Analogue wavelet transform with single biquad stage per scale

Abstract

A new approach to obtain the wavelet transform with easily constructed analogue circuits is presented. It is shown that the impulse response of a bandpass biquad filter satisfies the conditions to be considered a mother wavelet. Using this, an integrated circuit performing the wavelet transform at 16 scales along eight octaves has been designed. The design has CMOS transistors working in the subthreshold region with a total power consumption of 650 nW. On-chip measurements are reported.Introduction: Traditionally, the wavelet transform (WT) is implemented numerically or algorithmically. Recently, there have been significant advances in the analogue implementations of this transform and its practical applications [1,2]. The analogue realisation is attractive for low power signal processing in areas such as portable and bio-implantable devices. Currently, the common analogue approach is to obtain each scale of the WT by means of convolution in a bandpass continuous filter, the impulse response of which has the wavelet shape. References [1,2] have applied this principle with favourable results. These works show efforts to obtain good approximations of the known classical wavelets: the Morlet wavelet [1], and the first derivative of a Gaussian [2]. However, to obtain a good approximation, the filters have to be of high order, since the shape of classical wavelets is not the natural impulse response of the simpler analogue filters. In this Letter, we show that the impulse response of a bandpass biquad filter is a mother wavelet.

Approximations of the inverse wavelet transform for analogue circuits

“…We will denote this frequency range as L. For this approximation, the condition for perfect reconstruction is M m=0 χH(r m ω) = 1, for ω ∈ L. To validate the proposed approach, the simple approximation has been on-chip fabricated for H(ω), r, M , and χ given in Table I. In [2] it is shown that the IR of this H(ω) satisfies the mathematical conditions to be used as a mother wavelet. The χ value is selected to normalize the system gain.…”

“…3b. This OTA has a differential input range of ±29 mV within 1% of linearity [2]. The OTA transistors have the following dimensions in μm/μm: 36/4.5 for M 1 to M 4 , 17/4.5 for M 5 and M 6 , and 10/4.5 for M 7 and M 8 .…”

“…The OTA transistors have the following dimensions in μm/μm: 36/4.5 for M 1 to M 4 , 17/4.5 for M 5 and M 6 , and 10/4.5 for M 7 and M 8 . Transistors are biased in subthreshold region, where the current I b (and the OTA transconductance G that is proportional to I b ) has exponential dependence with respect to nearly linear changes in the bias voltage V b [2]. The bias voltages V b0 to V b15 are produced with the help of a ladder of 15 poly resistors of nearly equal value, R 1 to R 15 [2].…”

“…For example, in [1] a methodology is presented to design analog filters (intended for low power bioimplantable devices) whose Impulse Response (IR) is an approximation of a given mother wavelet. Moreover, in [2] it is shown that the IR of a biquadratic band-pass filter fulfils the admissibility and stability constraints to be considered as an exact wavelet. Thus, a complete WT system can be created with a set of biquad scaled filters, where the set of output signals is the wavelet domain representation of the input signal.…”

A Novel Method for Gaussian Wavelet Approximation

“…In order to overcome these difficulties, a number of attempts at analog hardware implementations of the continuous wavelet transform have been reported recently. Previous studies on analog hardware implementation of WT can be divided into two categories, namely time-domain approach [2][3][4] and frequency-domain approach [5]- [11]. The latter is now the mainstream approach and the performance of such implementation primarily depends on the accuracy of the corresponding wavelet approximations.…”