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We already know that Rule 4 holds when n = 1, Dx (x 1 ) = Dx (x) = 1 = 1 x 0 (Remember that x 0 = 1.) We can prove the rule by mathematical induction. This involves showing that if the rule holds for any particular positive integer k, then the rule also must hold for the next integer k + 1. Since we know that the rule holds for n = 1, it would then follow that it holds for all positive integers. Assume, then, that Dx (x k ) = kx k 1 . We have Dx (x k+1 ) = Dx (x k x) = x k Dx (x) + x Dx (x k ) = x k 1 + x(kx k 1 ) = x k + kx k = (1 + k)x k = (k + 1)x (k+1) 1 and the proof by induction is complete. [since x k+1 = x k x 1 = x k x] [by the product rule] [by the asumption that Dx (x k ) = kx k 1 ] [since x x k 1 = x 1 x k 1 = x k ]

conjugate pairs, and these complex conjugate factors may be combined to form second-order factors with real coefficients:

Fig. 7-39

A sixth-order 1IR filter implemented as a cascade of three second-order systems in direct form I1 is shown in Fig. 8-9.

The dashed perpendiculars in the diagrams are additional segments needed only for the solutions. Note how rectangles are formed by these added segments. (a) EF (b) b

Fig. 8-9. A sixth-order 11R filter implemented as a cascade of three direct form 11 second-order systems.

17 7

There is considerable flexibility in how a system may be implemented in cascade form. For example, there are different pairings of the poles and zeros and different ways in which the sections may be ordered.

CHAP. 12]

An alternative to factoring H ( z ) is to expand the system function using a partial fraction expansion. For example, with

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7.38 Applying the pythagorean theorem to a circle (a) Find the distance d from the center of a circle of radius 17 to a chord whose length is 30 [Fig. 7-40(a)]. (b) Find the length of a common external tangent to two externally tangent circles with radii 4 and 9 [Fig. 7-40(b)].

if p > q and ai # c r k (the roots of the denominator polynomial are distinct), H ( z ) may be expanded as a sum of p first-order factors as follows:

Fig. 7-40

where the coefficients At and a k are, in general, complex. This expansion corresponds to a sum of p first-order system functions and may be realized by connecting these systems in parallel. If h ( n ) is real, the poles of H ( z ) will occur in complex conjugate pairs, and these complex roots in the partial fraction expansion may be combined to form second-order systems with real coefficients:

CHAP. 12]

1 2(30)

Shown in Fig. 8-10 is a sixth-order filter implemented as a parallel connection of three second-order direct form I1 systems. If p 5 q, the partial fraction expansion will also contain a term of the form

A 30 -60 -90 triangle is one-half an equilateral triangle. Thus, in right nABC (Fig. 7-41), a that c 2; then a 1, and the Pythagorean Theorem gives b2 c2 a2 22 12 3 or b 23

which is an FIR filter that is placed in parallel with the other terms in the expansion of H(z).

1: 23:2

Fig. 8-10. A sixth-order IIR filter implemented as a parallel connection of three second-order direct form 11 structures.

12.14 A function f , de ned for all real numbers, is such that: (i) f (1) = 2; (ii) f (2) = 8; (iii) f (u + v) f (u) = kuv 2v 2 for all u and v, where k is some constant. Find f (x) for arbitrary x. 12.15 Let f (x) = 2x 2 + 3x for all x.

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