An analogue bandpass filter realisation of the Continuous Wavelet Transform

Abstract

Motivation• Lightweight, portable devices must be physically small.• They can thus have only small, limited capacity batteries making power consumption the key system design parameter.• In addition, it is often desirable to carry out onboard signal processing.• Low power, low computational complexity signal processing methods are thus required.• The Continuous Wavelet Transform (CWT) is one such signal processing technique. It is used by the portable system described in [1] and so a low power implementation requires investigation. Objectives• To illustrate a wavelet approximation procedure termed the Low Power CWT (LPCWT) as it is suitable for low power implementation. The method is similar to that in [2] and related papers, and can be implemented using any desired circuit topology.• To act as a step-by-step wavelet design guide, illustrating the trade-offs that are possible. This could be useful to engineers with similar signal processing requirements.• To give application specific performance metrics for the approximation method. This allows the performance of the LPCWT in a real usage situation to be found which is more meaningful than an arbitrary error norm between the wanted and approximated wavelet shapes. Wavelet design procedure• The CWT of signal x(t) at a scale a, time b and with mother wavelet ψ(t) is defined as• This is equivalent to the convolution of the signal x(t) with an impulse response• All that is needed to implement the CWT at scale a is thus a filter with impulse response h(t), or the corresponding H(s).The design procedure to generate H(s) is summarised as:1. Based on the application determine the mother wavelet required. The Mexican hat wavelet is the example used here.2. Choose the scales at which analysis must be carried out.Reducing the number of scales used will reduce the circuitry required and minimise the power consumption.3. Mathematically approximate the wavelet shape using the procedure outlined here, exploiting the mathematical form if possible.4. Determine a time delay T to ensure that the wavelet approximation is realisable.5. Use the dynamic range and other application specific criteria to select the order of the approximation required.6. Generate the required filter transfer functions and implement these in a suitable circuit.• The Mexican hat mother wavelet is considered here: • But ψ(t) is symmetrical around t = 0 so this is not stable.• To correct for this a time delay T is introduced shifting ψ(t) to the right. Letting jω → s:• This is in rational form, but requires a finite order denominator for implementation.• Thus a Maclaurin series expansion is used to give the final generalised transfer function:• T can be varied to help place the poles in desired positions. The figure below illustrates the case for a 3 rd order approximation. • Note that the bode plots only need to match over the dynamic range of the situation under consideration.• Full numerical transfer functions ready for implementation can now be found.• Substituting a = 0.1 and T = 0.4 into the 7 th order a...

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“…A stable transfer function approximation that would allow for the CWT to be implemented as a filter can be realized by introducing a time shift (T ) in (3) [24]. The introduced time shift would result in a delay in the output of the final filter implementation.…”

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“…The first phase: Convert EEG signal into delta, theta, alpha, beta and gamma waves using Mexican hat Wavelet transform by (1) and (2) [7].…”

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“…Instead, an approximation is required in order to make a finite order, stable filter. For conciseness, the approximation used here is not repeated in depth, but can be found in [27].…”