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By David A. Patterson, John L. Hennessy

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Synthesis is possible once either H(s) or K(s) is known since the Feldtkeller equation relates these two functions. 9 25 Input Impedance The filter reactive element values historically have been found from the input impedance of the network. The input impedance, reflection coefficient and network functions are related by (37) Solving for the input impedance in terms of the network function polynomials we have The actual element values are then found by a continued fraction expansion of Zi. 10 Synthesis Example Next, these concepts are applied to a lowpass filter with RR equal to 1 ohm and o, equal to 1 radian.

5 degrees are tabulated at the end of the chapter for orders up to 10. Even with phase-error ripple, selectivity is far worse than Butterworth. 23 dB respectively Notice both controlled phase filters are much less selective than the Butterworth and the equiripple phase-error is not significantly better than the Bessel. 0 dB respectively The equiripple phase-error approximation provides more rejection than the Bessel further into the stopband. Other controlled phase approximations include transitional responses where the amplitude is similar to the Bessel over much of the passband (down to 6 or 12 dB) but which have improved selectivity above the transition region.

Again the denominator is unity so F(s) = K(s). Next these results are substituted in the expression for the input impedance of the terminated network to be synthesized and we have Zi= s3+2s2+2s+1-s3 [ $+&2+2s+l+s3 ]=[ ,z~zi+1] (4% This expression for the impedance of the terminated network is used to find the network element values. First we rationalize the numerator Zi= 1 23+2s2+2s+l (43) 2s2+2s+1 Next we continually divide the lower order polynomial into the higher order polynomial and invert the remainder.

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Computer organization and design 4th ed solutions manual by David A. Patterson, John L. Hennessy

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