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Weighted Norms of Ambiguity Functions and Wigner Distributions

Peter Jung

Published Jan 1, 2006• 6 citations
OpenAlex

Abstract

Abstract — In this article new bounds on weighted p-norms of ambiguity functions and Wigner functions are derived. Such norms occur frequently in several areas of physics and engineering. In pulse optimization for Weyl–Heisenberg (WH) signaling in widesense stationary uncorrelated scattering (WSSUS) channels for example it is a key step to find the optimal waveforms for a given scattering statistics which is a problem also well known in radar and sonar waveform optimizations. The same situation arises in quantum information processing and optical communication when optimizing pure quantum states for communicating in WH quantum channels, i.e. optimizing the pure state channel fidelity. Due to the non-convex nature of this problem the optimum and the maximizers itself are general difficult find, numerically and analytically. Therefore upper bounds on the achievable performance are important which will be provided by this contribution. Based on a result due to E. Lieb [1], the main theorem states a new upper bound which is independent of the waveforms and becomes tight only for Gaussian weights and waveforms. A discussion of this particular important case, which tighten recent results on Gaussian quantum fidelity and coherent states, will be given. Another bound is presented for the case where scattering is determined only by some arbitrary region in phase space. I.

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Weighted Norms of Ambiguity Functions and Wigner Distributions

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In this article new bounds on weighted p-norms of ambiguity functions and Wigner functions are derived. Such norms occur frequently in several areas of physics and engineering. In pulse optimization for Weyl-Heisenberg signaling in wide-sense stationary uncorrelated scattering channels for example it is a key step to find the optimal waveforms for a given scattering statistics which is a problem also well known in radar and sonar waveform optimizations. The same situation arises in quantum information processing and optical communication when optimizing pure quantum states for communicating in bosonic quantum channels, i.e. find optimal channel input states maximizing the pure state channel fidelity. Due to the non-convex nature of this problem the optimum and the maximizers itself are in general difficult find, numerically and analytically. Therefore upper bounds on the achievable performance are important which will be provided by this contribution. Based on a result due to E. Lieb, the main theorem status a new upper bound which is independent of the waveforms and becomes tight only for Gaussian weights and waveforms. A discussion of this particular important case, which tighten recent results on Gaussian quantum fidelity and coherent states, will be given. Another bound is presented for the case where scattering is determined only by some arbitrary region in phase space

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In this article new bounds on weighted p-norms of am- biguity functions and Wigner functions are derived. Such norms occur frequently in several areas of physics and engineering. In pulse optimization for Weyl-Heisenberg signaling in wide-sense stationary uncorrelated scattering channels for example it is a key step to find the optimal waveforms for a given scattering statistics which is a problem also well known in radar and sonar waveform optimizations. The same situation arises in quantum information processing and optical communication when optimizing pure quantum states for communicating in bosonic quantum channels, i.e. find optimal channel input states maximizing the pure state channel fidelity. Due to the non-convex nature of this problem the optimum and the maximizers itself are in general difficult find, numerically and analytically. Therefore upper bounds on the achievable performance are important which will be provided by this contribution. Based on a result due to E. Lieb (1), the main theorem states a new upper bound which is independent of the waveforms and becomes tight only for Gaussian weights and waveforms. A discussion of this particular important case, which tighten recent results on Gaussian quantum fidelity and coherent states, will be given. Another bound is presented for the case where scattering is determined only by some arbitrary region in phase space.

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An upper bound is proved for the Lp norm of Woodward’s ambiguity function in radar signal analysis and of the Wigner distribution in quantum mechanics when p>2. A lower bound is proved for 1≤p<2. In addition, a lower bound is proved for the entropy. These bounds set limits to the sharpness of the peaking of the ambiguity function or Wigner distribution. The bounds are best possible and equality is achieved in the Lp bounds if and only if the functions f and g that enter the definition are both Gaussians.

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IEEE Transactions on Wireless Communications•Aug 25, 2020•9 citations
Abstract — In this article new bounds on weighted p-norms of ambiguity functions and Wigner functions are derived. Such norms occur frequently in several areas of physics and engineering. In pulse optimization for Weyl–Heisenberg (WH) signaling in widesense stationary uncorrelated scattering (WSSUS) channels for example it is a key step to find the optimal waveforms for a given scattering statistics which is a problem also well known in radar and sonar waveform optimizations. The same situation arises in quantum information processing and optical communication when optimizing pure quantum states for communicating in WH quantum channels, i.e. optimizing the pure state channel fidelity. Due to the non-convex nature of this problem the optimum and the maximizers itself are general difficult find, numerically and analytically. Therefore upper bounds on the achievable performance are important which will be provided by this contribution. Based on a result due to E. Lieb [1], the main theorem states a new upper bound which is independent of the waveforms and becomes tight only for Gaussian weights and waveforms. A discussion of this particular important case, which tighten recent results on Gaussian quantum fidelity and coherent states, will be given. Another bound is presented for the case where scattering is determined only by some arbitrary region in phase space. I.
In this article new bounds on weighted p-norms of ambiguity functions and Wigner functions are derived. Such norms occur frequently in several areas of physics and engineering. In pulse optimization for Weyl-Heisenberg signaling in wide-sense stationary uncorrelated scattering channels for example it is a key step to find the optimal waveforms for a given scattering statistics which is a problem also well known in radar and sonar waveform optimizations. The same situation arises in quantum information processing and optical communication when optimizing pure quantum states for communicating in bosonic quantum channels, i.e. find optimal channel input states maximizing the pure state channel fidelity. Due to the non-convex nature of this problem the optimum and the maximizers itself are in general difficult find, numerically and analytically. Therefore upper bounds on the achievable performance are important which will be provided by this contribution. Based on a result due to E. Lieb, the main theorem status a new upper bound which is independent of the waveforms and becomes tight only for Gaussian weights and waveforms. A discussion of this particular important case, which tighten recent results on Gaussian quantum fidelity and coherent states, will be given. Another bound is presented for the case where scattering is determined only by some arbitrary region in phase space
In this article new bounds on weighted p-norms of am- biguity functions and Wigner functions are derived. Such norms occur frequently in several areas of physics and engineering. In pulse optimization for Weyl-Heisenberg signaling in wide-sense stationary uncorrelated scattering channels for example it is a key step to find the optimal waveforms for a given scattering statistics which is a problem also well known in radar and sonar waveform optimizations. The same situation arises in quantum information processing and optical communication when optimizing pure quantum states for communicating in bosonic quantum channels, i.e. find optimal channel input states maximizing the pure state channel fidelity. Due to the non-convex nature of this problem the optimum and the maximizers itself are in general difficult find, numerically and analytically. Therefore upper bounds on the achievable performance are important which will be provided by this contribution. Based on a result due to E. Lieb (1), the main theorem states a new upper bound which is independent of the waveforms and becomes tight only for Gaussian weights and waveforms. A discussion of this particular important case, which tighten recent results on Gaussian quantum fidelity and coherent states, will be given. Another bound is presented for the case where scattering is determined only by some arbitrary region in phase space.
An upper bound is proved for the Lp norm of Woodward’s ambiguity function in radar signal analysis and of the Wigner distribution in quantum mechanics when p>2. A lower bound is proved for 1≤p<2. In addition, a lower bound is proved for the entropy. These bounds set limits to the sharpness of the peaking of the ambiguity function or Wigner distribution. The bounds are best possible and equality is achieved in the Lp bounds if and only if the functions f and g that enter the definition are both Gaussians.
The phase space characteristics of a quantum state are best captured by the Wigner distribution. This displays transparently the diagonality information of the density matrix. The complementary function offering transparently the off-diagonal elements is captured by a function called the S-function, or the ambiguity. In carrying the maximal information about the quantum coherences it represents the uncertainties or ambiguity of the diagonal information. Mathematically this is manifested in its role as the phase space moment generating function. Formally it complements the information in the Wigner function. These formal relations provide the starting point for the present investigations. As a measure of quantum uncertainties, ambiguity may be used to define a probability measure on the off-diagonality. The mathematical and physical consistency of this view is presented in this paper. For a pure state, we find the extraordinary result that such distributions are their own Fourier transforms. The physical interpretation of this distribution as a carrier of classical signal fuzziness suggests the introduction of heuristic approximations to the observational uncertainties. We illustrate the properties and interpretation of the ambiguity function by some specific examples. We find that for smooth, ‘Gaussian-like’ distributions, the heuristic considerations provide good approximations. On the other hand, representing quantum interferences, the ambiguity serves as the most positive probe for the ultimate quantum structures which have been called sub-Planckian. They are interesting because it has been argued that such structures are physically observable.
This article considers a two-way multiple-input multiple-output (MIMO) Rayleigh block fading wiretap channel with two full-duplex nodes (Alice and Bob) exchanging messages simultaneously and wiretapped by a $ {N}_{ {e}}$ -antennas eavesdropper (Eve). The channel state remains unchanged over a coherence interval T and is unknown to all terminals, including Alice, Bob, and Eve, at the beginning of the coherence interval. We first show the expression of the optimal signal waveform for Alice and Bob, which maximizes the sum secrecy rate, should be the products of independent random diagonal matrices and isotropically distributed unitary matrices. Then, conditioned on large T and small $ {N}_{ {e}}$ , the upper-bound for the secure degree of freedom (s.d.o.f.) of the two-way wiretap channel is derived, and a constant-norm-signaling (CNS) scheme is presented to achieve this theoretical bound. Finally, we formulate an optimization problem of the s.d.o.f. achieved by the CNS scheme for general cases with arbitrary T and $ {N}_{ {e}}$ , and solve this problem by exploring the monotonicity of the achievable s.d.o.f.. Our result shows that the optimal s.d.o.f. can be independent of coherence time T and becomes the product of the numbers of legitimate nodes's transmitter antennas (NLNTA) for large $ {N}_{ {e}}$ and small NLNTA.