DirectForm I
As mentioned in §5.5,
the difference equation
(10.1) 
specifies the DirectForm I (DFI) implementation of a digital filter [60]. The DFI signal flow graph for the secondorder case is shown in Fig.9.1.
The DFI structure has the following properties:
 It can be regarded as a twozero filter section followed in series
by a twopole filter section.
 In most fixedpoint 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 fixedpoint 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 DFI filter structure.
 There are twice as many delays as are necessary. As a result,
the DFI structure is not canonical with respect to delay. In
general, it is always possible to implement an thorder filter
using only delay elements.
 As is the case with all directform filter structures
(those which have coefficients given by the transferfunction coefficients),
the filter poles and zeros can be very sensitive to roundoff errors
in the filter coefficients. This is usually not a problem for a
simple secondorder section, such as in Fig.9.1, but it can
become a problem for higher order directform filters. This is the
same numerical sensitivity that polynomial roots have with respect to
polynomialcoefficient roundoff. 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 secondorder
sections, as discussed in §9.2 below.
It is a very useful property of the directform I implementation that it cannot overflow internally in two's complement fixedpoint 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 DFI 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 secondorder sections (§9.2 below), and normalized ladder forms [32,48,86].^{10.2}Also, we'll see that the transposed directform II (Fig.9.4 below) is a strong contender as well.
Two's Complement WrapAround
In this section, we give an example showing how temporary overflow in two's complement fixedpoint causes no ill effects.
In 3bit signed fixedpoint arithmetic, the available numbers are as shown in Table 9.1.

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 threebit 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 wraparound in the first addition is canceled by a negative wraparound in the second addition.
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Direct Form II
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OnePole Transfer Functions