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where UN(52/Qp) is a Jacobian elliptic function. The Jacobian elliptic function U N ( x ) a rational function of is order N with the following property:

Elliptic filters have an equiripple passband and an equiripple stopband. Because the ripples are distributed uniformly across both bands (unlike the Butterworth and Chebyshev filters, which have a monotonically decreasing

6.1 Matching test of circle vocabulary Match each part of Fig. 6-13 on the left with one of the names on the right: (a) OE (b) FG 1. Radius 2. Central angle

Theorem 7.1: If f (x) = an x n + an 1 x n 1 + + a1 x + a0 is a polynomial with integral coef cients (that is, the numbers an , an 1 , , a1 , a0 are integers), and if r is an integer that is a root of f , then r must be a divisor of the constant term a0 . EXAMPLE By Theorem 7.1, any integral root of x3 2x2 5x + 6 must be among the divisors of 6, which are 1, 2, 3, and 6. By actual substitution, it is found that 1, 2, and 3 are roots. Theorem 7.2: A number r is a root of the polynomial f (x) = an x n + an 1 x n 1 + + a1 x + a0 if and only if f (x) is divisible by the polynomial x r. For hints for the proofs of Theorems 7.1 and 7.2, see Supplementary Problems 7.26 and 7.27. EXAMPLES

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passband and/or stopband), these filters are optimum in the sense of having the smallest transition width for a given filter order, cutoff frequency Q,, and passband and stopband ripples. The frequency response for a 4th-order elliptic filter is shown in Fig. 9- 10.

Fig. 6-13

The design of elliptic filters is more difficult than the Butterworth and Chebyshev filters, because their design relies on the use of tables or series expansions. However, the filter order necessary to meet a given set of specifications may be estimated using the formula

(c) FH (d) CD (e) IJ (f) EF (g) FGH (h) FEG (i) j EOF (j) Circle O about EFGH (k) Circle O in ABCD (l) Quadrilateral EFGH (m) Quadrilateral ABCD

The design of a digital filter from an analog prototype requires that we transform ha(t) to h(n) or Ha(s) to H(z). A mapping from the s-plane to the z-plane may be written as

(a) 1 (b) 6 (c) 7 (d) 9 (e) 8 (f) 4 (g) 3 (h) 5

where s = m(z) is the mapping function. In order for this transformation to produce an acceptable digital filter, the mapping m(z) should have the following properties: The mapping from the jS2-axis to the unit circle, Izl = 1, should be one to one and onto the unit circle in order to preserve the frequency response characteristics of the analog filter. 2. Points in the left-half s-plane should map to points inside the unit circle to preserve the stability of the analog filter. 3. The mapping m(z) should be a rational function of z so that a rational Ha(s) is mapped to a rational H(z).

(a) Consider the polynomial f (x) = x 3 4x 2 + 14x 20. By Theorem 7.1, any integral root must be a divisor of 20; that is, 1, 2, 4, 5, 10, or 20. Calculation reveals that 2 is a root. Hence, by Theorem 7.2, x 2 must divide f (x); we carry out the division in Fig. 7-10(a). Since f (x) = (x 2)(x 2 2x + 10), its remaining roots are found by solving x 2 2x + 10 = 0

3. Semicircle 4. Minor arc 5. Major arc 6. Chord 7. Diameter 8. Secant 9. Tangent 10. Inscribed polygon 11. Circumscribed polygon 12. Inscribed circle 13. Circumscribed circle

Described below are two approaches that are commonly used to map analog filters into digital filters.

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