ABSTRACT
A digital filter with impulse response having infinite length (i.e., its values outside a finite interval cannot all be zero) is termed infinite impulse response (IIR) filter. The most important class of IIR filters can be described by the difference equation
y(n)¼ b0x(n)þ b1x(n 1)þ þ bMx(nM) a1y(n 1) a2y(n 2) aNy(n N) (19:1)
where x(n) is the input y(n) is the output of the filter fa1, a2, . . . , aNg and fb0, b1, . . . , bMg are constant coefficients
We assume that aN 6¼ 0. The impulse response is the output of the system when it is driven by a unit impulse at n ¼ 0, with the system being initially at rest, i.e., the output being zero prior to applying the input. We denote the impulse response by h(n). With x(0) ¼ 1, x(n) ¼ 0 for n 6¼ 0, and y(n) ¼ 0 for
n < 0, we can compute h(n), n 0, from Equation 19.1 in a recursive manner. Taking the z-transform of Equation 19.1, we obtain the system function
H(z) ¼ Y(z) X(z)
¼ b0 þ b1z 1 þ þ bMzM
1þ a1z1 þ þ aNzN (19:2)
where N is the order of the filter. The system function and the impulse response are related through the z-transform and its inverse, i.e.,
H(z) ¼ X1
n¼1 h(n)z1 h(n) ¼ 1
2pj
þ C
H(z)zn1dz (19:3)
where C is a closed counterclockwise contour in the region of convergence. See Chapter 5 of Fundamentals of Circuits and Filters for a discussion of z-transform. We assume that M N . Otherwise, the system function can be written as
H(z) ¼ c0 þ c1z1 þ þ cMNz(MN)
]þ b 0 0 þ b01z1 þ þ b0MzM 1þ a1z1 þ þ aNzN
(19:4)
which is a finite impulse response (FIR) filter in parallel with an IIR filter, or as
H(z) ¼ c00 þ c01z1 þ þ c0MNz(MN) b000 þ b001z1 þ þ b00MzM
1þ a1z1 þ þ aNzN
(19:5)
which is an FIR filter in cascade with an IIR filter. FIR filters are covered in Chapter 18. This chapter covers IIR filters.
