Elsevier

Signal Processing

Volume 94, January 2014, Pages 255-263
Signal Processing

Amplitudes of mono-component signals and the generalized sampling functions

https://doi.org/10.1016/j.sigpro.2013.06.034Get rights and content

Author-Highlights

  • The amplitude of a mono-component is determined by its phase.

  • The amplitude is represented by the so-called generalized sampling functions that are related to the phase.

  • The amplitude and its Hilbert transform must be continuous.

  • A new characterization of the band-limited functions in terms of its phase is given.

Abstract

There is a recent trend to use mono-components to represent nonlinear and non-stationary signals rather than the usual Fourier basis with linear phase, such as the intrinsic mode functions used in Norden Huang's empirical mode decomposition [12]. A mono-component is a real-valued signal of finite energy that has non-negative instantaneous frequencies, which may be defined as the derivative of the phase function of the given real-valued signal through the approach of canonical amplitude-phase modulation. We study in this paper how the amplitude is determined by its phase for a class of signals, of which the instantaneous frequency is periodic and described by the Poisson kernel. Our finding is that such an amplitude can be perfectly represented by a sampling formula using the so-called generalized sampling functions that are related to the phase. The regularity of such an amplitude is identified to be at least continuous. Such characterization of mono-components provides the theory to adaptively decompose non-stationary signals. Meanwhile, we also make a very interesting and new characterization of the band-limited functions.

Introduction

Any real-valued non-stationary signal f of finite energy, that is, f is in the space L2(R) of square integrable functions on the set R of real numbers, may be represented as an amplitude-phase modulation with a time-varying amplitude ρ and a time-varying phase ϕ where phase ϕ is, in general, nonlinear [14]. Specifically, the value of f at tR may be represented asf(t)=ρ(t)cosϕ(t).Unfortunately, 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 A(f) be the analytic signal associated with f with the characteristic property(A(f))(ω)={2f^(ω)ifω00ifω<0,where for any signal gL2(R), g^=Fg is the Fourier transform of g defined at ξR by the equationg^(ξ)=(Fg)(ξ)12πRg(t)eiξtdt.Eq. (1.2) is equivalent to for tR, A(f)(t)=f(t)+iHf(t),where the operator H:L2(R)L2(R) stands for the Hilbert transform, and for fL2(R), Hf at tR is defined through the principal value integral Hf(t)p.v.1πRf(x)txdx=limϵ0|xt|>ϵf(x)txdx.The value A(f)(t) at tR is complex which can be written into the quadrature form A(f)(t)=ρ(t)eiϕ(t).

Under the conditions that the derivative value ϕ(t) is non-negative for all tR, the quantities ρ(t) and ϕ(t) are called the instantaneous amplitude and instantaneous frequency at tR, 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 (ρ,ϕ) 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 (ρ,ϕ) such that for tR, the following equation holds true:H(ρ(·)cosϕ(·))(t)=ρ(t)sinϕ(t).We remark that (1.4) can be apparently considered as a special case of the Bedrosian identity H(fg)=fH(g).In [1], the author proved that, if both f,g belong to L2(R), f is of lower frequency, g is of higher frequency and f,g have no overlapping frequency, then H(fg)=fH(g). 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 L2(R) 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 Δ{z:zC,|z|1}, where C indicates the set of complex numbers. Specifically, for a(1,1), the Blaschke product at zC⧹{1/a} is given byBa(z)=za1az.Subsequently the non-linear phase function, denoted by θa, is defined at tR by the equationeiθa(t)Ba(eit).If we recall that the periodic Poisson kernel pa whose value at tR is given bypa(t)1a212acost+a2,then by taking the derivative of both sides of Eq. (1.6), we find that the phase θa is an anti-derivative of pa, and its derivative is always positive, that is, ddtθa(t)=pa(t)>0.

We shall in this paper characterize the amplitude function ρ when the phase function ϕ is chosen at tR by ϕ(t)=θa(t)=[0,t]pa(x)dxsuch 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 tR is given bysinca(t)sinθa(t)t.

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 sinca has many properties that are similar to the classic sinc defined at tR by the equation sinc(t)sintt.Those properties include cardinality, orthogonality, decaying rate, among others. In the special case a=0, the function sinca reduces to the classic sinc, which will become clear later. Let us first review the approach to obtain an explicit form of sinca.

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 setSa{f:fL2(R),H(fcosθa(·))=fsinθa(·)}.It is clear that the set Sa is a subspace of L2(R) due to the linearity of the Hilbert transform. The subspace Sa 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 tR, H(ρ(·)eiϕ·)(t)=iρ(t)eiϕt.

Proof

Applying the Hilbert transform H to both sides of Eq. (1.4) and utilizing Eq. (2.10) yieldsH(ρ(·)

An observation – how a linear phase determines the amplitude

In this section, we specifically consider the case when the phase ϕ 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 supp(f) for the set of real numbers on which the values f(x) at xR are nonzero.

Theorem 4.1

Suppose that γ is a positive real number and ρ is a non-zero real signal in L2(R). Then the following equation:H(ρ(·)cos(γ·))(t)=ρ(t)sin(γt)holds if and only if ρ is bandlimited with supp(

How does non-linear phase determine amplitude?

In this section, we shall completely characterize a real-valued function ρSa. We begin with introducing two one-sided filters that are related to the two-sided symmetric cascade filter Ha:Ha+(t)Ha(t)χR+(t),Ha(t)Ha(t)χR(t),where tR. The next lemma provides the Fourier transform pairs of Ha+ and Ha, respectively.

Lemma 5.1

The Fourier transform of Ha+ and Ha are given, respectively, by(Ha+(·))(ξ)=12π11aeiξ1eiξiξ,and (Ha(·))(ξ)=12π11aeiξ1eiξiξ.

Proof

We start with the observation Ha+(t)=kZ+akχ[k

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

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1

The author is partially supported by the Natural Science Foundation of Guangdong Province under Grant no. S2011010004986.

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The author is partially supported by the National Natural Science Foundation of China under Grant no. 11071058.

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