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Concept
Analytic signal
Summary
In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components. 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. The basic idea is that the negative frequency components of the Fourier transform (or spectrum) of a real-valued function are superfluous, due to the Hermitian symmetry of such a spectrum. These negative frequency components can be discarded with no loss of information, provided one is willing to deal with a complex-valued function instead. That makes certain attributes of the function more accessible and facilitates the derivation of modulation and demodulation techniques, such as single-sideband.
As long as the manipulated function has no negative frequency components (that is, it is still analytic), the conversion from complex back to real is just a matter of discarding the imaginary part. The analytic representation is a generalization of the phasor concept: while the phasor is restricted to time-invariant amplitude, phase, and frequency, the analytic signal allows for time-variable parameters.
If is a real-valued function with Fourier transform , then the transform has Hermitian symmetry about the axis:
where is the complex conjugate of .
The function:
where
is the Heaviside step function,
is the sign function,
contains only the non-negative frequency components of . And the operation is reversible, due to the Hermitian symmetry of :
The analytic signal of is the inverse Fourier transform of :
where
is the Hilbert transform of ;
is the binary convolution operator;
is the imaginary unit.
Noting that this can also be expressed as a filtering operation that directly removes negative frequency components:
Since , restoring the negative frequency components is a simple matter of discarding which may seem counter-intuitive.
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