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
f t
F w e dw
F w
f t e dt
jwt
jwt
( )
( )
( )
( )
ϭ
ϭ
ϩ
1
2
⇔
∞
∞
∞
∞
∫
∫
Ϫ
Ϫ
Ϫ
ϩ
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 characteristics of a signal or system. Note that the FT can also be used for periodic signals (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
∞
∞
∑
{ ( )}
(
)
f t
F w nw
n
ϭ
ϭ
ϩ
2 Ϫ
Ϫ
0
n
where
F
te
d t
n
jw nt
ϭ
1
0
T
f
T
( )
Ϫ 0
∫
Observe that the difference between the FS and the FT in the case of a periodic
function is given by 2π, a constant that equally affects all the frequencies, and
therefore, with no effect on the relative importance of the frequencies that constitute the spectrum.
CRC_47760_CH004.indd 332
CRC_47760_CH004.indd 332
7/28/2008 12:25:53 PM
7/28/2008 12:25:53 PM