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If s=σ+jΩ and z=re^jω and r=1, then which of the following inference is correct?(a) LHS of the s-plane is mapped inside the circle, |z|=1(b) RHS of the s-plane is mapped outside the circle, |z|=1(c) Imaginary axis in the s-plane is mapped to the circle, |z|=1(d) None of the mentionedThis question was addressed to me in final exam.My question is based upon Bilinear Transformations in division Digital Filters Design of Digital Signal Processing |
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Answer» RIGHT answer is (c) IMAGINARY axis in the s-plane is mapped to the circle, |z|=1 The explanation: We know that if =σ+jΩ and z=re^jω, then by substituting the VALUES in the below expression s = \(\frac{2}{T}[\frac{1-z^{-1}}{1+z^{-1}}]\) =>σ = \(\frac{2}{T}[\frac{r^2-1}{r^2+1+2rcosω}]\) When r=1 => σ = 0. This shows that the imaginary axis in the s-domain is mapped to the circle of unit radius centered at z=0 in the z-domain. |
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