A review and tutorial of the fundamental ideas and methods of joint time-frequency distributions is presented. The objective of the field is to describe how the spectral content of a signal changes in time and to develop the physical and mathematical ideas needed to understand what a time-varying spectrum is. The basic gal is to devise a distribution that represents the energy or intensity of a signal simultaneously in time and frequency. Although the basic notions have been developing steadily over the last 40 years, there have recently been significant advances. This review is intended to be understandable to the nonspecialist with emphasis on the diversity of concepts and motivations that have gone into the formation of the field. >

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Time-frequency distributions-a review

The concept of instantaneous frequency (IF), its definitions, and the correspondence between the various mathematical models formulated for representation of IF are discussed. The extent to which the IF corresponds to the intuitive expectation of reality is also considered. A historical review of the successive attempts to define the IF is presented. The relationships between the IF and the group-delay, analytic signal, and bandwidth-time (BT) product are explored, as well as the relationship with time-frequency distributions. The notions of monocomponent and multicomponent signals and instantaneous bandwidth are discussed. It is shown that these notions are well described in the context of the theory presented. >

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Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals

For pt.I see ibid., vol.80, no.4, p.520-38 (1992). The concept of instantaneous frequency (IF) is extended to discrete-time signals. The specific problem explored is that of estimating the IF of frequency-modulated (FM) discrete-time signals embedded in Gaussian noise. Well-established methods for estimating the IF include differentiation of the phase and smoothing thereof, adaptive frequency estimation techniques such as the phase locked loop (PLL), and extraction of the peak from time-varying spectral representations. More recently, methods based on a modeling of the signal phase as a polynomial have been introduced. These methods are reviewed, and their performance compared on both simulated and real data. Guidelines are given as to which estimation method should be used for a given signal class and signal-to-noise ratio (SNR). >

Estimating and Interpreting The Instantaneous Frequency of a Signal

This paper discusses the application of least-mean-squares approximate inverse filtering techniques to radar range sidelobe reduction. The method is illustrated by application to the 13-element Barker code. The performance of the least-mean-square inverse filter is compared with the matched filter and with the simplified sidelobereducing filters of Rihaczek and Golden. A filter which completely suppresses the range sidelobes of a 13-element Barker sequence is only 0.2 dB worse than a matched filter in noise.

Optimum Mismatched Filters for Sidelobe Suppression

In 1945, Dirac attempted to develop a “formal probability” distribution to describe quantum operators in terms of two noncommuting variables, such as position x and momentum p [Rev. Mod. Phys. 17, 195 (1945)]. The resulting quasiprobability distribution is a complete representation of the quantum state and can be observed directly in experiments. We measure Dirac’s distribution for the quantum state of the transverse degree of freedom of a photon by weakly measuring transverse x so as to not randomize the subsequent p measurement. Furthermore, we show that the distribution has the classical-like feature that it transforms (e.g., propagates) according to Bayes’ law.

http://www.photonicquantum.info/Research/publications/Bamber%20PRL%20112,%20070405%20%282014%29.pdf

Observing Dirac's classical phase space analog to the quantum state.

Several definitions of 'time-varying', 'short-time' and 'instantaneous' spectra exist. The paper relates these to the time-frequency energy distribution of a signal and to the time-varying power spectrum of a non-stationary random process. The treatment emphasizes physical interpretation, rather than mathematical rigour.

Instantaneous and time-varying spectraߞan introduction

The paper reviews the theory by which amplitude and phase modulated (a.m.ph.m.) pulse trains can be represented by complex number sequences and the means by which they can be generated and processed. The synthesis of pulse trains to have a desired autocorrelation function is described, some useful short Huffman sequences derived from Barker sequences are presented. Finally, the z-transform is applied to the study of combination codes formed by combining shorter sequences.

M.H. Ackroyd

Papers

Instantaneous and time-varying spectraߞan introduction

Amplitude and phase modulated pulse trains for radar