1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

9.3 • DIGITAL SIGNAL FILTERS 379

x{n]

2

y[n]

4

Figure 9.8 The input x(n] = cos( in)+ sin(fn) + sin(2.60n) and output y[n).

where {a.} :=I and {b.} ~=o are constants. Note carefully tba.t the terms involved

are of the form y[n -p] and y [n - q] where p ~ 0 and q ~ 0, which makes these

terms t ime-delayed. The compact form of writing t he difference equation is

p Q

y[n] + L apy[n -p] = L bqx[n -q], (9-3 0)

p=O q=O

where the input signal Xn = x [n] is modified to obtain the output signal Yn = y[n]

using the recursion formula

Q p

y(n] = L bqx[n -q] - L apy[n -p]. (9-31)
q= O p=l

The portion L:~=O bqx[n - q] will "zero out" signals and L::=l apy[n - p] will
"boost up" signals.

Remark 9.13

Formula (9-31) is called the recursion equation and the recursion coefficients

are {ap}:=l and {bq}~=o· It explicitly shows that the present output y[n ] is a

function of the past values y [n-p], for p = 1, 2, .. ., P, the present input x [n], and

the previous inputs x (n - q] for q = 1, 2, .. ., Q. The sequences can be regarded

as signals and they are zero for negative indices. With this information we can

now define the general formula for the transfer function H (z). Using the time

delayed-shift property for causal sequences and taking the z-transform of each

term in (9-31), we obtain

p Q

Y(z) = - l:apY(z)z-P + LbqX(z)z-q. (9-32)

p=l q=O