specifies the Direct-Form I (DF-I) implementation of a digital filter . The DF-I signal flow graph for the second-order case is shown in Fig.9.1.
The DF-I structure has the following properties:
- It can be regarded as a two-zero filter section followed in series
by a two-pole filter section.
- In most fixed-point arithmetic schemes (such as two's complement,
the most commonly used
there is no possibility of internal filter overflow. That is,
since there is fundamentally only one summation point in the filter,
and since fixed-point overflow naturally ``wraps around'' from the
largest positive to the largest negative number and vice versa, then
as long as the final result is ``in range'', overflow is
avoided, even when there is overflow of intermediate results in the sum
(see below for an example). This is an important, valuable, and
unusual property of the DF-I filter structure.
- There are twice as many delays as are necessary. As a result,
the DF-I structure is not canonical with respect to delay. In
general, it is always possible to implement an th-order filter
using only delay elements.
- As is the case with all direct-form filter structures
(those which have coefficients given by the transfer-function coefficients),
the filter poles and zeros can be very sensitive to round-off errors
in the filter coefficients. This is usually not a problem for a
simple second-order section, such as in Fig.9.1, but it can
become a problem for higher order direct-form filters. This is the
same numerical sensitivity that polynomial roots have with respect to
polynomial-coefficient round-off. As is well known, the sensitivity
tends to be larger when the roots are clustered closely together, as
opposed to being well spread out in the complex plane
[18, p. 246]. To minimize this sensitivity, it is common to
factor filter transfer functions into series and/or parallel second-order
sections, as discussed in §9.2 below.
It is a very useful property of the direct-form I implementation that it cannot overflow internally in two's complement fixed-point arithmetic: As long as the output signal is in range, the filter will be free of numerical overflow. Most IIR filter implementations do not have this property. While DF-I is immune to internal overflow, it should not be concluded that it is always the best choice of implementation. Other forms to consider include parallel and series second-order sections (§9.2 below), and normalized ladder forms [32,48,86].10.2Also, we'll see that the transposed direct-form II (Fig.9.4 below) is a strong contender as well.
Two's Complement Wrap-Around
In this section, we give an example showing how temporary overflow in two's complement fixed-point causes no ill effects.
Let's perform the sum , which gives a temporary overflow (, which wraps around to ), but a final result () which is in the allowed range :10.3
Now let's do in three-bit two's complement:
In both examples, the intermediate result overflows, but the final result is correct. Another way to state what happened is that a positive wrap-around in the first addition is canceled by a negative wrap-around in the second addition.
Direct Form II
One-Pole Transfer Functions