### A new type of mathematics for signal processing

Beginning in 2004, a new mathematical theory called "compressed sensing" (CS) has turned conventional signal processing upside-down. CS has applications outside of signal processing, but this project focuses on just applying it to analog-to-digital converters (ADC). Conventional ADC sample analog signals at twice the bandwidth contained in the signal, according to the Nyquist sampling theorem that is taught in freshman electrical engineering courses. The drawback of this approach is that the bandwidth of a signal may be a poor proxy for how much information is in the signal, and so ADC sample at unnecessarily high rates. Unfortunately, high-rate sampling is inherently more difficult than low-rate sampling. The approach of compressed sensing is to change how we sample, and the benefit is that it is now possible to sample at the information-rate, which can lead to sampling at rates that are orders of magnitude slower than the Nyquist rate.

There are three things that make CS possible. The first is a new type of sampling scheme, which is that the A2I project is about. The second is powerful modern mathematics to provide conditions when the approach works, and the third item is efficient algorithms to convert the samples back to the original signal. More information on the mathematics and algorithms can be found in the references contained on this website.

### From mathematics to silicon

The A2I project is a result of the foresight of Dennis Healy, who started the investigation after hearing about the new compressed sensing theory being developed in 2004 by Emmanuel Candès and Justin Romberg (along with David Donoho and Terence Tao).

The theoretical performance of CS devices was well-understood by 2006, but practical questions remained, and no integrated-circuit design of a CS device was developed. Are CS devices more sensitive to noise? Are they robust to calibration errors? The A2I project was proposed to answer these questions. Through clever design, we have shown that prototype CS integrated circuits can exceed the performance of state-of-the-art ADC that have been perfected over the past half-century, and we are happy to conclude that the A2I designs are indeed robust.

#### RMPI

Compressed sensing suggests a general paradigm of signal acquisition, and there are many particular systems to realize this. The random modulator pre-integrator (RMPI) is one such system, and it is extremely powerful and universal, meaning that it can acquire and decode signals that are compressible in many domains (e.g. compressible in frequency, or compressible in wavelets, compressible by the short-time Fourier transform, etc.). Our team fabricated two sets of RMPI IC, one in 90 nm CMOS (led by the Caltech sub-team) and one in InP (led by the Northrop Grumman Space Technologies sub-team). For more information, see RMPI

#### NUS

The non-uniform sampler (NUS) is another realization of a compressed sensing system, and it is tailored specifically for signals that have a sparse spectrum and therefore benefits from considerable simplicity in design. The NUS chips were fabricated in InP by Northrop Grumman Space Technologies. For more information, see NUS