## Contents

## Properties of nonlinear systems

The system description by means of the frequency response $H(f)$ and/or the impulse response $h(t)$ is only possible for an »**LTI system**«. However, if the system contains nonlinear components, as it is assumed for this chapter, no frequency response and no impulse response can be stated. The model must be designed in a more general way.

Also in this nonlinear system, we denote the signals at the input and the output by $x(t)$ resp. $y(t)$ and the corresponding spectral functions by $X(f)$ and $Y(f)$.

An observer will note the following here:

- The transmission characteristics are now also
**dependent on the amplitude of the input signal**.

If $x(t)$ results in the output signal $y(t)$, it can now no longer be concluded that the input signal $K · x(t)$ will always result in the signal $K · y(t)$.

- This also implies that the
**superposition principle is no longer applicable**.

Consequently, the result $x_1(t) + x_2(t)$ ⇒ $y_1(t) + y_2(t)$ cannot be reasoned from the two correspondences $x_1(t) ⇒ y_1(t)$ and $x_2(t) ⇒ y_2(t)$.

- Due to nonlinearities
**new frequencies occur**.

If $x(t)$ is a harmonic oscillation with frequency $f_0$, the output signal $y(t)$ also contains components at multiples of $f_0$. In communications engineering, these are referred to as »harmonics«.

- In practice, an information signal $x(t)$ usually contains many frequency components.

The harmonics of the low-frequency signal components now fall into the range of higher-frequency components. This results in »**nonreversible signal falsifications**«.

Before mentioning »constellations which result in nonlinear distortions« at the end of this section, the problem of nonlinear distortions is captured mathematically.

$\text{Definition:}$
We assume here that **the system has no memory** so that

- the output value $y = y(t_0)$ depends only on the instantaneous input value $x = x(t_0)$,

- but not on the signal curve $x(t)$ for $t < t_0$.

## Description of nonlinear systems

$\text{Definition:}$
A system is said to be »**nonlinear**« if the following relationship exists between the signal value $x = x(t)$ at the input and the output value $y = y(t)$ :
$$y = g(x) \ne {\rm const.} \cdot x.$$
The curve shape $y = g(x)$ is called the »**nonlinear characteristic curve**« of the system.

- As an example, in the diagram the green curve is the nonlinear characteristic curve $y = g(x)$ which is shaped according to the first quarter of a sine function.

- In this diagram the special case of a linear system with the characteristic curve $y = x$ can be seen dashed in red.

Since every characteristic curve can be developed into a Taylor series around the operating point the output signal can also be represented as follows:
$$y(t) = \sum_{i=0}^{\infty}\hspace{0.1cm} c_i \cdot x^{i}(t) = c_0 + c_1 \cdot x(t) + c_2 \cdot x^{2}(t) + c_3 \cdot x^{3}(t) + \hspace{0.05cm}\text{...}$$
If $x(t)$ has a unit, e.g. "Volt”, then the coefficients of the Taylor series also have appropriate and different units:

- $c_0$ ⇒ "Volt”,
- $c_1$ without unit,
- $c_2$ ⇒ "1/Volt”, etc.

In the above diagram, the operating point is identical to the zero point and $c_0 = 0$ holds.

$\text{Example 1:}$ The properties of nonlinear systems listed in the »first section of this page« are illustrated here using the characteristic curve $y = g(x) = \sin(x)$ shown in the center of the diagram.

- Here, the direct (DC) signal $x(t) = 0.5$ results in the constant output signal $y(t) = 0.479$.
- For $x(t) = 1$ the input signal results in the output signal $y(t) = 0.841 ≠ 2 · 0.479$.
- Thus, doubling $x(t)$ does not cause the doubling of $y(t)$ ⇒ the superposition principle is violated.

The outer diagrams show – each in blue – cosine-shaped input signals $x(t)$ with different amplitudes $A$ and the corresponding distorted output signals $y(t)$ in red.

It can be seen that the nonlinear distortions increase with increasing amplitude, which are quantified by the distortion factor $K$ defined in the next section.

- The diagram on the upper right-hand corner for $A = 1.5$ clearly shows that $y(t)$ is no longer cosine-shaped; the half-waves run rounder than the blue ones of the cosine function.

- But also for $A = 0.5$ and $A = 1.0$ the signals $y(t)$ deviate - although less strongly - from the cosine form due to the harmonics. That is, new frequency components at multiples of the cosine frequency $f_0$ arise.

- In the figure on the bottom right-hand corner the characteristic curve is operated unilaterally due to an additional direct component. Now an unbalance in the signal $y(t)$ can be seen, too. The lower half-wave is more peaked than the upper one. The distortion factor here is $K \approx 22\%$.

## The distortion factor

To quantitatively capture the nonlinear distortions we assume:

- The input signal $x(t)$ is cosine-shaped with amplitude $A_x$.

- The output signal $y(t)$ contains harmonics due to the nonlinear distortions and the following is generally true:

- $$y(t)\hspace{-0.03cm} =\hspace{-0.03cm} A_0 \hspace{-0.03cm}+\hspace{-0.03cm} A_1 \cdot \cos(\omega_0 t) \hspace{-0.03cm}+\hspace{-0.03cm} A_2 \cdot \cos(2\omega_0 t) \hspace{-0.03cm}+\hspace{-0.03cm} A_3 \cdot \cos(3\omega_0 t) \hspace{-0.03cm}+\hspace{-0.03cm} \hspace{0.05cm}\text{...}$$

$\text{Definition:}$
With these amplitude values $A_i$ the equation for the »**distortion factor**« is:

- $$K = \frac {\sqrt{A_2^2+ A_3^2+ A_4^2+ \hspace{0.05cm}\text{...} } }{A_1} = \sqrt{K_2^2+ K_3^2+K_4^2+ \hspace{0.05cm}\text{...} }.$$

In the second equation:

- $K_2 = A_2/A_1$ denotes the distortion factor of second order,
- $K_3 = A_3/A_1$ denotes the distortion factor of third order, etc.

It is expressly pointed out that the amplitude $A_x$ of the input signal is not taken into account when computing the distortion factor. Also a resulting direct (DC) component $A_0$ remains unconsidered.

In $\text{Example 1}$ the distortion factors were specified with values between about $1\%$ and $20\%$ .

- These values are already significantly above the distortion factors of low-cost audio equipment, for which $K < 0.1\%$ applies.

- In HiFi equipment, particular emphasis is placed on linearity and a very low distortion factor is also reflected in the price.

A comparison with the section »Consideration of attenuation and runtime« reveals that for the special case of a cosine-shaped input signal the signal–to–distortion–power ratio is equal to the reciprocal of the distortion factor squared:

- $$\rho_{\rm V} = \frac{ \alpha^2 \cdot P_{x}}{P_{\rm V}} = \left(\frac{ A_{1}}{A_x} \right)^2 \cdot \frac{ {1}/{2} \cdot A_{x}^2}{{1}/{2} \cdot (A_{2}^2 + A_{3}^2 + A_{4}^2 + \hspace{0.05cm}...) } = \frac{1}{K^2}\hspace{0.05cm}.$$

$\text{Example 2:}$ We now consider an averaged cosine signal:

- $$x(t) = {1}/{2} + {1}/{2}\cdot \cos (\omega_0 \cdot t).$$

$x(t)$ takes values between $0$ and $1$, and is drawn as the blue curve. The signal power is

- $$P_x = 1/4 + 1/8 = 0.375.$$

If we apply this signal to a nonlinearity with the characteristic curve

- $$y=g(x) = \sin(x) \approx x - {x^3}/{6} \hspace{0.05cm},$$

then the output signal is:

- $$y(t) = A_0 + A_1 \cdot \cos (\omega_0 \cdot t)+ A_2 \cdot \cos (2\omega_0 \cdot t)+ A_3 \cdot \cos (3\omega_0 \cdot t)\hspace{0.05cm},$$
- $$\Rightarrow \hspace{0.3cm} A_0 = {86}/{192},\hspace{0.3cm}A_1 = {81}/{192},\hspace{0.3cm}A_2 = - {6}/{192},\hspace{0.3cm}A_3 = - {1}/{192}\hspace{0.05cm}.$$

The trigonometric transformations for $\cos^2(α)$ and $\cos^3(α)$ were used to calculate the Fourier coefficients. The distortion factor is thus given by

- $$K = \frac {\sqrt{A_2^{\hspace{0.05cm}2} + A_3^{\hspace{0.05cm}2} } }{A_1}\approx 7.5\%\hspace{0.05cm}.$$

It can be further seen that the signal $y(t)$ sketched in red is almost equal to the signal $α · x(t)$ sketched in green with $α = \sin(1) ≈ 5/6$ .

- Defining the error signal as $ε_1(t) = y(t) - α · x(t)$, with its power

- $$P_{\varepsilon 1} = \frac {(80-86)^2}{192^2} + \frac {6^2 + (-1)^2}{2 \cdot 192^2}\approx 1.48 \cdot 10^{-3}$$
- the following is obtained for the signal–to–noise–power ratio:
- $$\rho_{{\rm V} 1} = \frac {\alpha^2 \cdot P_x}{P_{\varepsilon 1} } = \frac {(5/6)^2 \cdot 0.375}{1.48 \cdot 10^{-3} }\approx 176 = {1}/{K^2}\hspace{0.05cm}.$$

- In contrast, the SNR is significantly lower if we do not consider the attenuation factor, that is, if we assume the error signal $ε_2 = y(t) - x(t)$ :

- $$P_{\varepsilon 2} = \frac {(86-96)^2}{192^2} + \frac {(81-96)^2 + 6^2 + (-1)^2}{2 \cdot 192^2}\approx 6.3 \cdot 10^{-3} \hspace{0.3cm} \Rightarrow \hspace{0.3cm}\rho_{{\rm V} 2} = \frac { P_x}{P_{\varepsilon 2}}= \frac {0.375}{6.3 \cdot 10^{-3}} \approx 60 \hspace{0.05cm}.$$

## Clirr measurement

A major disadvantage of the distortion factor definition is the thereby specification to cosine-shaped test signals, i.e. to conditions remote from reality.

- In the so-called »clirr measurement« the signal $x(t)$ to be transmitted is modelled by white noise with noise power density ${\it \Phi}_x(f)$.

- In addition, a narrow band-stop filter $\rm (BS)$ with center frequency $f_{\rm M}$ and (very small) bandwidth $B_{\rm BS}$ is introduced into the system.

In a linear system, the output spectrum ${\it \Phi}_y(f)$ would not be wider than $B_x$ and also in the region around $f_{\rm M}$ there would be no components.

These result solely from frequency conversion products (»intermodulation components«) of different spectral components, i.e. from nonlinear distortions.

By varying the center frequency $f_{\rm M}$ and integrating over all these small interfering components the distortion power can thus be determined. More details on this method can be found, for example, in [Kam04]^{[1]}.

## Constellations which result in nonlinear distortions

As an example of the occurrence of **nonlinear distortions in analog transmission systems** some constellations which result in such distortions shall be mentioned here. In terms of content, this anticipates the book »Modulation Methods«.

Nonlinear distortions of the sink signal $v(t)$ with respect to the source signal $q(t)$ occur when

- nonlinear distortions already occur on the channel – i.e. with respect to the transmitted signal $s(t)$ and the received signal $r(t)$,
- an envelope demodulator is used for »double-sideband amplitude modulation« $\rm (DSB–AM)$ with modulation factor $m > 1$,
- for DSB–AM and envelope demodulation there is a linearly distorting channel, even with a modulation factor $m < 1$,
- the combination of »single-sideband modulation« and »envelope demodulation« is used $($regardless of the sideband–to–carrier ratio$)$,
- an »angle modulation« $($generic term for frequency and phase modulation$)$ is applied and the available bandwidth is finite.

## Exercises for the chapter

Exercise 2.3: Sinusoidal Characteristic

Exercise 2.3Z: Asymmetrical Characteristic Operation

Exercise 2.4: Distortion Factor and Distortion Power

Exercise 2.4Z: Characteristics Measurement

## References

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