Oxford English for electrical and mechanical engineering by Eric H. Glendinning, Norman Glendinning

By Eric H. Glendinning, Norman Glendinning

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This design process involves finding the filter coefficients -the a;s and b;s of G(Z), thereby yielding a pulse transfer function which is a rational function in z- 1 . A number of useful design methods are discussed in this chapter; each one is basically a mathematical method of obtaining a solution to the problem of approximating to a desired filter characteristic. For example, it may be required that the frequency response of the filter approximates a lowpass brickwall characteristic, which would involve one of the frequency·domain design methods: bilinear Z·transform, matched Z·transform, frequency sampling, and direct approach using squared magnitude functions.

For a digital filter to be stable all the poles (roots of the characteristic equation) of G(Z) must be within the unit-circle in the Z-plane. If one or more poles are located outside this circle the filter is unstable. 35), may result in one or more poles moving outside the unit-circle. This small change is brought about by inaccurate representation of the coefficients using finite processor word-length. This later point is discussed more fully in chapter 4. 3 Steady-state Sinusoidal Response It is well known that the amplitude and phase response of an electronic system provides valuable information in the design and analysis of transmission circuits, for example, in the design of filter networks.

The frequency-domain approach to the design of recursive digital filters may be subdivided into two main techniques of solution. The first method is an indirect approach, which requires that a suitable prototype continuous (analogue) filter transfer function, G(S), is designed, and subsequently this is transformed via an appropriate S-plane to Z-plane mapping to give a corresponding digital filter pulse transfer function, G(Z). The mappings used in this chapter are the standard Z-transform (impulse-invariant design method), the bilinear Z-transform and the 52 Design of Recursive Digital Filters 53 matched Z-transform.

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