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Application of compressive sensing to the design of wideband signal acquisition receivers

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Compressive sensing (CS) exploits the sparsity present in many signals to reduce the number of measurements needed for digital acquisition. With this reduction would come, in theory, commensurate reductions in the size, weight, power consumption,
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  APPLICATION OF COMPRESSIVE SENSING TOTHE DESIGN OF WIDEBAND SIGNAL ACQUISITION RECEIVERS  John Treichler  Applied Signal Technology, Inc.Sunnyvale, California  Mark Davenport, Richard Baraniuk  Rice UniversityHouston, Texas ABSTRACT Compressive sensing (CS) exploits the sparsity present inmany signals to reduce the number of measurements neededfor digital acquisition. With this reduction would come, intheory, commensurate reductions in the size, weight, powerconsumption, and/or monetary cost of both signal sensors andany associated communication links. This paper examines theuse of CS in environments where the input signal takes theform of a sparse combination of narrowband signals of un-known frequencies that appear anywhere in a broad spectralband. We formulate the problem statement for such a receiverand establish a reasonable set of requirements that a receivershould meet to be practically useful. The performance of aCS receiver for this application is then evaluated in two ways:using the applicable (and still evolving) CS theory and usinga set of computer simulations carefully constructed to com-pare the CS receiver against the performance expected froma conventional implementation. This sets the stage for work in a sequel that will use these results to produce comparisonsof the size, weight, and power consumption of a CS receiveragainst an exemplar of a conventional design. 1. INTRODUCTION Compressive sensing (CS) [1–3] exploits the sparsity presentin many signals to reduce the number of measurementsneeded for acqusition. It has been shown theoretically that,under the right set of circumstances, CS can dramaticallyreduce the number of measurements needed to detect, char-acterize, and/or extract signals, and therefore can reduce bythe same factor the storage and/or transmission rate neededto handle the signal at the sensing point. Conversely, signalswith much larger bandwidths could be accepted by existingacquisition systems. If these reductions were found to pro-portionally reduce the size, weight, and power consumption(SWAP) and cost of operational signal acquisition systems,then the practical impact could be transformative.This paper examines the potential practicality of CS tobuild signal acquisition receivers for the specific, but impor-tant, case where the receiver’s input signal takes the form of a sparse combination of narrowband signals of unknown fre- Front-EndSensor Forwarding Link OutputsBack-EndProcessor  Fig. 1 . A wideband signal acquisition receiver.quencies that can appear anywhere in a broad spectral band.Our objective here is to examine the specific application of the acquisition receiver, thus providing the opportunity to testthe robustness of CS techniques to imperfect match with itsunderlying theoretical assumptions.Section 2 formulates the problem statement for the signalreceiver and establishes a set of requirements that a receivershould meet to be highly attractive for practical use. Sec-tion 3 applies the existing CS theory to determine how well aCS-based receiver should work in this case, while Section 4presents the interim results of a set of simulation-based eval-uations designed to measure the expected performance of aCS-based receiver in comparison with theory and with a con-ventional receiver design. These results are discussed in Sec-tion 5, and recommendations for additional study and investi-gation appear in Section 6. 2. PUTATIVE REQUIREMENTS Our objective in this paper is to explore the attributes and ca-pabilities of CS by examining how it might be applied to meeta specific set of requirements. The particular application ad-dressed is a wideband radio frequency (RF) signal acquisitionreceiver, a device commonly used in both commercial andmilitary systems to monitor a wide band of radio frequen-cies for the purposes of  (i) detecting the presence of signals, (ii) characterizing them, and, where appropriate, (iii) extract-ing a specific signal from the several that might be presentwithin that band. A high-level system diagram incorporatingsuch a receiver is shown in Figure 1. While many types of acquisition receivers have been designed, built, and sold overthe years, we will choose here a set of putative requirementsfor such a receiver so that comparisons and analysis can bedone. The reader is free to, and in fact invited to, choose1  Table 1 . A putative set of specifications for an advanced RFsignal acquisition receiver.Instantaneous bandwidth B 500 MHzInstantaneous dynamic range D 96 dBSNR degradation/noise figure NF  12 dBMaximum signal bandwidth W  200 kHzother operational parameters and repeat the comparison.The attributes that characterize an acquisition receivertypically fall into two categories: technical specifications —such as instantaneous bandwidth — and various “costs”—such as SWAP and monetary cost. In this paper we willaddress only the few most important technical specifications: • Instantaneous bandwidth — the RF range over whichsignals will be accepted by the receiver and handledwith their full fidelity; • Instantaneous dynamic range — the ratio of the max-imum to minimum signal power level with which re-ceived signals can be handled with full fidelity; • SNR degradation — sometimes termed “noise figure”,a measure of the tendency of the receiver to lower theinput signal-to-noise ratio (SNR) of a received signal,usually measured in dB. The root cause of this degrada-tion has historically depended on the technology usedto build the receiver. • Maximum signal bandwidth — the maximum com-bined bandwidth of the constituent signals in the acqui-sition bandwidth of the receiverThese requirements must be met subject to many constraints,including, at least, SWAP and monetary cost. There arealso typically system-level constraints, such as the bandwidthavailable for communicating what the receiver has discoveredto other assets or a central processing facility.Historically RF signal acquisition receivers were firstbuilt using purely analog technology, then, more recently,with analog technology conditioning the signal environmentsufficiently to employ a high-rate analog-to-digital converter(ADC) followed by digital processing, storage, and/or trans-mission. When and if it can be applied, CS offers the promiseto (i) increase the instantaneous input bandwidth, (ii) lowerall of the cost attributes, and (iii) move the computationallyintensive portions of the acquisition process away from thesensor and toward a central processing facility.For the purposes of the comparisons to be made in thispaper, we will assume a set of requirements, listed in Table 1,for an acquisition system that are quite audacious and wouldstress, at the least, conventional implementations. To meetthe bandwidth and dynamic range requirements, conventionaldesigns would typically be forced to use techniques based on Sensor OutputState of RandomizedSensor DetectionCharacterizationExtractionGeolocationSupportBack-EndProcessor Front-EndCompressiveSensor  Fig. 2 . The “processing asymmetry” assumed in a CS signalacquisition receiver.scanning narrowband receivers across the band. If CS-basedsystems can be shown to work in such cases without the needfor scanning at the receiver, then they would have broad ap-plication.We make two last, but important, assumptions:1. Signal sparsity — In order to meet the first-order as-sumption of all CS techniques that the input signal besparse in some way, we assume that the sum of thebandwidths of all signals present in the full acquisitionband, which we denote W  , is no more than 200 kHz.Note that this is significantly smaller than the instanta-neous bandwidth B of 500 MHz. Thus we are assum-ing that the RF input to the receiver is quite sparse inthe frequency domain (the instantaneous bandwidth isonly 1 / 2500 occupied in this case). While inputs withthis level of spectral sparsity are not common, they ex-ist often enough to make a solution useful if it can befound. To test the impact of the sparsity assumption forthis application, we will evaluate the performance, boththeoretical and in simulation, for both the case wherethe input is noise-free, in which case the input signal istruly sparse, and in the more practical case where theinput is contaminated with additive white noise.2. Processing asymmetry — Our presumption is that thereis no cost to the computation done to process the out-put of the compressive sensor. Our objective is to min-imize all costs, including the transmission bandwidthbetween the sensor and the processor, and, for the pur-poses of this paper, we are prepared to do as muchprocessing as needed to detect, characterize, and/or re-cover the signal of interest. This assumed separation of functions is illustrated in Figure 2. 3. COMPRESSIVE SENSING AND ITSAPPLICATION TO WIDEBAND RADIO RECEIVERS3.1. CS theory In laying out the putative requirements in Section 2, we as-sumed that the real-valued, continuous-time signal of interest,which we will denote by x ( t ) , has instantaneous bandwidth B and maximum signal bandwidth W  . Thus, each segment of 2  x ( t ) of duration T  = 1 seconds has the Fourier representation x ( t ) = Ψ ( α ) = 2 B − 1  k =0 α k ψ k ( t ) , (1)where ψ k ( t ) = e j 2 πkt/ (2 B ) are the Fourier basis functionsand where α = [ α 0 ,α 1 ,...,α 2 B − 1 ] is a complex-valuedvector of length 2 B that has 2 W  nonzeros corresponding tothe active frequencies of the constituent signals in the acquisi-tion bandwidth B . 1 For the moment, we assume that α k = 0 exactly outside of these 2 W  nonzeros, in which case we saythat the vector α is 2 W  - sparse . Finally, note that while wefocus on Fourier-sparse signals in this paper, the CS theory isgeneral and extends to signals sparse in arbitrary bases.The Shannon-Nyquist sampling theorem tells us that 2 B uniform samples of  x ( t ) per T  = 1 second contain all of the information in x ( t ) . Our goal in CS is to do better: toacquire x ( t ) via a set of  2 B/Q measurements with Q ≥ 1 aslarge as possible. The subsampling factor  Q strongly affectsthe various costs (i.e., SWAP and monetary cost) described inSection 2. Observe that if the locations of the 2 W  non-zerosof  α are known a priori , then by filtering and decimation, wecoulddrive Q aslargeas Q f  = B/W  . OuraimistoshowthatCS-based techniques will enable us to acquire x ( t ) via a setof  2 B/Q fixed, nonadaptive, linearmeasurementsthatrequireno a priori knowledge of the locations of the non-zeros of  α ,with Q nearly as large as Q f  .Towards this end, we take the measurements y = Φ ( x ( t )) + e , (2)where Φ isalinear measurementoperator  thatmapscontinuous-time functions defined on the time interval t ∈ [0 , 1] to alength 2 B/Q vector y of measurements, and where e is alength 2 B/Q vector that represents measurement noise gen-erated by the acquisition hardware. The central theoreticalquestion in CS is how to design the measurement operator Φ to ensure that we will be able to recover the signal x ( t ) fromthe measurements y . While there are many approaches tosolving this problem, the most common method is to split thisquestion into two parts: (i) What properties of  Φ will ensurethat there exists some algorithm that can recover x ( t ) from y ? and (ii) What algorithms can perform this recovery in anefficient manner?The answer to the first question is rather intuitive. Whilealternative properties have been studied, the majority of thework in CS assumes that the measurement operator Φ satis-fies the so-called restricted isometry property (RIP) [4]. Inour setting, where x ( t ) has a 2 W  -sparse representation withrespect to the basis Ψ , the RIP requires that there exists a 1 We set the segment-length to T  = 1 second for notational convenience;for snippets of other lengths one can merely replace B and W  below with BT  and WT  , mutatis mutandis . constant δ ∈ (0 , 1) such that √ 1 − δ ≤ Φ ( Ψ ( α )) − Φ ( Ψ ( β ))  2  α − β  2 ≤√ 1 + δ, (3)holds for all 2 W  -sparse α and β . In words, ΦΨ preservesthe Euclidean distance between vectors that are 2 W  -sparse.Intuitively, the RIP means that for any particular set of mea-surements y , there exists at most one possible 2 W  -sparse sig-nal consistent with these measurements, since if there weretwo distinct 2 W  -sparse signals that mapped to the same set of measurements, then this would violate the lower inequality of (3). In principle, this means that it should be possible (in thenoise-free setting at least) to exactly recover any 2 W  -sparsesignal. Furthermore, if a small amount of measurement noiseis added to the measurements y as in (2), then the RIP pro-vides a guarantee that any 2 W  -sparse signal consistent withthe perturbed measurements will be close to the srcinal sig-nal, and so the RIP ensures that the system has a degree of stability and robustness to measurement noise.We now consider how to design an operator Φ satisfyingthe RIP. An important result from the CS theory is that for anygiven basis Ψ (not just the Fourier basis we focus on here),if we draw a random matrix R  of size 2 B/Q × 2 B whoseentries r ij are independent realizations from a Gaussian,Rademacher ( ± 1 -valued), or more generally, any bounded,zero-mean distribution, then with overwhelmingly high prob-ability Φ = R  Ψ − 1 will satisfy (3) for 2 W  -sparse signalsprovided that Q ≤ Q c = κ 0 Q f  ln Q f  (4)where κ 0 < 1 is a constant that depends on B and the prob-ability with which (3) holds [5]. From this we concludethat CS-based measurement operators Φ pay a small penalty,quantified by κ 0 / ln( Q f  ) , for not exploiting any a priori knowledge of the locations of the nonzero frequencies.In general, the theoretical analysis is somewhat lacking interms of the precise value of  κ 0 . For instance, an asymptoticanalysis of the results in [5] would suggest that κ 0 ≈ 1 / 50 would be sufficient, but this value seems to be far more con-servative than what is required in practice. Thus, if a specificvalue for κ 0 is required, one must determine this value ex-perimentally. This is typically accomplished via Monte Carlosimulations that identify how many measurements are suffi-cient to ensure exact recovery in the noise-free setting on atleast, say, 99% of trials. As an example, it is shown in [6]that Q < ≈ 0 . 6 Q f  / ln( Q f  ) is sufficient to enable exact re-covery in the noise-free setting. Note that this is not the sameas demonstrating that Q < ≈ 0 . 6 Q f  / ln( Q f  ) is sufficient toensure that Φ satisfies the RIP, but it is highly suggestive thatthe true value of  κ 0 is much greater than the conservative es-tiamtes provided by the theory. We will observe this phe-nomenon for ourselves in Section 4.Since the random matrix approach is somewhat impracti-cal to build in hardware, several hardware architectures have3  Pseudorandom Number Generator  Seed Integrator Sample-and-Hold y [ n ] Quantizer  Fig. 3 . Random demodulator for obtaining compressive mea-surements of analog signals.been implemented and/or proposed that enable compressivesamples to be acquired in practical settings. Examples in-clude the random demodulator [6], random filtering [7], andrandom convolution [8,9].We briefly describe the random demodulator as an exam-ple of such a system; see Figure 3 for a block diagram. Thefour key components are a pseudo-random ± 1 “chipping se-quence” p c ( t ) operating at the Nyquist rate 2 B or higher, alow pass filter, represented by an ideal integrator with reset,and a low-rate ADC consisting of a sample-and-hold circuitand a quantizer. The input analog signal x ( t ) is modulated bythe chipping sequence and integrated. The output of the inte-grator is sampled and quantized, and the integrator is reset af-ter each sample. The output measurements from the sample-and-hold circuit are then quantized. The subsampling factor Q c determines the sampling rate C  at which the sample-and-hold circuit and quantizer operate. Specifically, we must have C  ≥ 2 B/Q c .Mathematically, the linear measurement operator Φ of the random demodulator can be decomposed into the product Φ = R  Ψ − 1 , where R  is a matrix of size of size 2 B/Q × 2 B .The resulting R  matrix, while randomized, typically containssome degree of structure. For example, the random convo-lution architecture [8,9] endows R  with a Toeplitz structure.While theoretical analysis of structured random matrices re-mains a topic of active study in the CS community, there doexist theoretical guarantees for some architectures [6,8]. Theamount of subsampling possible ( Q c ) with these construc-tions is generally consistent with fully random measurementsas given in (4), although the denominator is sometimes raisedto a small power (e.g., 2 or 4) for provability. 3.2. CS recovery algorithms We now address the question of how to recover the signal x ( t ) from the measurements y . The srcinal CS theory proposed  1 -minimization as a recovery technique when dealing withnoise-free measurements [1,2]. Noisy measurements as in(2) can be easily handled using similar techniques providedthat the noise e is bounded, meaning that  e  2 ≤  . In thiscase, assuming that Φ = R  Ψ − 1 satisfies the RIP, in whichcase we can write y = R  Ψ − 1 ( Ψ ( α )) = Rα , the convexprogram   α = argmin α  α  1 s.t.  Rα − y  2 ≤  (5)can recover any sparse signal α . Thus by setting  x ( t ) = Ψ (   α ) , we can recover x ( t ) . More specifically, this is madeprecise in [10] which establishes that for 2 W  -sparse signals,the recovery error can be bounded by    x ( t ) − x ( t )  2 ≤ κ 1 , (6)where κ 1 ≥ 2 is a constant that depends on the subsamplingfactor Q . This constant is similarly difficult to determine the-oretically, but in practice it should be very close to 2 providedthat Q < Q c . Thus, measurement noise has a controlled im-pact on the amount of noise in the reconstruction. A similarguarantee can be obtained for approximately sparse, or com- pressible , signals 2 .While convex optimization techniques like (5) are power-ful methods for CS signal recovery, there also exist a varietyof alternative algorithms that are commonly used in practiceand for which performance guarantees comparable to that of (6) can be established. In particular, iterative algorithms suchas CoSaMP, iterative hard thresholding (IHT), and variousother thresholding algorithms are known to satisfy similarguarantees to (6) [11–13]. Most of these algorithms are builton similar techniques and can be easily understood by break-ing the recovery problem into two separate sub-problems:identifying the locations of the nonzero coefficients of  α and estimating the values of the nonzero coefficients of  α .The former problem is clearly somewhat challenging, butonce solved, the latter is relatively straightforward and can besolved using standard techniques like least squares. In partic-ular, again using the factorization of  Φ = R  Ψ − 1 , supposethat we have identified the indices of the nonzero coefficientsof  α (its support  ), denoted by the index set J  , and let R  J  denote the submatrix of  R  that contains only the columnscorresponding to the index set J  . Then an optimal recoverystrategy is to solve the problem:   α = argmin α  y − R  J  α  2 , (7)which is a standard least-squares problem that can be solvedvia the pseudo-inverse of  R  J  , denoted R  † J  :   α = R  † J  y = ( R  T J  R  J  ) − 1 R  J  y . (8)Note that in the noise-free case, if  α is 2 W  -sparse and thesupport estimate J  is correct , then y = R  J  α , and so plug-ging this into (8) yields  α = α . Thus, the central challenge 2 By compressible, we mean signals that are well approximated by asparse signal. The guarantee on the recovery error for compressible signals issimilar to (6) but includes an addititional additive component that quantifiesthe error incurred by approximating x ( t ) with a sparse signal. Therefore, if a signal is very close to being sparse, then this error is negligible, while if asignal is not sparse at all, then this error can be quite large. 4  in recovery is to identify the correct locations of the nonze-ros. CoSaMP and many related algorithms solve this problemby iterativelyidentifying likely nonzeros, estimating theirval-ues, and then improving the estimate of which coefficients arenonzero. 3.3. Noise folding and CS measurements Most of the CS literature focuses on ensuring the stability of the CS acquisition and recovery process in the face of  mea-surement noise e in (2) [10–13]. In this section, we take acareful look at the effect of  signal noise that is present in thesignal before acquisition. Specifically, we now assume thatthe Fourier representation of  x ( t ) consists of a 2 W  -sparsesignal corrupted by additive Gaussian noise n . We will as-sume that the noise n is zero-mean and spectrally white withcovariance matrix Σ n = σ 2 I  2 B and added across the en-tire 2 B -dimensional Fourier spectrum α and not just the 2 W  non-zeros of  α . Thus, we acquire the measurements y = Φ ( Ψ ( α + n )) = Rα + Rn . (9)The noise situation is subtly different from (2), because thenoise in the measurements y is now scaled by the matrix R  .Our chief interest here is to understand how R  impacts thesignal noise and how it manifests itself in the final recoveredsignal.We first observe that Rn is zero mean with covariancematrix Σ Rn = σ 2 RR  T  . To further simplify this expression,we make some relatively intuitive assumptions concerning R  : (i) each column of  R  has norm (energy) approximately equalto 1 , 3 (ii) the rows of  R  have approximately equal norm, and (iii) the rows of  R  are orthogonal. The first assumption sim-ply means that the columns of  R  are weighted equally, whichmeans that the noise n k on each coefficient α k will contributeroughly an equal amount to the total noise. If the locationsof the nonzeros were known a priori , then we could do bet-ter by assigning more weight to the columns corresponding to α k  = 0 and less weight to columns corresponding to α k = 0 (which are purely noise). However, in the absence of thisknowledge, if we want the impact of the noise to be indepen-dent of the location of the nonzeros, then there is no alterna-tive but to weight the columns equally. Similarly, the secondassumptionrequiresthateachmeasurementshouldhaveequalweight. Finally, the third assumption is not strictly necessary,butitseemsreasonablethatifwewishtotakeasfewmeasure-ments as possible, then each measurement should provide asmuch new information about the signal as possible. Note thatthese assumptions hold for randomly generated R  matrices,as well the R  matrices corresponding to more practical archi-tectures such as the random demodulator [6].By combining (i) and (ii) , we can infer that each row of  R  should have norm of approximately   2 B/ (2 B/Q ) = √ Q . 3 Note that this property is implied by the RIP by considering signals withonly one nonzero coefficient. Combining this with (iii) , we obtain the approximation RR  T  ≈ Q I  2 B/Q . (10)Thus, the covariance matrix of  Rn is approximately Σ Rn ≈ RR  T  ( σ 2 I  2 B ) = σ 2 Q I  2 B/Q . (11)From (11) we see that the noise Rn is spectrally white butamplified by the subsampling factor Q . This makes senseintuitively, since we are projecting all of the noise in the 2 B -dimensional input signal ( α or equivalently x ( t ) ) downinto the 2 B/Q -dimensional measurements y , and all of thenoise power must be preserved. In the literature, this effect isknown as noise folding .Noise folding has a significant impact on the amount of noise present in CS measurements. Specifically, when acquir-ing a 2 W  -sparse signal, as we double the subsampling factor Q (a one octave increase), the signal-to-noise ratio (SNR) of the measurements y decreases by 3dB. In words, for the ac-quisition of a noisy signal of fixed sparsity, the SNR of the CSmeasurements decreases by 3 dB for every octave increase inthe subsampling factor  .We note that alternative signal acquisition techniques like bandpass sampling (sampling a narrowband signal uniformlyat a sub-Nyquist rate to preserve the values but not the loca-tionsofitslargeFouriercoefficients)areaffectedbyanidenti-cal 3dB/octave SNR degradation. However, in practice band-pass sampling suffers from the limitation that it is impossibleto determine the srcinal component center frequencies, andfurthermore, if there are multiple narrowband signals presentthen it causes irreversible aliasing, in which case the com-ponents can overlap and will be impossible to separate. Incontrast to bandpass sampling, however, CS acquisition pre-serves sufficient information to enable the recovery of boththe values and the locations of the large Fourier coefficients. 3.4. Noise folding and CS reconstruction We now examine the impact of noise folding on CS signalreconstruction. Rather than directly analyzing a particularreconstruction algorithm, we will instead consider the perfor-mance of an oracle -based recovery algorithm that has perfectknowledge of the true location of the 2 W  nonzeros of  α ,which we have denoted by J  . While an oracle is clearly im-practical, it characterizes the best that we can hope to achieveusing any practical algorithm. And, in fact, we find thatpractical algorithms like CoSaMP typically perform almostas well as the oracle-based recovery algorithm.Given J  , the optimal estimate of  α is given by (8), result-ing in   α = α + R  † J  Rn . (12)Thus, we recover the signal α plus the additive noise nowscaled by the matrix R  † J  R  . To understand the properties of 5
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