### Non-Uniform Sampler

The non-uniform sampler (NUS) is an approach to digitize an analog signal that is sparse in the frequency domain. Conceptually, we can think of it like a standard analog-to-digital converter (ADC) that only keeps some of the samples; crucially, we keep a *random* subset of the samples, since otherwise there is aliasing error. Recovery is non-linear and requries a computer.

#### Concept

Fig 2: simplified block diagram

Fig 1: NUS conceptual diagram.

The idea behind the NUS is explained in Figure 1. For our setup, the maximum EIBW is 2.2 GHz because of an underlying "Nyquist rate" clock with a frequency of fs = 4.4 GHz. For the sake of explanation, assume there is a Nyquist rate ADC which samples the input signal---the actual implementation does not use a Nyquist rate ADC, since the point of the NUS is to avoid a high-rate ADC. A pseudo-random bit sequence (PRBS), generated off-chip, controls which of these samples are collected and which samples are ignored. Of every 8192 Nyquist-rate samples, only 440 are collected. Note that our method of "on grid" non-uniform sampling is very different from allowing arbitrarily spaced samples that are not integer multiples of the underlying Nyquist rate, since the latter approach would be nearly impossible to calibrate.

The actual implementation, shown in Figure 2, replaces the theoretical sub-sampled Nyquist-rate ADC with a nonuniformly clocked sample-and-hold (S/H). The sample times of the S/H are controlled by the PRBS sequence, and the same sequence controls a single low-rate ADC which performs the final quantization step. The custom S/H is necessary because the ADC is not designed for 2.2 GHz bandwidth signals.

Reconstruction---that is, interpolation of the omitted Nyquist-rate samples---is performed on a desktop personal computer.

#### Design

The NUS IC is designed in an InP HBT technology. A TI ADS5474 14-bit 400 Msps ADC (10.9 ENOB at 230 MHz) is used to digitize the samples, and the data is transferred to a computer for processing

#### Results

Fig 3: Example GSM test spectra with 100 MHz clutter. Spectra are plotted from 700 MHz to 2.1 GHz. The GSM signal is located at 1.595 GHz and is indicated with a green marker. For such a signal, the NUS gives a BER of 1 in a million for input power of -60 dB below full scale, and falls to a BER of approximately 1 in a hundred for input power of -67 dB below full scale.

See Fig 3 for a sample GSM signal. Using test signals of this form, the bit error-rate (BER) of the NUS was tested by varying the input powe r of the signal. The test signal was designed to mimic a real-world GSM signal in a background of other communication signals ("clutter") as well as noise.

#### Further reading

A Non-Uniform Sampler for Wideband Spectrally-Sparse Environments Michael Wakin, Stephen Becker, Eric Nakamura, Michael Grant, Emilio Sovero, Daniel Ching, Juhwan Yoo, Justin Romberg, Azita Emami-Neyestanak, Emmanuel Candès, IEEE J. on Emerging and Selected Topics in Circuits and Systems, 2012.