Signals and Systems - Electrical Engineering

90 C H A P T E R 1: Continuous-Time Signals

n The impulseδ(t)is impossible to generate physically, but characterizes very brief pulses of any shape. It

can be derived using pulses or functions different from the rectangular pulse (see Eq. 1.22). In Problem

1.7 at the end of the chapter it is indicated how it can be derived from either a triangular pulse or a sinc

function of unit area.

n Signals with jump discontinuities can be represented as the sum of a continuous signal and unit-step

signals at the discontinuities. This is useful in computing the derivative of these signals.

Ramp Signal

The ramp signal is defined as

r(t)=t u(t) (1.29)

Its relation to the unit-step and the unit-impulse signals is

dr(t)

dt

=u(t) (1.30)

d^2 r(t)

dt^2

=δ(t) (1.31)

The ramp is a continuous function and its derivative is given by

dr(t)

dt

=

dtu(t)

dt

=u(t)+t

du(t)

dt

=u(t)+tδ(t)

=u(t)+ 0 δ(t)=u(t)

nExample 1.15

Consider the discontinuous signals

x 1 (t)=cos( 2 πt)[u(t)−u(t− 1 )]

x 2 (t)=u(t)− 2 u(t− 1 )+u(t− 2 )

Represent each of these signals as the sum of a continuous signal and unit-step signals, and find

their derivatives.

Solution

The signalx 1 (t)is a period of a cosine of periodT 0 =1, 0≤t≤1, with a discontinuity of 1 att= 0

andt=1. Subtractingu(t)−u(t− 1 )fromx 1 (t)we obtain a continuous signal, but to compensate

we must add a unit pulse betweent=0 andt=1, giving

x 1 (t)=(cos( 2 πt)− 1 )[u(t)−u(t− 1 )]+[u(t)−u(t− 1 )]=x 1 a(t)+x 1 b(t)