Frequency-Domain Analysis - Introduction to Digital Signal Processing and Filter Design - page 140

Example 3.10
Consider a rectangular pulse

1

| n | ≤ N

x r (n) =

>N
which is plotted in Figure 3.20. It is also known as a rectangular window (of
length 2 N

0 | n |

+ 1 ) and will be used in Chapter 5 when we discuss the design of
FIR filters. Its DTFT is derived as follows:

N

X r (e jω ) e − jωn =

n =− N

To simplify this summation, we use the identity 5
N

r N + 1 r − N

−

r n = ; r = 1 (3.52)

r − 1

n =− N

= 2 N + 1 ; r = 1 (3.53)

and get

jNω

e − j(N + 1 )ω − e

X r (e jω ) =

e − jω − 1

e − j 0 . 5 ω e − j(N + 0 . 5 )ω e j(N + 0 . 5 )ω −

=

e − j 0 . 5 ω (e − j 0 . 5 ω − e j 0 . 5 ω )

⎨
⎩

sin[ (N 0 . 5 )ω ]

sin[0 . 5 ω ]

+

ω = 0

=

2 N + 1 ω = 0

which is shown in Figure 3.21.

X r (n)

− 5 0 1 2 3 4 5 n

Figure 3.20 A rectangular pulse function.

5 Proof:
n =− N (r n + 1

− r n ) = (r − N + 1 + r − N + 2 +···+ r − 1 + + r + r 2 +···+ r N + r N + 1 ) − (r − N +

1

r − N . Therefore n =− N (r n + 1 r − N + 1 r − N + 2 r 2 r N ) r N + 1 r n )

+ +···+ 1 + r + +···+ = − − =

− 1 ) n =− N r n and n =− N r n (r N + 1 r − N )/r (r = − − 1 ; r = 1 = 2 N + 1 ; r = 1.