In mathematics and processing, the concept of an analytical signal (for brevity, C, AC) is a complex function that does not have negative frequency components. The real and imaginary parts of this phenomenon are real functions related to each other by the Hilbert transform. An analytical signal is a fairly common phenomenon in chemistry, the essence of which is similar to the mathematical definition of this concept.
Representation
The analytical representation of a real function is an analytical signal containing 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 function are redundant due to the Hermitian symmetry of such a spectrum. These negative frequency components can be dropped without loss of information, provided that instead you want to deal with a complex function. This makes certain function attributes more accessible and makes it easier to derive modulation and demodulation methods, such as a single-band band.
Negative components
As long as the manipulated function does not have negative frequency components (that is, it is still analytical), the conversion from complex to real is just a matter of discarding the imaginary part. The analytical representation is a generalization of the concept of the vector: while the vector is limited by the amplitude, phase and frequency unchanged in time, a qualitative analysis of the analytical signal allows parameters that are variable in time.
The instantaneous amplitude, instantaneous phase and frequency in some applications are used to measure and detect local features of C. Another application of the analytical representation relates to the demodulation of modulated signals. Polar coordinates conveniently separate the effects of amplitude modulation and phase (or frequency) modulation and effectively demodulate certain types.
Then a simple low-pass filter with real coefficients can cut off the part of interest. Another motive is to reduce the maximum frequency, which reduces the minimum frequency for sampling without aliases. The frequency shift does not undermine the mathematical suitability of the representation. Thus, in this sense, the down-converted is still analytic. However, restoring a material representation is no longer a simple matter of simply extracting the real component. Upconversion may be required, and if the signal is sampled (discrete time), interpolation (upsampling) may also be required to avoid overlap.
Variables
The concept is clearly defined for the phenomena of one variable, which is usually temporary. This temporality confuses many beginning mathematicians. For two or more variables, analytical C can be defined differently, and two approaches are presented below.
The real and imaginary parts of this phenomenon correspond to two elements of a vector-valued monogenic signal, as defined for similar phenomena with one variable. Nevertheless, monogenic can be expanded to an arbitrary number of variables in a simple way, creating an (n + 1) -dimensional vector function for the case of n-variable signals.
Signal conversion
You can convert a real signal into an analytical one by adding an imaginary (Q) component, which is the Hilbert transform of the real component.
By the way, this is not new for its digital processing. One of the traditional methods of generating AM with one sideband (SSB) - the phasing method - involves creating signals by generating a Hilbert transform of an audio signal in an analog resistor-capacitor network. Since it has only positive frequencies, it is easy to convert it to a modulated RF signal with only one sideband.
Definition formulas
The analytical expression of the signal is a holomorphic complex function defined on the boundary of the upper complex half-plane. The boundary of the upper half-plane coincides with the randomness, therefore, C is defined by the mapping fa: R → C. Since the middle of the last century, when in 1946 Denis Gabor proposed using this phenomenon to study constant amplitude and phase, the signal has found many applications. The peculiarity of this phenomenon was emphasized [Vak96], where it was shown that only a qualitative analysis of the analytical signal corresponds to the physical conditions for amplitude, phase, and frequency.
Recent Achievements
Over the past few decades, interest has arisen in the study of signals in many dimensions, motivated by problems that arise in fields ranging from image / video processing to multidimensional oscillatory processes in physics, such as seismic, electromagnetic and gravitational waves. It was generally accepted that for the correct generalization of analytical C (qualitative analysis) for the case of several measurements, one should rely on an algebraic construction that extends the usual complex numbers in a convenient way. Such constructions are usually called hypercomplex numbers [SKE].
Finally, it should be possible to construct a hypercomplex analytical signal fh: Rd → S, where some general hypercomplex algebraic system is presented, which naturally expands all the required properties to obtain instantaneous amplitude and phase.
The study
A number of works are devoted to various issues related to the correct choice of the hypercomplex number system, the definition of the hypercomplex Fourier transform and fractional Hilbert transforms to study the instantaneous amplitude and phase. Mostly these works were based on the properties of various spaces, such as Cd, quaternions, Cliron algebras, and Cayley-Dickson constructions.
Further, we list only a few of the works devoted to the study of the signal in many dimensions. As far as we know, the first works on the multidimensional method were obtained in the early 1990s. These include the work of Ell [Ell92] on hypercomplex transformations; Bülow's work on the generalization of the analytical reaction method (analytical signal) to many measurements [BS01] and the work of Felsberg and Sommer on monogenic signals.
Further perspectives
It is expected that the hypercomplex signal will expand all the useful properties that we have in the one-dimensional case. First of all, we must be able to extract and generalize the instantaneous amplitude and phase to the measurements. Secondly, the Fourier spectrum of a complex analytical signal is supported only at positive frequencies, so we expect that the hypercomplex Fourier transform will have its own hypervalued spectrum, which will be supported only in some positive quadrant of the hypercomplex space. Therefore, it is very important.
Thirdly, the conjugate parts of the complex concept of the analytic signal are related to the Hilbert transform, and we can expect that the conjugate components in the hypercomplex space should also be connected by some combination of the Hilbert transforms. And, finally, indeed, a hypercomplex signal should be defined as a continuation of some hypercomplex holomorphic function of several hypercomplex variables defined on the boundary of some form in the hypercomplex space.
We solve these problems in a consistent manner. First of all, we start by considering the Fourier integral formula and show that the Hilbert transform in 1-D is related to the modified Fourier integral formula. This fact allows us to determine the instantaneous amplitude, phase, and frequency without any reference to hypercomplex number systems and holomorphic functions.
Modification of integrals
We continue by generalizing the modified formula of the Fourier integral to several dimensions and determine all the necessary phase-shifted components that we can assemble into instantaneous amplitude and phase. Secondly, we turn to the question of the existence of holomorphic functions of several hypercomplex variables. After [Sch93], it turns out that the commutative and associative algebra of a hypercomplex generated by a set of elliptic (e2i = −1) generators is a suitable space so that a hypercomplex analytic signal can live, we call such a hypercomplex algebra the Sheffers space and denote it Sd.
Therefore, the hypercomplex of analytical signals is defined as a holomorphic function at the polydisk / upper half-plane boundary in some hypercomplex space, which we call the general Schäfers space, and denote it by Sd. Then we observe the validity of the Cauchy integral formula for functions Sd → Sd, which are calculated from the hypersurface inside the polydisk in Sd and derive the corresponding fractional Hilbert transforms that connect the hypercomplex conjugate components. Finally, it turns out that the Fourier transform with values in the Schäfer space is supported only at non-negative frequencies. Thanks to this article, you learned what is an analytical signal.