The Four Direct Forms
As mentioned in §5.5, the difference equation
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.
- 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 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.
Two's Complement Wrap-AroundIn this section, we give an example showing how temporary overflow in two's complement fixed-point causes no ill effects. In 3-bit signed fixed-point 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 IIThe signal flow graph for the Direct-Form-II (DF-II) realization of the second-order IIR filter section is shown in Fig.9.2.
- It can be regarded as a two-pole filter section followed by a two-zero filter section.
- It is canonical with respect to delay. This happens because delay elements associated with the two-pole and two-zero sections are shared.
- In fixed-point arithmetic, overflow can occur at the delay-line input (output of the leftmost summer in Fig.9.2), unlike in the DF-I implementation.
- As with all direct-form filter structures, the poles and zeros are sensitive to round-off errors in the coefficients and , especially for high transfer-function orders. Lower sensitivity is obtained using series low-order sections (e.g., second order), or by using ladder or lattice filter structures .
Transposed Direct-FormsThe remaining two direct forms are obtained by formally transposing direct-forms I and II [60, p. 155]. Filter transposition may also be called flow graph reversal, and transposing a Single-Input, Single-Output (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. 176-177]. Transposition of filters in state-space 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 branch-points to summers, and summers to branch-points. 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 left-right flipped. Figure 9.3 shows the Transposed-Direct-Form-I (TDF-I) structure for the general second-order IIR digital filter, and Fig.9.4 shows the Transposed-Direct-Form-II (TDF-II) structure. To facilitate comparison of the transposed with the original, the inputs and output signals remain ``switched'', so that signals generally flow right-to-left instead of the usual left-to-right. (Exercise: Derive forms TDF-I/II by transposing the DF-I/II structures shown in Figures 9.1 and 9.2.)
transposed direct-form 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 elliptic-function-filter example on page ; in such filters, the sharp transitions are achieved using near pole-zero cancellations close to the unit circle in the plane.10.4
Series and Parallel Filter Sections
Pole-Zero Analysis Problems