By Jones D.W.

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The problem now is merely to ﬁnd all the poles of G(s)G(Às), then to sort them. 36. 38 in the form This has the solution and f¼ p p þ (2k À 1) 2 2n (2:42) where k is any integer. Of course, only those values of f between 0 and 2p rad are to be considered unique. Approximation 2-11 As an example, suppose that n ¼ 2; that is, one is interested in determining the transfer function of a second-order Butterworth ﬁlter. 42 are p=4, 3p=4, 5p=4, and 7p=4. All other values are simply these four with integer multiples of 2p added.

For instance, one can transform from an lpp with, say, a passband cutoff frequency of 1 rad=s to a low-pass ﬁlter with a passband cutoff of perhaps 1 kHz. ) As one can readily see, a ﬁrst-order lpp is transformed into a second-order brf. In general, for bandpass and bandreject design, the object transfer function is of twice the order of the lpp. 16. 15 the 1-V resistor maps into a 1-V resistor, but the 1 F capacitor maps into a combination of elements having the admittance Y( p) ¼ p ¼ 1 s B (2:18) v2 þ Bso But this is simply the series connection of a capacitor of value B=v2o farads and an inductor of value 1=B henrys.

Assign the left-half s-plane zeros and the original poles to the minimum-phase transfer function HM(s). 3. Assign the right-half s-plane zeros and the left-hand s-plane poles generated in step 1 to the allpass transfer function HAP(s). 7. 7 Allpass filter Decomposition of nonminimum-phase transfer function. 8 Introduction to the Design Process The design of ﬁlters starts with a set of speciﬁcations and ends with the implementation of a prototype. It comprises four general steps, as follows: 1. 2.

### Arithmetic tutorials by Jones D.W.

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