374 CHAPTER 9 • z-TRANSFORMS AND APPLICATIONS9.3.2 The Basic Filters
The following three simplified basic filters serve as illustrations.
(i) Zeroing Out Filter y[n] = box[n] + b 1 x [n -1] + bzx[n-2J + bax[n -3]
(Note that a1 = 0, and a2 = O).
(ii) Boosting Up Filter y[n] = box(nJ - a 1 y [n - l] -a2y[n -2).
(Note that b1 = 0, b2 = 0 , and ba = 0.)(iii) Combination Filter y[n) = box[n ] + b1x(n - 1) + b2x[n - 2] +
bax(n -3) - a 1 y [n -1) - a2y(n - 2).The transfer function H ( z ) for these model filters has the following general
formH (z) = Y(z) = bo + b1z-^1 + bzz-^2 + baz-^3
X(z) 1 + a1z-^1 + a2z-^2 '
(9-27)where the z-transforms of the input and output sequences are X (z) = 3[x,.) and
Y(z) = 3[Yn), respectively. In Section 9.2 we mentioned that the general solution
to a homogeneous difference equation y[n) + a 1 y [n - 1) + a2y[n -2] = 0 is stable
only if the zeros of the characteristic equation lie inside the unit circle. Similarly,
if a filter is stable, then the poles of the transfer function H ( z) must all lie inside
the unit circle.
Before d eveloping the general theory, we would like to investigate t he am-
plitude response A(B) when the input signal is a linear combination of cos(Bn)
and sin(Bn). The amplitude response for t he frequency 8 uses the complex unit
signal z = ei^8 , and is defined to be(9-28)T he formula for A(B) will be rigorously explained after a few introductory ex-
amples.- EXAMPLE 9.21 Given the filter y [n] = x [n] - J2x(n -1] + x [n - 2].
(a) Show that it is a zeroing out filter for the signals cos(1n) and sin( in)
and calculate the amplitude response A(~)= A(0.785398).(b) Calculate the amplitude responses A(0.10) and A(0.77) and investi-gate the the filtered signal for x(n] = cos(O.lOn) + sin(0.77n).
(c) Calculate the amplitude responses A(0.10) and A(~p and investigate
the filtered signal for x[n] = cos(O.lOn) + 0.20sin( ; n).Solution(a) In Section 9.2, Example 9.19, we established that t he difference equa-
tion x[n + 2] - J2x[n + l ] + x [n ) = 0 with initial conditions x[O] = 2