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Digital Filters Design for Signal and Image Processing by Mohamed Najim

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6.2. Synthesizing IIR filters

6.2.1. Impulse invariance method for analog to digital filter conversion

In the I970s, the development of digital processing of filters led to implementing previously established methods for synthesizing continuous filters. In this section, we will discuss the impulse invariance method for analog to digital data conversion. Then we will present the bilinear transformation, which enables us to use all available techniques for synthesizing continuous filters.

The impulse invariance method for analog to digital filter conversion is based on the fact that the impulse response of a digital filter must correspond to the sampling of the impulse response of a continuous filter.

The method is described below.

– we write the transfer function of the continuous filter in the form of a development in basic elements:

images

– from this we deduce that

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The sampling frequency Ts being fixed, we obtain:

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which produces, by using the z-transform:

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We see that the poles of H(z) are in the form exp (piTs). If the original continuous-time filter is stable, the poles pi are partly real ...

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