PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

332 Practical MATLAB® Applications for Engineers

R.4.41 Let f(t) be a real function of t, then

F(w) = F*(w)

(Recall that the character * denotes the complex conjugate of ).

R.4.42 Since F(w) is in general a complex function, then two equations that are translated
into plots are required to fully specify F(w). They are referred as
a. The magnitude spectrum
b. The phase spectrum

R.4.43 The magnitude spectrum is an even function of w, that is, F(w) = F(−w); whereas
the phase spectrum is an odd function of w, that is, θ(w) = −θ(−w). Recall that
those conditions are similar to the case of a periodic function, in which the FS was
employed. Recall that in the series case

Fn = F − n

whereas

θ(nw 0 ) = −θ(−nw 0 )

R.4.44 Table 4.1 summarizes some of the most frequently used time/frequency function
transforms, where

ft()

Fwe dw()jwt Fw() fte dt()jwt



1

2 

⇔

∞

∞

∞

∞

∫ ∫



 



R.4.45 The functions f(t) and F(w) constitute an FT pair, indicated by the following
notation:

f(t) ↔ F(w) or f(t) ⇔ F(w)

R.4.46 The FT is one of the most common forms of describing the frequency domain char-
acteristics of a signal or system. Note that the FT can also be used for periodic sig-
nals (Table 4.1, transform No. 17), instead of the more traditional FS expansion.

R.4.47 Note that the FT of a periodic function f(t) is given by

ℑ

∞

∞

{()}ft
 ∑ F w nwn( )



2 




0

n

where

Ftedtn
 jw nt

1

T 0

f

T

∫ ()^0

Observe that the difference between the FS and the FT in the case of a periodic
function is given by 2 π, a constant that equally affects all the frequencies, and
therefore, with no effect on the relative importance of the frequencies that consti-
tute the spectrum.