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204 Part II Continuous-time signals and systems

Table 5.2. CTFT pairs for elementary CT signals

Time domain Frequency domain

CT signals x(t) =1

2

X()ejtdt X() =

x(t)ejtdt Comments

(1) Constant 1 2()

(2) Impulse function (t) 1

(3) Unit step function u(t) ()+1

j

(4) Causal decaying

exponential function

eatu(t)1

a +ja > 0

(5) Two-sided decaying

exponential function

ea|t|2a

a2 + 2a > 0

(6) First-order time-rising

causal decaying

exponential function

teatu(t)1

(a +j)2a > 0

(7) Nth-order time-rising

causal decaying

exponential function

tn eatu(t) n!(a +j)n+1

a > 0

(8) Sign function sgn(t) =

1 t > 0

1 t < 0

2

j

(9) Complex exponential ej0 t 2( 0)

(10) Periodic cosine function cos(0t) [( 0)+ ( + 0)]

(11) Periodic sine function sin(0t)

j[( 0) ( + 0)]

(12) Causal cosine function cos(0t)u(t)

2[( 0)+ ( + 0)]+

j

20 2

(13) Causal sine function sin(0t)u(t)

2j [( 0) ( + 0)]+

0

20 2

(14) Causal decaying

exponential cosine

function

eat cos(0t)u(t)a +j

(a +j)2 + 20a > 0

(15) Causal decaying

exponential sine function

eat sin(0t)u(t)0

(a +j)2 + 20a > 0

(16) Rectangular function rect

t

=

1 |t| /2

0 |t| > /2 sinc

2

= 0

(17) Sinc functionW

sinc

W t

rect

2W

=

1 || W

0 || > W

(18) Triangular function t

=1 |t|

|t|

0 otherwise sinc

22

> 0

(19) Impulse train

k=

(t kT0) 0

m=

( m0) angular

frequency

0 = 2/T0

(20) Gaussian function et2/22

2e

22/2

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Table

5.3.

Magnitudeand

phasespectraforselectedelementaryCTfunctions

Magnitudespectraaresho

wnaslinesandphasespectraaresh

ownasdashedlines

Function

Time-domainwavef

orm

Magnitudeandphasespectra

(1)Constantx(t)=

1

t

0

x(t)=1

1

w

0

X(w)=

2pd(w)

2p