Digital Signal Processing Notes eBook

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onto the unit circle in the z-plane. The price paid for this, however, is the introduction of a distortion in the frequency axis. 12
Figure 4.6 Mapping of the s-plane into the z-plane using the bilinear transformation
The Matched-Z Transform
Another method for converting an analog lter into an equivalent digital lter is to map the poles and zeros of Ha s directly into poles and zeros in the z-plane. For analog lter Ha s
M k 1 s N k 1 s
sk pk
esk T z 1 epk T z 1
the corresponding digital lter is H z
M k 1 1 N k 1 1
where T is the sampling interval. Thus each factor of the form s-a in Ha s is mapped into the factor 1 eaT z 1 .
Characteristics Of Commonly Used Analog Filters
From the previous discussion it is clear that, IIR digital lters can be obtained by beginning with an analog lter. Thus the design of a digital lter is reduced to designing an 13
appropriate analog lter and then performing the conversion from Ha s to H z . Analog lter design is a well - developed eld, many approximation techniques, viz., Butterworth, Chebyshev, Elliptic, etc., have been developed for the design of analog low pass lters. Our discussion is limited to low pass lters, since, frequency transformation