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Quantum Electronics

Measurement of signals and instrumental functions

V. A. Zubov

Published Apr 30, 1996• 9 citations
Publisher

Abstract

The solution is obtained of the problem of simultaneous determination of the characteristics of an amplitude—phase signal and of the instrumental function, which acts on the signal amplitude and phase. This avoids the need for separate determination of the instrumental function. The solution follows from an analysis of measurements of the intensities in the spectral representations of signals acted upon by the instrumental function and subjected to additional modulation created in a specified manner.

DOI
10.1070/qe1996v026n04abeh000673

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The solution is obtained of the problem of simultaneous determination of the characteristics of an amplitude—phase signal and of the instrumental function, which acts on the signal amplitude and phase. This avoids the need for separate determination of the instrumental function. The solution follows from an analysis of measurements of the intensities in the spectral representations of signals acted upon by the instrumental function and subjected to additional modulation created in a specified manner.
The possibilities of obtaining information on the time dependence of the amplitude and phase of an optical signal by the spectral modulation method are analysed. Three variants of the signal analysis are considered: a direct analysis of the signal whose parameters are changing in time; studies of the time-dependent optical characteristics of an object or a medium under study by probing them with radiation of the known structure; and measurements of the parameters of radiation propagated through the object and simultaneous determination and analysis of time-dependent structures of the signal and transfer function describing the effect of the transfer medium, the object, or an optical system on the propagated signal.
In this paper we motivate and develop the analytic theory of measurement, in which autonomously specified algebras of quantities (together with the resources of mathematical analysis) are used as a unified mathematical framework for modeling (a) the time-dependent behavior of natural systems, (b) interactions between natural systems and measuring instruments, (c) error and uncertainty in measurement, and (d) the formal propositional language for describing and reasoning about measurement results. We also discuss how a celebrated theorem in analysis, known as Gelfand representation, guarantees that autonomously specified algebras of quantities can be interpreted as algebras of observables on a suitable state space. Such an interpretation is then used to support (i) a realist conception of quantities as objective characteristics of natural systems, and (ii) a realist conception of measurement results (evaluations of quantities) as determined by and descriptive of the states of a target natural system. As a way of motivating the analytic approach to measurement, we begin with a discussion of some serious philosophical and theoretical problems facing the well-known representational theory of measurement. We then explain why we consider the analytic approach, which avoids all these problems, to be far more attractive on both philosophical and theoretical grounds.
An electronic modulation can retain a substantial local amplitude while its translational coherence is limited by spatial disorder. Its coupling to superconductivity therefore cannot be characterized by a single modulation intensity. We develop a phenomenological formulation of a finite-wave-vector complex electronic envelope coupled to a charged superconducting field. Translation and gauge symmetries constrain the interaction terms, while amplitude, modulation-gradient and condensate-gradient couplings act on distinct responses. The resulting variational equations give uniform coexistence conditions, local stability criteria and phase-gradient coefficients. We define the helicity modulus by a boundary twist, distinguish phase-only from amplitude-weighted correlations, and give an explicit minimal spectral Hamiltonian illustrating how the electronic texture may enter momentum-resolved measurements. The proposed tests use distinct spatial, spectral and stiffness observables; none assumes that a tunnelling gap directly measures condensate amplitude. An explicit kernel expansion yields the leading displacement of the selected wave vector, while the thermodynamic twist response determines phase rigidity independently of charge intensity. The analytical relations and selected numerical responses distinguish amplitude, spatial organization and condensate rigidity within the specified coupled-field model.