Shannon Wavelet Approach to Sub-Band Coding

It is well known that the Shannon Sampling Theorem allows us to fully recover a continuous-time bandlimited signal from its digital samples, as long as the sampling rate to be chosen is not smaller than the Nyquist frequency. This theory applies to all bandlimited signals, which may or may not occupy the entire frequency band. Hence, it is intuitively convincing that for continuous-time signals, such as those in speech, that do not fully utilize the entire frequency intervals, less digital samples are required for their full recovery. Current techniques in sub-band coding are used for achieving this goal. The objective of this paper is to present a wavelet theory for establishing the mathematical foundation of this sub-band coding approach. A wavelet packet decomposition of the signal provides the optimal sub-band coding bit-rate by using the Shannon wavelet library introduced in this paper.

Download Full-text

On the uncertainty inequality as applied to discrete signals

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms/2006/48185 ◽

2006 ◽

Vol 2006 ◽

pp. 1-22 ◽

Cited By ~ 6

Author(s):

Y. V. Venkatesh ◽

S. Kumar Raja ◽

G. Vidyasagar

Keyword(s):

Discrete Time ◽

Continuous Time ◽

Sampling Theorem ◽

Discrete Signals ◽

Bandlimited Signal ◽

Time Signals ◽

Shannon Sampling Theorem ◽

Heisenberg Uncertainty ◽

Comparison Of The Results ◽

Uncertainty Inequality

Given a continuous-time bandlimited signal, the Shannon sampling theorem provides an interpolation scheme forexactly reconstructingit from its discrete samples. We analyze the relationship between concentration (orcompactness) in thetemporal/spectral domainsof the (i) continuous-time and (ii) discrete-time signals. The former is governed by the Heisenberg uncertainty inequality which prescribes a lower bound on the product ofeffectivetemporal and spectral spreads of the signal. On the other hand, the discrete-time counterpart seems to exhibit some strange properties, and this provides motivation for the present paper. We consider the following problem:for a bandlimited signal, can the uncertainty inequality be expressed in terms of the samples, using thestandard definitions of the temporal and spectral spreads of the signal?In contrast with the results of the literature, we present a new approach to solve this problem. We also present a comparison of the results obtained using the proposed definitions with those available in the literature.

Download Full-text

The Sampling Theorem

Handbook of Fourier Analysis & Its Applications ◽

10.1093/oso/9780195335927.003.0010 ◽

2009 ◽

Author(s):

Robert J Marks II

Keyword(s):

Sampling Rate ◽

Finite Energy ◽

Frequency Component ◽

Sampling Theorem ◽

The Other ◽

Interpolation Theory ◽

Analog To Digital ◽

Bandlimited Signal ◽

Two Samples ◽

Shannon Sampling Theorem

Much of that which is ordinal is modelled as analog. Most computational engines, on the other hand, are digital. Transforming from analog to digital is straightforward: we simply sample. Regaining the original signal from these samples or assessing the information lost in the sampling process are the fundamental questions addressed by sampling and interpolation theory. This chapter deals with understanding, generalizing and extending the cardinal series of Shannon sampling theory. The fundamental form of this series states, remarkably, that a bandlimited signal is uniquely specified by its sufficiently close equally spaced samples. The cardinal series has many names, including the Whittaker-Shannon sampling theorem [514], the Whittaker-Shannon-Kotelnikov sampling theorem [679], and the Whittaker- Shannon-Kotelnikov-Kramer sampling theorem [679]. For brevity, we will use the terms sampling theorem and cardinal series. If a signal has finite energy, the minimum sampling rate is equal to two samples per period of the highest frequency component of the signal. Specifically, if the highest frequency component of the signal is B hertz, then the signal, x(t), can be recovered from the samples by The frequency B is also referred to as the signal’s bandwidth and, if B is finite, x(t) is said to be bandlimited [1283].

Download Full-text

Shannon Wavelets for the Solution of Integrodifferential Equations

Mathematical Problems in Engineering ◽

10.1155/2010/408418 ◽

2010 ◽

Vol 2010 ◽

pp. 1-22 ◽

Cited By ~ 40

Author(s):

Carlo Cattani

Keyword(s):

Hypergeometric Series ◽

Sampling Theorem ◽

Integrodifferential Equations ◽

Series Connection ◽

Wavelet Representation ◽

Connection Coefficients ◽

Shannon Sampling Theorem ◽

Shannon Wavelet ◽

Definition Of ◽

Shannon Wavelets

Shannon wavelets are used to define a method for the solution of integrodifferential equations. This method is based on (1) the Galerking method, (2) the Shannon wavelet representation, (3) the decorrelation of the generalized Shannon sampling theorem, and (4) the definition of connection coefficients. The Shannon sampling theorem is considered in a more general approach suitable for analysing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction ofL2(ℝ)functions. Shannon wavelets areC∞-functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series (connection coefficients).

Download Full-text

Shannon Wavelets Theory

Mathematical Problems in Engineering ◽

10.1155/2008/164808 ◽

2008 ◽

Vol 2008 ◽

pp. 1-24 ◽

Cited By ~ 56

Author(s):

Carlo Cattani

Keyword(s):

Hypergeometric Series ◽

Sampling Theorem ◽

Connection Coefficients ◽

Shannon Sampling Theorem ◽

Shannon Wavelet ◽

Differential Properties ◽

Derivatives Of ◽

Shannon Wavelets

Shannon wavelets are studied together with their differential properties (known as connection coefficients). It is shown that the Shannon sampling theorem can be considered in a more general approach suitable for analyzing functions ranging in multifrequency bands. This generalization coincides with the Shannon wavelet reconstruction ofL2(ℝ)functions. The differential properties of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon wavelets areC∞-functions and their any order derivatives can be analytically defined by some kind of a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction of the derivatives of theCℓ-functions.

Download Full-text