Direct-Form I

As mentioned in §5.5,
the difference equation

(10.1) |

specifies the

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 [84]^{10.1}) 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].^{10.2}Also, we'll see that the *transposed direct-form II*
(Fig.9.4 below) is a strong contender as well.

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