K mitra 30 j 1 d x re e j dx im e 2 d d x e

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Unformatted text preview: If ln X (e jω ) exits, then its derivative with respect to ω also exists and is given by 1 ⎡ dX (e jω ) ⎤ d ln X (e jω ) = ⎢ ⎥ dω X (e jω ) ⎣ dω ⎦ ⎡ dX re (e jω ) dX (e jω ) ⎤ + j im ⎢ ⎥ dω dω X ( e jω ) ⎣ ⎦ 1 = )} 27 0.8 The Unwrapped Phase Function • The conditions under which the phase function will be a continuous function of ω is next derived • Now ln X (e jω ) = X (e jω ) + jθ(ω) jω 0.6 Copyright © 2005, S. K. Mitra The Unwrapped Phase Function where 0.4 ω/π 26 Copyright © 2005, S. K. Mitra 28 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra The Unwrapped Phase Function The Unwrapped Phase Function • From ln X (e jω ) = X (e jω ) + jθ(ω) , d ln X (e jω ) / dω is also given by • Thus, dθ(ω) / dω is given by the imaginary part of jω d ln X (e jω ) d X (e ) d θ(ω) = +j dω dω dω 29 1 X (e ⎡ dX re (e jω ) dX (e jω ) ⎤ + j im ⎥ ⎢ dω dω )⎣ ⎦ • Hence, 30 Copyright © 2005, S. K. Mitra jω jω 1 dθ(ω) [ X re (e jω ) dX im (e ) = 2 dω dω X ( e jω ) dX...
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This document was uploaded on 11/09/2013.

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