Signals and Systems - Electrical Engineering

1.5 What Have We Accomplished? Where Do We Go from Here? 107

Table 1.1Basic Signals

Signal Definition

Complex exponential |A|ert[cos( 0 t+θ)+jsin( 0 t+θ)] −∞<t<∞

Sinusoid Acos( 0 t+θ)=Asin( 0 t+θ+π/ 2 ) −∞<t<∞

Unit impulse δ(t)= 0 t6= 0 , undefined att= 0

∫t

−∞

δ(τ)dτ=1, t> 0

∞∫

−∞

f(τ)δ(t−τ)dτ=f(t)

Unit step u(t)=

{

1 t> 0

0 t< 0

Ramp r(t)=tu(t)=

{

t t> 0

0 t< 0

δ(t)=du(t)/dt

u(t)=

∫t

−∞

δ(τ)dτ

r(t)=

∫t

−∞

u(τ)dτ

Rectangular pulse p(t)=A[u(t)−u(t− 1 )]=

{

A 0 ≤t≤ 1

0 otherwise

Triangular pulse 3(t)=A[r(t)− 2 r(t− 1 )+r(t− 2 )]=







At 0 ≤t≤ 1

A( 2 −t) 1 <t≤ 2

0 otherwise

Sampling δTs(t)=

∑

kδ(t−kTs)

Sinc S(t)=sin(πt)/(πt)

S( 0 )= 1

S(k)= 0 k6= 0 integer

∞∫

−∞

S^2 (t)dt= 1

we began to see how some of these operations lead to practical applications, such as amplitude, fre-
quency, and phase modulations, which are basic in the theory of communications. Very importantly,
we have also begun to represent signals in terms of basic signals, which in later chapters will allow us
to simplify the analysis and will give us flexibility in the synthesis of systems. These basic signals are
used as test signals in control systems. Table 1.1 displays basic signals.

Our next step is to connect signals with systems. We are particularly interested in developing a
theory that can be used to approximate, to some degree, the behavior of most systems of inter-
est in engineering. After that we consider the analysis of signals and systems time and frequency
domains.