Improved Arbitrary Waveform Synthesis for Tri-State Transmitters by an Impulse Response Factorization Enabling Use of the Viterbi Algorithm

Abstract

While tri-state, class-D transmitters enable cost-effective HW with large channel counts and high power capacity, performance of encoders for useful waveform synthesis is critical in achieving acoustic fidelity. Examples of waveforms with complex specifications include LFM chirps, and those with minimal time-bandwidth products such as discrete prolate spheroidal (DPSS) pulses. One previously disclosed encoder uses knowledge of the transducer impulse response (IR) to enable a linear deconvolution process allowing memoryless quantization of the processed signal, suitable for tri-state transmitter hardware control. Though low in complexity, these linear methods are suboptimal to nonlinear constrained optimization methods, such as Dynamic Programming (DP). DP methods can estimate sequenced symbols jointly, rather than individually as in a memoryless scheme. One candidate for a DP approach is the Viterbi Algorithm (VA), which reduces computation cost of optimal sequence inference from being exponential in signal length, to being exponential in the length $L$ of the IR. Unfortunately, the typically long IR of ultrasound transducers makes direct use of the VA intractable. The contribution of this paper is to show that the VA can be successfully employed by defining an abstracted two-stage IR model. Here, we illustrate the factorization of the IR into a shorter prototype IR, b, of length suitable for the VA, and a longer abstracted convolutional factor g to be processed linearly. In this way, an abstracted signal, and a shortened impulse response, is presented to the VA for tractable inference of transmit symbols through b and subsequently g. Further, we show how an explicit IR factorization can be avoided by first estimating an abstracted intermediate signal as a prefilter output, rendering the IR factorization implicit and eliminating a deconvolution operation. Using the empirical IR measured for commercially available transducers, we demonstrate examples including LFM and DPSS waveform encodings for a tri-state transmitter. Fidelity improvement of 10 or more dB NRMSE is demonstrated for these two waveforms, compared to the performance of a previously disclosed algorithm. We also demonstrate examples of transmit pulse scaling by specified amplitudes, mimicking a change in transmitter power supply voltage.