16 C H A P T E R 0: From the Ground Up!

which is a lot better result. In general, we have that the integral can be computed quite accurately

using a very small value ofTs, indeed

( (^10) ∑/Ts)− 1
n= 0
p[n]=
( 10 /∑Ts)− 1
n= 0
nTs^2
=Ts^2
( 10 /Ts)×(( 10 /Ts)− 1 )
2

10 ×( 10 −Ts)
2
which for very small values ofTs(so that 10−Ts≈10) gives 100/ 2 =50, as desired.
Derivatives and integrals take us into the processing of signals by systems. Once a mathematical model for a
dynamic system is obtained, typically differential equations characterize the relation between the input and
output variable or variables of the system. A significant subclass of systems (used as a valid approximation in
some way to actual systems) is given by linear differential equations with constant coefficients. The solution
of these equations can be efficiently found by means of the Laplace transform, which converts them into
algebraic equations that are much easier to solve. The Laplace transform is covered in Chapter 3, in part to
facilitate the analysis of analog signals and systems early in the learning process, but also so that it can be
related to the Fourier theory of Chapters 4 and 5. Likewise for the analysis of discrete-time signals and systems
we present in Chapter 9 the Z-transform, having analogous properties to those from the Laplace transform,
before the Fourier analysis of those signals and systems.

0.3.4 Differential and Difference Equations

A differential equation characterizes the dynamics of a continuous-time system, or the way the system

responds to inputs over time. There are different types of differential equations, corresponding to

different systems. Most systems are characterized by nonlinear, time-dependent coefficient differential

equations. The analytic solution of these equations is rather complicated. To simplify the analysis,

these equations are locally approximated as linear constant-coefficient differential equations.

Solution of differential equations can be obtained by means of analog and digital computers. An

electronicanalog computerconsists of operational amplifiers (op-amps), resistors, capacitors, voltage

sources, and relays. Using the linearized model of the op-amps, resistors, and capacitors it is possible

to realize integrators to solve a differential equation. Relays are used to set the initial conditions on

the capacitors, and the voltage source gives the input signal. Although this arrangement permits the

solution of differential equations, its drawback is the storage of the solution, which can be seen with

an oscilloscope but is difficult to record. Hybrid computers were suggested as a solution—the analog

computer is assisted by a digital component that stores the data. Both analog and hybrid computers

have gone the way of the dinosaurs, and it is digital computers aided by numerical methods that are

used now to solve differential equations.

Before going into the numerical solution provided by digital computers, let us consider why inte-

grators are needed in the solution of differential equations. A first-order (the highest derivative

present in the equation); linear (no nonlinear functions of the input or the output are present) with