[1.6 SYSTEMS](#page-7-0)

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1.6 SYSTEMS

As mentioned in Sec. 1.1, systems are used to process signals to allow modification or extraction of additional information from the signals. A system may consist of physical components (hardware realization) or of an algorithm that computes the output signal from the input signal (software realization).

Roughly speaking, a physical system consists of interconnected components, which are characterized by their terminal (input–output) relationships. In addition, a system is governed by laws of interconnection. For example, in electrical systems, the terminal relationships are the familiar voltage-current relationships for the resistors, capacitors, inductors, transformers, transistors, and so on, as well as the laws of interconnection (i.e., Kirchhoff’s laws). We use these laws to derive mathematical equations relating the outputs to the inputs. These equations then represent a mathematical model of the system.

A system can be conveniently illustrated by a “black box” with one set of accessible terminals where the input variables x1(t), x2(t), …, xj(t) are applied and another set of accessible terminals where the output variables y1(t), y2(t),…, yk(t) are observed (Fig. 1.25).

The study of systems consists of three major areas: mathematical modeling, analysis, and design. Although we shall be dealing with mathematical modeling, our main concern is with

x1(t)	y1(t)
x2(t)	y2(t)
• •	• •
• •	• •
• •	• •
xj(t)	yk(t)	Figure 1.25 Representation of a system.

analysis and design. The major portion of this book is devoted to the analysis problem—how to determine the system outputs for the given inputs and a given mathematical model of the system (or rules governing the system). To a lesser extent, we will also consider the problem of design or synthesis—how to construct a system that will produce a desired set of outputs for the given inputs.

DATA NEEDED TO COMPUTE SYSTEM RESPONSE

To understand what data we need to compute a system response, consider a simple RC circuit with a current source x(t) as its input (Fig. 1.26).

The output voltage y(t) is given by

$$y(t) = Rx(t) + \frac{1}{C} \int_{-\infty}^{t} x(\tau) d\tau$$

\n(1.18)

The limits of the integral on the right-hand side are from −∞ to t because this integral represents the capacitor charge due to the current x(t) flowing in the capacitor, and this charge is the result of the current flowing in the capacitor from −∞. Now, Eq. (1.18) can be expressed as

$$y(t) = Rx(t) + \frac{1}{C} \int_{-\infty}^{0} x(\tau) d\tau + \frac{1}{C} \int_{0}^{t} x(\tau) d\tau$$

The middle term on the right-hand side is vC(0), the capacitor voltage at t = 0. Therefore,

$$y(t) = v_C(0) + Rx(t) + \frac{1}{C} \int_0^t x(\tau) d\tau \qquad t \ge 0$$

This equation can be readily generalized as

$$y(t) = v_C(t_0) + Rx(t) + \frac{1}{C} \int_{t_0}^t x(\tau) d\tau \qquad t \ge t_0$$

\n(1.19)

From Eq. (1.18), the output voltage y(t) at an instant t can be computed if we know the input current flowing in the capacitor throughout its entire past (−∞ to t). Alternatively, if we know the input current x(t) from some moment t⁰ onward, then, using Eq. (1.19), we can still calculate y(t) for t ≥ t⁰ from a knowledge of the input current, provided we know vC(t0), the initial capacitor voltage (voltage at t0). Thus vC(t0) contains all the relevant information about the circuit’s entire

Figure 1.26 Example of a simple electrical system.

past (−∞ to t0) that we need to compute y(t) for t ≥ t0. Therefore, the response of a system at t ≥ t⁰ can be determined from its input(s) during the interval t⁰ to t and from certain initial conditions at t = t0.

In the preceding example, we needed only one initial condition. However, in more complex systems, several initial conditions may be necessary. We know, for example, that in passive RLC networks, the initial values of all inductor currents and all capacitor voltages† are needed to determine the outputs at any instant t ≥ 0 if the inputs are given over the interval [0,t].