Amplitudes of mono-component signals and the generalized sampling functions
Introduction
Any real-valued non-stationary signal f of finite energy, that is, f is in the space of square integrable functions on the set of real numbers, may be represented as an amplitude-phase modulation with a time-varying amplitude and a time-varying phase where phase is, in general, nonlinear [14]. Specifically, the value of f at may be represented asUnfortunately, this type of representation is not unique because the modulation is obtained through a complex signal that can have various choices of the imaginary part. However, one can determine a unique such factorization (1.1) by using the approach of analytic signals [10]. Indeed, let be the analytic signal associated with f with the characteristic propertywhere for any signal , is the Fourier transform of g defined at by the equationEq. (1.2) is equivalent to for , where the operator stands for the Hilbert transform, and for , at is defined through the principal value integral The value at is complex which can be written into the quadrature form
Under the conditions that the derivative value is non-negative for all , the quantities and are called the instantaneous amplitude and instantaneous frequency at , of the real signal f, respectively. The corresponding modulation (1.1) is then called the canonical amplitude-phase modulation, or canonical modulation for short [8]. The signal f with such defined non-negative instantaneous frequencies is thus called a mono-component. A large body of literature addresses this problem, see for example, [1], [2], [16], [17], [9], [21].
The notions of instantaneous amplitude and frequency are fundamental in many applications involving modulations of signals that appear especially in communications or information processing. Thus constructing the canonical pair of the instantaneous amplitude and phase is important in the theories of analytic signals and in order to facilitate the modulation and demodulation techniques such as processing speech signals [15] or signals in electrical and radio engineering [11]. It is equivalent to the problem of seeking the function pair such that for , the following equation holds true:We remark that (1.4) can be apparently considered as a special case of the Bedrosian identity In [1], the author proved that, if both belong to , f is of lower frequency, g is of higher frequency and have no overlapping frequency, then . This classic result of Bedrosian is not useful for constructing a mono-component. The reason lies in that the requirement of both f and g in is invalid.
Recently, an important phase function that renders mono-components was given in [18]. The phase function is defined through the boundary values of a Blaschke product on a unit disk , where indicates the set of complex numbers. Specifically, for , the Blaschke product at is given bySubsequently the non-linear phase function, denoted by , is defined at by the equationIf we recall that the periodic Poisson kernel pa whose value at is given bythen by taking the derivative of both sides of Eq. (1.6), we find that the phase is an anti-derivative of pa, and its derivative is always positive, that is,
We shall in this paper characterize the amplitude function when the phase function is chosen at by such that Eq. (1.4) is satisfied. Our main result indicates that such kind of amplitude can be perfectly reconstructed in terms of a sampling formula using the generalized sampling function whose value at is given by
In Section 2, we review the construction of the generalized sampling function and discuss some properties pertaining to it. In Section 3, we introduce the concept of Bedrosian subspace of the Hilbert transform and investigate some properties of functions in this space. In Section 4, we make an important observation when a linear phase is chosen, the amplitude function must be bandlimited in order to satisfy equation (1.4). In Section 5, we present our main result in Theorem 5.7.
Section snippets
Generalized sampling functions
Not very surprisingly the function has many properties that are similar to the classic sinc defined at by the equation Those properties include cardinality, orthogonality, decaying rate, among others. In the special case a=0, the function reduces to the classic , which will become clear later. Let us first review the approach to obtain an explicit form of .
The classic sinc function is fundamentally significant in digital signal processing due to the
Bedrosian subspace of the Hilbert transform
In this section, we pay attention to the setIt is clear that the set is a subspace of due to the linearity of the Hilbert transform. The subspace shall be called the Bedrosian subspace of the Hilbert transform. We first make a simple observation that we shall need frequently later. Lemma 3.1 Eq. (1.4) is true if and only if for , Proof Applying the Hilbert transform to both sides of Eq. (1.4) and utilizing Eq. (2.10) yields
An observation – how a linear phase determines the amplitude
In this section, we specifically consider the case when the phase in Eq. (1.4) is a linear phase and investigate the representation of the corresponding amplitude. The result is given in the following theorem. We use the notation for the set of real numbers on which the values f(x) at are nonzero. Theorem 4.1 Suppose that is a positive real number and is a non-zero real signal in . Then the following equation:holds if and only if is bandlimited with
How does non-linear phase determine amplitude?
In this section, we shall completely characterize a real-valued function . We begin with introducing two one-sided filters that are related to the two-sided symmetric cascade filter Ha:where . The next lemma provides the Fourier transform pairs of and , respectively. Lemma 5.1 The Fourier transform of and are given, respectively, byand Proof We start with the observation
Conclusions
In signal analysis, researchers have been trying to understand, for a given signal, what are its instantaneous amplitude, instantaneous phase, and instantaneous frequency. The rigorous mathematical definitions for those notions, however, have not been well agreed by signal analysts. The divergent understandings, as a matter of fact, have created many controversies. We give in this paper rigorous mathematical characterizations of an instantaneous amplitude when the instantaneous frequency is
References (21)
- et al.
A sampling theorem for non-bandlimited signals using generalized sinc functions
Computational and Applied Mathematics
(2008) On nontrivial analytic signals with positive instantaneous frequency
Signal Processing
(2003)On the uniqueness of the definition of the amplitude and phase of the analytic signal
Signal Processing
(2003)- et al.
Instantaneous envelope and phase extraction from real signaltheory, implementation, and an application to EEG analysis
Signal Processing
(1980) - et al.
Analytic unit quadrature signals with nonlinear phase
Physica DNonlinear Phenomena
(2005) A product theorem for Hilbert transform
Proceedings of IEEE
(1963)Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals
Proceedings of IEEE
(1992)- et al.
The Shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines
SIAM Journal of Applied Mathematics
(1986) - et al.
A necessary and sufficient condition for a Bedrosian identity
Mathematical Methods in Applied Sciences
(2010) - et al.
Functions with spline spectra and their applications
International Journal of Wavelets, Multiresolution and Information Processing
(2010)
Cited by (11)
Hypersampling of pseudo-periodic signals by analytic phase projection
2018, Computers in Biology and MedicineCitation Excerpt :Monocomponent signals have a monotonically increasing phase. In other words, in a monocomponent signal the instantaneous frequency or time derivative of its phase is non-negative at any time [6]. This phase monotonicity is needed later on in the phase projection step, which requires the phase to be a piecewise invertible function.
Multi-scale analysis of influencing factors for soybean futures price risk: Adaptive Fourier decomposition mathematical model applied for the case of China
2021, International Journal of Wavelets, Multiresolution and Information ProcessingReconstruction of spline spectra-signals from generalized sinc function by finitely many samples
2021, Banach Journal of Mathematical AnalysisA new method to reduce electromagnetic interference using signal modeling
2020, Archives of Electrical EngineeringDesign and implementation of the subsystem subject to emission of multicomponent high-frequency signals to ensure its reliability
2019, International Journal of Electrical and Computer Engineering
- 1
The author is partially supported by the Natural Science Foundation of Guangdong Province under Grant no. S2011010004986.
- 2
The author is partially supported by the National Natural Science Foundation of China under Grant no. 11071058.