Page 208 - Mechanical Engineers' Handbook (Volume 2)

3 Basic Digital Filter 197

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2
2
(G; ;p) [
H(j )
F( )] d (19)
D
subject to
e(p ) 0 for all k 1,..., N (20)
k
Unfortunately, this results in a nonlinear system of equations. Furthermore, the topography
of this objective function is generally complicated with many local minima, making standard
gradient-descent methods unfeasible. It is usually best to use finite-impulse-response (FIR)
filtering or filtering in the frequency domain for these types of problems. If an infinite-
impulse-response (IIR) filter is desired, the reader is referred to Prony’s method, which
linearizes this system and finds an approximate optimal solution. 4

2.6 Circuit Prototypes for Pole and Zero Placement for Realization of Filters Designed
from Rational Functions

The voltage–current relation for a resistor (R), inductor (L), and capacitor (C)is
di v t
1
v iR v L c idr (21)
r
l
dt C
These relationships are represented in the S-domain as
I(s)
V (s) RI(s) V (s) sLI(s) V (22)
r
c
l
sC
Thus, the general transfer function for any linear circuit involving standard passive, active,
and reactive devices is a rational function, that is, a ratio of polynomials:
M 1 i
i 0 as
i
H(s)
N 1 k
k 0 bs
k
V Z
0 2 (23)
V Z
i 1
Thus one can construct any arbitrary transfer function through a serial placement of this
building block circuit prototype shown in Fig. 8. Table 1 gives circuit elements for Z and
1
Z to construct the basic prototype circuits.
2
3 BASIC DIGITAL FILTER
Basic linear digital filters are of two types: those that have a finite response to an impulse,
or FIR, and those that have an infinite response to an impulse (IIR). The general form of a
linear digital filter is
y by by b y ax ax ax (24)
k 1 k 1 2 k 2 k mk m 0 k 1 k 1 nk n
where k i represents the ith delay. That is, the kth output from a linear digital filter is
some linear combination of previous inputs and outputs. The filter will have a finite response
to an impulse if b b b 0; otherwise, the filter is of type IIR.
2
1
m