K mitra d d d 0 if the above constraint is not

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Unformatted text preview: (e jω ) − X im (e jω ) re dω Copyright © 2005, S. K. Mitra 5 The Unwrapped Phase Function The Unwrapped Phase Function • The phase function can thus be defined unequivocally by its derivative dθ(ω) / dω: ω θ(ω) = ∫[ 0 dθ( η) dη • The phase function defined by ω θ(ω) = ∫[ dη, 0 dθ( η) dη dη is called the unwrapped phase function of X (e jω ) and it is a continuous function of ω • ⇒ ln X (e jω ) exists with the constraint θ(0) = 0 31 32 Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra The Unwrapped Phase Function The Frequency Response • Moreover, the phase function will be an odd function of ω if 1 π 2π ∫[ 0 dθ( η) dη dη = 0 • If the above constraint is not satisfied, then the computed phase function will exhibit absolute jumps greater than π 33 34 • Most discrete-time signals encountered in practice can be represented as a linear combination of a very large, maybe infinite, number of sinusoidal discrete-time signals of different angular frequencies • Thus, knowing the response of the LTI system to a single sinusoidal signal, we can determine its response to more complicated signals by making use of the superposition property Copyright © 2005, S. K. Mitra Copyright © 2005, S. K. Mitra The Frequency Response...
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