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The real and imaginary parts of an analytic signal are real-valued functions related to each other by the Hilbert transform . The analytic representation of a real-valued function is an analytic signal, comprising the original function and its Hilbert transform. This representation facilitates many mathematical manipulations.
How is the frequency of an analytic signal determined?
Analytic signals are often shifted in frequency (down-converted) toward 0 Hz, possibly creating [non-symmetrical] negative frequency components: is an arbitrary reference angular frequency. This function goes by various names, such as complex envelope and complex baseband. The complex envelope is not unique; it is determined by the choice of .
Why is the integration of analytic signals important?
Integration of analytic signals may be also of importance for signal processing. A complex function of a real variable t is an integral of the analytic signal if the real and the imaginary parts of the function are corresponding integrals forming a pair of the HT. Papia Ray,
Which is the analytic representation of a real valued function?
The analytic representation of a real-valued function is an analytic signal, comprising the original function and its Hilbert transform. This representation facilitates many mathematical manipulations.
Is the down converted signal still an analytic signal?
A frequency shift does not undermine the mathematical tractability of the complex signal representation. So in that sense, the down-converted signal is still analytic. However, restoring the real-valued representation is no longer a simple matter of just extracting the real component.
How is the spectrum of a discrete time signal obtained?
The spectrum of discrete-time signal can be obtained by Discrete-Time Fourier Transform (DTFT). At this point, we would like to construct a discrete-time analytic signal from the real-valued sampled signal . We wish the analytic signal is complex valued and should satisfy the following two desired properties
How is the analytic signal represented in time domain?
Since the spectrum of the analytic signal is one-sided, the analytic signal will be complex valued in the time domain, hence the analytic signal can be represented in terms of real and imaginary components as .