626 ieee journal on emerging and selected topics in ... · 628 ieee journal on emerging and...

626 IEEE JOURNAL ON EMERGING AND SELECTED TOPICS IN CIRCUITS AND SYSTEMS, VOL. 2, NO. 3, SEPTEMBER 2012

A Compressed Sensing Parameter ExtractionPlatform for Radar Pulse Signal Acquisition

Juhwan Yoo, Christopher Turnes, Eric B. Nakamura, Chi K. Le, Stephen Becker, Emilio A. Sovero,Michael B. Wakin, Michael C. Grant, Justin Romberg, Azita Emami-Neyestanak, and Emmanuel Candés

Abstract—In this paper we present a complete (hardware/software) sub-Nyquist rate ( 13) wideband signal acquisitionchain capable of acquiring radar pulse parameters in an instanta-neous bandwidth spanning 100 MHz–2.5 GHz with the equivalentof 8 effective number of bits (ENOB) digitizing performance.The approach is based on the alternative sensing-paradigm ofcompressed sensing (CS). The hardware platform features afully-integrated CS receiver architecture named the random-mod-ulation preintegrator (RMPI) fabricated in Northrop Grumman’s450 nm InP HBT bipolar technology. The software back-end con-sists of a novel CS parameter recovery algorithm which extractsinformation about the signal without performing full time-domainsignal reconstruction. This approach significantly reduces thecomputational overhead involved in retrieving desired informationwhich demonstrates an avenue toward employing CS techniquesin power-constrained real-time applications. The developed tech-niques are validated on CS samples physically measured by thefabricated RMPI and measurement results are presented. Theparameter estimation algorithms are described in detail and acomplete description of the physical hardware is given.

Index Terms—Compressed sensing (CS), indium-phosphide,parameter estimation, random-modulation pre-integration(RMPI).

Manuscript received February 27, 2012; revised June 30, 2012; accepted July31, 2012. Date of publication September 24, 2012; date of current version De-cember 05, 2012. This work was supported by the DARPA/MTO Analog-to-In-formation Receiver Development Program under AFRL contract FA8650-08-C-7853. The views expressed in this paper are those of the authors and do not re-flect the official policy or position of the Department of Defense or the U.S.Government. This is in accordance with DoDI 5230.29, January 8, 2009. State-ment “A” (Approved for Public Release, Distribution Unlimited) [DISTAR case#18837].J. Yoo and A. Emami-Neyestanak are with the Department of Electrical Engi-

neering, California Institute of Technology, Pasadena, CA 91125 USA (e-mail:[email protected]).C. Turnes and J. Romberg are with the School of Electrical and Computer

Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA (e-mail:[email protected]).E. B. Nakamura and C. K. Le are with the Northrop Grumman Corporation,

Redondo Beach, CA 90278 USA (e-mail: [email protected]).S. Becker is with the Department of Applied and Computational Mathe-

matics, California Institute of Technology, Pasadena, CA 91125 USA, and alsowith the Laboratoire Jacques-Louis Lions, Paris 6 University, 75005 Paris,France.E. A. Sovero is with the Northrop Grumman Corporation, Redondo Beach,

CA 90278 USA, and also with Waveconnex Inc., Westlake Village, CA 91361USA (e-mail: [email protected]).M. B. Wakin is with the Department of Electrical Engineering and Computer

Science, Colorado School of Mines, Golden, CO 80401 USA.M. C. Grant is with the Department of Applied and Computational Mathe-

matics, California Institute of Technology, Pasadena, CA 91125 USA.E. Candés is with the Departments of Mathematics and Statistics, Stanford

University, Stanford, CA 94305 USA.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/JETCAS.2012.2214634

I. INTRODUCTION

A PRINCIPAL goal in the design of modern electronicsystems is to acquire large amounts of information quickly

and with little expenditure of resources. In the wireless tech-nology sector, the goal of maximizing information throughput isillustrated by the strong interest in radio-frequency (RF) sensingand spectral applications that require instantaneous bandwidthsofmany gigahertz. Such systems have applications ranging fromscientific instrumentation to electronic intelligence. Althoughsome solutions already exist, their large size, weight, and powerconsumption make more efficient solutions desirable.At present, realizing high bandwidth systems poses two pri-

mary challenges. The first challenge comes from the amountof power required to operate back-end analog-to-digital con-verter (ADCs) at the necessary digitization rate. This issue isso significant that the remaining elements of the signal chain(RF front-end, digital signal processing (DSP) core, etc.) areoften chosen based upon an ADC that is selected to be com-patible with the available power budget [1]. The second chal-lenge comes from need to store, compress, and postprocess thelarge volumes of data produced by such systems. For example,a system that acquires samples at a rate of 1 Gs/s with 10 bitsof resolution will fill 1 Gb of memory in less than 1 s. In lightof the ever growing demand to capture higher bandwidths, itwould seem that a solution at the fundamental system level isneeded to address these challenges.Some promise for addressing these challenges comes from

the theory of compressed sensing (CS) [2]–[6]. CS has recentlyemerged as an alternative paradigm to the Shannon-Nyquistsampling theorem, which at present is used implicitly in the de-sign of virtually all signal acquisition systems. In short, the CStheory states that signals with high overall bandwidth but com-paratively low information level can be acquired very efficientlyusing randomized measurement protocols. The requisite sam-pling rate is merely proportional to the information level, andthus CS provides a framework for sub-Nyquist rate signal ac-quisition. As we discuss further in Section II, aliasing is avoidedbecause of the random nature of the measurement protocol.The emergence of the CS theory is inspiring a fundamental

reconception of many physical signal acquisition and pro-cessing platforms. The beginning of this renaissance hasalready seen the redesign of cameras [7], medical imagingdevices [8], and RF transceivers [9]–[11]. However, the bene-fits of CS are not without their costs. In particular, the task ofreconstructing Nyquist-rate samples from CS measurementsrequires solving an inverse problem that cannot be addressedwith simple linear methods. Rather, a variety of nonlinear

2156-3357/$31.00 © 2012 IEEE

YOO et al.: A COMPRESSED SENSING PARAMETER EXTRACTION PLATFORM FOR RADAR PULSE SIGNAL ACQUISITION 627

algorithms have been proposed (see, e.g., [12]–[14]). Whilethe speed of these methods continues to improve, their com-putational cost can still be appreciably greater than manyconventional algorithms for directly processing Nyquist-ratesamples. This matter of computation, if not addressed, poten-tially limits the widespread application of CS architectures inpower constrained real-time applications.In this paper, we address these issues by presenting a com-

plete, novel signal acquisition platform (both hardware and soft-ware) that is capable—in certain applications—of estimatingthe desired signal parameters directly from CS measurements[15]. In the spirit of compressive signal processing [16], ourapproach takes the principal motivation of CS one step furtherand aims to eliminate the overhead of first reconstructing theNyquist-rate signal samples before applying conventional DSPtechniques for parameter extraction.On the hardware side, we present a fully integrated wide-

band CS receiver called the random-modulation pre-integrator(RMPI) [9], [10], [12], [17]. We fabricate this device withNorthrop Grumman’s 450 nm InP HBT bipolar process. Onthe software side, we focus on signal environments consistingof radar pulses and present a novel algorithm for extractingradar pulse parameters—carrier frequency (CF), phase ,amplitude , time-of-arrival (TOA), and time-of-departure(TOD)—directly from CS measurements. (The exact signalmodel is described in Section IV.)Our complete system is capable of recovering radar pulse pa-

rameters within an effective instantaneous bandwidth (EIBW)spanning 100 MHz–2.5 GHz with a digitizing performance ofeight effective number of bits (ENOB). We validate the systemby feeding the fabricated RMPI with radar pulses and using thephysically digitized CS measurements to recover the parame-ters of interest.An outline of this paper is as follows. Section II provides a

brief background on CS and a description of the high-level op-eration of the RMPI, Section III provides a complete descrip-tion of the hardware platform used to encode the CS samples,Section IV provides details of the parameter estimation algo-rithms, and Section V presents measurement results.

II. THE RMPI

A. Compressed Sensing

CS at its heart relies on two concepts: sparsity and incoher-ence [5]. Sparsity captures the idea that many high-dimensionalsignals can be represented using a relatively small set of coeffi-cients when expressed in a properly chosen basis. Incoherencecaptures the idea of dissimilarity between any two representa-tions; two bases are said to be incoherent if any signal havinga sparse expansion in one is dense in the other. An example ofan incoherent pair comes from the classical time-frequency du-ality. A sparse signal in time—e.g., a Dirac-delta function—hasa dense spectrum. Similarly, a single tone is sparse in the Fourierdomain but dense in time.The key observation underlying CS is that when a signal is

sparse in some basis, it can be acquired by taking a small numberof measurements that are incoherent with its sparse basis [18].Often this incoherence is achieved by incorporating randomness

into the measurement process. There are many possibilities forimplementing incoherent random measurements; a convenientand admissible choice for hardware implementation is to cor-relate the input signal (in our case, a time-windowed version ofthe input signal) with a pseudo-random binary sequence (PRBS)[4]. We refer the reader to [5] and references therein for addi-tional information about the mathematical theory of CS.

B. A Brief History and Description of the RMPI

Almost simultaneously with the introduction of CS [2], anumber of CS-based signal-acquisition architectures were pro-posed. Some of the more well-known proposals include: therandom demodulator (RD) [17], [19], [20], the random-modula-tion preintegrator (RMPI) [9], [10], [21], the nonuniform sam-pler (NUS) [4], [22], random convolution [23], the modulatedwideband converter (MWC) [24], and many others [25] (for acomprehensive overview see [12]). The basic function that all ofthese systems implement is to correlate of the input signalwith an incoherent, randomly generated set of “basis” elementsover a fixed time window.The RMPI is one of the most direct physical implementa-

tions of the CS concept; it is composed of a parallel set of RDsdriven by a common input (see Fig. 1(a), which is describedmore fully in Section III). Each RD is driven by a distinct PRBS

; it uses this PRBS to modulate the incoming signal ,integrates the result over a time interval of duration , andthen digitizes the output at rate . Inour RMPI, with .Thus, ; the aggregateback-end sampling rate , which correspondsto undersampling the Nyquist rate by a factor of 13. Aliasing isavoided in this measurement scheme because 1) the modulationwith the PRBS will spread the spectrum of any tone (includinghigh-frequency ones) across the entire band so that one can ef-fectively subsample, and 2) the input signal is again assumed toobey some model (aside from merely being bandlimited).Letting denote a time-windowed vector of Nyquist-rate

samples of the input signal , we can implicitly model theRMPI measurement process as multiplication of by a matrixhaving 13 fewer rows than columns. Each row of this

matrix corresponds to a portion of the PRBS sequence used ina specific integration window from a specific channel. As anexample, if we consider a sample vector of length ,the matrix will be block-diagonal, with each block havingfour rows (representing the parallel operation of the fourchannels) and columns (representing an inte-gration window of 52 Nyquist bins). The rows of each blockcontain 1 entries, and the overall matrix will be composed of

blocks (one for each integration window).Denoting the vector of measurements as , the RMPI mode ofacquisition can be modeled as where .We point out that high-fidelity recovery/extraction of infor-

mation from CS measurements requires precise knowledge ofthe system transfer function . Thus, practical deviations fromthe block-diagonal 1 model described above must be takeninto account. For the measurements presented in this paper, weconstruct a model of our system’s matrix by feeding in sinu-soidal tones and using the output measurements to characterizethe system’s impulse response.


Fig. 1. (a) Simplified block diagram of 4-channel RMPI. The analog-signalpath of each RD channel is identical, however the timing signals they receivein operation are different. (b) Functional diagram of the mixer and integratorcircuits. (a) System block diagram. (b) RD channel block diagram.

III. HARDWARE IMPLEMENTATION DESCRIPTION

A. Architecture and Operation

The RMPI presented in this work was realized with theproprietary Northrop Grumman (NG) 450 nm InP HBT bipolarprocess [26]. The process features a four-layer metal stack withan and . Fig. 1(a) shows the block diagramof the integrated circuit (IC) containing the input buffer drivingthe common node of the four RD channels and the timing gen-erator. The timing generator is responsible for generating thepseudo-random bit sequences (PRBS) and the clocking wave-forms to coordinate the track-and-hold (T/H) and integrationoperations. All analog and digital signal paths are implementeddifferentially to improve common-mode noise rejection andincrease linearity of the system. The analog path up to theintegrator was designed for a 2.5 GHz bandwidth. The ensuingintegration reduces the bandwidth containing significant energycontent. The circuits following the integrator are designed tomeet the settling requirements of the reduced bandwidth. A5 GHz master clock reference (CLKin) is used to toggle thePRBS generators and is chosen to be the Nyquist-rate of theinput bandwidth [12], [17]. The T/H operates at 1/52 the master

Fig. 2. Simplified schematic of interleaved switched capacitor integrator. Thediodes act as switches to configure capacitors for integration or reset it basedon the level of the control signal (SEL). When SEL is asserted, integrator A isresetting and integrator B is integrating. When SEL is de-asserted, integrator Bis resetting and integrator A is integrating.

clock frequency (96.154 MHz). A switched-capacitor inter-leaving integrator [27] is used so that one capacitor can be resetwhile the second integrates the mixer output. Finally, an outputbuffer is designed to drive the ADC with the correct swing andcommon-mode voltage. The chip was designed for a full-scaleinput amplitude of 0.5 differential and 1 differentialoutput. In operation, the RMPI circuit takes the analog inputsignal, buffers it, and distributes the buffered signal to eachof the four channels. In each channel, the signal is multipliedby one of four orthogonal PRBS—each of which is a 3276 bitlong Gold code [28]. The resulting product is integrated by oneof two sets of interleaved capacitors for exactly one frame (52CLKin cycles). At the end of the integration period the signal issampled and then held for 26 CLKin cycles to allow the externalADC to digitize the signal for postprocessing. Immediatelyafter the signal is sampled, the capacitor begins dischargingand the second capacitor begins integrating the next frame [seeFig. 1(b)]. The interleaved integration capacitors are used toavoid missing frames due to the reset operation. Additionally,the sampling instants for each channel are staggered to createmore diversity in the windowed integrations obtained.

B. Analog Signal Path

The input buffer is a differential pair with emitter degener-ation and 50 termination at each single-ended input. It hasa gain of 3 dB, a 2.5 GHz bandwidth, 70 dB SFDR, and afull-scale differential input amplitude of 0.5 . The randommodulation is performed by a standard differential Gilbertmixer with the PRBS generator driving the top two pairs andthe analog input driving the bottom differential pair. Emitterdegeneration is used on the bottom differential pair to improvelinearity. To reduce noise, the mixer was designed to haveabout 20 dB gain to compensate for the attenuation from the in-tegrator. The output of the mixer is integrated using interleavedswitched capacitors as shown in Fig. 1(b) and Fig. 2 and hasa pole/cutoff frequency located at 12.5 MHz. Diode switchesroute the mixer output current to the integration capacitor, readout the capacitor voltage, and reset it to zero at the end of a


Fig. 3. Simplified block diagram of PRBS/timing generator. Shown are (4)quadrature clocks for the S/H and (4) control signals for the interleaved capaci-tors. Combinational logic is used to prevent the illegal start up condition of thePN generators and to generate divide by 52 and divide by 104 used as controlsignals in the S/H and in the interleaved integrators respectively.

cycle. Operation of the switched integrator is controlled bya capacitor select signal (SEL). When SEL is asserted isreset and acts as the integrating capacitor, when SEL isde-asserted the roles of and are reversed. At the endof each integration cycle one frame , the output ofthe integrator is sampled by the T/H and held for 5.2 ns. Thisensures that the external ADC has enough time to digitize theheld voltage.The T/H was implemented using a switched emitter fol-

lower topology with gain 1. To minimize the hold-modefeed-through, small feed-forward capacitors were inserted [29].The switched emitter follower was chosen in favor of the moreconventional diode bridge because of its smaller footprint andcomparatively low parasitic capacitance. The amplifier after theT/H has a gain of 2. In addition to emitter degeneration, diodeconnected transistors are used in the output load to cancel theinput differential pair modulation and improve linearity.The output driver was designed to be dc-coupled to the externalADC and has 70 dB SFDR and a 1 swing.

C. PRBS and Timing Generator

A master clock is applied to CLKin, from which all requiredtiming signals are generated. The input clock buffer was biasedwith a relatively high power to reduce jitter. In addition, it hasfour separate output emitter followers to mitigate the deleteriouseffects of cross-talk on the clock jitter. Each emitter followerprovides a low jitter signal to reclock the PRBS input before itis mixed with the RF input in each channel.The PRBS signals are generated with two 6 bit PRBS gener-

ating linear-feedback shift-registers (LFSR(s)) [30]. One PRBSgenerator (PN6A in Fig. 3) is programmed to cycle every 52CLKin cycles while the second (PN6B in Fig. 3) is allowed tocycle through all 63 states. The 2 PRBS generator outputs arecombined to generate four orthogonal longGold code sequences. PN6A is also used to generate the T/Hclocks (divide by 52) and select signal (divide by 104) for theswitched capacitor integrator. Both PN generators also outputa sync pulse used to synchronize the system. The output pulsefrom PN6B is reclocked with the pulse from PN6A to produce

Fig. 4. RMPI IC die photo. Die size is 4.0 mm 4.4 mm.

a sync pulse that is 52 CLKin cycles long once every 3276cycles. The synchronization pulse is essential to provide pre-cise knowledge of the chipping sequence used in each integra-tion window. This relative alignment information is crucial forsignal-recovery and parameter estimation.CLKin and the RF input are located on opposite sides of the

chip to minimize coupling. Special attention was paid to therouting of the PRBS, T/H clocks, and select signals to mini-mize clock/data coupling among the four channels. A simplifiedblock diagram of the PRBS/timing generator is shown in Fig. 3.The timing generator block was designed to operate at speedsin excess of 5 GHz and consumes 2.8 W when operated at thedesigned rate.

D. Performance Analysis

Simulation validation was done by performing tran-sient-based two-tone inter-modulation distortion simulationsin the Cadence design environment. Noise simulations wereperformed using the periodic steady-state (PSS) mode ofspectre. The RMPI sampling system, including the off-chipADCs consumes 6.1 W of power. We point out that this systemwas designed as a proof-of-concept and was not optimized forpower. Thus, caution should be used when comparing the CSsystem in this work to conventional counterparts. For example,the use of an InP process in this work leads to power penaltiescompared to the CMOS RMPI (which consumes 0.5 W) re-ported in [9] and [10], which is also similarly unoptimized, dueto the availability of static logic. A die photo of the fabricatedchip is shown in Fig. 4.

IV. PULSE-DESCRIPTOR WORD EXTRACTION

Having described the hardware, we now present algorithmsfor detecting radar pulses and estimating their parameters, re-ferred to as pulse-descriptor words (PDW), from RMPI sam-ples. The detection process is based on familiar principles em-ployed by detectors that operate on Nyquist samples. Our algo-rithms use a combination of template matching, energy thresh-olding, and consistency estimation to determine the presence of


pulses. By using all three of these methods, we gain robust de-tection at the cost of a number of tunable parameters that areset based upon the specific application and sensing equipment.The general procedure consists of three steps. First, we estimatethe carrier frequency and energy of a potential pulse segment atvarious time shifts. Second, based on consistency in frequencyestimates and large enough pulse energies, we apply criteria todetermine if a pulse is present. Finally, for detected pulses weuse our parameter estimation methods to refine our carrier fre-quency, amplitude, phase, time-of-arrival, and time-of-depar-ture estimates.The remainder of the section elaborates on the procedure and

is arranged as follows. First, we describe our methods of para-metric estimation, focusing in particular on carrier frequencyestimation. After describing how we can reliably estimate thecarrier frequency of a signal from compressive measurements,we then explain howwe use such estimations to form a detectionalgorithm that jointly uses energy detection and consistency ofour frequencymeasurements.We then describe howwe performparametric estimation on compressive samples while simulta-neously removing a band in which a known interfering signalis present. Finally, we combine the detection algorithm we haveformulated with the cancellation technique to present an algo-rithm capable of detecting multiple overlapping pulses.

A. General Parametric Estimation

Our general parameter estimation problem can be stated asfollows. We consider signals drawn from one of a col-lection of (low-dimensional) subspaces indexed by a pa-rameter set . Given the measurements

, we search for the set of parameters corre-sponding to the subspace which contains a signal which comesclosest to explaining the measurements . We solve

(1)

The inner optimization finds the signal in that is most consis-tent with the measurements for a fixed ; the outer optimizationcompares these best fits for different .The inner optimization program, which is the classical

“closest point in a subspace” problem, has a well-known closedform solution as it is easily formulated as a least-squaresproblem.Let be a basis for the space ,

meaning that

for some unique . If we define to be thematrix containing the inner products between each pair

of RMPI test functions and basis functions

......

...(2)

then we can rewrite (1) as

(3)

where is the orthogonal projector ontothe column space of . It is worth mentioning that when themeasurement noise consists of independent and identically dis-tributed Gaussian random variables, the result in (3) is themaximum likelihood estimate (MLE). When the noise is cor-related, we may instead pose the optimization in terms of aweighted least squares problem.In Sections IV-B and IV-C below, we will discuss the partic-

ular cases of frequency estimation for an unknown tone, andtime-of-arrival estimation for a square pulse modulated to aknown frequency. In both of these cases, we are trying to es-timate one parameter and the underlying subspaces have di-mension . Moreover, the functional can beefficiently computed over a fine grid of values for using thefast Fourier transform (FFT).When there are multiple parameters, performing the joint

minimization over can be computationallyprohibitive. In Section IV-D, we present a heuristic algorithmfor estimating the key parameters of a Doppler pulse (carrierfrequency, time-of-arrival, and pulse length) that operates bylooking for consistent frequency estimates over consecutivewindows of time. The frequency and time-of-arrival estimatorsplay a central role in this algorithm.In Section IV-F, we show how the algorithm can be modi-

fied to accommodate overlapping pulses and strong narrowbandinterferers.

B. Carrier Frequency, Amplitude, and Phase

In this section, we consider the task of estimating the fre-quency of a pure (Doppler) tone from the observed RMPI mea-surements. The algorithms developed here play a central role inthe detection process as well, as the frequency estimation pro-cedure is fundamental to determining the presence of pulses.We also describe how to estimate the amplitude and phase oncethe carrier frequency (CF) is known. A similar technique can beused to estimate the amplitude and phase while estimating thetime of arrival (TOA) and time of departure (TOD) as well.For our CF estimation task, we suppose that we observe an-vector (consisting of a concatenation of all samples from

the RMPI device over a certain interval) given by, where consists of Nyquist samples of a sinusoid

and is the Nyquist frequency. The amplitude , phase, and frequency of the sinusoid are a priori unknown, and

we make the implicit assumption that the TOA corresponds toand the TOD corresponds to .


Fig. 5. Plot of the energy function for measurements derived from anoisy Doppler tone at 1.4567 GHz over (a) the entire range of allowable fre-quencies and (b) frequencies close to the true CF. In (a) we see that the energyfunctional is clearly maximized in an area near the true CF, and (b) shows thatthe maximum of the energy function occurs at 1.4571 GHz. For this example,we used a total of 3315 Nyquist samples with a sampling rate of 5 GHz, so ourintrinsic frequency resolution is on the order of . Theestimate of the carrier frequency is well within this resolution.

It is convenient to rewrite as a weighted sum of a cosineand sine with zero phase

(4)

where we can relate to by realizing that they are thereal and complex parts of the phasor .The subspaces we search are thus spanned by two vectorsand with and .Since we have discretized the signal through its Nyquistsamples, our measurement process is modeled through a matrix, the rows of which are the basis elements . For a given fre-quency , we define as in (2) and solve (3) to obtain theMLEestimate of the basis expansion coefficients in the subspace .Rather than dealing with continuously variable frequency, we

define a fine grid of frequencies between 0 and . As anequivalent alternative to minimizing the outer optimization in(1), for each in the grid (and corresponding subspace ) wemay instead compute the quantity

and choose the frequency that maximizes .In practice, since we obtain values for on a grid of fre-

quencies, we can realize tangible gains by treating these valuesas samples of a continuous function and interpolating inbetween the samples of once the maximum has been lo-calized; that is, we can “super-resolve” using cubic in-terpolation. Fig. 5 illustrates the function for an exampleDoppler tone.Once the carrier frequency is estimated as , we estimate the

tone amplitude as and the tone phase as

.

C. Time of Arrival/Departure

Next we describe how the TOA of a pulse can be estimated.We now assume that we have RMPI samples of a pure tone ata known frequency windowed by a step function. That is, we

Fig. 6. Plot of the energy function formeasurements derived from a noisyDoppler tone at 1.4567 GHz arriving at Nyquist sample over (a) theentire range of sample indices and (b) sample indices close to the true TOA. In(a) we see that the energy functional is clearly maximized in an area near thetrue TOA, and (b) shows that the maximum of the energy function occurs at

. Since the sampling rate for this example is 5 GHz, this correspondsto an error of 400 ps.

observe an -vector , where has theform

(5)

and is the unit step, for and is zero for. While we assume that the frequency is known (or has

been estimated as in Section IV-B), the amplitude, phase, andTOA are unknown. The processes we describe are perfectlyanalogous for TOD estimation if we consider a “flipped” versionof .As in (4), we can write as a weighted sum of a windowed

sine and cosine

Then for any given candidate TOA , the subspace isspanned by the vectors

and

We can again construct the matrix of (2) and complete theMLE estimate of by solving (3) and subsequently (1)Again, rather than deal with continuously variable we de-

fine a grid of times . In practice, using the grid of Nyquistsample locations is sufficient. As in the case of car-rier frequency, rather than choose the subspace with the smallestvalue of the norm in (1), we instead solve (3) for each grid pointand then select the subspace yielding the largest value of

which is an equivalent solution. We may once again super-re-solve by treating these values as samples of a continuous func-tion. Fig. 6 illustrates the function for an example Dopplertone.


Fig. 7. Fraction of measurement energies that are explained by frequencies upto 2.5 GHz for the case where (a) there is a 1.581 GHz tone and noise presentand (b) there is only noise present. The noise energy is equally spread out overthe band, where the tone energy is concentrated at one frequency.

D. Pulse Detection

With our estimation techniques explained we next describea pulse detection algorithm that takes a stream of RMPI sam-ples and classifies each as either having a “pulse present” or“no pulse present.” We start by assuming that only one pulse ispresent at any given time. In the next section, we will describean extension of this algorithm that can account for multiple si-multaneous pulses.We model the pulses we are trying to detect as continuous

functions of the form

where is the pulse amplitude, the TOA, the TOD,is the rectangular window

, the CF, and the phase.We assumethat the input to our acquisition system consists of the Nyquistsamples of some number of these pulses; that is

where is the sampling period. The vector of RMPI measure-ments we receive is the result of a linear matrix applied to thisinput together with additive noise: noise. Our task isto determine how many pulses are present and to estimate theparameters of each pulse we find.We use the CF estimation method of Section IV-B as a

building block. The general approach of the detection algo-rithm is to divide the RMPI samples up into overlapping blocksand estimate how well the measurements corresponding to eachblock can be explained by the presence of a single tone. This isdone by assuming a pulse is present in each block, estimatingits CF, and determining how well the observed measurementsagree with measurements generated by this pulse. As is shownin Fig. 7, if a pulse is indeed present at the estimated CF, itshould account for a reasonably large portion of the measure-ment energy.Once we obtain CF estimates, we look for consistency from

block to block. If neighboring blocks have CF estimates thatare: consistent in value and account for a considerable portionof their measurement energies, then we declare a pulse present.If the CF estimates vary across neighboring blocks or contain

Fig. 8. Carrier frequency estimates and measurement energy percentages forblock shifts of RMPI measurements. In this simulation, several pure-tone pulsesare present at various times. The blocks corresponding to RMPI samples thatcover the time support of the pulse contain consistent CF estimates which ac-count for a reasonably large portion of the RMPI measurement energy. Whenpulses are absent, the CF estimates are erratic and account for considerably lessenergy in the measurements.

insufficient energy in a single frequency, we consider there tobe insufficient evidence that a pulse is present. Fig. 8 illus-trates consistency in frequency estimation and energy propor-tion when pulses are present. In practice, the blocks near the endof a pulse may account for a smaller percentage of their mea-surement energies with a single tone, since they may containsome RMPI samples that correspond only to noise. To coun-teract this effect we use a weighted average across blocks con-sistent in their CF estimates.Once we have obtained pulse detections, we can further re-

solve their parameters. For example, suppose we have detecteda pulse that starts at RMPI sample index and has length . Wecreate a vector with the following RMPI samples:

While we are uncertain of the exact TOA and TOD of thepulse, the middle portion of the detected pulse (correspondingto samples to ) almost certainly containsonly samples where the pulse is active. We use these samplesto refine and super-resolve our carrier frequency estimate.Empirically, we find that our initial estimates for the TOAand TOD can often dramatically under/overestimate the correctvalues. Accordingly, we use the samples toto refine and super-resolve our TOA estimate and to

to refine and super-resolve our TOD estimate.For longer pulses (large ), this means that we check a largernumber of potential locations for our TOA and TOD estimates.We could instead fix a certain number of samples before andafter each pulse detection to check, invariant of the detectedpulse length, but we have found experimentally that makingthe search length dependent on the detected pulse length resultsin more accurate estimation. Once the TOA, TOD, and CFestimates have been refined, we calculate our amplitude andphase estimates.Supposing we have a stream of RMPI measurements

over some time period, where each is a vector


of RMPI samples at a given sample time, the completedetection algorithm is outlined in Algorithm 1.

Algorithm 1 RMPI Pulse Parameter Extraction Algorithm

1: Choose a set of block lengths, each of which correspondsto a certain number of RMPI samples per channel.

2: For fixed block length and for each sample time, find the measurement vector containing

the samples from timeand the corresponding portion of the measurement matrix.

3: Estimate the CF for each block by following the procedurein Section IV-B, calling the frequency estimate andthe corresponding fraction of the energy it accounts for

.4: Determine for which blocks is below somethreshold. Call each sequence of blocks with consistentfrequency estimates a segment.

5: In each segment, take a weighted average of thevalues. Keep all segments for which this average exceedsa certain threshold.

6: Repeat steps 2–5 for different values of , creating a listof potential pulse segments for each block length.

7: Merge any segments that are close together in time andhave similar CF estimate.

8: Keep only the remaining segments in the merged list thatare longer than a predetermined minimum signal length.

9: Super-resolve the amplitude, phase, TOA, and TODcan then using the techniques in Section IV-B andSection IV-C.

E. Complexity

For the measurement system we consider, consists ofshifted repetitions of a matrix that produces 63 separatefour-channel RMPI samples (for a total of 252 rows). For CFestimation, we can precompute the product foreach frequency in our grid by taking the real and imaginaryparts of the FFT of each row of . In fact, since containsrepetitions of we need only take the FFT of the 252 rowsof , which we may then shift through complex modulation.This calculation does not depend on the measurement data, andtherefore can be done offline as a precomputation.For each window length (typically between 5–11 RMPI

samples) and each window shift, we have to compute foreach in the grid. For a given frequency , we have precom-puted the matrix . We have samples (since there are 4channels per measurement), so is and is a 22 system. We can explicitly invert using the 2 2 ma-trix inverse formula, and all other calculations involve a smallnumber of -point inner products. The number of frequencieswe test is proportional to the number of Nyquist samples ,and thus the cost of the frequency estimation for each slidingwindow shift is . Since the number of window shifts isequal to the number of RMPI samples, this is our per-sample

computational cost. If we use multiple window lengths, ourcomplexity is .

F. Cancellation and Multiple Pulses

In this section, we focus on how to remove contributions fromcertain frequency bands in the RMPImeasurements. This allowsus to remove the effects of interferers that occupy a known, fixedfrequency band. It also allows us to remove the contributions ofpulses that we have already detected in order to detect additionalpulses that may occur at simultaneous points in time.We assume that we have a signal interfering with the under-

lying signal whose parameters we wish to estimate, and that thisinterferer has energy only in a specific, fixed, and known fre-quency band. Our procedure for nulling the interfering signal in-volves the computation of discrete prolate spheroidal sequences(DPSS) [31], [32].1 A DPSS is essentially a set of functions thatcan best express signals of specified duration whose frequencycontent is restricted to a certain bandwidth. These can be mod-ulated to be centered at any frequency, and thus can serve asa basis for compact signals whose energy occupies a specificband.Suppose we have a DPSS with elements that serve as a

basis for signals of length . We can express the DPSS as anmatrix , whose columns are the elements of the

sequence. Then the interfering signal can be written as a linearcombination of these elements for some . Thecontribution from the interfering band in a set of measurementscan be modeled as

Then we can estimate the portion of the measurements thatcorrespond to frequency content outside the interfering band as

The operator can be used to remove the contributions of theinterfering band in the measurements when we run our esti-mation methods. This allows us to effectively operate as if theinterferer is absent. The matrix does not depend on the mea-surements but merely the RMPI matrix , and therefore canbe precomputed. Fig. 9 shows how the use of the nulling oper-ator aids in removing interfering bands.We can use similar concepts to detect multiple overlapping

pulses. After the detection algorithm of Section IV-D, we canrun a second pass of the algorithm. During this second pass,when we calculate the CF estimates for each block we do so byfirst constructing a nulling operator that cancels the contri-butions of any of the previously detected pulse segments that arewithin the block and then perform CF estimation using the mod-ified measurements .When we compute the metric ,we use in place of so that the quantity expresses thefraction of the energy of the potential pulse relative to the nulledmeasurements.The nulling operators that we construct will cancel a band,

rather than a single frequency. The bandwidth and block lengthaffect the number of elements in the DPSS that we require to

1These are also known as Slepian sequences.


Fig. 9. Fraction of measurement energies that are explained by frequencies upto 2.5 GHz for the case where (a) no nulling is used and there is strong interferingsignal between 1.976 and 2.026 GHz and (b) the nulling operator is used tocancel out the interfering band. The nulling operator allows us to estimate the CFof the underlying tone, which is 1.367 GHz. In (b) the energy in the interferingband has virtually disappeared.

Fig. 10. Stages of the pulse detection and parameter estimation algorithm foroverlapping pulse data: (a) source pulses; the pulse heights indicate their ampli-tude and the color of the pulse is reflective of the CF of the pulse, with red closerto 2.5 GHz and blue closer to 0 Hz; (b) CF estimates for blocks using 5 RMPIsamples; (c) segments detected based on consistent CF estimates for blocks ofsize 5 with no phase or amplitude estimation; (d) CF estimates for blocks of size9; (e) segments detected based on consistent CF estimates for blocks of size 9with no phase or amplitude estimation; (f) the merged segments; (g) the mergedsegments after the second pass is completed; (h) the detected pulses with refinedestimates of their parameters.

cancel within a certain accuracy. Since the DPSS is designed tocancel a band rather than a single frequency, we must make anassumption as to how close in CF two simultaneously overlap-ping pulses are allowed to be; for our 5 GHz system we assumethat overlapping pulses are at least 10 MHz apart in their carrierfrequencies. Fig. 10 shows an example of the two-stage detectorfor overlapping pulse data.The additional cost introduced by nulling detected pulses

is dominated by the computation of the DPSS functions. This

Fig. 11. Assembled RMPI IC/digitizer interface. The board is 5 in 5 in. TheADC board has 4 12 bit ADCs with output bits routed to 4 data-connectorsthat are acquired with a logic analyzer. (a) RMPI IC Mount. (b) RMPI digitizerboard.

Fig. 12. Block diagram of RMPI test setup.

TABLE IMAXIMUM, MINIMUM, AND STANDARD DEVIATION OF ESTIMATION ERRORSFOR CF, TOA, AND TOD OVER 686 TRIALS (

IS THE LENGTH OF ONE INTEGRATION WINDOW)

computation is dependent on the detected pulse length. How-ever, if it is known that the pulses we are to detect are of aspecific length, it may be possible to precompute the DPSSfunctions. In this case, the nulling produces introduces fewextra calculations.

V. VERIFICATION OF HARDWARE

A. Measurement Test Setup Description

In order to test the performance of the radar-pulse param-eter estimation system-composed of the RMPI sampling hard-ware and the PDW extraction algorithm Section IV, we ran a


Fig. 13. Parameter estimation errors for (a) CF, (b) TOA, and (c) TOD over 686 trials.

TABLE IIDETECTION PERFORMANCE AS A FUNCTION OF THE INTERFERER STRENGTH

TABLE IIIDETECTION RATE AND STANDARD DEVIATION OF THE PARAMETER ESTIMATE ERRORS AS A

FUNCTION OF PULSE AMPLITUDES Frame

TABLE IVDETECTION RATE AND STANDARD DEVIATION OF THE PARAMETER ESTIMATE ERRORS AS A

FUNCTION OF PULSE LENGTHS Frame

set of over 686 test radar pulses composed of permutations of, , CF, TOA, and TOD through the RMPI and estimated

the varied parameters from the compressed-samples digitizedby the RMPI.Fig. 12 shows a block diagram for the test setup used for

the RMPI. The input clock and data to the RMPI were drivendifferentially and ac-coupled. An arbitrary waveform generator(AWG) with an output sampling rate of 10 Gs/s was used tooutput the pulses of interest. The stimulus was input into theRMPI whose outputs were then sampled by external ADCs lo-cated on a custom digitizing PCB shown in Fig. 11(b): the RMPIIC is mounted on a low-temperature co-fired ceramic (LTCC)substrate shown in Fig. 11(a) which is placed in the center of thedigitizing board. The digitized samples were then transferred toa PC where the PDW extraction algorithm was used to estimatethe signal parameters.

B. Parameter Estimation

For each pulse, we estimated the CF only frommeasurementscorresponding to times when the signal was active.We then esti-mated the TOA from RMPI samples corresponding to noise fol-lowed by the front end of the pulse. We repeated the procedurefor theTOD,usingRMPI samples corresponding to the endof thepulse followedbynoiseonly.Fig.13shows thedistributionofourestimation errors for CF, TOA, and TOD. Additionally, Table Ishows statistics on the errors for each of the three parameters.

C. Pulse Detection

We tested the pulse detection system by generating 60 testcases containing 12 pulses each (for a total of 720 pulses) withvarying amplitudes (ranging over 60 dB), phases, durations (100ns–1 ), carrier frequencies (100 MHz–2.5 GHz), and over-laps. All pulse rise times were approximately 10 ns. The pulse


amplitudes were taken from a set of six discrete values, with 120pulses at each amplitude level, while the durations were takenfrom a set of three discrete values, with 240 pulses at each pulselength. We then ran the detector on each data capture and col-lected the detection statistics. The detector successfully detected691 of the 720 pulses for a detection rate of 95.97%, while alsoallowing 23 false positives for a false positive rate of 3.22%.Table III shows the detection rate and standard deviation of the

parameter estimate errors as a function of the pulse amplitudes.Aside from the lowest-amplitude pulses, the detector’s perfor-mance is relatively invariant to the pulse amplitude. Table IVsimilarly shows the detection rate and standard deviation of theparameter estimateerrors asa functionof thepulse length.As is tobe expected, the pulse detection rate improves as the pulse lengthgrows. The TOA and TOD estimates are worse for longer pulsesthan shorter pulses, but this is to be expected; since the pulses arelonger, there are more potential locations to check, and thereforethepossibility of error increases.However, it is surprising that theCF estimates get slightly worse as the pulse length increases.

D. Interferer Cancellation

To test the robustness of the detection and estimation system,we repeated our detection experiment and included a constant-frequency interferer at set amplitudes in each experiment. Wetested six interferer strengths, running 60 experiments with 12pulses per experiment, for a total of 720 pulses per interfererstrength. In each case, all of the pulse amplitudes were the same(to keep the relative interferer strengths well-defined) and thepulses were allowed varying amounts of overlap. For each ex-periment, we assumed that we knew the center frequency of theinterfering bandwidth, and that the interferer occupied a bandwith a 25 MHz width. Table II shows the detector performanceas a function of interferer strength. When the interference isvery small in magnitude, the estimation is predictably better. Asthe interferer grows in strength, the performance only degradesslightly.

VI. CONCLUSION

We have presented a detailed overview of the design of bothhardware and software used in a novel radar-pulse receiver inwhich information is extracted without performing full signalreconstruction. This novel approach obtains desired informationwith high accuracy while considerably reducing the back-endcomputational load. The reduced computational load for param-eter extraction potentially expands the applicability of CS-basedsystems, particularly for real-time processing.The system was validated using parameter estimates obtained

from testing with a large and exhaustive set of realistic radarpulses spanning the parameter space. The physically measuredresults generated from this prototype proof-of-concept systemdemonstrates the feasibility of the approach. In addition, the dataobtained provides ample motivation for further investigation ofthe merit of CS-based signal acquisition schemes in general.

ACKNOWLEDGMENT

The authors would like to thankDr. D. Healy for his foresight,encouragement, and support of this endeavour.

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Juhwan Yoo received the B.S. and Ph.D. degrees inelectrical engineering from the California Instituteof Technology, Pasadena, in 2006 and 2012, respec-tively. He is currently a postdoctoral researcher in theDepartment of Electrical and Computer Engineeringat Duke University, Durham, NC.During the summer of 2004 he held a summer in-

ternship at Finisar Corporation, where he worked onmulti-mode optical transceivers. His current researchinterests include: mixed-signal circuit design, analogsignal processing, and the application of dimension-

ality reduction techniques to hardware signal processing platforms.Mr. Yoo was the recipient of the Analog Devices Outstanding Student De-

signer award in 2009.

Christopher Turnes received the B.S. degreein electrical and computer engineering from theUniversity of Maryland, College Park, in 2003. Heis currently a graduate student in the Electrical andComputer Engineering Department at the GeorgiaInstitute of Technology, Atlanta.His research interests include structured linear al-

gebra for signal processing and 3-D MRI.

Eric B. Nakamura received the B.S., M.S., andPh.D. in electrical engineering, all from the Univer-sity of Hawaii, Manoa.He joined Northrop Grumman Aerospace Systems

(NGAS)—Mixed Signal and Power Center in 2002and is the current principle investigator for the NGASA-to-I program.

Chi K. Le received the B.S. degree in electrical engi-neering from California State University, Northridge,and theM.S. degree fromUniversity of Southern Cal-ifornia, Los Angeles.She has worked on several mixed signal circuits

like ADC, Frequency Synthesizer at NorthropGrumman, Mixed Signal and Power Product Groupsince 1995. Her particular interest is the hardwareimplementation of compressive sensing receivers.

Stephen Becker received the B.A. degree in math-ematics and in physics from Wesleyan University,Middletown, CA, in 2005, and the Ph.D. degree inapplied and computational mathematics from theCalifornia Institute of Technology, Pasadena, in2011.He is currently a postdoctoral researcher hosted

at the Laboratoire Jacques-Louis Lions in Paris6, thanks to a fellowship by Foundation SciencesMathnieur from the Ecole Polytechnique in 1993.His research interests are in computational harmonic

analysis, mathematical signal processing, information theory, mathematicaloptimization, statistical estimation and detection with applications to theimaging sciences, scientific computing, and inverse problems.

Emilio A. Sovero received the B.S. degree in engi-neering, the M.S. degree in mechanical engineering,and the Ph.D. degree in applied physics and informa-tion science, all from the California Institute of Tech-nology, Pasadena.He is currently VP of Technology at Waveconnex

Inc. (October 2011). Prior to his current position, in2006 he joined Northrop Grumman Aerospace Sys-tems (NGAS)—Mixed Signal and Power Center asChief Scientist. He was the Program anager at NGASof the A-to-I development.

Michael B.Wakin received the B.A. degree in math-ematics, in 2000, and the B.S., M.S., and Ph.D. de-grees in electrical engineering, in 2000, 2002, and2007, all from Rice University, Houston, TX.He was an NSF Mathematical Sciences Postdoc-

toral Research Fellow at the California Institute ofTechnology, from 2006 to 2007 and an Assistant Pro-fessor at the University of Michigan from 2007 to2008. He is now an Assistant Professor in the Depart-ment of Electrical Engineering and Computer Sci-ence at the Colorado School of Mines.

Dr. Wakin shared the Hershel M. Rich Invention Award from Rice Universityfor the design of a single-pixel camera based on compressive sensing in 2007.In 2008, he received the DARPA Young Faculty Award for his research in com-pressive multi-signal processing and in 2012, he received the NSF CAREERAward for research into dimensionality reduction techniques for structured datasets.

Michael C. Grant received the B.S. degree in elec-trical and computer engineering from the Universityof Texas, Austin, and the M.S. and Ph.D. degreesin electrical engineering from Stanford University,Stanford, CA.He is a principal at CVX Research, Inc., and the

primary developer of CVX, a popular modelingframework for disciplined convex programming.The research described in this paper was conductedwhile he served as Staff Scientist in the Departmentof Applied and Computational Mathematics at the

California Institute of Technology. He has served as a Guest Lecturer at theUniversity of Texas at Austin, as a Consulting Assistant Professor in theInformation Systems Laboratory at Stanford University, and as a ResearchAssociate in the Department of Energy Resources Engineering at StanfordUniversity.


Justin Romberg received the B.S.E.E., M.S., andPh.D. degrees from Rice University, Houston, TX, in1997, 1999, and 2004, respectively.He is an Associate Professor in the School of Elec-

trical and Computer Engineering at the Georgia Insti-tute of Technology, Atlanta. From Fall 2003 until Fall2006, he was a Postdoctoral Scholar in Applied andComputational Mathematics at the California Insti-tute of Technology, Pasadena. He spent the Summerof 2000 as a researcher at Xerox PARC, the Fall of2003 as a visitor at the Laboratoire Jacques-Louis

Lions in Paris, and the Fall of 2004 as a Fellow at UCLA’s Institute for Pure andAppliedMathematics. In the Fall of 2006, he joined the Electrical and ComputerEngineering faculty as a member of the Center for Signal and Image Processing.During 2006 to 2007, he was a consultant for the TV show “Numb3rs”.Dr. Romberg received an ONR Young Investigator Award, in 2008, a

PECASE award and a Packard Fellowship, in 2009, and he was named aRice University Outstanding Young Engineering Alumnus, in 2010. From2008 to 2011, he was an Associate Editor for the IEEE TRANSACTIONS ONINFORMATION THEORY.

Azita Emami-Neyestanak (S’97–M’04) was born inNaein, Iran. She received the B.S. degree with honorsfrom Sharif University of Technology, Tehran, Iran,in 1996, and the M.S. and Ph.D. degrees in electricalengineering from Stanford University, Stanford, CA,in 1999 and 2004, respectively.She is currently anAssistant Professor of Electrical

Engineering at the California Institute of Technology,Pasadena. From July 2006 to August 2007 she waswith Columbia University, New York, as an Assis-tant Professor in the Department of Electrical Engi-

neering. She also worked as a Research Staff Member at IBM T. J. Watson Re-search Center, Yorktown Heights, NY, from 2004 to 2006. Her current researchareas are high-performance mixed-signal integrated circuits and VLSI systems,with the focus on high-speed and low-power optical and electrical interconnects,clocking, biomedical implant, and compressed sensing.

Emmanuel Candes received the Diplôme d’In-génieur from the Ecole Polytechnique, in 1993,and the Ph.D. degree in statistics from StanfordUniversity, Stanford, CA, in 1998.He is the Simons Chair in Mathematics and Sta-

tistics, and Professor of Electrical Engineering (bycourtesy) at Stanford University, Stanford, CA. Priorto this, he was the Ronald and Maxine Linde Pro-fessor of Applied and Computational Mathematicsat the California Institute of Technology, Pasadena.His research interests are in computational harmonic

analysis, mathematical signal processing, information theory, mathematical op-timization, statistical estimation and detection with applications to the imagingsciences, scientific computing, and inverse problems.Prof. Candes received numerous awards, most notably the 2006 Alan T. Wa-

terman Medal, which is the highest honor bestowed by the National ScienceFoundation and which recognizes the achievements of scientists who are noolder than 35, or not more than seven years beyond their doctorate. Other awardsinclude an Alfred Sloan Fellowship in 2001, the Popov Prize in ApproximationTheory in 2001, the 2005 James H. Wilkinson Prize in Numerical Analysis andScientific Computing awarded by the Society of Industrial and Applied Math-ematics (SIAM), the 2008 Information Theory Society Paper Award, the 2010George Polya Prize awarded by SIAM. In 2011, he received the Collatz Prizeawarded every four years by the International Council for Industrial and Ap-plied Mathematics (ICIAM) established to “provide international recognitionto individual scientists under 42 years of age for outstanding work on industrialand applied mathematics”.

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