Gabor's expansion and the Zak transform for continuous-time and discrete-time signals

A good echo cancellation algorithm should have a fast convergence rate, small steady-state residual echo, and less implementation cost. The normalized least mean square (NLMS) adaptive filtering algorithm may not achieve this goal. We show that using the Gabor expansion is a way to achieve this goal. For direct digital signal processing compatibility the Gabor expansion introduced in this paper is for discrete-time signals, although the Gabor expansion also can be used for continuous-time signals. The Gabor expansion can be defined as a discrete-time signal representation in the joint time-frequency domain of a weighted sum of the collection of functions (known as the synthesis functions). There are several design issues in the echo canceller based on the Gabor expansion: the design of the analysis functions for the far-end speech, the design of the analysis functions for the near-end signal containing the echo plus the near-end speech, the design of the adaptive filters in the subsignal path, and the design of the synthesis functions. All the adaptive filters are designed using identical NLMS adaptive filtering algorithms.

Gabor's (1946) signal expansion and the Gabor transform are formulated on a non-orthogonal time-frequency lattice instead of on the traditional rectangular lattice. The reason for doing so is that a non-orthogonal sampling geometry might be better adapted to the form of the window functions (in the time-frequency domain) than an orthogonal one. Oversampling in the Gabor scheme, which is required to have mathematically more attractive properties for the analysis window, then leads to better results in combination with less oversampling. The new procedure presented in this paper is based on considering the non-orthogonal lattice as a sub-lattice of a denser orthogonal lattice that is oversampled by a rational factor. In doing so, Gabor's signal expansion on a non-orthogonal lattice can be related to the expansion on an orthogonal lattice (restricting ourselves, of course, to only those sampling points that are part of the non-orthogonal sub-lattice), and all the techniques that have been derived for rectangular sampling can be used, albeit in a slightly modified form.

It is proposed that the Gabor (1946) expansion coefficients of a periodic (or finite) discrete signal in both the critical sampling case and the oversampling case can be computed by TH neural networks. Theoretical analysis and simulations show that this network will certainly provide approximate solutions arbitrarily close to the accurate one within the time of the order of time constants of this network without any programming complexity.

Gabor's (1946) signal expansion and the Gabor transform are formulated on a quincunx lattice instead of on the traditional rectangular lattice; the representation of the quincunx lattice is based on the rectangular lattice via either a shear operation or a rotation operation. A modified Zak (1972) transformation is defined with the help of which Gabor's signal expansion and the Gabor transform can be brought into product forms that are identical to the ones that are well known for the rectangular sampling geometry.

Chapter 9 - Discrete-Time Signals and Systems

We address the use of sampling representations for functions expressed in a Gabor expansion. Assuming that the window function of the expansion is bandlimited and that the Gabor coefficients vanish for a sufficiently high index in frequency, one easily obtains a Gabor sampling theorem from the classical Shannon-Kotelnikov (1949) result. This result is then explored to give insight into the relationships between the time bandwidth products of the signal and window, and the number of Gabor coefficients retained in a finite expansion. >

Multi-window Gabor expansion for evolutionary spectral analysis

In this paper, the problem of sampling a continuous-time generalized almost-cyclostationary (GACS) signal is addressed. The class of such nonstationary signals includes, as a special case, the almost-cyclostationary (ACS) signals. ACS signals filtered by some linear time-variant channels are further examples. It is shown that the discrete-time signal constituted by the samples of a GACS signal is a discrete-time ACS signal. Thus, the nonstationarity kind of a continuous-time GACS signal cannot be deducted from that of the discrete-time signal of its samples. However, in the paper it is shown how, starting from the sampled signal, the GACS or ACS nature of the continuous-time signal can be conjectured, provided that analysis parameters such as sampling period, padding factor, and data-record length are properly chosen.

Advanced Topics in Shannon Sampling and Interpolation Theory is the second volume of a textbook on signal analysis solely devoted to the topic of sampling and restoration of continuous time signals and images. Sampling and reconstruction are fundamental problems in any field that deals with real-time signals or images, including communication engineering, image processing, seismology, speech recognition, and digital signal processing. This second volume includes contributions from leading researchers in the field on such topics as Gabor's signal expansion, sampling in optical image formation, linear prediction theory, polar and spiral sampling theory, interpolation from non-uniform samples, an extension of Papoulis' generalized sampling expansion to higher dimensions, and applications of sampling theory to optics and to time-frequency representations.

Chapter 1 - Continuous-Time Signals

The basic concepts and relationships of the theory of discrete-time signals and systems are analogous to those for continuous-time signals and systems. In some respects, however, they are more simply derived and perhaps easier to visualize in the discrete-time case. In this chapter, we will introduce these basic concepts and relationships, developing them further in subsequent chapters.

Hybrid systems modelers like Simulink come with a rich collection of discrete-time and continuous-time blocks. Most blocks are not defined in terms of more elementary ones—and some cannot be—but are instead written in imperative code and explained informally in a reference manual. This raises the question of defining a minimal set of orthogonal programming constructs such that most blocks can be programmed directly and thereby given a specification that is mathematically precise, and whose compiled version performs comparably to handwritten code. In this paper, we show that a fairly large set of blocks of a standard library like the one provided by Simulink can be programmed in a precise, purely functional language using stream equations, hierarchical automata, Ordinary Differential Equations (ODEs), and deterministic synchronous parallel composition. Some blocks cannot be expressed in our setting as they mix discrete-time and continuous-time signals in unprincipled ways that are statically forbidden by the type checker. The experiment is conducted in Zélus, a synchronous language that conservatively extends L ustre with ODEs to program systems that mix discrete-time and continuous-time signals.

Continuous-time or analog signals are processed using analog devices such as amplifiers, filters, etc. It is impossible to process signals multiplexed from various sources using a single hardware system in the analog domain. On the other hand, digital signals can be processed using both special-purpose hardware and software systems. Worldwide use of Internet, mobile communications, etc. demands all kinds of data such as video, audio, graphics, etc. In order to receive this information on a single device, computer, for instance, it is impossible to use analog signals and techniques. In order to be able to design and implement digitally based systems, it is absolutely necessary to have an understanding of digital signals and systems. Digital signals are discrete in time and amplitude. However, we will assume discrete-time signals to have a continuum of amplitude in order to be able to analyze such signals and systems mathematically. In this chapter we will describe typical discrete-time signals mathematically and then use them to describe and analyze linear time-invariant discrete-time systems. To help the readers understand the mathematical details, we will work out examples followed by MATLAB-based examples. Since digital signals are obtained from analog sources, we will also discuss the conversion of continuous-time signals into digital signals using analog-to-digital converters (ADC).

This chapter deals with the minimum sampling interval \(T_s\) needed to correctly represent an analog signal by samples extracted periodically from it, so as to be able to reconstruct the continuous-time signal from its discrete-time version. The sampling theorem prescribes this lower limit and highlights the fact that a representative sampling is possible if, and only if, the analog signal does not contain frequencies higher than the Nyquist frequency \(1/(2 T_s)\): no finite-rate sampling can capture the variations of an analog signal which is not bandlimited. Other issues related to analog signals, such as the signal’s concentrations in the time and frequency domains and their mutual inverse dependence (uncertainty principle), as well as the definition of bounded support in both domains, are also discussed. An appendix provides a summary of the relations among the variables used to express the concept of frequency in the continuous-time and discrete-time cases.

A discrete time signal is one that has a value only for a finite or infinite number of time instants whereas a continuous time signal has a value for every (real) time instant. For example, consider: The sound pressure wave of a speech signal The electrical that is the output of a transducer that is used to mesdsure the cound pressure wave of a speech signal The sequence of numbers that is obtained from connecting an analg to digital converter (ADC) to the electrical signal sound pressure wave of a speech signal KeywordsSpeech SignalAnalog SignalAnalog InputLinear Predictive CodeDiscrete Time SignalThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.