B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

3.2 Transforms of Some Useful Functions 71
Figure 3.8
Sine pulse.
(a)
sine (3;)
(b)
This function plays an important role in signal processing. We define
sin x
sinc (x) = -­ x
(3.18)
Inspection of Eq. (3.18) shows that
1. sine (x) is an even function of x.
2. sine (x) = 0 when sin x = 0 except at x = 0, where it is indeterminate. This means that
sinc (x) = 0fort = ±n, ±2n,±3n, . ...
3. Using L' Hospital's rule, we find sine (0) = 1.
4. sine (x) is the product of an oscillating signal sin x (of period 2n) and a monotonically
decreasing function 1/x . Therefore, sine (x) exhibits sinusoidal oscillations of period 2n ,
with amplitude decreasing continuously as 1/x.
5. In summary, sine (x) is an even oscillating function with decreasing amplitude. It has a unit
peak at x = 0 and zero crossings at integer multiples of ;r.
Figure 3.8a shows sine (x). Observe that sine (x) = 0 for values of x that are positive and
negative integral multiples of n. Figure 3.8b shows sine (3w/7). The argument 3w/7 = n
when w = 7n /3 or f = 7 /6. Therefore, the first zero of this function occurs at w = 7n /3
(f = 7 /6).