20
Biomedical Signal and Image Processing
Example 2.3
Another useful function in signal processing is the unit pulse p Π (t) as shown in Figure 2.4.
Calculating the FT of the unit pulse function, we have
+∞
P f
( ) =
∫
p (t)e
− j f
2 p t
Π
Π
dt
−∞
∫
T
= Ae
− j f
2 p t dt
0
⎡
t T
A
⎤
=
= −
⎢
e
− j f
2 p t ⎥ ⎥
⎣ j f
2p
⎦ t =0
A
= −
⎡e
− j f
2p T − 1⎤
j f ⎣
⎦
2p
A
= −
⎡e
− j f
p T − e
j f
p T
⎤ e
− jp fT
j f ⎣
⎦
2p
A A ⎡ e
j f
p T − e
− jp fT
⎤
=
⎢
⎥ e
− j f
p T
p f ⎣
2j
⎦
A
=
sin(p fT)e
− j f
p T
(2.13)
p f
which means
sin(pft)
P f
Π ( ) = AT pfT
= ATsinc(pfT)
(2.14)
t
T
P Π (t)
A
FIGURE 2.4 Unit pulse function.