered, which is always present in real systems. For the description of these phenomena we refer to the book »Theory of Stochastic Signals«.

- About the system »
**no detailed knowledge**« is assumed. In the following of this chapter, all system properties are derived from the signals $x(t)$ and $y(t)$ or their spectra alone.

- In particular, no specifications are made here with regard to »
**linearity**«. The »system« can be »linear« $($prerequisite for the application of the superposition principle$)$ or »non-linear«.

- Not all system properties are discernible from a single test signal $x(t)$ and its response $y(t)$ . Therefore,
**sufficiently many test signals**must be used for evaluation.

In the following, we will classify transmission systems in more detail in this respect.

## Ideal and distortion-free system

$\text{Definition:}$
One deals with an »**ideal system**« if the output signal $y(t)$ is identical with the input signal $x(t)$:

- $$y(t) \equiv x(t).$$

- It should be noted that such an ideal system does not exist in reality even if statistical disturbances and noise processes $($that always exist but are not considered in this book$)$ are disregarded.
- Every transmission medium exhibits losses $($»attenuation«$)$ and »transit times». Even if these physical phenomena are very small, they are never zero. Therefore it is necessary to introduce a somewhat less strict quality characteristic.

$\text{Definition:}$
A »**distortion-free system**« exists if the following condition is fulfilled:

- $$y(t) = \alpha \cdot x(t - \tau).$$

- Here, $α$ describes the »attenuation factor« and $τ$ the »transit time«.
- If this condition is not met, the system is said to be »
**distortive**«.

$\text{Example 1:}$ The following diagram shows the input signal $x(t)$ and the output signal $y(t)$ of a nonideal but distortion-free system. The system parameters are $α = 0.8$ and $τ = 0.25 \ \rm ms$.

$\text{Note:}$

- The attenuation factor $α$ can be completely reversed by a receiver-side gain of $1/α = 1.25$, but it must be taken into account that this also increases any noise.

- However, the transit time $τ$ cannot be compensated due to »causality reasons«. It now depends on the application whether such a transit time is subjectively perceived as disturbing or not.

- For example, even with a transit time of one second the $($unidirectional$)$ TV broadcast of an event is still described as "live".
- In contrast to this, transit times of $\text{300 ms}$ are already perceived as very disturbing in bidirectional communication – e.g. a telephone call.
- You either wait for the other person to react or both participants interrupt each other.

## Quantitative measure for the signal distortions

We now consider a distortive system on the basis of the input and output signal.

- We assume that apart from the signal distortions there is no additional frequency-independent attenuation factor $α$ and no additional transit time $τ$. These conditions are fulfilled for the signal sections sketched on the right.

- In addition to the signals $x(t)$ and $y(t)$, the difference signal is shown in the diagram:

- $$\varepsilon(t) = y(t) - x(t).$$

As a quantitative measure of the strength of distortions, the »mean square value of this difference signal« is applicable:

- $$\overline{\varepsilon^2(t)} = \frac{1}{T_{\rm M}} \cdot \int_{ 0 }^{ T_{\rm M}} {\varepsilon^2(t) }\hspace{0.1cm}{\rm d}t\hspace{0.4cm} \left( = P_{\rm V} \right).$$

The following should be noted about this equation:

- The measuring time $T_{\rm M}$ must be chosen sufficiently large. Actually, this equation should be formulated as a limit process.
- This expression is called »
**mean squared error**« $\rm (MSE)$ or »**distortion power**« $P_{\rm V}$ $($because of "distortion" ⇒ German: "Verzerrung" ⇒ subscript "$\rm V$"$)$. - If $x(t)$ and $y(t)$ are voltage signals, then $P_{\rm V}$ has the unit of ${\rm V}^2$, meaning the power is related to the resistance $R = 1 \ Ω$ according to the above definition.

$\text{Definition:}$ Making use of the power $P_x$ $($based on $R = 1 \ Ω)$ of the input signal $x(t)$ the »**signal–to–distortion (power) ratio**« can be given as:

- $$\rho_{\rm V} = \frac{ P_{x} }{P_{\rm V} } \hspace{0.3cm} \Rightarrow \hspace{0.3cm} 10 \cdot \lg \hspace{0.1cm}\rho_{\rm V} = 10 \cdot \lg \hspace{0.1cm}\frac{ P_{x} }{P_{\rm V} }\hspace{0.3cm} \left( {\rm in \hspace{0.15cm} dB} \right).$$

For the signals shown in the diagram above ⇒ $P_x = 4 \ {\rm V}^2$, $P_{\rm V} = 0.04 \ {\rm V}^2$;

- $$10 \cdot {\rm lg} \ ρ_{\rm V} = 20 \ \rm dB.$$

We reference the interactive applet »Linear Distortions of Periodic Signals«.

## Elimination of attenuation factor and transit time

The equations given in the last section do not result in applicable statements if the system is additionally affected by an attenuation factor $α$ and/or a transit time $τ$. The diagram shows the attenuated, delayed and distorted signal

- $$y(t) = \alpha \cdot x(t - \tau) + \varepsilon_1(t).$$

- Here, instead of the »distortion power« the »distortion energy« must be considered because $x(t)$ and $y(t)$ are energy-limited signals.

- The term $ε_1(t)$ summarizes all distortions. It can be seen from the green area that the difference signal ⇒ »error signal« $ε_1(t)$ is relatively small.

In contrast to this, if the attenuation factor $α$ and the transit time $τ$ are unknown, the following should be noted:

- In the second example the difference signal $ε_2(t) = y(t) - x(t)$ determined in this way is relatively large despite small distortions $ε_1(t)$.

- The distortion energy is obtained by varying the unknown quantities $α$ and $τ$ and thus finding the minimum of the »mean squared error«:

- $$E_{\rm V} = \min_{\alpha, \ \tau} \int_{ - \infty }^{ + \infty} {\big[y(t) - \left(\alpha \cdot x(t - \tau) \right) \big]^2}\hspace{0.1cm}{\rm d}t.$$

- The energy of the attenuated and delayed signal $α · x(t - τ)$ is $E_{\rm V} =α^2 · E_x$ independent of the transit time $τ$. Thus for the signal-to-distortion $($energy or power$)$ ratio the following is applicable:

- $$\rho_{\rm V} = \frac{ \alpha^2 \cdot E_{x}}{E_{\rm V}}\hspace{0.3cm}{\rm or}\hspace{0.3cm}\rho_{\rm V}= \frac{ \alpha^2 \cdot P_{x}}{P_{\rm V}} .$$

- The first of these two equations applies to time-limited and thus energy-limited signals, the second one to time-unlimited and thus power-limited signals according to the section »Energy-limited and power-limited signals« in the book »Signal Representation«.

## Linear and nonlinear distortions

A distinction is made between »linear distortions« and »nonlinear distortions«:

If the system is linear and time-invariant $(\rm LTI)$, then it is fully characterized by its $\text{frequency response}$ $H(f)$ and the following can be stated:

- According to the $H(f)$ definition the following holds for the output spectrum: $Y(f)=X(f) · H(f)$.
- As a consequence according to the calculation rules of multiplication, $Y(f)$ cannot contain any frequency components that are not already contained in $X(f)$.
- The inverse implies: The output signal $y(t)$ can include any frequency $f_0$ already contained in the input $x(t)$ . The prerequisite is therefore that $X(f_0) ≠ 0$.
- For an LTI system the absolute bandwidth $B_y$ of the output signal is never greater than the bandwidth $B_x$ of the input signal: $B_y \le B_x .$

$\text{Conclusion:}$ The differences between linear and non-linear distortions are illustrated by the following diagram:

$\rm (A)$ In the upper diagram $B_y = B_x$ holds. There are »**linear distortions**« because in this frequency band $X(f)$ and $Y(f)$ differ.

- A band limitation $(B_y < B_x)$ is a special form of linear distortions, which will be discussed in the »chapter after next«.

$\rm (B)$ The lower diagram shows an example of »**non-linear distortions**« $(B_y > B_x)$. For such a system no frequency response $H(f)$ can be given.

- Descriptive quantities applicable for nonlinear systems will be explained in the next chapter »Non-linear Distortions« .

In most real transmission channels both linear and nonlinear distortions occur. However, for a whole range of problems the precise separation of the two types of distortions is essential. In [Kam04]^{[1]} a corresponding substitute model is shown.

We refer here to the $($German language$)$ learning video »Lineare und nichtlineare Verzerrungen ⇒ »Linear and nonlinear distortions«.

## Exercises for the chapter

Exercise 2.1: Linear? Or Non-Linear?

Exercise 2.1Z: Distortion and Equalisation

Exercise 2.2: Distortion Power

Exercise 2.2Z: Distortion Power again

## References

- ↑ Kammeyer, K.D.: Nachrichtenübertragung. Stuttgart: B.G. Teubner, 4. Auflage, 2004.