PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 333

R.4.48 It is useful to know the relations and the effects that operations in the time domain

have in the frequency domain, and vice versa, to be able to get an insight of the

transformation process, and in many cases to avoid the integral defi nition of the

transform.

Let the FT pair time/frequency be represented using the short notation

f(t) ↔ F(w)

Then the most important properties that relate the two domains are summarized

as follows:

a. Linearity

Let

a 1 f 1 (t) ↔ a 1 F 1 (w) and a 2 f 2 (t) ↔ a 2 F 2 (w)

Then a 1 f 1 (t) + a 2 f 2 (t) ↔ a 1 F 1 (w) + a 2 F 2 (w), where a 1 and a 2 are arbitrary (may be

complex) constants, and the addition is complex.

b. Time scaling

fat aF

w

()↔ a









1

TABLE 4.1

Time Frequency Transformations

Signal/Time Domain Transform/Frequency Domain

Aδ(t) A

A 2 πδ(w)

A[u(t + a/2) − u(t− a/2)] Aa

wa

wa

()

sin( )





2

2

Aa

a

a

()

sin( )









2

2 2 πA[u(w + a/2)−u(w − a/2)]

Aejwaut 2 πAδ(w − w 0 )

Acos(w 0 t) πA[δ(w − w 0 ) + δ(w + w 0 )]

Asin(w 0 t) −jπA[δ(w − w 0 ) + δ(w + w 0 )]

Asgn(t)2A/jw

Au(t) πAδ(w) + (A/jw)

Ae−at ⋅^ u(t), a > 0 A(jw + a)

Ate−at ⋅ u(t)1(jw + a)^2

Ate−at ⋅ u(−t), a > 0 A/(−jw + a)

Ae−a|t| 2 Aa(w^2 + a^2 )

Ae−b|t|cos(w 0 t)u(t), b > 0 A

bjw

bww jwb



(^22)  02  2
Ae−b|t|sin(w 0 t)u(t), b > 0 A
w
bww jbw
0
(^2)  022  2
A|t| − 2 A∕w^2
Fenjnw t
n
0



∞
∞
∑^2 
n
Fn()wnw^0

∞
∞
∑

()tk
k


T



∞
∞
∑
22 

T
w k
k

 T



∞ 
∞
∑