The Analytic Representation of Band-Pass Signals |
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| Authors: | Enrique A González-Velasco Emilio Sanvicente |
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| Institution: | Department of Mathematics, University of Lowell, Lowell, MA 01854, USA;Polytechnic University of Barcelona, Barcelona, Spain |
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| Abstract: | The analytic representation of band-pass signals, which is extremely useful in information transmission and radar, is usually obtained by first introducing the Hilbert transform of a signal. We consider this procedure artificial, and propose here to follow the opposite path, which leads, in a natural way, to a motivated definition of the Hilbert transform. The key to this procedure is to show how, as a simple application of the Fourier integral, every band-pass signal can be expressed as the real part of a low frequency signal which modulates an exponential. Such a representation, called the analytic signal, is then used to define the Hilbert transform in a natural manner. Finally, we show that the term analytic signal is motivated by the fact that this signal is the restriction to the real axis of a function defined and analytic (holomorphic) in the upper half of the complex plane. |
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