Basic Linear Systems

Linear system theory plays a key role in the engineering analysis of electrical and mechanical systems. Engineers model a wide variety of physical systems as linear transformations, including circuit behavior, signal propagation, coupling and radiation. Therefore, it is important to clarify exactly what we mean by a linear system so that we recognize when and how to use the powerful linear system analysis tools available to us.

Figure 4.4 illustrates a system with one input, x(t), and one output, y(t)=H{x(t)}. If an input x₁(t) produces an output y₁(t), and an input x₂(t) produces an output y₂(t), then the system is linear if and only if,

$a{y}_1\left(t\right)+b{y}_2\left(t\right)=H\left\{a{x}_1\left(t\right)+b{x}_2\left(t\right)\right\}$

where a and b are constants. In other words, scaling the input by a constant will produce an output scaled by the same constant, and combining (summing) two inputs will produce an output that is the sum of the outputs produced by each individual input.

Figure 4.4. A linear system with input x(t) and output y(t).

Of the choices presented in this question, only a, b and g are linear system transformations. y(t)=0 is not a very interesting system, because its output is always zero, but it is linear. Simple derivative and integral operators are linear because they satisfy the conditions in Equation (4.1). The remaining choices are not linear operations. Note that y=8x+3 is the equation of a straight line, but it does not describe a linear system because it has a non-zero output when there is no input.

At first, it may appear that very few real electrical or mechanical systems of interest behave this way. However, many non-linear systems can be approximated as linear over a limited subset of possible input values. Most engineering analysis depends on modeling real devices and circuits as linear systems.

Linear systems have the unique property that any sinusoidal input will produce a sinusoidal output at exactly the same frequency. In other words, if the input is of the form,

x(t) = A_x cos(ωt + φ_x)

then the output will have the form,

y(t) = A_y cos(ωt + φ_y)

In general, the magnitude and phase of the sinusoidal signal may change, but the frequency must be constant. This provides us with a very powerful analysis tool for analyzing linear systems. If we represent an input signal as the sum of its components in the frequency domain, then we can express the output as a simple scaling of the component magnitudes and shifting of the component phases.

Phasor Notation

To facilitate the analysis of linear system responses to sinusoidal inputs, it is convenient to represent signals in an abbreviated form known as phasor notation. Consider an input of the form,

x(t) = A cos(ωt + φ)   (4.4)

This can be represented as,

x(t) = Re{A e^jφ e^jωt}   (4.5)

where Re{ } indicates the real part of a complex quantity. Recognizing that the frequency ω will be the same throughout the system, we don’t need to write the term e^jωt explicitly, as long as we remember that it’s there. The same applies to the Re{ } notation. This allows us to express a sinusoidal signal simply in terms of its magnitude and phase as,

x = A e^jφ.   (4.6)

The expression in (4.6) is the signal in (4.4) represented using phasor notation. Note that we must know the frequency of a signal in order to convert from phasor notation to the time-domain representation.

The first signal expressed in phasor notation is simply x = 5 volts. To obtain the phasor notation for the second signal, we recognize that sin(ωt) = cos(ωt + π/2), so y = 5∠90° amps. The third signal is not a sinusoid and therefore cannot be expressed using phasor notation.