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Authors: Vlašić, Tin; Pavić, Ivan; Sever, Karlo; Oštrić, Lucija; Papa, Vito; Seršić, Damir

Title: A system for compressive sensing of analog signals

Affiliation: Faculty of Electrical Engineering and Computing, University of Zagreb, Croatia and Ericsson Nikola Tesla d. d., Croatia

Extended Abstract:

Digital signal processing is the basis for most modern consumer electronics, medical devices, smartphones, etc. As majority sources of information are of analog nature, continuous-time signals must be converted to discrete sequences of numbers, while preserving the information contained in those signals. Sampling theorems are an important part of signal processing providing the connection between the continuous and the discrete-time worlds. The standard sampling theorem, often attributed to Shannon and Nyquist, states that the sampling rate has to be at least twice the maximum frequency present in the signal. While it is extremely elegant, there are several drawbacks associated with it. First, it is an idealization: real-world signals are rarely truly bandlimited and are often better represented in alternative bases other than Fourier basis. Second, a high sampling rate is needed whenever the signal has a wide bandwidth, even if the information rate of the signal is small. Third, the standard reconstruction formula is rarely used because of the slow decay of the \textit{sinc} function. In practice, much simpler techniques are used, such as linear interpolation. In order to overcome the limitations, a few sampling paradigms were developed over the years. The paradigms that have received growing attention recently are those that leverage signal's sparsity, whether in time or transform domain, to go beyond the Nyquist limit.

Compressive sensing (CS) is a technique for recovering an input signal from a reduced set of measurements that are linear projections of the signal. CS relies on signal's sparsity when expressed in a proper basis. Consider a discrete-time signal $\mathbf{x} \in \mathbb{R}^N$ and a transform basis $\mathbf{\Psi}$. We can express vector $\mathbf{x}$ as a weighted sum of the basis vectors as $\mathbf{ x {=} \Psi s}$. Vector $\mathbf{s}$ is considered as sparse if only a small number of coefficients are non-zero. The signal is measured through linear projections by an $M {\times} N$ measurement matrix $\mathbf{\Phi}$, where $M{<}N$. Measurement vector $\mathbf{y} \in \mathbb{R}^M$ can be written as $\mathbf{y {=} \Phi \Psi s}$. The problem is ill-posed and the vector of coefficients $\mathbf{s}$ is recovered by solving the $\ell_1$ optimization problem.

CS framework has focus primarily set on discrete-time signals. However, there were several frameworks developed which leverage CS to the analog domain. The main idea of these frameworks is analog compression that narrows down the input bandwidth prior to sampling and following by recovery. Tropp et al. in [1] propose a random demodulator (RD) which is a type of the data acquisition system used to acquire sparse, bandlimited signals from a reduced set of measurements. The signal is demodulated by a multiplication with a high-rate pseudo-random sequence. The sequence is a piece-wise constant signal of numbers that take values $\pm 1$ with equal probability. Then a low-pass filter is applied and finally the signal is sampled at the rate lower than the Nyquist. The transform basis in which input signals are assumed to be sparse is the discrete Fourier transform basis. The authors in [2] propose a framework for acquisition of multiband analog signals where the sparsity is modeled by treating the case in which only a few out of all possible bands are active. They developed a modulated wideband converter (MWC) that is slightly different from the RD. The front-end converter consists of $M$ channels. In each channel the signal is multiplied by a periodic waveform. The goal of the converter is to alias the spectrum of the signal into the baseband. The output is low-pass filtered and sampled at a rate at twice a width of the baseband.

We propose a system for CS of analog signals based on the front-end that is a hybrid of the RD and the MWC. The front-end is made of $M$ parallel channels. Each channel consists of a mixer, an integrator and an analog-to-digital converter (ADC). An input signal is multiplied by a periodic pseudo-random sequence with period $T$. The sequence is a piecewise constant signal with $W$ different pseudo-random values. We propose the reconstruction on intervals of duration $T$. Thus, the output of the mixer is integrated and at the end of each interval a value of integration is sampled. The sampled value represents a single CS measurement. Each of the $M$ parallel channels differs from others in pseudo-random values of the piecewise constant signal. The outputs of the channels are organized in a vector. That is, the vector of CS measurements contains $M$ different sampled values. This vector is then used in the reconstruction procedure. The proposed front-end can be used for analog compression and sampling prior to a recovery of signals represented as proposed in [3]. The authors in [3] propose a spline-like Chebyshev polynomial representation of analog signals on intervals recovered from CS measurements. The idea is to obtain a parametric model directly from CS measurement without converting them into samples. A polynomial representation is much better adapted to perform operations such as differentiation or integration than computing discrete approximations of these mathematical constructions. To avoid discontinuities of the model at the intervals' ends, neighboring intervals are linked in a spline-like fashion by equating a desired number of function derivatives.

The proposed front-end can be used in an acquisition system with the CS reconstruction procedure of low complexity that is suited for embedded devices. Statistical CS uses statistical models to develop effective acquisition of signals that follow some statistical distribution. The proposed front-end can be tailored so that the input signal is multiplied with deterministic instead of pseudo-random values. We created a dictionary of one-dimensional signals with a specific statistical distribution. The principal component analysis (PCA) was applied on the dictionary, and eigenvectors and eigenvalues were obtained. Values of the eigenvectors with the $M$ largest eigenvalues are the values that mix an input signal. These eigenvectors constitute the rows of the CS measurement matrix. This can be seen as the projection of the signal onto the eigenvectors in the analog domain and the sampled values are results of the projection. The beauty of the PCA lies in the unitary projection matrix. The signal is reconstructed by mapping it back with the transpose of the measurement matrix. Instead of computationally complex, time consuming solving of the $\ell_1$ optimization problem, the signal is recovered only by the multiplications which are perfectly suited for embedded devices.

The acquisition system consists of the proposed front-end part and the back-end part realized on the Zynq-7000 System-on-Chip (SoC). The front-end has six parallel channels. The mixer in a channel is achieved by a programmable-gain amplifier (PGA) realized with an 8-bit multiplying analog-to-digital converter (MDAC). The PGA is configured to provide $4$-quadrant multiplying operation with $256$ different gain levels from $-1$ to $1$. This way we can realize a multiplication with quantized values of the eigenvectors. An output of the integrator is sampled by a $12$-bit ADC with up to $1$ \textit{mega sample per second} sampling rate. The front end is designed on a standalone PCB that can be used as an extension for ZedBoard development board. The Zynq-7000 SoC integrates the software programmability of an ARM-based processor with the hardware programmability of a field-programmable gate array (FPGA). The FPGA part of the chip has a task of quickly changing the gains of the PGAs by driving the MDACs. The gains are determined before every measurement session and the values are written by the processing system (PS) in the shared registers. This makes the system adaptive to different distributions of signals and to possible different analog compression approaches. The second task of the FPGA part is to manage a communication with the ADCs via serial peripheral interface (SPI) and to prepare sampled values to be fetched by the PS. The CS reconstruction is done on the PS.

Reference 1: 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, vol. 56, no. 1, pp. 520--544, Jan 2010.

Reference 2: M. Mishali and Y. C. Eldar, "From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals," IEEE Journal of Selected Topics in Signal Processing, vol. 4, no. 2, pp. 375--391, April 2010.

Reference 3: T. Vlašić, J. Ivanković, A. Tafro and D. Seršić, "Spline-like Chebyshev polynomial representation for compressed sensing," in Proceedings of the 11th International Symposium on Image and Signal Processing and Analysis (ISPA), Sep 2019., pp. 135-140