Measurement matrix design for compressed sensing based time delay estimation

In this paper we study the problem of estimating the unknown delay(s) in a system where we receive a linear combination of several delayed copies of a known transmitted waveform. This problem arises in many applications such as timing-based localization or wireless synchronization. Since accurate delay estimation requires wideband signals, traditional systems need high-speed AD converters which poses a significant burden on the hardware implementation. Compressive sensing (CS) based system architectures that take measurements at rates significantly below the Nyquist rate and yet achieve accurate delay estimation have been proposed with the goal to alleviate the hardware complexity. In this paper, we particularly discuss the design of the measurement kernels based on a frequency-domain representation and show numerically that an optimized choice can outperform randomly chosen functionals in terms of the delay estimation accuracy.