In the theory of Walsh-Fourier series on the dyadic group G developed so far, such essential concepts as continuity, Lipschitz conditions modulus of continuity ,etc., have been employed. However, with the exception of several articles by J. E. Gibbs, no attempt has been made to define a derivative for functions f given on G. In this paper such a derivative D [1] f is defined which has most properties in common with the ordinary derivative. D [1] is a linear, closed operator, the inverse operator or integral is defined, the fundamental theorem of the calculus is valid for these two notions, The derivative enables one to estimate the order of magnitude of the Walsh-Fourier coefficients, the degree of approximation of f by the partial sums of the Walsh-Fourier series of f, etc. Moreover the Walsh functions are the non-trivial eigensolutions of a first-order linear differential equation. The proofs depend upon a Walsh-Fourier "coefficient method".

Walsh-fourier series and the concept of a derivative †

We consider analog to digital (A/D) conversion, based on the quantization of coefficients obtained via the projection of a continuous time signal over a set of basis functions. The framework presented here for A/D conversion is motivated by the sampling of an input signal in domains which may lead to significantly less demanding A/D conversion characteristics, i.e., lower sampling rates and lower bit resolution requirements. We show that the proposed system efficiently parallelizes the analog to digital converter (ADC), which lowers the sampling rate requirements by increasing the number of basis functions on which the continuous time signal is projected, leading to a tradeoff between sampling rate reduction and system complexity. Additionally, the A/D conversion resolution requirements can be reduced by optimally assigning the available number of bits according to the variance distribution of the coefficients obtained from the signal projection over the new A/D conversion domain. In particular, we study A/D conversion in the frequency domain, where samples of the continuous signal spectrum are taken such that no time aliasing occurs in the discrete time version of the signal. We show that the frequency domain ADC overcomes some of the difficulties encountered in conventional time-domain methods for A/D conversion of signals with very large bandwidths, such as ultra-wideband (UWB) signals. The proposed A/D conversion method is compared with conventional ADCs based on pulse code modulation (PCM). Fundamental figures of merit in A/D conversion and system tradeoffs are discussed for the proposed ADC. The signal-to-noise and distortion ratios of the frequency domain ADC are presented, which quantify the impact of the most critical impairments of the proposed ADC technique. We also consider application to communications receivers, and provide a design example of a multi-carrier UWB receiver.

/pdf/ultra-wideband-analog-to-digital-conversion-via-signal-om1oezm3h9.pdf

Ultra-wideband analog-to-digital conversion via signal expansion

Previous work on empirical mode decomposition in two dimensions typically generates a residue with many extrema points. In this paper we propose an improved method to decompose an image into a number of intrinsic mode functions and a residue image with a minimum number of extrema points. We further propose a method for the variable sampling of the two-dimensional empirical mode decomposition. Since traditional frequency concept is not applicable in this work, we introduce the concept of empiquency, shortform for empirical mode frequency, to describe the signal oscillations. The very special properties of the intrinsic mode functions are used for variable sampling in order to reduce the number of parameters to represent the image. This is done blockwise using the occurrence of extrema points of the intrinsic mode function to steer the sampling rate of the block. A method of using overlapping 7 × 7 blocks is introduced to overcome blocking artifacts and to further reduce the number of parameters required to represent the image. The results presented here shows that an image can be successfully decomposed into a number of intrinsic mode functions and a residue image with a minimum number of extrema points. The results also show that subsampling offers a way to keep the total number of samples generated by empirical mode decomposition approximately equal to the number of pixels of the original image.

http://aquador.vovve.net/IEMD/emd-samp.pdf

Variable sampling of the empirical mode decomposition of two-dimensional signals

The current status of ultra-wideband technology and its main applications such as radars and telecom are presented. The main system approaches for radars and communications are discussed and compared.

Ultra-wideband systems

This chapter is devoted to the study of the Walsh-Hadamard transform (WHT), which is perhaps the most well-known of the nonsinusoidal orthogonal transforms. The WHT has gained prominence in various digital signal processing applications, since it can essentially be computed using additions and subtractions only. Consequently its hardware implementation is also simpler.

Walsh-Hadamard Transform

H. F. Harmuth, Transmission of Information by Orthogonal Functions. XI + 322 S. m. 110 Fig. Berlin/Heidelberg/New York 1969. Springer‐Verlag. Preis geb. DM 40,—

G. Maibaum

Papers

H. F. Harmuth, Transmission of Information by Orthogonal Functions. XI + 322 S. m. 110 Fig. Berlin/Heidelberg/New York 1969. Springer‐Verlag. Preis geb. DM 40,—