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

A tutorial review of both linear and quadratic representations is given. The linear representations discussed are the short-time Fourier transform and the wavelet transform. The discussion of quadratic representations concentrates on the Wigner distribution, the ambiguity function, smoothed versions of the Wigner distribution, and various classes of quadratic time-frequency representations. Examples of the application of these representations to typical problems encountered in time-varying signal processing are provided. >

Linear and quadratic time-frequency signal representations

the of the magnitude, i.e., intensity, of the field.) It is shown that Rayleigh statistics govern the fist-order behavior of the magnitude; and the autocorrelation of the resulting image speckle is obtained by the methodof Middleton. The corresponding power spectrum follows immediately by Fourier transformation. Theoretical and experimentally determined autocorrelation functions and power spectra derived from B-scans of a scattering phantom containing many scatterers per resolution cell are presented. These functions lead naturally to the definition of the average speckle spot or cell sue, and this inturn is comparable to the resolution cell. Each independent speckle servesas a degreeof freedom that determines the number of samples of tissue available over a target.As the speckle cell size decreases this number increases in a manner predictable from the physical parameters of the cell size. However, it is found that the speckle cellis broadened, the degrees of freedom diminished, when the object structureis correlated. This yields the possibilityof deducing information about the object structure from the second-order statistics of the speckle texture, in addition to that obtainable from the fiistorder statistics.

Statistics of Speckle in Ultrasound B-Scans

Since speckle plays an important role in many physical phenomena, it is essential to fully understand its statistical properties. Starting from the basic idea of a random walk in the complex plane, we derive the first-order statistics of the complex amplitude, intensity and phase of speckle. Sums of speckle patterns are also considered, the addition being either on an amplitude or on an intensity basis, with partially polarized speckle being a special case. Next we consider the sum of a speckle pattern and a coherent background, deriving the first-order probability density functions of intensity and phase. Attention is then turned to second-order statistics. The autocorrelation function and power spectral density are derived, both for a free-space propagation geometry and for an imaging geometry. In some cases the recorded speckle pattern may be spatially integrated or blurred, and accordingly consideration is given to the statistics of such patterns. Finally, the relationship between detailed surface structure and the resulting speckle pattern is explored, with emphasis on the effects of the surface autocorrelation function and the effects of finite surface roughness.

Statistical Properties of Laser Speckle Patterns

Localized fluid flow measurements with an He-Ne laser spectrometer

Effect of surface roughness on the statistical distribution of image speckle intensity

As one of the interesting optical techniques for measurements of the velocity, the spatial-filtering method is treated briefly in this review. The basic theory of the method is examined by analyzing the filtering characteristics of a spatial filter using the power spectral density function. The signal analyses are summarized as one of the techniques used in the spatial filtering method. Various configurations of the spatial filtering velocimeter are classified into typical four groups and outlined according to a variety of spatial filters employed in the velocimeters.

Principles and development of spatial filtering velocimetry

Measurement of surface roughness properties by means of laser speckle techniques

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T. Asakura

Papers

Effect of surface roughness on the statistical distribution of image speckle intensity

Principles and development of spatial filtering velocimetry

Measurement of surface roughness properties by means of laser speckle techniques

Dispersion, Complex Analysis and Optical Spectroscopy

A contrast variation of image speckle intensity under illumination of partially coherent light