Nonlinear relationship of frequencies in bilinear transformation
In the bilinear transformation the digital and analog frequencies are non-linearly related. Setting
First we sketch Ω (the analog frequency) as a method of ω (the digital frequency) as provided by Ω = 2/T tan(ω/2) to describe qualitatively the distortion of the frequency scale that happens
Because of the nonlinear behavior of the relationship.
Equally spaced pass bands are rolled back together or warped on the higher frequency end of the digital frequency measurement scale. This effect is usually compensated for by pre-warping the analog filter before giving bilinear transformation.
Because of taking the relationship between Ω1 and Ω2 on the one hand and ω1 and ω2 on the other is not linear. The digital and analog frequencies ω1 and ω2 are pushed in tends the origin (ω = 0). In that method Ω = ¥ is transformed to ω =Π.
If the bilinear transformation is given to the process Ha(s) with critical frequency Ωc, the digital filter will occupy a critical frequency ωc = 2 tan-1 (Ω T/2). If the output H(z) is used in an A/D-H(z)-D/A structure, the similar critical frequency (of the equivalent analog filter) is calculated by replacing ωc with ΩceqT:
If (ΩcT/2) is so similar that tan-1 (ΩT/2) » ΩcT/2, then they have
If this condition is not fulfilled, then the warping of the final frequency (in the bilinear design) is compensated for by pre-warping.
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