Signals and Systems - Electrical Engineering

684 C H A P T E R 11: Introduction to the Design of Discrete Filters

Examples of windows that are smoother than the rectangular window are:

Triangular or Barlett window: w[n]=

{

1 −N^2 |−n 1 | −(N− 1 )/ 2 ≤n≤(N− 1 )/ 2

0 otherwise

(11.58)

Hamming window: w[n]=

{

0.54+0.46 cos( 2 πn/(N− 1 )) −(N− 1 )/ 2 ≤n≤(N− 1 )/ 2

0 otherwise

(11.59)

Kaiser window: This window has a parameterβthat can be adjusted. It is given by

w[n]=







I 0

(

β

√

1 −(n/(N− 1 ))^2

)

I 0 (β) −(N−^1 )/^2 ≤n≤(N−^1 )/^2

0 otherwise

(11.60)

whereI 0 (x)is the zero-order Bessel function of the first kind, which can be computed by the series

I 0 (x)= 1 +

∑∞

k= 1

(

(0.5x)k

k!

) 2

(11.61)

Whenβ=0 the Kaiser window coincides with a rectangular window, sinceI 0 ( 0 )=1. Asβincreases

the window becomes smoother.

The above definitions are for windows symmetric with respect to the origin. Figures 11.21 and 11.22

show the causal rectangular, Barlett, Hamming, and Kaiser windows, and their magnitude spectra.

FIGURE 11.21

(a) Rectangular and (b) Barlett

causal windows and (c) their

spectra.

0 10 20 30

0

0.5

1

n

w^1

[n

]

0 10 20 30

0

0.5

1

n

w^2

[n

]

(a) (b)

0.2 0.4 0.6 0.8 1

− 100

− 80

− 60

− 40

− 20

0

ω/π

Gain(dB)

Rectangular

Barlett

0

(c)