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The magic of the Hilbert transform: how does it relate to analytic signals?
In mathematics and signal processing, an analytic signal is a complex-valued function with no negative frequency components. The real and imaginary parts of the signal are real-valued functions related to each other and are converted into each other through the Hilbert transform. This relationship brings higher flexibility, which becomes particularly important in signal analysis and processing.
The core of the Hilbert transform is to transform the signal into its analytical form, which further facilitates the convenience of many mathematical operations.
For a real-valued function s(t), its Fourier transform S(f) is usually symmetric, that is, it has Hermitian symmetry. The frequency components that occur at f = 0 are uniquely real-valued, while negative frequencies are redundant, allowing us to discard this part of the data without losing information. Based on this, analytic signals provide an efficient processing solution because negative frequency components are not required during the conversion process.
This also explains why parsing the signal is so important for techniques such as single sideband modulation, as it simplifies the mathematical derivation of these techniques.
Furthermore, analytic signals ensure that the treatment of their complex-valued functions is reversible, as allowed by Hermitian symmetries. When it is necessary to convert an analytic signal back to a real-valued signal, the imaginary part is simply discarded. In such an operation, the changes in parameters are dynamic, and compared with the traditional phasor concept, the analytical signal has better flexibility.
The definition of analytic signal is simple and clear: it is composed of a real-valued signal and its Hilbert transform. This characteristic makes analytical signals have numerous applications in the field of signal processing, such as sound processing, communications, and image analysis. Its practical value cannot be underestimated.
In cybernetics and engineering, analytic signals are surprisingly widely used to reduce computational burden and improve computational efficiency.
On the other hand, negative frequency components also play a key role in reconstructing the original signal. Although we discard them in the expression of the analytic signal, we can easily recover these negative frequency components when needed by taking advantage of its Hermitian symmetry. This feature makes signal recovery more flexible.
With the continuous advancement of signal processing technology, the application of Hilbert transform and its analytical signals is not limited to theoretical research, but is also becoming more and more common in practical engineering. As the demand for data analysis and processing continues to expand, this technology is expected to usher in greater development. Does this mean that there will be new breakthroughs in this field in the future?
