Analog Wavelet Filters: The Need for Approximation

From its definition, we stated that the implementation of the wavelet transform is based on the design of a bandpass filter that presents an impulse response equal to the desired wavelet base. In order to obtain a synthesizable transfer function of a particular wavelet filter, mathematical approximation techniques are required. In Chapter 4, we present several methods to obtain good approximations in the time domain of wavelet bases functions. One important objective of the introduced approaches is that the resulting approximated function should be rational and stable in the Laplace domain. This entails that the approximating function leads to a physically realizable network. Nevertheless, due to limitations in chip area, power consumption and coefficient matching, there is a trade-off between the approximation accuracy versus the order of the filter to be implemented. Thus, the design challenge is to obtain a low-order system while preserving a good approximation to the intended function.