There are a number of ways to define the phase of real snusold with unknown amplitude frequency. and initial phase. One way, as already discussed, is to define it as the fractional part of the period that has been completed. This is a valld and intuitively pleasing and one that can be read' generalized to periodic signals that contain not only a sinusoid, but also a number of harmonucs. however be elegantly generalized to allow for slow variations in the frequency of the a scenario occurs in communicatons with phase and frequency modulation Gabor put forward a definition in 1946 that can be used for signais with slowly varying frequency. He proposed a mathematlcal definition for generaung the complex phasor, 20, associated with the real signal, sto. The so-called analytic signal zdo is defined according to the following definition(41.3) where denotes the Hilbert transform and is given by(41.4) with signifying the Cauchy principal value or the integral 12 The imaginary part the analytic signal can be generated pracucally by passing the original signa through a Hilbert transform filter From 41.3 that this filter has impul response given by 1/rL The filter can b implemented, for example, with one the HSP43xx series of ICs Harris Semiconductors. Details of how determine the filter coellicients can be found in izl Having formally defined the analytic signal, it is posible to provide definitions for plas frequency and amplitude as functions of time. They are given below
There are a number of ways to define the phase of real snusold with unknown amplitude frequency. and initial phase. One way, as already discussed, is to define it as the fractional part of the period that has been completed. This is a valld and intuitively pleasing and one that can be read' generalized to periodic signals that contain not only a sinusoid, but also a number of harmonucs. however be elegantly generalized to allow for slow variations in the frequency of the a scenario occurs in communicatons with phase and frequency modulation Gabor put forward a definition in 1946 that can be used for signais with slowly varying frequency. He proposed a mathematlcal definition for generaung the complex phasor, 20, associated with the real signal, sto. The so-called analytic signal zdo is defined according to the following definition(41.3) where denotes the Hilbert transform and is given by(41.4) with signifying the Cauchy principal value or the integral 12 The imaginary part the analytic signal can be generated pracucally by passing the original signa through a Hilbert transform filter From 41.3 that this filter has impul response given by 1/rL The filter can b implemented, for example, with one the HSP43xx series of ICs Harris Semiconductors. Details of how determine the filter coellicients can be found in izl Having formally defined the analytic signal, it is posible to provide definitions for plas frequency and amplitude as functions of time. They are given below
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