Nonlinear Systems in Amplification of Signals

by Barry Van Veen

In the previous post we discussed the use of nonlinear systems for enhancing faint details in digital photographs of the night sky.  Another application where nonlinear systems are frequently encountered is amplification of signals, the topic of this post.

Nonlinear Systems in Amplification

An amplifier is an analog system that takes a small amplitude signal as an input and produces a large amplitude signal as output.  They are used in measurement of weak signals, like the electric fields at the scalp due to brain activity (electroencephalograpy) and reception of communication signals, e.g., from a space probe or cellular telephone tower.  The are also used to increase the power of a signal prior to transmission, such as when your cell phone transmits to a base station, or playing music through a loudspeaker.

An ideal amplifier has the characteristic $y(t) = Gx(t)$ where $G$ is the constant gain applied to the input signal $x(t)$. Clearly this idealized system satisfies the principle of superposition and is both linear and time invariant.  However, physical devices can only approximate this ideal characteristic over a limited range of input amplitudes. The output amplitude for any physical system is limited.  If the input amplitude is such that the output of the ideal system exceeds the physical system limit, then the output of the physical system saturates.

An example nonlinear amplification characteristic is illustrated in Figure 1.  For sufficiently small inputs, say $|x(t)|<0.05$, this system is linear with $G=7$.  The output amplitude gradually begins to saturate for larger inputs, with input amplitudes $|x(t)|>0.5$ producing nearly constant output amplitudes.

Problems with Nonlinear Amplification

Linear, time-invariant systems have an extremely useful and intuitive property.  If the input is a sinusoid of a particular frequency, the output is a sinusoid of the same frequency with an amplitude and phase change reflecting the characteristics of the system.  An input consisting of a weighted sum of sinusoids produces an output consisting of a weighted sum of sinusoids of the same frequencies with the amplitudes and phases modified by the system.

Nonlinear systems do not satisfy this important property.  Inputs consisting of sums of sinusoids produce outputs consisting of sinusoids at frequencies in addition to those of the input. We can see this by using a Taylor series expansion for the nonlinear amplification characteristic:

$y(t)=Gx(t)+c_3x^3(t)+c_5x^5(t)+c_7x^7(t)+\ldots$

Note that we assume the characteristic has odd symmetry (e.g., Figure 1) and thus have omitted all the even order polynomial terms.  The first term $Gx(t)$ is the linear system characteristic and dominates for $x(t)\approx 0$. The remainder of the terms reflect the nonlinearity and have an impact for larger input amplitudes.

Consider the third-order polynomial term $c_3x^3(t)$. Suppose the input is a sum of two sinusoids with frequencies $f_1$ and $f_2$ Hz

$x(t)=A_1\cos(2\pi f_1t)+A_2\cos(2\pi f_2t)$

The output due to the third-order term is

$c_3x^3(t)=c_3A_1^3\cos^3(2\pi f_1t)+3c_3A_1^2A_2\cos^2(2\pi f_1t)\cos(2\pi f_2t)+3c_3A_1A_2^2\cos(2\pi f_1t)\cos^2(2\pi f_2t)+c_3A_2^3\cos^3(2\pi f_2t)$

Application of trigonometry identities for products of cosines reveals that the output due to the third-order term has frequencies $f_1,2f_1,3f_1,2f_1-f_2,2f_1+f_2,2f_2-f_1,2f_2+f_1,f_2,2f_2,3f_2$.  The higher order terms in the Taylor series expansion produce even more frequencies. The sum and difference frequencies produced as a result of nonlinear characteristics are known as intermodulation distortion.

Intermodulation distortion is generally undesirable.  For example, it can create audible effects in audio systems and cause interference in communication systems.

Examples of Nonlinear Amplification

A synthetic 'C' chord is provided here.

It consists of a sum of three sinusoids of frequencies 523.25, 659.25, and 783.99 Hz.   These correspond to the fundamental frequencies of the music notes 'C', 'E', and 'G', respectively.  A 20 ms segment of the waveform is depicted in Figure 2. This chord is applied to the input of the nonlinear amplification characteristic shown in Figure 1.

The resulting sound is

Note that we have scaled the amplitude of the sound by a factor of 0.5 so that the perceived loudness is similar to the original synthetic chord.  This allows you to better hear the qualitative differences between the original and amplified sounds. The 20 ms segment of the output of the nonlinear amplifier corresponding to Figure 2 is shown in Figure 3.

The nonlinearity introduces intermodulation distortion that has a buzzing quality.  The distortion dominates the sound of the underlying chord.  The effect of the nonlinear saturation is evident in the peaks of Figures 2 and 3.

Useful Nonlinear Amplification

One application where nonlinear amplification is desirable is producing sound effects with electric guitars in rock music.  In the early days these effects were obtained by driving vacuum tube amplifiers into saturation, while now they are also generated digitally. Vibrating strings produce sinusoidal signals at the fundamental frequency - determined by the length, tension, and string properties - and harmonics of the fundamental. Nonlinear amplification of these sums of sinusoids produces a sound that is sometimes said to be dirty in relation to the clean sound involving only the fundamental and harmonics.  This character is due to the presence of intermodulation distortion in the amplifier output.  An example of such an electric guitar sound is at the end of this short clip taken from a rock piece.

 

Equalization of Nonlinear Amplification

An important signal-processing problem is design of systems that equalize or reverse nonlinear effects.  These can be applied either prior to the nonlinear system, or afterwards depending on the application.  Equalization of nonlinearities is a very challenging problem due to the lack of analytical tools for analysis of nonlinear systems.