Fixed instantaneous nonlinear systems

Fig. 532 Blue: ideal static instantaneous linear input-output relation Red: realized static instantaneous nonlinear input output relation.

In this section, we will introduce the basic concepts for characterization of nonlinear time-invariant instantaneous systems. Information processing systems are often intended to behave linearly. Deviations from this linear behavior will be modeled in this section. There are many different description methods for nonlinear behavior. Harmonic distortion , intermodulation , differential-gain , and gain compression are terms that are often used to express the perception of nonlinear effects. In general, description methods that show the best correspondence to the observer’s error perception must be used. This perception strongly depends on the way in which the information is embedded in the signal and how this information is interpreted by the observer.

Fig. 533 Definition of the desired operating point for linearization.

A single-input single-output nonlinear instantaneous fixed system can be characterized by a curve in its so-called input-output plane. Fig. 532 shows an example of a static nonlinear instantaneous input-output relation, with \(x\) and \(y\) representing the input and the output signal values, respectively. One could measure such a curve through application of an input signal that varies with time and plot the output signal values as a function\ of the input signal values. Fig. 532 also shows the intended ideal input-output relation of this system.

The total error shown in the figure is the difference between the intended system output and the actual output. It depends on the value of the input signal.

This total error can be decomposed in a number of contributions. These contributions are defined in the following sections.

Operating point, input and output offset

Fig. 534 Modified source-load relation.

The static nonlinear curve from Fig. 532 can mathematically be described as

(205)\[y=y(x). \label{eq-offset-nonlin}\]

In order to specify the deviations from the ideal behavior, we have to select our intended origin, also called the*\ quiescent operating point* \(Q\). This is shown in Fig. 533.

The curve is shifted to this new origin through application of the so-called input and output offset quantities \(x_{off}\) and \(y_{off}\), respectively. For a voltage amplifier, \(x_{off}\) and \(y_{off}\) are the voltages of a voltage source in series with the signal source and a voltage source in series with the load, respectively. The quescent operating point \(Q\) can be selected on various grounds. At this stage, we will not discuss this in detail, but just introduce the concept of an operating point.

After application of \(x_{off}\) and \(y_{off}\), the operating point \(Q\) is the origin of the modified source-to-load relation that is written as:

(206)\[\tilde{y}=y_{Q}(\tilde{x}) \label{eq-offset-free-nonlin}\]

In which \(\tilde{x}\) and \(\tilde{y}\) are the deviations of the input and output quantities from the operating point \(Q\). This is illustrated in Fig. 534.

Fig. 535 Definition of small-signal gain and inaccuracy.

The new function \(\tilde{y}=y_{Q}(\tilde{x})\) is free of offset. It can be approximated by a Taylor series expansion as

(207)\[\tilde{y}=\sum_{n=1}^{n=\infty}g_{n}\tilde{x}^{n},\label{ex-tailor}\]

with

(208)\[g_{n}=\frac{1}{n!}\left. \frac{d^{n}\{y_{Q}(\tilde{x})\}}{d\tilde{x}^{n} }\right\vert _{\tilde{x}=0}.\label{gain-factors-1}\]

Small-signal gain and inaccuracy

For small signals, the system can be linearized by approximating the input-output relation by a straight line through the (new) origin. This is shown in Fig. 535. The tangent of the angle between this line and the \(x\)-axis is the small-signal gain of the system. It is found as \(g_{1}\) from expression gain-factors-1:

Fig. 536 Definition of the nonlinearity

(209)\[\tilde{y}=g_{1}\tilde{x},\text{ with }g_{1}=\left. \frac{d\{y_{Q}(\tilde{x})\}}{d\tilde{x}}\right\vert _{\tilde{x}=0}. \label{no-offset}\]

The inaccuracy \(\delta\) of the linearized system is defined as the difference between the actual gain \(g_{1}\) and the required gain of the system, while the relative inaccuracy \(\delta_{rel}\) equals the quotient of the inaccuracy and the desired gain.

Nonlinearity

Fig. 537 Definition of the differential gain.

For large signal excursions, the approximation by a straight line is not accurate. The nonlinearity \(\Delta\) is defined as the difference between the actual output signal and the output signal that would be obtained from the linearized system. Fig. 536 shows \(\Delta\) as a function of the relative input excursion:

\[\Delta=y_{Q}(\tilde{x})-g_{1}\tilde{x}.\]

The relative nonlinearity \(\Delta_{rel}\) equals the ratio of the nonlinearity and the ideal output value \(gx_{1}:\)

\[\Delta_{rel}=\frac{y_{Q}(\tilde{x})-g_{1}\tilde{x}}{g_{1}\tilde{x}}.\]

Fig. 538 Differential gain as a function of the source signal deviation from the operating point.

Differential gain

Due to nonlinearity, the gain changes with the operating point. At an arbitrary operating point, for example, at \(\tilde{x}=x_{1}\), the differential gain error \(\epsilon\) is defined as the difference between the gain at \(\tilde{x}=x_{1}\) and the gain at \(\tilde{x}=0\), divided by the gain at \(\tilde{x}=0:\)

\[\epsilon(\tilde{x})=\frac{\left. \frac{d\{y_{Q}(\tilde{x})\}}{d\tilde{x} }\right\vert _{\tilde{x}=x_{1}}-\left. \frac{d\{y_{Q}(\tilde{x})\}}{d\tilde{x}}\right\vert _{\tilde{x}=0}}{\left. \frac{d\{y_{Q}(\tilde{x} )\}}{d\tilde{x}}\right\vert _{\tilde{x}=0}}.\]

Harmonic distortion

Fig. 539 Determination of \(\epsilon^{+}\) and \(\epsilon^{-}\) for harmonic and intermodulation distortion.

We have seen that in linear fixed dynamic systems, sinusoidal signals retain their shape. The nonlinearity of a system is therefore often characterized by the amount of distortion of the response to a sinusoidal input signal. The response of a fixed nonlinear instantaneous system to a sinusoidal input signal is a periodic signal whose period equals the period of the sine wave. The nonlinearity of the system generally introduces a change in the output offset and in the harmonic contents. The total harmonic distortion THD is defined as the relative \(RMS\)-value of all the harmonics in the output signal that have a frequency larger than the fundamental frequency:

\[THD=\frac{1}{d_{1}}\sqrt{\sum_{n=2}^{n=\infty}d_{n}^{2}},\]

where \(d_{n}\) equals the amplitude of the n-th harmonic in the output signal.

For weak nonlinear, instantaneous systems, the \(THD\) is related to the differential gain. To illustrate this, we consider a nonlinear system with an input-output relation as given by expression (207), and apply an input signal \(\tilde{x}(t)\) given by

\[\tilde{x}(t)=X\cos\omega t.\]

The output signal \(\tilde{y}(t)\) is then obtained as

\[\tilde{y}(t)=g_{1}X\cos\omega t+g_{2}X^{2}\cos{}^{2}\omega t+g_{3}X^{3}\cos{}^{3}\omega t+.....\ ,\]

which can be approximated by

\[\tilde{y}(t)=g_{1}X\cos\omega t+\frac{g_{2}}{2}X^{2}\cos{}2\omega t+\frac{g_{3}}{4}X^{3}\cos{}3\omega t+.....~.\]

The relative amplitude \(d_{2}\) of the second harmonic is thus found as \(g_{2}X/2g_{1}.\) The relative amplitude of the third harmonic \(d_{3}\) is found as \(g_{3}X^{2}/4g_{1}.\)

In order to find the relation between the THD and the differential gain , we will write the differential gain as a function of \(g_{1}.....g_{n}\). Since the differential gain depends on the large signal excursion, we use approximations for the differential gain at maximum and minimum signal excursion. The differential gain at the largest positive signal excursion (\(\tilde{x}(t)=X\)) will be denoted as \(\epsilon^{+}\) and is obtained from the definition as

(211)\[\epsilon^{+}=\frac{2g_{2}X+3g_{3}X^{2}}{g_{1}}. \label{eps+}\]

In a similar way, the differential gain at the largest negative signal excursion (\(x(t)=-X\)), denoted by \(\epsilon^{-}\), is found to be

(211)\[\epsilon^{-}=\frac{-2g_{2}X+3g_{3}X^{2}}{g_{1}}. \label{eps-}\]

With the aid of eps+ and eps-, we are able to calculate the second and order third harmonic distortion from the differential gain values. We obtain

(212)\[d_{2}=\frac{\epsilon^{+}-\epsilon^{-}}{8},\text{ and }d_{3}=\frac{\epsilon ^{+}+\epsilon^{-}}{24}, \label{ex-diffGainTHD}\]

in which \(\epsilon^{+}\) and \(\epsilon^{-}\) are the differential gain errors at \(\tilde{x}_{\max}\) and \(\tilde{x}_{\min}\), respectively. This is shown in Fig. 539.

Intermodulation distortion

Let us consider a situation in which the excitation of a fixed nonlinear instantaneous system consists of two sinusoidal components with different frequencies \(\omega_{1}\) and \(\omega_{2}\), but equal amplitudes. Due to the nonlinearity of the system the output signal will have components at multiples of \(\omega_{1}\) and \(\omega_{2}\) (known as harmonic distortion) and at frequencies \(m\omega_{1}\pm n\)\(\omega_{2}\) (\(m\) and \(n\) are integers). The latter effect is called intermodulation distortion. The amplitudes of the components at these frequencies will be denoted as \(A_{m\omega_{1}\pm n\omega_{2}}.\) The second order intermodulation distortion \(IM_{2\text{ }}\)is defined as the relative \(RMS\)-value of the component of the output signal with \(m=1\) and \(n=1,\) when the components at \(\omega_{1}\) and \(\omega_{2}\) have equal amplitudes:

\[IM_{2}=\frac{\left\vert A_{\omega_{1}\pm\omega_{2}}\right\vert }{\left\vert A_{\omega_{1,2}}\right\vert }.\]

The third order intermodulation distortion \(IM_{3}\) is defined as the relative \(RMS\)-value of the component with \(m=2\) and \(n=1\) or \(m=1\) and \(n=2,\) when both the components at \(\omega_{1}\) and \(\omega_{2}\) have equal amplitudes:

\[IM_{3}=\frac{\left\vert A_{\omega_{1}\pm2\omega_{2}}\right\vert }{\left\vert A_{\omega_{1,2}}\right\vert }=\frac{\left\vert A_{2\omega_{1}\pm\omega_{2} }\right\vert }{\left\vert A_{\omega_{1,2}}\right\vert }.\]

For weak nonlinear, instantaneous systems, the second and third order intermodulation distortion are related to the differential gain. In order to find this relation, we apply

\[x(t)=X\{\cos\omega_{1}t\ +\cos\omega_{2}t\}\\]

to the input of the system. The second order term from the Taylor approximation of the output signal is then obtained as

\[\begin{split}\begin{array} [c]{c} Y_{2}=g_{2}X^{2}\{\frac{1}{2}+\frac{1}{2}\cos(2\omega_{1}t\ )\ +\cos (\omega_{1}t\ +\omega_{2}t\ )\ +\\ \cos(\omega_{1}t\ -\omega_{2}t\ )\ +\frac{1}{2}+\frac{1}{2}\cos(2\omega _{2}t\ )\}, \end{array}\end{split}\]

which yields

(213)\[IM_{2}=\frac{g_{2}}{g_{1}}X=\frac{\epsilon^{+}-\epsilon^{-}}{4}. \label{ex-IM2}\]

In a similar way, we find

(214)\[IM_{3}=\frac{3g_{3}}{4g_{1}}X^{2}=\frac{\epsilon^{+}+\epsilon^{-}}{8}, \label{ex-IM3}\]

where \(\epsilon^{+}\) and \(\epsilon^{-}\) are the differential gain errors at \(\tilde{x}_{\max}\) and \(\tilde{x}_{\min}\), respectively. This is shown in Fig. 539.