Complex Wavelet Transform for Analog Signal Processing

For signal processing, the Wavelet transform (WT) has been shown to be a very promising mathematical tool. WT is efficient for local analysis of nonstationary and fast transient signals due its good estimation of time and frequency localization. In the complex wavelet transform analysis, the modulus maxima and the ± π/2 phase crossings point out the locations of sharp signal transitions. The phase information reveals isolated singularities in a signal more accurately than does the modulus. Unfortunately, in an ultra low power environment it’s not favorable to implement the WT by means of digital signal processing because of the high power consumption associated with the required A/D converter. In [1], we proposed a method for implementing the WT in an analog way. An analog complex wavelet transform filter was proposed, of which the impulse responses are approximated Gaussian window functions. This complex wavelet filter, subsequently, was implemented with Complex First Order Systems (CFOS). However, besides the derivatives of the Gaussian wavelet presented in [1], there are several families of wavelets that have proven to be especially useful. Therefore, a more general procedure based on the Padé approximant to obtain various types of wavelet bases was presented in [2]. Moreover, the Padé method yields a much better approximation than the method using CFOS for a filter of the same order. This paper presents an analog implementation of the complex wavelet transform using the Padé approximation. The Padé approximation is introduced to calculate the transfer functions of both the real and the imaginary parts, whose impulse responses are the second and the first derivatives of Gaussian, respectively. The complex filter design is based on the combination of the state space descriptions that implement the transfer functions. Simulations demonstrate an excellent approximation of the complex Gaussian wavelet bases using a tenth order filter.