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.
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}
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 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}Next Section:
Series and Parallel Filter Sections
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PoleZero Analysis Problems