(d)
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This is a second-order system (N = 2) with b0 = 0, b1 = 4, b2 = 28, a1 = 6, a2 = 5. Figure 5.12 shows the canonic direct and transposed canonic direct realizations.
Figure 5.12 Realization of (4z +28)/(z2 +6z+5): (a) canonic direct form and (b) its transpose.
(c)
DR ILL 5.16 Realization of a Second-Order Transfer Function
Realize the transfer function
REALIZATION OF FINITE IMPULSE RESPONSE (FIR) FILTERS
So far we have been quite general in our development of realization techniques. They can be applied to infinite impulse response (IIR) or FIR filters. For FIR filters, the coefficients ai = 0 for all i = 0.† Hence, FIR filters can be readily implemented by means of the schemes developed so far by eliminating all branches with ai coefficients. The condition ai = 0 implies that all the poles of a FIR filter are at z = 0.
EXAMPLE 5.9 Realization of an FIR Filter
Realize H[z] = (z3 +4z2 +5z+2)/z3 using canonic direct and transposed forms.
† This statement is true for all i = 0 because a0 is assumed to be unity.
For H[z], b0 = 1, b1 = 4, b2 = 5, and b3 = 2. Hence, we obtain the canonic direct realization, shown in Fig. 5.13a. We have shown the horizontal orientation because it is easier to see that this filter is basically a tapped delay line. That is why this structure is also known as a tapped delay line or transversal filter. Figure 5.13b shows the corresponding transposed implementation.
CASCADE AND PARALLEL REALIZATIONS, COMPLEX AND REPEATED POLES
The considerations and observations for cascade and parallel realizations as well as complex and multiple poles are identical to those discussed for LTIC systems in Sec. 4.6-3.
DR ILL 5.17 Cascade and Parallel Realizations of a Transfer Function
Find canonic direct realizations of the following transfer function by using the cascade and parallel forms. The specific cascade decomposition is as follows:
DO ALL REALIZATIONS LEAD TO THE SAME PERFORMANCE?
For a given transfer function, we have presented here several possible different realizations (DFI, canonic form DFII, and its transpose). There are also cascade and parallel versions, and there are many possible grouping of the factors in the numerator and the denominator of H[z], leading to different realizations. We can also use various combinations of these forms in implementing different subsections of a system. Moreover, the transpose of each version doubles the number. However, this discussion by no means exhausts all the possibilities. Transforming variables affords limitless potential realizations of the same transfer function.
Theoretically, all these realizations are equivalent; that is, they lead to the same transfer function. This, however, is true only when we implement them with infinite precision. In practice, finite wordlength restriction causes each realization to behave differently in terms of sensitivity to parameter variation, stability, frequency response distortion error, and so on. These effects are serious for higher-order transfer functions, which require correspondingly higher numbers of delay elements. The finite wordlength errors that plague these implementations are coefficient quantization, overflow errors, and round-off errors. From a practical viewpoint, parallel and cascade forms using low-order filters minimize the effects of finite wordlength. Parallel and certain cascade forms are numerically less sensitive than the canonic direct form to small parameter variations in the system. In the canonic direct form structure with large N, a small change in a filter coefficient due to parameter quantization results in a large change in the location of the poles and the zeros of the system. Qualitatively, this difference can be explained by the fact that
526 CHAPTER 5 DISCRETE-TIME SYSTEM ANALYSIS USING THE Z-TRANSFORM
in a direct form (or its transpose), all the coefficients interact with each other, and a change in any coefficient will be magnified through its repeated influence from feedback and feedforward connections. In a parallel realization, in contrast, a change in a coefficient will affect only a localized segment; the case of a cascade realization is similar. For this reason, the most popular technique for minimizing finite wordlength effects is to design filters by using cascade or parallel forms employing low-order filters. In practice, high-order filters are realized by using multiple second-order sections in cascade, because second-order filters not only are easier to design but are less susceptible to coefficient quantization and round-off errors, and their implementations allow easier data word scaling to reduce the potential overflow effects of data word-size growth. A cascaded system using second-order building blocks usually requires fewer multiplications for a given filter frequency response [1].
There are several ways to pair the poles and zeros of an Nth-order H[z] into a cascade of second-order sections, and several ways to order the resulting sections. Quantizing error will be different for each combination. Although several papers published provide guidelines in predicting and minimizing finite wordlength errors, it is advisable to resort to computer simulation of the filter design. This way, one can vary filter hardware characteristic, such as coefficient wordlengths, accumulator register sizes, sequencing of cascaded sections, and input signal sets. Such an approach is both reliable and economical [1].