1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

378 CHAPTER 9 • z-TRANSFORMS AND APPLICATIONS

A(8)

2.5

2

1.5

0.5

Figure 9.1 The amplitude response A( 8) = I (^1) - i V1 .;:^1 1 +9• 1 , I for the boosting u p filter
y{n] = x[n] + ~v/2y[n - 1) - ~y(n - 2).
Solution
(b) Calculate the amplitude responses A(i) and A(2.60) and invest igate

the filtered signal for x[n] = cos(in) + sin(fn) + sin(2.60n).

A(i) = 1/ Ii - ~h (elf )-

1

+ ~ (elf )-

2

1 = 12.231585 -0.234260il

= 2.243847,

and

A(2.60) = 1/ Ii - ~J2 (e26oir 1 + ~ (e2.60;)- 2I

= I0.4 16831 - o.181664il = o.454698.

Using these calculations we conclude that the components cos(in)

and sin(fn) will be boosted up by the factors A(i) = 2.243847 and

A(f) = 2.49615, respectively, and the component sin(2.60n) will be

attenuated by the factor A(2.60) = 0.454698. The situation is shown

in Figure 9.8.

9.3.3 The General Form

The general form of a Pth order filter difference equation is

y[n] + a1y[n - 1] + a2y(n - 2] + ... + ap-1y[n - P + l ] + apy[n - P ] (9-29)

= box(n] + b1x[n - 1] + b2x[n - 2] + ... + bQ-1X[n - Q + l] + bQx[n - Q],