# A Compressed Sensing Parameter Extraction Platform for Radar Pulse Signal Acquisition

TL;DR: A complete (hardware/ software) sub-Nyquist rate (× 13) wideband signal acquisition chain capable of acquiring radar pulse parameters in an instantaneous bandwidth spanning 100 MHz-2.5 GHz with the equivalent of 8 effective number of bits (ENOB) digitizing performance is presented.

Abstract: In this paper we present a complete (hardware/ software) sub-Nyquist rate (× 13) wideband signal acquisition chain capable of acquiring radar pulse parameters in an instantaneous bandwidth spanning 100 MHz-2.5 GHz with the equivalent of 8 effective number of bits (ENOB) digitizing performance. The approach is based on the alternative sensing-paradigm of compressed sensing (CS). The hardware platform features a fully-integrated CS receiver architecture named the random-modulation preintegrator (RMPI) fabricated in Northrop Grumman's 450 nm InP HBT bipolar technology. The software back-end consists of a novel CS parameter recovery algorithm which extracts information about the signal without performing full time-domain signal reconstruction. This approach significantly reduces the computational overhead involved in retrieving desired information which demonstrates an avenue toward employing CS techniques in power-constrained real-time applications. The developed techniques are validated on CS samples physically measured by the fabricated RMPI and measurement results are presented. The parameter estimation algorithms are described in detail and a complete description of the physical hardware is given.

## Summary (5 min read)

### Introduction

- A principal goal in the design of modern electronic systems is to acquire large amounts of information quickly and with little expenditure of resources.
- In light of the ever growing demand to capture higher bandwidths, it would seem that a solution at the fundamental system level is needed to address these challenges.
- The requisite sampling rate is merely proportional to the information level, and thus CS provides a framework for sub-Nyquist rate signal acquisition.
- The authors complete system is capable of recovering radar pulse parameters within an effective instantaneous bandwidth (EIBW) spanning 100 MHz—2.5 GHz with a digitizing performance of 8 ENOB.

### A. Compressed Sensing

- CS at its heart relies on two concepts: sparsity and incoherence [5].
- Incoherence captures the idea of dissimilarity between any two representations; two bases are said to be incoherent if any signal having a sparse expansion in one of them must be dense in the other.
- Fig. 1a shows the block diagram of the IC containing the input buffer driving the common node of the four RD channels, and the timing generator.
- After the integrator the signal bandwidth is reduced and the circuits are design to meet settling requirements.
- Finally, an output buffer is designed to the drive the ADC with the correct swing and common-mode voltage.

### B. A Brief History and Description of the RMPI

- Almost simultaneously with the introduction of CS [2], a number of CS-based signal-acquisition architectures were proposed.
- The rows of each block contain ±1 entries, and the overall matrix will be composed of NTNyq/Tint = 20 blocks (one for each integration window).
- For the measurements presented in this paper, the authors construct a model of their system’s Φ matrix by feeding in sinusoidal tones and using the output measurements to characterize the system’s impulse response.

### A. Architecture and Operation

- The RMPI presented in this work was realized with the proprietary Northrop Grumman (NG) 450 nm InP HBT bipolar process [26].
- The timing generator is responsible for generating the pseudo-random bit sequences (PRBS) and the clocking waveforms to coordinate the track-and-hold (T/H) and integration operations.
- The circuits following the integrator are designed to meet the settling requirements of the reduced bandwidth.
- In operation, the RMPI circuit takes the analog input signal, buffers it, and distributes the buffered signal to each of the 4 channels.
- Immediately after the signal is sampled, the capacitor begins discharging and the second capacitor begins integrating the next frame (see Fig. 1b).

### B. Analog Signal Path

- The input buffer is a differential pair with emitter degeneration and 50 Ω termination at each single-ended input.
- Emitter degeneration is used on the bottom differential pair to improve linearity.
- At the end of the integration period the signal is sampled and then held for 26 CLKin cycles to allow the external ADC to digitize the signal for post-processing.
- The diodes act as switches to configure capacitors for integration or reset it based on the level of the control signal (SEL).
- The input clock buffer was biased with a relatively high power to reduce jitter and it has 4 separate output emitter followers to limit cross-talk.

### C. PRBS & Timing Generator

- A master clock is applied to CLKin, from which all required timing signals are generated.
- The output pulse from PN6B is re-clocked with the pulse from PN6A to pr duce a sync pulse that is 52 CLKin cycles long once every 3276 cycles.
- Special attention was paid to the routing of the PRBS, T/H clocks, and select signals to inimize clock/data coupling among th four channels.
- Shown are (4) quadrature clocks for the T/H and (4) select signals for the interleaved capacitors.

### D. Performance Analysis

- Simulation validation was done by performing transientbased two-tone inter-modulation distortion simulations in the Cadence design environment.
- Noise simulations were perform d using the peri i steady-state (PSS) mode of spectre.
- The RMPI sam ling system, including the off-chip ADCs consumed 6.1 W of power.
- The authors point out that this system was designed as a proof-of-concept and was not optimized for power.
- 4 Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841].

### IV. PULSE-DESCRIPTOR WORD (PDW) EXTRACTION

- Having described the acquisition system, the authors now present algorithms for detecting radar pulses and estimating their parameters, referred to as pulse-descriptor words (PDW), from randomly modulated pre-integrated (RMPI) samples.
- The detection process is based on familiar principles employed by detectors that operate on Nyquist samples.
- The authors algorithms use a combination of template matching, energy thresholding, and consistency estimation to determine the presence of pulses.
- The remainder of the section elaborates on the procedure and is arranged as follows.
- After describing how the authors can reliably estimate the carrier frequency of a signal from compressive measurements, they then explain how they use such estimations to form a detection algorithm that jointly uses energy detection and consistency of their frequency measurements.

### A. General Parametric Estimation

- The authors general parameter estimation problem can be stated as follows.
- Given the measurements y = Φ[x0]+noise, the authors search for the set of parameters corresponding to the subspace which contains a signal which comes closest to explaining the measurements y.
- Tα is the orthogonal projector onto the column space of Vα.
- When the noise is correlated, the authors may instead pose the optimization in terms of a weighted least squares problem.
- In Sec. IV-B and Sec. IV-C below, the authors will discuss the particular cases of frequency estimation for an unknown tone, and time-of-arrival estimation for a square pulse modulated to a known frequency.

### B. Carrier Frequency, Amplitude, and Phase

- The authors consider the task of estimating the frequency of a pure tone from the observed RMPI measurements.
- The algorithms developed here play a central role in the detection process as well, as the frequency estimation procedure is fundamental to determining the presence of pulses.
- The authors also describe how to estimate the amplitude and phase once the carrier frequency (CF) is known.
- The subspaces Since the authors have discretized the signal x(t) through its Nyquist samples, their measurement process is modeled through a matrix Φ, the rows of which are the basis elements φk.
- Rather than dealing with continuously variable frequency, the authors define a fine grid of frequencies between 0 and fnyq/2.

### D. Pulse Detection

- With their estimation techniques explained the authors next describe a pulse detection algorithm that takes a stream of RMPI samples and classifies each as either having a “pulse present” or “no pulse present.”.
- The authors task is to determine how many pulses are present and to estimate the parameters of each pulse the authors find.
- If neighboring blocks have CF estimates that are consistent in value and tones at these frequencies account for a considerable portion of their measurement energies, then the authors are confident that a pulse is indeed present.
- Once the TOA, TOD, and CF estimates have been refined, the authors calculate their amplitude and phase estimates.
- Merge any segments that are close together in time and have similar CF estimate, also known as 7.

### E. Complexity

- In fact, since Φ contains repetitions of Φ0 the authors need only take the FFT of the 252 rows of Φ0, which they may then shift through complex modulation.
- This calculation does not depend on the measurement data, and therefore can be done offline as a precomputation.
- For each window length L (typically between 5-11 RMPI samples) and each window shift, the authors have to compute.
- The authors can explicitly invert V Tf Vf using the 2×2 matrix inverse formula, and all other calculations involve a small number of 4L-point inner products.
- The number of frequencies the authors test is proportional to the number of Nyquist 8 Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841] samples N , and thus the cost of the frequency estimation for each sliding window shift is O(NL).

### F. Cancellation and multiple pulses

- The authors focus on how to remove contributions from certain frequency bands in the RMPI measurements.
- The operator Ψ can be used to remove the contributions of the interfering band in the measurements y when the authors run their estimation methods.
- Fig. 9 shows how the use of the nulling operator aids in removing interfering bands.
- During this second 1These are also known as Slepian sequences.
- Fig. 10 shows an example of the two-stage detector for overlapping pulse data.

### A. Measurement Test Setup Description

- In order to test the performance of the radar-pulse parameter estimation system-composed of the RMPI sampling hardware and the PDW extraction algorithm §IV, the authors ran a set of over 686 test radar pulses composed of permutations of A0, θ0, CF, TOA and TOD through the RMPI and estimated the varied parameters from the compressed-samples digitized by the RMPI.
- 9 Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841] Fig. 12 shows a block diagram for the test setup used for the RMPI.
- An Arbitrary Waveform Generator AWG with an output sampling rate of 10 Gsps was used to output the pulses of interest.
- The stimulus was input into the RMPI whose outputs were then sampled by external ADCs located on a custom digitizing PCB shown in Fig. 11b: the RMPI IC is mounted on a a low-temperature co-fired ceramic (LTCC) substrate shown in Fig. 11a which is placed in the center of the digitizing board.
- The digitized samples were then transferred to a PC where the PDW extraction algorithm was used to estimate the signal parameters.

### B. Parameter Estimation

- For each pulse, the authors estimated the CF only from measurements corresponding to times when the signal was active.
- The authors then estimated the TOA from RMPI samples corresponding to noise followed by the front end of the pulse.
- The authors repeated the procedure for the TOD, using RMPI samples corresponding to the end of the pulse followed by noise only.
- Fig. 13 shows the distribution of their estimation errors for CF, TOA, and TOD.
- Additionally, Table I shows statistics on the errors for each of the three parameters.

### C. Pulse Detection

- The authors tested the pulse detection system by generating 60 test cases containing 12 pulses each (for a total of 720 pulses) with varying amplitudes (ranging over 60 dB), phases, 10 Distribution Statement “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case #18841] durations (100 ns—1 µs), carrier frequencies (100 MHz— 2.5 GHz), and overlaps.
- All pulse rise times were approximately 10 ns.
- Table III shows the detection rate and standard deviation of the parameter estimate errors as a function of the pulse amplitudes.
- To test the robustness of the detection and estimation system, the authors repeated their detection experiment and included a constant-frequency interferer at set amplitudes in each experiment.
- The authors tested 6 interferer strengths, running 60 experiments with 12 pulses per experiment, for a total of 720 pulses per interferer strength.

### VI. CONCLUSION

- The authors have presented a detailed overview of the design of both hardware and software used in a novel radar-pulse receiver in which information is extracted without performing full signal reconstruction.
- This novel approach obtains desired information with high accuracy while considerably reducing the back-end computational load.
- The system was validated using parameter estimates obtained from testing with a large and exhaustive set of realistic radar pulses spanning the parameter space.
- The physically measured results generated from this prototype proof-ofconcept system demonstrates the feasibility of the approach.
- In addition, the data obtained provides ample motivation for further investigation of the merit of CS-based signal acquisition schemes in general.

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...In short, the CS theory states that signals with high overall bandwidth but comparatively low information level can be acquired very efficiently using randomized measurement protocols....

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