Analog Continuous-Time Filter Designing for Morlet Wavelet Transform Using Constrained L2-Norm Approximation

Abstract

In this paper, a novel methodology is proposed to implement the Morlet wavelet transform in an analog circuit. Under the proposed scheme, the impulse response of the linear time-invariant system is used to approximate the Morlet wavelet function. The approximation accuracy is guaranteed by the constrained L 2-norm method, which reduces the approximation error by decreasing free parameters in searching routines. Due to the complex wavelet function, a common-pole strategy is presented for filter construction. With the common poles in real and imaginary parts, partial components can be shared during the analog implementation of the wavelet transform. With the reasonably good approximation for Morlet wavelet function, an analog continuous-time filter is designed based on the Gm-C integrator and orthonormal ladder topology. Application examples are introduced to illustrate the superior performance of the proposed filter. It is shown that the designed analog filter approximates the ideal Morlet wavelet function well in both time and frequency.

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“…where ψ(t) is the wavelet base, and a and b are the scale factor and the translation factor, respectively. 32 Since the output of the filter is the convolution of the impulse response of the filter circuit and the input signal, equation ( 1) can be written as…”

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