1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

9.3 • D IGITAL SIGNAL FILTERS 373

9.3 Digital Signal Filters

9.3.1 Introduction to Filtering

In the field of signal processing, the design of digital signal filters involves the

process of suppressing certa.in frequencies a.nd boosting others. A simplified filter

model is

y[n] + a1y(n -1] + a2y[n - 2) = box[n] + b1x[n -1) + b2x[n - 2) + bax[n - 3),
(9-22)

where the input signal is X n = x{n] is modified to obtain the output signal

Yn = y[n) using the recursion formula

y[n] = box[n] + b1x[n -l] + b2x[n - 2) + bsx[n - 3) - a 1 y [n -1) - a2y[n -2].
(9-23)

The implementation of (9- 23 ) is straightforward and only requires starting values,
then Yn = y(n) is obtained by simple iteration. Since the signals must have a
starting point , it is common to require that Xn = 0 and Yn = 0 for n < 0. We
emphasize this concept by making the following definition.

Definition 9.3 (Causal Sequence) Given the input {xn}::"=-oo and output

{Yn}::°=-oo sequences. If Xn = 0 and Yn = 0 for n < 0, the sequence is said

to be causal.

Given the causal sequence { Xn = x[n]} :, 0 , it is easy to calculate the solution

{Yn = y[n]} ::"=o to (9-23). Use the fact that these sequences are causal:

X _ 3 = 0, X- 2 = 0, X-1 = 0 and Y- 2 = 0, Y- 1 = 0. (9-24)

Then compute

Yo= boxo, (9-^25 )

Y1 = box1 + bixo - a1yo,

Y2 = box2 + b1x1 + b2xo -a1y1 - a2yo,

Y3 = boxs + b1x2 + b2x 1 + b3xo -a 1Y2 - a2Y1·

The general iterative step is

(9-26)