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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 11 53 Copyright © Orchard Publications Digital Filters Figure 11.38. Non recursive digital filter realization Figure 11.39. Components of recursive and non recursive digital filter realization 3. The Bilinear Transformation which uses the transformation * (11.80) to transform the left half of the plane into the interior of the unit circle in the plane. We will discuss only the bilinear transformation. We recall from (9.67) of Chapter 9, Page 9 22, that (11.81) But the relation is a multi valued transformation and, as such, cannot be used to derive a rational form in . It can be approximated as (11.82) Substitution (11.82) into (11.81) yields * is the sampling period, that is, the reciprocal of the sampling frequency + a 3 a 2 a 0 a 1 Z 1 xn [] yn Z 1 Z 1 + A Constant Multiplier Unit Delay Adder/Subtractor ± Z 1 vn v ± n = xn 1 = Ax n = s 2 T s ----- z1 + ----------- = T s f s sz Fz () Gz Gs s 1 T S z ln = == s 1 T S z ln = z s 1 T S z ln 2 T --- + 1 3 -- +   3 1 5 + 5 1 7 + 7 ++++ 2 T S +

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Chapter 11 Analog and Digital Filters 11 54 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications (11.83) and with the substitution , we obtain (11.84) Since the transformation maps the unit circle into the axis on the plane, the quan- tity and must be equal to some point on the axis, that is, or or (11.85) We see that the analog frequency to digital frequency transformation results in a non linear map- ping; this condition is known as warping . For instance, the frequency range in the ana- log frequency is warped into the frequency range in the digital frequency. To express in terms of , we rewrite (11.85) as Then, and for small , Therefore, (11.86) that is, for small frequencies, (11.87) Gz () Gs s 2 T S ------- z 1 z 1 + ------------- = = ze j ω d T S = Ge j ω d T S G 2 T S ----- e j ω d T S 1 e j ω d T S 1 + ------------------------    = zs j ω s 2 T s e j ω d 1 e j ω d 1 + ------------------- j ωω ω a = j ω j ω a 2 T S e j ω d T S 1 e j ω d T S 1 + = ω a 1 j -- 2 T s e j ω d T S 1 e j ω d T S 1 + ⋅⋅ 2 T S 1 j2 12 ---------------- e j ω d T S 2 e j ω d T S 2 e j ω d T S 2 e j ω d T S 2 + ---------------------------------------------- 2 T S ω d T S 2 sin ω d T S 2 cos --------------------------------- == = ω a 2 T S ω d T S 2 ------------ tan = 0 ω a ≤≤ 0 ω d π T s ω d ω a ω d T S 2 tan ω a T S 2 = ω d T S 2tan 1 ω a T S 2 = ω a T s 2 tan 1 ω a T S 2 ω a T S 2 ω d T S 2 ω a T S 2 ω a T S ≈≈ ω d ω a
Signals and Systems with MATLAB Computing and Simulink

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