58 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL ...58 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL...

58 IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 6, NO. 1, FEBRUARY 2012

Cyclic Feature Detection With Sub-Nyquist Samplingfor Wideband Spectrum Sensing

Zhi Tian, Senior Member, IEEE, Yohannes Tafesse, and Brian M. Sadler, Fellow, IEEE

Abstract—For cognitive radio networks, efficient and robustspectrum sensing is a crucial enabling step for dynamic spectrumaccess. Cognitive radios need to not only rapidly identify spectrumopportunities over very wide bandwidth, but also make reli-able decisions in noise-uncertain environments. Cyclic spectrumsensing techniques work well under noise uncertainty, but requirehigh-rate sampling which is very costly in the wideband regime.This paper develops robust and compressive wideband spectrumsensing techniques by exploiting the unique sparsity property ofthe two-dimensional cyclic spectra of communications signals.To do so, a new compressed sensing framework is proposed forextracting useful second-order statistics of wideband randomsignals from digital samples taken at sub-Nyquist rates. Thetime-varying cross-correlation functions of these compressivesamples are formulated to reveal the cyclic spectrum, which isthen used to simultaneously detect multiple signal sources over theentire wide band. Because the proposed wideband cyclic spectrumestimator utilizes all the cross-correlation terms of compressivesamples to extract second-order statistics, it is also able to recoverthe power spectra of stationary signals as a special case, permittinglossless rate compression even for non-sparse signals. Simulationresults demonstrate the robustness of the proposed spectrumsensing algorithms against both sampling rate reduction and noiseuncertainty in wireless networks.

Index Terms—Cognitive radio (CR), compressive sampling,cyclic spectrum, feature detection, wideband spectrum sensing.

I. INTRODUCTION

I N cognitive radio (CR) applications, secondary CR usersshall be able to sense a wide spectrum and adapt their trans-

mission parameters dynamically to utilize the available spec-trum while avoiding interference to any primary users (PU). Toaccomplish these tasks, fast and accurate spectrum sensing isa key enabler [1], [2]. Different from traditional spectral es-timation problems for narrowband systems, CRs face distinctimplementation challenges in wideband processing, which ren-ders Nyquist-rate sampling costly or even impractical. Mean-while, the CR channel environment is subject to frequency se-

Manuscript received March 29, 2011; revised August 02, 2011, November16, 2011; accepted December 11, 2011. Date of publication December 28, 2011;date of current version January 18, 2012. This work was supported in part by Na-tional Science Foundation Grants #ECS-0925881 and #CNS-1017887. A shortversion of this paper appeared in the IEEE International Conference on Com-munications (ICC), June 2011. The associate editor coordinating the review ofthis manuscript and approving it for publication was Dr. Octavia Dobre.

Z. Tian and Y. Tafesse are with the Electrical and Computer EngineeringDepartment, Michigan Technological University, Houghton, MI 49931 USA(e-mail: [email protected]).

B. M. Sadler is with the Army Research Laboratory, AMSRD-ARL-CI-CN,Adelphi, MD 20783 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSTSP.2011.2181940

lective fading as well as dynamically varying noise and interfer-ence levels, which perplex those spectrum sensing techniquesbased on known or fixed signal and noise statistics. Further, thesensing time has to be short to ensure network responsivenessand efficiency.

Traditional spectrum sensing techniques have major limita-tions for wideband CR networks. The matched filter is optimalfor signal detection under Gaussian noise, but requires priorknowledge of the signal waveforms to be detected, whichcan be infeasible to acquire in practice. An energy detectordoes not require prior knowledge of PU signals and is simpleto implement, but its detection accuracy is very sensitiveto noise uncertainty and becomes unacceptable under lowsignal-to-noise-ratio (SNR) conditions [3]–[5]. Cyclic featuredetectors are quite robust to both noise uncertainty and strongnoise, and can differentiate various modulation types andhence potentially separate PUs from CRs [6]–[8]. These salientproperties have been recognized and utilized in recent develop-ment on CR sensing [9]–[17]. On the other hand, it typicallyrequires sampling at a higher-than-Nyquist rate in order toinduce cyclostationarity, and generally takes a long sensingtime to acquire reliable cyclic statistics [7], [8]. Due to the highsampling rate requirements and the limited dynamic range ofwideband receivers, these techniques are typically carried outon a channel-by-channel basis over small bandwidth, in combi-nation with a frequency shifter that scans one channel at a timeover a wide band, or using a parallel set of multiple narrowbandprocessors. However, for fast and reliable adaptation to highlydynamic wideband fading channels, it is desired to performwideband spectrum sensing that simultaneously detects allsignal sources over a wide band of interest.

This paper aims to develop robust wideband spectrumdetection techniques that are efficient in computation andfeasible in hardware implementation. The spectrum pool undermonitoring is very wide, say, from 300 MHz to several GHz,which is known to have low instantaneous utilization accordingto extensive spectrum occupancy measurements [18]. In thiswork, the cyclic feature detection principle is adopted in thewideband regime to simultaneously recover the entire spec-trum with robustness against noisy uncertainty and low SNRconditions, and the unique sparsity property of the 2-D cyclicspectrum of communication signals is exploited to alleviatethe sampling rate requirements with robustness to rate reduc-tion. The key technical innovation is the development of newcompressive sampling and sparse signal recovery techniquesthat directly recover the desired cyclic statistics instead of the(random) signal waveforms, which not only solve the mainhindrance of wideband cyclic feature detection in terms of

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TIAN et al.: CYCLIC FEATURE DETECTION WITH SUB-NYQUIST SAMPLING FOR WIDEBAND SPECTRUM SENSING 59

high sampling rate requirements but also alleviates the longsensing time compared with traditional techniques. It has beenrecognized that a signal which is sparse in a certain domaincan be recovered using computationally feasible algorithmsfrom a reduced number of samples [19], [20]. Such samplescan be collected effectively via random linear projections[19]–[24]. Interestingly, the use of compressive random sam-plers can potentially enhance the dynamic range of widebandsystems as well [25]. For a CR user, the wide band of interesttypically possesses sparse features in the frequency domain,due to spectrum under-utilization of legacy radio systems.Accordingly, compressive sampling techniques have beendeveloped for wideband sensing in CR networks [26]–[30],which effectively reconstruct the sparse frequency spectrumfrom sub-Nyquist-rate samples. Afterward, spectrum occu-pancy is estimated based on the recovered frequency or powerspectrum, which are still prone to estimation errors causedby noise variations. In this work, our main contribution is toreconstruct the 2-D cyclic spectrum of a sparse signal directlyfrom sub-Nyquist-rate compressive samples, without havingto recover the signal itself or its frequency response. In doingso, an additional dimension of sparsity arises owing to themodulation-dependent structure in cyclic-frequency domain, inaddition to sparsity in frequency domain. Exploitation of thesparsity structure in the added cyclic domain not only offersrobust measures against time-varying yet non-cyclic noiseand interference, but also makes it possible to further reducethe sampling rate while still reliably recovering the desiredstatistics for sensing.

This work exploits the signal sparsity in both dimensions toextract second-order statistics of random processes, which de-parts from existing compressed sensing techniques for recov-ering one-dimensional deterministic signals. It is worth empha-sizing that this conceptual departure constitutes a major chal-lenge in utilizing existing sparse signal recovery techniques forlinear samplers, because the compressive samples collected inthe time domain do not have a direct linear relationship withthe 2-D sparse cyclic spectrum to be recovered. To circumventthis difficulty, this paper derives a transformed linear systemthat connects the time-varying cross-correlations of compres-sive measurements to the desired cyclic statistics. This key re-formulation allows the 2-D cyclic spectrum to be feasibly solvedvia convex -norm minimization. The proposed compressivecyclic spectrum estimator can simultaneously capture multiplecommunication signals over a very wide band, and meanwhileremove non-cyclic noise and interference. As a special case, thedeveloped compressive estimator is also instantiated to recon-struct the power spectral density (PSD) of stationary signals,which has a reduced dimension from the 2-D cyclic spectrum.Similar to [31], [32], because all the time-varying cross-correla-tion terms of compressive samples are utilized to recover a sta-tionary power spectrum, the developed PSD estimator permitscompressive sampling with considerable sampling-rate reduc-tion, even for non-sparse signals.

This paper also contributes to estimating the spectrum oc-cupancy of wideband signals from the cyclic spectrum. On asingle channel, binary signal detection based on the cyclic spec-trum has been extensively studied. Subject to unknown noise

levels, a family of cyclic feature detectors have been developedand compared based on various design principles [6], [7]. Themaximum-SNR single cycle detector has been described in [7].However, the goal here is to use the cyclic spectrum for si-multaneous estimation of the spectrum occupancy of multiplesignals spreading on a wide band, which is motivated by thesensing needs of wideband CR networks but has not been in-vestigated by traditional techniques. This paper extends the tra-ditional single-channel setting to perform cyclic feature basedspectrum occupancy estimation when multiple signal sourcesare present over a wide band. A band-by-band detection ap-proach is proposed, where a multi-cycle generalized likelihoodratio test (GLRT) is used on each channel to detect PU presence[11]. To obtain the noise statistics used in the GLRT detector, ablind estimator is derived to estimate the noise variance from theavailable compressive samples. Further, when the modulationtype of the signals of interest is known, simple and fast tech-niques can be developed to detect the signal presence and es-timate their cyclic features that reveal their carrier frequenciesand transmission rates. Such a technique is illustrated for BPSKsignals.

The rest of the paper is organized as follows. Section II pro-vides the problem statement and signal model. Section III de-scribes the sparsity structure of the two-dimensional cyclic spec-trum, the relationship between the sub-Nyquist-rate samples andthe cyclic spectrum, the optimization formulation for recoveringthe sparse 2-D cyclic spectrum, as well as the practical imple-mentations with robustness to finite sample effects. Section IVpresents a compressive PSD estimator for stationary signals as aspecial case, and illustrates the mechanism of rate compressionfor non-sparse signals. Section V develops spectrum occupancyestimators based on the recovered cyclic spectrum. Corrobo-rating simulation results are provided in Section VI, followedby a concluding summary in Section VII.

II. SIGNAL MODEL AND PROBLEM STATEMENT

Consider a wide band of interest in the frequency range, where is very large, e.g., over-GHz. There

are active PU signals emitting over this wide band, wherethe th signal is denoted by . We haveno information regarding the waveform, bandwidth and carrierfrequency of each signal, nor the number of signals present. ACR is equipped with an (ideal) wideband antenna that passes allsignal components within . Hence, the receivedsignal is given by

(1)

where is the additive ambient noise. We suppose that pri-mary user signals are non-overlapping in frequency, sincethey could represent emitters from different services. As in-dicated by extensive spectrum occupancy measurements [1],[18], a good portion of the frequency range is idle due to spec-trum under-utilization, suggesting that the frequency spectrumof is sparse. Our goal is to estimate the spectrum occu-pancy of the composite signal over the entire wide band,


whose nonzero support regions reveal the frequency locationsand bandwidths of individual signal components . This isdifferent from most existing work on cyclostationary signal pro-cessing, in which only one signal or signal mixture occupies thefrequency band of interest.

We are motivated to investigate the cyclic spectrumof , where is the frequency and is the

cyclic frequency. Defining the cyclic autocorrelation func-tion , thecyclic spectrum, also termed the spectral correlation function(SCF), is the Fourier transform of with respect to thetime-delay [7]:

It essentially indicates the correlation of spectral componentsthat are separated by in frequency, and exhibits nonzero valuesonly at a discrete set of cyclic frequencies that reveals the in-herent second-order periodicity of . It is known thatcan be nonzero only for [7], which suggestsa diamond-shaped region around the origin of the bifrequencyplane, as shown in Fig. 1(a). In a traditional analog-to-digitalconversion (ADC) process, is sampled uniformly every

seconds to yield digital samples ,where is the sampling rate in Hz. The cyclic spectrum of thedigital samples contains folded replicas of the original cyclicspectrum, with folding intervals being integer multiples ofon both directions [[7], Eq. (62)]. To avoid any aliasingin the cyclic spectrum, the minimum sampling rate should be

[6], that is,

(2)

Because is very large in our CR sensing task, the requiredsampling rate has to be very high, causing large energy con-sumption and high hardware costs in ADC. On the other hand,we observe that the cyclic spectrum of the signal is in fact sparse,which motivates us to utilize the compressive sampling (CS) ap-proach to reduce the sampling rate. In addition to the sparsity infrequency due to spectrum under-utilization, the signal isalso sparse in cyclic frequency since some of the cyclic frequen-cies may not be occupied by any active signals. We refer asa sparse signal by focusing on its sparse cyclic spectrum.

In the CS framework, is not directly sampled, butis passed through a set of properly designed analog fil-ters and then down-sampled at a reduced rate

, yielding compressive samples( denotes convolution). In

the discrete-time domain, we can stack successive sam-ples into , which spans asensing time block of seconds with . Withinthis block, a traditional uniform-rate ADC would yieldsamples , where islarger than . Hence, the sampling rate of the CS receiver is

, where is the rate-compres-sion ratio. This linear compressive sampling process can bedescribed as

(3)

Fig. 1. Original cyclic spectrum �� can be obtained by wrapping the re-covered cyclic frequency values in � �� . (a) Original cyclicspectrum �� of the analog signal ��. (b) The folded cyclic spectrum�� of digital samples � , shown within � � �� .

where the rows of the sampling matrix are givenby the digital representations of the sampling filters ,respectively. Advances in CS assert that , or equivalently

, can be recovered error-free with high probability fromeven when is lower than the Nyquist rate, as long as sat-isfy certain restricted isometric property [19], [20]. The min-imal number of samples needed (hence the sampling rate )is determined by both and the sparsity order of the signal ofinterest. Examples of include the parallel compres-sive samplers in [22]–[24] that can implement various forms ofrandom matrices such as Gaussian, Fourier and Bernoulli ma-trices, and the serial analog-to-information converter (AIC) in[21].

In this paper, we first reconstruct the 2-D cyclic spectrumfor , directly from the compressive

measurements . Then, based on the recovered cyclic spectrum,we extract useful features to estimate the spectrum occupancyof the wide band of interest. In contrast, a conventional CS ap-proach may have to use a two-step approach to cyclic spectrumestimation: it first estimates from by recovering the sparsefrequency response of , and then compute the cyclic spec-trum from the estimated . In that case, the sparsity propertyin cyclic spectrum cannot be utilized, and both the computa-tional costs and the required minimal sampling rate are high.


Comparison will be detailed later on in Section VI-B. It sufficesto remark here that the cyclic spectrum as signal statistics hasless degrees of freedom than the random signal itself, and henceallows for faster recovery at lower sampling costs.

III. RECOVERY OF SPARSE CYCLIC SPECTRUM

Consider a real-valued communication signal for clearexposition, while the complex-valued case can be derived ac-cordingly. The corresponding digital sequence is zero-meancyclostationary, which means that its time-varying covariancefunction, defined as

, is periodic in with some integer as thecyclic period ( denotes expectation). That is,

being integers. The cyclic period re-veals the hidden periodicity in the signal, such as carrier fre-quency, symbol rate, chip rate, etc. [6]. It is assumed that thedigital cyclic periods of all the signal components of be-long to a permissible set , corresponding to ac-tual cyclic frequencies . That is, for any compo-nent whose cyclic spectrum can be sensed from , thereexists an integer for some positive integer such that

. For other signal components whosecyclic periods do not fall within this permissible set , it is stillpossible to reconstruct its cyclic spectrum via some interpola-tion formula deduced from Nyquist sampling theorem, whichwill not be derived in this paper.

Let denote the digital counterpart of the actual bi-fre-quency pair , with and , re-spectively. The cyclic covariance function of is given bythe Fourier series of with respect to time . Sincehas a finite length , its cyclic covariance function can be esti-mated as

(4)

where is the digital cyclic frequency, and theadjustment term makes the sequence symmetric withrespect to . This is a biased estimate, but its estimation varianceis smaller than that of an unbiased one [8].

The cyclic spectrum is the Fourier transform of the cycliccovariance with respect to time-lag , given by

(5)

where is the digital frequency. The sparsity orderof is given by that of its cyclic spectrum , whichis the fraction of nonzero points on the 2-D cyclic spectrum.We assume that some loose upper bound of the sparsity order isknown a priori in order to choose an appropriate compressionratio.

Let us represent the 2-D quantities and

as matrices and , respectively, eachof size . Meanwhile, is represented bythe time-varying covariance matrix , whoseelements are shown in (38) in Appendix A. The support regionof the 2-D cyclic spectrum is shown in Fig. 1(b), whichcorresponds to the folded cyclic spectrum within the square

region . Once is acquired, it can be wrappedinto the original cyclic spectrum in the diamond-shapedregion of Fig. 1(a), by properly mapping the three subregionsmarked by I, II, and III. The problem of cyclic spectrumrecovery amounts to estimating fromcompressive samples .

A. Linear Relationships

Reconstruction of the cyclic spectrum can benefit from someexplicit linear relationships among second-order statistics. Inthis subsection, we first derive a linear relationship between theunknown of interest in (5) and in (4). Then,because both and are not available, we express

as a linear function of, which is the time-varying data covariance of the compres-

sive samples . Finally, a linear relationship betweenand is established, which simplifies the reconstructionprocess as sparse linear regression, as shown in the next subsec-tion.

1) Relating to : When is real-valued,it holds that ,and hence is symmetric positive semi-definite. For

, there are only unique terms ofsatisfying , which represent the degrees of

freedom for the second-statistics of . We organize them intoa vector of length , as follows:

(6)

To facilitate the representation of (4) using , we form anauxiliary covariance-related matrix :

......

......

(7)which contain the same elements of besideszeros. After some algebraic operations, it can be shown that

(8)

where stacks all columns of a matrix into a vector,and maps the entries ofto those in . Specifically, is constructed by re-taining columns of the identitymatrix, for any column index satisfying

, or equivalently satisfying.

Following (4) and (5), the cyclic covariance matrix andthe cyclic spectrum matrix are related to through Fourierseries transformations, as follows:

(9)

(10)


where is the -point DFT matrix,, and is an

matrix with only its th diagonal element being 1 and allother elements being 0.

From (8) and (9), and using the property( denotes Kronecker product), we can vec-

torize and express it in a linear form of as

(11)

where is of size .Meanwhile, we can also express in terms of

from (10), as follows:

(12)

where and is the size- identity matrix.From (11) and (12), it holds that

(13)

where is the (pseudo) inverse of .2) Relating to and : Next, we

aim to relate the unknown with the data vector by de-riving the relationship between and the time-varying data co-variance matrix . Since is sym-metric, its elements can be represented by the following vectorof length :

(14)

It holds evidently from (3) that . Meanwhile,and are linearly related to their vectorized counterparts

and as follows:

(15)

where andare some specific mapping

matrices derived in Appendix A. Accordingly, it can be shownthat

(16)

where is of size.

Finally, using (13) and (16), we can express the measurementvector as a linear function of the vector-form cyclic spectrum

as

(17)

where is of size .

3) Further Simplifications: Since is conjugate sym-metric about the origin for real-valued signals, all its usefulinformation is contained in the subregion I of the cyclic spec-trum plots in Fig. 1; as a result only the shaded region inFig. 1(b) needs to be recovered in order to reconstruct the orig-inal complex-valued cyclic spectrum of real-valued signals. Let

denote the cyclic spectrum matrix for the shaded region,which is of size ( denotes floor operation).It is straightforward to show that is related to as

(18)

where... is an identity matrix of size

and of size is formedby removing the first and last columns of an reverseidentity matrix. Vectorizing on both sides yield

(19)

where and is of size.

Plugging (19) into (17), we obtain the equivalent expressionfor the real-valued case:

(20)

where is of size. Here, the number of complex-valued unknowns in (20)

has been reduced to . The estimate can beacquired from the inverse problem of (20). Afterward,

can be found based on (19).

B. Reconstruction Formulation

Given the available data covariance vector , the cyclicspectrum recovery problem boils down to solving from(17), which is an under-determined linear system. To accuratelysolve this problem, we make use of the prior knowledge aboutwideband communication signals. We observe that the 2-Dcyclic spectrum is highly sparse, which can be incorporated intothe spectrum estimator by penalizing the -norm of the cyclicspectrum vector to induce a sparse solution. Hence, weformulate an unconstrained -norm regularized least squares(LR-LS) problem for reconstructing , as follows:

(21)

In (21), is a weighting scalar that balances between thesparsity of the solution induced by the -norm term and the datareconstruction error reflected by the -norm LS term.

The optimization formulation in (21) is attractive since itaddresses the sparsity of the cyclic spectrum and is a convexproblem, which can be solved by convex solvers such as cvx toefficiently reach the global optimum [34].

C. Implementation

In practical implementations with finite data samples, the datavector in (21) is replaced by its finite-sample estimate .


Specifically, the steps for sparse cyclic spectrum recovery areas follows:

1) Observe the received analog signal for a total time of, which is chosen according to the desired sensing time

and the time needed to produce reliable statistics.2) Divide the total time into blocks, each of time-block

length . Within each block , the same randomsampler (e.g., the AIC [21]) is used to obtain a compres-sive sample vector directly from at a reduced rateof , whereand , with a compression ratio . Es-timate the covariance matrix of the compressive sam-ples by the finite-sample average across all blocks as

, and form .3) Replace with , and solve (21) to recover .The choice of reflects the design tradeoff, via

for given and . As increases, the estimation variance ofis reduced and the finite-sample effect is alleviated, but the

frequency resolution of the recovered suffers, because theresolution increases linearly in .

IV. RECOVERY OF POWER SPECTRUM FOR

STATIONARY SIGNALS

Besides cyclic spectrum estimation, the developed formula-tions can be used to efficiently estimate the power spectrum (akaPSD) in the frequency domain, which is a special case of the 2-Dcyclic spectrum when the signal is (treated as) stationary.Power spectrum estimation is useful in many well-known spec-tral estimators, such as energy detectors.

When is stationary, the covariance function be-comes time invariant, which means that

(22)

Let denote the discrete power spectrum with respect tofrequency . It is given by the Fourier series of , inthe form

(23)

Defining andresults in

(24)

where is the -point DFT matrix.

A. Power Spectrum Reconstruction

Because of the stationarity property in (22), the time-varyingcovariance matrix in (38) is reduced to a stationary one asfollows:

.... . .

...

(25)

Mapping between (22) and (25) reveals that

(26)

Apparently, can be represented linearly by througha binary-valued mapping matrix as follows:

(27)

where can be deduced from (26) as: for any column, there are elements of 1’s at row indices

and , whileall the rest elements are zero; for the column , there are

elements of 1’s at row indices ,while the rest elements on this column are zero.

Although is stationary, is not, because each elementis generated by filtering with an individual filter

indicated by the corresponding row of , and aremutually incoherent as explained for (3). Therefore, there arestill degree of freedom in , represented by

as in (15). It follows from (16) and (27)that

(28)

where is of size .Putting together (24) and (28), we reach

(29)

where is of size . In (29),there are linear equations and unknowns. When

, this is an over-determined system that canbe solved using simple least squares. This is a revealing observa-tion, suggesting that sub-Nyquist-rate sampling withcan be employed to permit a unique least squares solution to thePSD vector , even when is non-sparse. Strongest compres-sion is attained when , whichsuggests

(30)

This result owes to the use of all cross-correlations of thecompressive samples in recovering the second-order statistics,which is also exploited in [31], [32]. With (29), the PSD can beestimated as

(31)

By setting , (31) reduces to a sparsity-agnostic LS esti-mator that can be used when the PSD is non-sparse and

; on the other hand, adopting results in a spar-sity-aware LR-LS estimator useful when the PSD is sparse, asin CR sensing applications.

B. Sampler Design and Minimal Sampling Rates

It is of interest to investigate the issues of sampler design andminimal sampling rates when (31) is employed for power spec-


trum estimation. First, we address a special case when the un-known PSD is non-sparse and a special form of nonuniformrandom ADCs is employed [22]. The sampling matrix

has a single nonzero element of one in each rowwhile all other elements are zero [22], which is essentially an

matrix constructed by randomly selecting rows outof the identity matrix. The sampler design problemboils down to the row selection problem. When the pair-wisedifferences of the selected row-indices span over all the pos-sible time-lags , the resulting sampler en-sures that all elements of be captured by the cross-cor-relation terms of the compressive samples, and hence is in-vertible. Similar to [32], we deduce that solutions to the minimallength- sparse ruler problem yield the desired row selec-tions. The minimal sparse ruler problem has been studied in theliterature, see, e.g., [33], and has been used to design minimalredundant linear arrays [36]. Many exact and approximate so-lutions for this problem have been precalculated and tabulated.Some examples of the minimal sparse rulers are shown in [32]for various values of . The examples in [32] indicate that theminimal compression ratio can be lower than 50%for , and goes to below 25% for , while en-suring lossless reconstruction of any non-sparse PSD with lim-ited bandwidth .

Since the aforementioned sparse sampling matrices are de-generated versions of full (random) matrices, we conjecture thatthe compression ratio can be made even lower when full randommatrices are utilized.

Further, when the PSD is sparse and the sparsity-aware for-mulation in (31) is used, the inverse problem of (29) can besolved with high probability even when it is an under-deter-mined system, which means that the compression ratio can bemade even lower than that in (30) or that given by the minimalsparse ruler problem for non-sparse signals.

V. CYCLIC-BASED SPECTRUM OCCUPANCY ESTIMATION

After recovering the sparse 2-D cyclic spectrum inits vectorized form , we now want to simultaneously esti-mate the spectrum occupancy of all frequency sub-bands withinthe monitored wide band. We develop two spectrum detectionalgorithms: one adopts a band-by-band multi-cycle generalizedlikelihood ratio test (GLRT) framework that works for all typesof modulation and waveform patterns, and the other is tailoredto known modulation types for simple and fast estimation, forwhich we take BPSK signals as an example.

A. Band-by-Band Approach Based on Multi-Cycle GLRT

The goal of spectrum occupancy estimation is to decidewhether a specific frequency location is occupied or not.We set , according to thefrequency resolution of the discrete cyclic spectrumin (5). The sensing task amounts to a band-by-band inspectionof the spectrum occupancy over the entire frequency range

, with . It is important to note that sucha band-by-band inspection is a computational approach forprocessing the data collected simultaneously from a widebandantenna, which is fundamentally different from a narrowband

approach in which narrowband antennas scan the wide spec-trum one by one along with frequency shifters and narrowbandfilters/processors.

We now focus on the occupancy decision on a single band. An active signal on this band would occupy a region

of the 2-D cyclic spectrum map defined by the cyclic-frequencyand frequency pairs satisfying [7]:

and (32)

In the discrete-time domain, this region is represented by dis-crete points , which correspond to a set of in-teger-valued indices , that is,

. Because and by def-inition, the index set can be deduced from (32) as

and

(33)

We stack the estimated into a vector

of length , which is formed by selected en-tries of the vectorized cyclic spectrum in (21), that is,

. This row selection operationcan be expressed as

(34)

where the binary-valued selection matrixis obtained from the identity matrix by retaining itsrows with indices only. Apparently, .

To test for the presence of a PU signal at the frequency ,the following binary hypothesis test is formulated:

(35)

where is the nonrandom true vector of cyclic spectrumvalues, and is asymptotically Gaussian distributed, i.e.,

, where is the asymp-totic covariance matrix. Because is not readily available,we derive a blind estimator for using the available re-duced-rate measurements , shown in Appendix B.Replacing by results in a data-adaptive GLRTdetector.

Treating as an unknown nuisance parameter, the adaptiveGLRT can be derived for (35), which yields the following teststatistic:

(36)

Using the noise statistics derived in Appendix B, (36) is theoptimal detector in the presence of colored Gaussian noise[35]. Specifically, is compared with a threshold tomake decisions on the occupancy at frequency , whereis chosen to comply with the allowable false alarm rate [35].The detection performance is not sensitive to the threshold,because the sparsity-enforcing -norm operation has shrank


Fig. 2. Nonzero region of support for the cyclic spectrum of BPSK.

small values of to be near zero and much separable fromtrue signals [19]. This is a multi-cycle GRLT, becausecontains multiple cyclic frequencies for . Whenthe test is done for all , we will have identified not onlythe carrier frequency but also the bandwidth of each occupiedfrequency band. Putting together all the occupied bands, weare able to depict the spectrum occupancy over the entire wideband. Overall, the proposed spectrum occupancy estimationalgorithm is implemented by the following steps:

1) Set . Form the cyclic spectrum matrix from.

2) Let . Find the cyclic spectrum values ofinterest from (32), and calculate from (34).

3) Calculate the test statistic in (36) using calculatedfrom (45) and (47). If it is larger than a predeterminedthreshold , then a PU is declared present at frequency

, otherwise PU absence is declared.4) If , increase by 1 and go to Step 2).

B. Fast Algorithm for Known Modulation Types

When the modulation type of the signal sources are known,their cyclic features on the bi-frequency plane can be utilizedto quickly identify the key signal paprameters. For example,consider that consists of multiple BPSK signals, each withcarrier frequency , symbol rate and a full-width rectan-gular pulse shaper. The number of signal components and theirmodulation parameters are unknown, but the modulation type isknown. Ignoring the weak sidelobes in the frequency domain,the major cyclic feature is a main lobe at , spanningover to (Fig. 14(a) in [7]). The nonzeroregion of support for such a signal is shown in Fig. 2.

Our idea is to identify the modulation-dependent parametersand by finding the lobe locations on the estimated 2-D

cyclic spectrum. First, we can simply search over the cyclic fre-quency along the axis . If ( is a preset

threshold), then we claim that there is a BPSK signal with esti-mated carrier frequency .

Next, we let . For each , wesearch along the line to find the double-sided width ofthe lobe, denoted by , such that all the points in the lobe haveabsolute values equal to or greater than . The bandwidth of thecorresponding BPSK signal is also , and hence we can claimthat the frequency band is occupied, andthe estimated symbol period is .

By now, we have rapidly identified not only the carrier fre-quency but also the bandwidth of a BPSK signal. Theprocedure is applicable to the entire wide band to identify allBPSK signals. Combining all the occupied frequency bands, weare able to draw the spectrum occupancy map.

Extensions of the fast approach to other types of modulatedsignals are possible, using the features of the modulation types.For instance, for an SQPSK signal [7] with carrier frequencyand symbol rate , two peaks with similar heights will appearat and , and this feature can beused for fast detection.

VI. SIMULATIONS

This section presents Monte Carlo simulation results that tes-tify the effectiveness of the proposed algorithms under samplingrate reduction and noise uncertainty. The frequency band undermonitoring has MHz, in which two signal sources

and appear at the center frequencies 187.5 MHz and375 MHz, respectively. Two setups are considered: 1) bothand are BPSK signals, 2) is BPSK and is QPSK.The symbol period for each source is fixed at s.The reference (Nyquist) sampling rate is GHz, andthe non-compressed discrete-time representation is of length

in each time block1. Although both setups 1) and 2)have the same nonzero spectral occupancy of 8%, their nonzerocyclic spectral occupancies are not the same, due to the dif-ferent cyclic features of BPSK and QPSK signals. In fact, Setup1) has a 0.75% cyclic spectral occupancy, and Setup 2) has acyclic spectral occupancy of 0.625%. These occupancy ratiosalso demonstrate that the 2-D cyclic spectrum is much moresparse than the power spectrum.

A. Robustness to Rate Reduction

A key parameter of interest is the compression ratiofor sampling rate reduction. Its impact on the error

performance of the proposed cyclic spectrum estimationalgorithm is tested on Setup 2) with a mixture of BPSKand QPSK signals, in a noise-free case. The performancemetric is the normalized mean-square error (MSE) of thereconstructed cyclic spectrum with respect to the uncom-pressed one , for a given compression ratio . Thatis, . Similarly, the MSEperformance for the proposed PSD reconstruction is also eval-uated as: , where representsthe power spectrum reconstructed from compressive samples

1Here� is set to be small for simple illustration, but shall take a proper largevalue in practice according to the desired frequency resolution � �� .


Fig. 3. Normalized MSE of reconstruction versus compression ratio�� .

Fig. 4. Probability of detection � versus compression ratio�� , for a fixedfalse alarm ratio of 0.01.

with compression ratio of and stands for the uncom-pressed one. Fig. 3 depicts the MSE versus the compressionratio , for and data blocks. The MSEperformance of the PSD reconstruction algorithm is also shownin the same figure for . The figure shows that theMSE curves start to flatten out when , which isthe rate-reduction region where the cyclic spectrum and PSDrecovery algorithms are robust. The estimation performance for

outperforms that of , because of more effectivealleviation of the finite-sample effect. On the other hand, theestimation performance for is quite close to that for

, indicating that the performance improvement offeredby increasing eventually saturates. This is because the finitesample effect has been adequately alleviated when is largeenough, such that the compression ratio has started to dominatethe MSE performance.

After cyclic spectrum recovery, the GLRT-based approach isutilized to estimate the frequency occupancy. Fig. 4 depicts theprobability of detection versus , for a false alarm rate

. The decision threshold is set to satisfy thefalse alarm rate via simulations. It shows that the compressionratio can go as low as 0.25 while keeping . As a re-sult, when the frequency occupancy is 8%, the sampling ratecan be reduced by three-quarter, while still maintaining good es-timation performance. When the frequency occupancy is moresparse, the effective sampling rate can be reduced even further.

B. Robustness to Noise Uncertainty

To testify the robustness of spectrum sensing under noiseuncertainty, we compare the receiver operating characteristic(ROC) of the proposed compressive cyclic feature detector tothat of an energy detector (ED). A traditional ED can be com-bined with existing compressive sampling techniques through atwo-step procedure. First, the signal realization within each datablock is treated as a deterministic signal, and the frequency re-sponse is reconstructed from compressive sam-ples using the LR-LS formulation as

(37)

Then, the estimated over all blocks are used to computethe power spectrum as the sample average, as follows:

where returns the diagonal of a matrix. In energy detec-tion, each element of serves as the energy-related test statisticfor the signal component at the corresponding frequency band,which is passed through a decision threshold to detect the fre-quency occupancy band by band. Note that the existing com-pressive sampling solution in (37) is computationally expensive,because inverse problems have to be solved before computingthe sample average . In contrast, the proposed compressivePSD approach in Section IV only solves one inverse problemby directly extracting the signal statistics rather than the signalitself. Besides, (37) is inferior in terms of recovery performance,because it does not make use of the fact that the frequency re-sponses across all blocks share the same nonzero support.

During spectrum detection, the noise power level is varyingand hardly known exactly, due to thermal noise change, ampli-fier gain change, calibration error, and fluctuating interferencelevels in CR networks. The lack of accurate knowledge of thenoise variance is known as noise uncertainty [4]. In the simula-tions, the SNR is defined as the in-band signal to noise ratio ofeach signal source. We use a nominal SNR of dB with noiseuncertainty levels of and dB, which means thatthe actual SNR levels vary randomly between of the nom-inal value. The impact of the noise uncertainty is assessed usingthe method of robust statistics [5]. In this method, the upperlimit of the noise PSD level that corresponds to the lower limitof the SNR is used to calculate the probability of false alarm,whereas the lower limit of the noise PSD corresponding to theupper limit of the SNR is used to calculate the probability of de-tection. Such robust statistics assess the detection performanceunder worst case scenarios. A total of data blocks


Fig. 5. ROC of the compressive cyclic feature approach (“Cyclo”), PSD ap-proach (“PSD”) and energy detection (“ED”).

are used for noise averaging, and the number of compressivemeasurements in a block is , which corresponds to acompression ratio of 0.5. Fig. 5 depicts the ROC of the pro-posed compressive cyclic feature detector and PSD detector,compared with ED. The figure shows that the proposed cyclicfeature detector is quite insensitive to noise uncertainty, becausethe varying yet non-cyclic noise does not appear at . TheROC performance considerably outperforms that of energy de-tection, while the latter degrades noticeably as the noise uncer-tainty level increases.

C. Comparison of Spectrum Occupancy Estimation Techniques

Here the GLRT and the fast cyclic-based spectrum occupancyestimation techniques are compared, using Setup 1) where bothPUs are BPSK signals. The test parameters are: the SNR isdB without uncertainty, (i.e., %) and

. Fig. 6 shows the ROC curves for both algorithms. Thefigure shows that the GLRT approach provides slightly betterperformance. Considering that the GLRT approach applies togeneral modulation types, it is preferred whenever enough com-putational resources are available.

D. Influence of the Number of Blocks Used

The total sensing time is an important parameter that de-termines network responsiveness and performance. When theblock size and the sampling rate are fixed, the total sensingtime depends on the number of blocks . Increasing meanslonger sensing time and shorter time for transmission, but itcould also lead to improved sensing performance. For Setup2), Fig. 7 shows the impact of on sensing performance, for

dB without noise uncertainty, and 50% rate com-pression with . It shows that has better perfor-mance than , but increasing further to doesnot offer much additional performance improvement. Similarto Fig. 3, this phenomenon of diminishing performance gain isbecause the finite sample effect has been adequately alleviatedwhen is large enough.

Fig. 6. ROC performance of the two spectrum occupancy estimationapproaches.

Fig. 7. ROC of the proposed sensing method for different � values:� � ��

� � ��.

VII. CONCLUSION

This paper has proposed a new method for recovering thesparse 2-D cyclic spectrum from a reduced number of compres-sive samples. The vectorized cyclic spectrum is reformulatedto take a linear relationship with the covariance function of thecompressive samples, which is a key step in enabling effectiverecovery of the 2-D cyclic spectrum via convex -norm mini-mization. As a special case of the compressive cyclic spectrumestimator, a new power spectrum estimator for stationary signalsis also developed, which allows sub-Nyquist rate sampling evenfor non-sparse signals. From the recovered cyclic spectrum, twotechniques have been developed to estimate the spectrum oc-cupancy of a wide band hosting an unknown number of activesources: a band-by-band multi-cycle GLRT detector, and a fastthresholding technique for signals with known modulation typessuch as BPSK signals. The proposed spectrum occupancy esti-mation techniques demonstrate salient robustness to samplingrate reduction and noise uncertainty.


APPENDIX AMAPPING MATRICES

This Appendix shows the mapping matrices and thatenable linear connections between the data covariance matrix

and the vector in (6). The results apply for any value of.The time-varying covariance matrix has the

form

.... . .

...

(38)which is symmetric for real-valued signals.

To relate with , we inspect , where the oper-ator stacks all the columns into a vector. Bothand contain the same set of nonzero elements, and hence canbe linearly related to each other as

and (39)

where and are specific mapping matrices.To show the expressions for and , let us consider

the th element of in (7), denoted as , which isnonzero for any and . Its value

appears at two symmetric locations inside of (38)for , and appears at one location on the diagonal ofwhen . Specifically,

(40)

Note that stacks all the columns of after removing thosezero entries at the lower-right triangle; hence, the th ele-ment of shows up as the th element of with

. That is, , where

(41)

Meanwhile, and in (40) appear atthe th and th rows of respectively, where

(42)

Linking the above relationships among and, we now summarize the expressions for and

used in (39). For to hold, hasonly two nonzero entries of value one at the th and

th rows while the rest row entries are zero, for a givencolumn indexed by . When , the two nonzerorow locations coincide at ,on the column . In mathematical form,

can be expressed as

(43)

To construct , we note from (40) thatgiven , the th entry of is equal to both the

th and th entries of , and hencecan be computed by the averaged sum of these two entries.When , these two entries degenerate to one entry at

. Letting denote the Kro-necker delta, can be deducedfrom (40) as

(44)

APPENDIX BNOISE COVARIANCE ESTIMATION ON CYCLIC SPECTRUM

This Appendix derives a non-data-aided estimator for the noisecovariance matrix used in the cyclic feature detector in(35).

First, we evaluate the error covarianceof the finite-sample estimate obtained

from . Define and formfor each block .

Then, the true covariance vector of the compressive samplescan be estimated as . Each summandcan be expressed as , where is theestimation error from the th block. An unbiased estimate of

can be obtained from as follows [35]:

(45)

Next, we estimate the error covariancewhere the estimate results from via

(17). The inverse problem for (17) is solved by (21), but it isnot convenient for performance analysis. Alternatively, we relaxthe -norm term in (21) by its -norm, which yields

. Defining , theestimation error covariance of can be obtained from that of

as

(46)

Finally, noting the linear relationship in (34),the error covariance of can be estimated as

(47)

With (45) and (47), we have found the error covariance esti-mate that can be computed from the available compressivesample vectors .

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[35] H. L. Van Trees, Detection, Estimation, and Modulation Theory, PartI. New York: Wiley-Interscience, 2001.

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Zhi Tian (M’98–SM’06) received the B.E. degree inelectrical engineering from the University of Scienceand Technology of China, Hefei, in 1994, and theM.S. and Ph.D. degrees from George Mason Univer-sity, Fairfax, VA, in 1998 and 2000, respectively.

Since August 2000, she has been on the facultyof Michigan Technological University, Houghton,where she is a Professor. Her general interests are inthe areas of signal processing for wireless commu-nications, estimation, and detection theory. Currentresearch focuses on cognitive radio networks and

distributed wireless sensor networks.Prof. Tian served as an Associate Editor for the IEEE TRANSACTIONS

ON WIRELESS COMMUNICATIONS and the IEEE TRANSACTIONS ON SIGNAL

PROCESSING. She received a CAREER award from the US National ScienceFoundation in 2003.

Yohannes Tafesse received the M.S. degree in elec-trical engineering from Michigan Technological Uni-versity, Houghton, in 2011.

His research interests lie in the areas of digital andwireless communications.

Brian M. Sadler (S’81–M’81–SM’02–F’07) re-ceived the B.S. and M.S. degrees from the Universityof Maryland, College Park, and the Ph.D. degreefrom the University of Virginia, Charlottesville, allin electrical engineering.

He is a senior research engineer at the Army Re-search Laboratory (ARL), Adelphi, MD.

Dr. Sadler is Fellow of the Army Research Lab-oratory. He is an Associate Editor for the EURASIPSignal Processing, was an Associate Editor for theIEEE TRANSACTIONS ON SIGNAL PROCESSING and

IEEE SIGNAL PROCESSING LETTERS, and has been a guest editor for severalIEEE journals. He is a member of the IEEE Signal Processing Society SensorArray and Multi-channel Technical Committee, and has been active in IEEEconference planning and organization for many years. He received a Best PaperAward (with R. Kozick) from the Signal Processing Society in 2006. He hasreceived several ARL and Army R&D awards, as well as a 2008 OutstandingInvention of the Year Award from the University of Maryland (with J. Barasand P. Yu). His research interests include information science, networked andautonomous systems, acoustics, optics, and mixed-signal integrated circuitarchitectures.

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