PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 335

or the more general relation given by

()()

()

jt f t

dFw

dw

n

n

↔ n

h. Time integration

fd

jw

Fw F w

t

() ↔ () 
()()







∫∞ 

1

0





Note that differentiation of f(t) in the time domain has the effect of multiply-

ing F(w) by jw in the frequency domain, similarly integration of f(t) in the time

domain has the effect of dividing F(w) by jw in the frequency domain.

i. Frequency integration

1

jt 

ft F d

w

()↔ ( )

∞

∫^

j. Multiplication in time

ft ft 12 F1 2Fw d Fw Fw 1 2

1

2

1

2

()⋅↔() ( )⋅−=( ) [ ( )⊗( )]

∞

∞

∫^ 





This property states that multiplying two time domain functions given by f 1 (t)

and f 2 (t) in the time domain has the effect of evaluating the convolution of their

spectrums F 1 (w) with F 2 (w) in the frequency domain times ___^1

2 π

. This property is
used extensively in linear controls and communication system analysis.
k. Convolution in time

ft 12 ()⊗⋅−↔ft() f 12 ( )ft( )d [ ( ) ( )]FwFw 12

∞

∞


∫





This property states that the convolution of the time functions f 1 (t) with f 2 (t) (in

the time domain) has the effect of multiplying their spectrums F 1 (w) with F 2 (w)

(in the frequency domain). Observe that the convolution integral (indicated by

the character ⊗) is in general a process not easy to evaluate, and that is precisely

the reason why the transform is used just to avoid it. Note that the convolution

process in one domain is translated into a product in the other domain.

l. Cross correlation in time

fft d FwFw FwFw 12 () ( ) 1 2*( ) ( ), 1 ( ) ( ) 2





∞

∞

∫ ↔^