We develop construction techniques in the context of Riesz bases which facilitate the conception of a new family of multi-channel, multi-sampling rate theorems. This family takes advantage of the dilation property of the Fourier transform. In each channel a simple transform is taken which merely involves the addition and subtraction of subsets of the given samples. Each channel is then multiplied by a scalar and added together to construct the Fourier transform. This process can be characterised as the parallel decomposition of the Fourier transform into arrays of step, or binary, transforms. We then generalize this construction to include realizations of other Riesz basis families. Each family is seen to quantize uniquely an orthonormal basis by using different constructions. The dynamic range, or resolution, required to implement the basis coefficients depends upon the relative divisibility of the range of the basis. A measure which evaluates the required resolution is introduced. This leads to the construction of the uniform step transform which minimizes the dynamic range of the basis coefficients. This results in a coefficient quantization strategy which offers an optimal balance between programme memory, speed, and efficacy. We also show how the constructions are used to develop a novel method which refines wavelet bases. The refinement method modifies an existing wavelet such that the efficacy of the wavelet filter characteristic is increased. The ratios of main lobe and side lobe power and of maximal main lobe and side lobe magnitude of a wavelet filter characteristic are used as measures of efficacy. We extend these measures and propose the localized ratio of main lobe and side lobe power and the globalized ratio of maxima main lobe and side lobe magnitude. When these generalized measures are taken it is again

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The construction of some Riesz basis families and their application to coe-cient quantization, sampling theory, and wavelet analysis.

A family of function bases is devised from a superposition of unit characteristic functions. It is shown that these simple bases can be used to perform a quantized Fourier transform. Furthermore, the transform can be implemented in an iterative scheme such that system resources are balanced in accordance with performance.

On the coefficient quantization of the Fourier basis

Approximate frequency analysis in structural dynamics

In an earlier paper, (see ibid., vol.136, no.4, p.223-8 (1989)) Hughes and Heron have reported on a method of spectral analysis using truncated approximations to the cosine and sine functions. The spectral estimates are transformed to the Fourier estimates via the matrices M/sub c/ and M/sub s/. In this paper the analytic inverses of these matrices are found.

https://ieeexplore.ieee.org/iel1/2202/1902/00052809.pdf

Transformation of square-wave spectral estimates to Fourier coefficients

Structural health monitoring systems are becoming increasingly popular to detect and evaluate faults in structural systems Many of these systems are applied to large structures and wireless sensor systems (often incorporating energy harvesting devices) avoid the significant problems of long cable runs in these cases In wireless systems energy management is very important, and data transmission and computation are significant drains on the available energy Often it is sufficient to transmit frequency domain data rather than time domain, and this reduces the bandwidth of the communication and hence reduces power demand Thus, this paper considers the efficient calculation of the frequency response of the structure

/pdf/efficient-frequency-analysis-for-wireless-structural-health-6chbvhph7z.pdf

Efficient frequency analysis for wireless structural health monitoring systems

Power spectra may be computed via a discrete fourier transform with severely rounded-off trigonometric terms, and recently the ultimate round-off – that is to rectangular waves – has been proposed.1 A comparison is made between this and a 3 level round-off, +1, −1 and 0.

J.S.D. Mason

Papers

Power spectra from approximate transforms