### Random Modulator Pre-Integrator

Fig 1: Block diagram of the RMPI. Each row is a channel. The incoming signal has 2.5 GHz bandwidth, then it is split among the channels and mixed with a pseudo-random bit sequence (PRBS), integrated, and then sampled at a low rate.

The random modulator pre-integrator (RMPI) is designed to capture all types of compressible signals, although our work has focused on radar pulses. It can also be applied to frequency sparse signals, like the NUS. The cost of this universality, compared to the NUS, is more challenging hardware.

#### Concept

The RMPI can be conceptualized in the finite discrete domain. Consider the class of band-limited signals, which we can approximate arbitrarily well by vectors of large enough dimension. Conventional analog-to-digital converters take periodic measurements, which are represented in the discrete setting by a diagonal measurement matrix. The RMPI, on the other hand, uses a measurement matrix that is not diagonal, and has energy spread out in all the entries. Specifically, each entry is randomly assigned either +1 or -1. The benefit to this approach is that compressed sensing theory allows us to take many fewer measurements than conventionally approaches allow.

The actual RMPI is slightly more complicated, since it is hard to make an arbitrary large matrix with all +1 and -1 entries. Instead, the large matrix is composed of blocks of these +/- 1 matrices. There are also eight channels, which allows these sub-blocks to be larger, and thus the matrix more closely resembles a full +/- 1 matrix.

#### Further reading

- MICS lab at Caltech
- Stephen's RMPI page
- J. Yoo, S. Becker, M. Loh, M. Monge, E. Candès, A. Emami-Neyestanak, A 100MHz-2GHz 12.5x sub-Nyquist Rate Receiver in 90nm CMOS, 2012 IEEE Radio Frequency Integrated Circuits Symposium (RFIC), Montreal, Canada.
- J. Yoo, S. Becker, M. Monge, M. Loh, E. Candès, A. Emami-Neyestanak, Design and implementation of a fully integrated compressed-sensing signal acquisition system, ICASSP 2012 (Kyoto, Japan, March 2012).
- J. Yoo, C. Turnes, E. Nakamura, C. Le, S. Becker, E. Sovero, M. Wakin, M. Grant, J. Romberg, A. Emami-Neyestanak, and E. Candès, A Compressed Sensing Parameter Extraction Platform for Radar Pulse Signal Acquisition. IEEE Journal on Emerging and Selected Topics in Circuits and Systems, 2012
- Stephen Becker, Practical compressed sensing : modern data acquisition and signal processing, thesis, Caltech, 2011.
- E. J. Candès and M. B. Wakin, An introduction to compressive sampling, in IEEE Signal Processing Magazine, vol. 25, no. 2, pp. 21--30, March 2008 (PDF)
- J. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, Beyond Nyquist: efficient sampling of sparse bandlimited signals, IEEE Transactions on Information Theory, 2010