PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

334 Practical MATLAB® Applications for Engineers

where a is a real nonzero constant. If a < 1, then f(t) is expanded in time, whereas

F(w) is compressed in frequency by a similar factor.

Note that if a > 1, then the function f(at) is compressed in time, whereas F(w) is

expanded in frequency by a similar factor.

c. Duality or symmetry

Ft()↔^2 f w( )

obtained by the following subtitution t → w and w → t.

Observe that this property can be used to double the table of transform pairs,

by simply interchanging the time and frequency variables.

d. Time shifting

ft t Fwe
()() jt w
0 ↔^0

where a time shift translates as a linear phase shift in frequency.

e. Frequency shifting

fte Fw w
()jw t^0 ↔ (  0 )

This property is also referred to as the modulation property. It is a funda-

mental property in communication theory, and is used to prove the modulation

theorem that states

ft()coswt 000 ↔ { (Fw w)Fw w( )}

1

2

Observe that multiplying an arbitrary time function f(t) by cos(w 0 t) shifts the

spectrum of f(t), so that half the original spectrum is centered at w 0 , and the other

half is centered at −w 0. The signal f(t) is referred to as the modulating signal,

whereas cos(w 0 t) is referred as the carrier signal (in practical applications the

frequency w 0 must be much higher than the highest frequency of f(t)).

f. Time differentiation

df t

dt

jwF w

()

↔ ()

or the more general relation given by

dft

dt

jw F w

n

n

()↔()()n

g. Frequency differentiation

()()

()

jt f t

dF w

dw

↔