The Four Direct Forms
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.
Direct Form II
The signal flow graph for the DirectFormII (DFII) realization of the secondorder IIR filter section is shown in Fig.9.2.
The difference equation for the secondorder DFII structure can be written as
which can be interpreted as a twopole filter followed in series by a twozero filter. This contrasts with the DFI structure of the previous section (diagrammed in Fig.9.1) in which the twozero FIR section precedes the twopole recursive section in series. Since LTI filters in series commute (§6.7), we may reverse this ordering and implement an allpole filter followed by an FIR filter in series. In other words, the zeros may come first, followed by the poles, without changing the transfer function. When this is done, it is easy to see that the delay elements in the two filter sections contain the same numbers (see Fig.5.1). As a result, a single delay line can be shared between the allpole and allzero (FIR) sections. This new combined structure is called ``direct form II'' [60, p. 153155]. The secondorder case is shown in Fig.9.2. It specifies exactly the same digital filter as shown in Fig.9.1 in the case of infiniteprecision numerical computations.
In summary, the DFII structure has the following properties:
 It can be regarded as a twopole filter section followed by a twozero
filter section.
 It is canonical with respect to delay. This happens because
delay elements associated with the twopole and twozero sections are
shared.
 In fixedpoint arithmetic, overflow can
occur at the delayline input (output
of the leftmost summer in Fig.9.2), unlike in the DFI
implementation.
 As with all directform filter structures, the poles
and zeros are sensitive to roundoff errors in the coefficients
and , especially for high transferfunction orders. Lower
sensitivity is obtained using series loworder sections (e.g., second
order), or by using ladder or lattice filter structures
[86].
More about Potential Internal Overflow of DFII
Since the poles come first in the DFII realization of an IIR filter, the signal entering the state delayline (see Fig.9.2) typically requires a larger dynamic range than the output signal . In other words, it is common for the feedback portion of a DFII IIR filter to provide a large signal boost which is then compensated by attenuation in the feedforward portion (the zeros). As a result, if the input dynamic range is to remain unrestricted, the two delay elements may need to be implemented with highorder guard bits to accommodate an extended dynamic range. If the number of bits in the delay elements is doubled (which still does not guarantee impossibility of internal overflow), the benefit of halving the number of delays relative to the DFI structure is approximately canceled. In other words, the DFII structure, which is canonical with respect to delay, may require just as much or more memory as the DFI structure, even though the DFI uses twice as many addressable delay elements for the filter state memory.
Transposed DirectForms
The remaining two direct forms are obtained by formally transposing directforms I and II [60, p. 155]. Filter transposition may also be called flow graph reversal, and transposing a SingleInput, SingleOutput (SISO) filter does not alter its transfer function. This fact can be derived as a consequence of Mason's gain formula for signal flow graphs [49,50] or Tellegen's theorem (which implies that an LTI signal flow graph is interreciprocal with its transpose) [60, pp. 176177]. Transposition of filters in statespace form is discussed in §G.5.
The transpose of a SISO digital filter is quite straightforward to find: Reverse the direction of all signal paths, and make obviously necessary accommodations. ``Obviously necessary accommodations'' include changing signal branchpoints to summers, and summers to branchpoints. Also, after this operation, the input signal, normally drawn on the left of the signal flow graph, will be on the right, and the output on the left. To renormalize the layout, the whole diagram is usually leftright flipped.
Figure 9.3 shows the TransposedDirectFormI (TDFI) structure for the general secondorder IIR digital filter, and Fig.9.4 shows the TransposedDirectFormII (TDFII) structure. To facilitate comparison of the transposed with the original, the inputs and output signals remain ``switched'', so that signals generally flow righttoleft instead of the usual lefttoright. (Exercise: Derive forms TDFI/II by transposing the DFI/II structures shown in Figures 9.1 and 9.2.)
Numerical Robustness of TDFII
An advantage of the transposed directform II structure (depicted in Fig.9.4) is that the zeros effectively precede the poles in series order. As mentioned above, in many digital filters design, the poles by themselves give a large gain at some frequencies, and the zeros often provide compensating attenuation. This is especially true of filters with sharp transitions in their frequency response, such as the ellipticfunctionfilter example on page ; in such filters, the sharp transitions are achieved using near polezero cancellations close to the unit circle in the plane.^{10.4}
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Series and Parallel Filter Sections
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PoleZero Analysis Problems