Projected Barzilai-Borwein Methods Applied to Distributed Compressive Spectrum Sensing

Projected Barzilai-Borwein Methods Applied toDistributed Compressive Spectrum Sensing

Le Thanh Tan∗, Hyung Yun Kong∗, Vo Nguyen Quoc Bao∗∗School of Electrical Engineering

University of Ulsan, San 29 of MuGeo Dong, Nam-Gu, Ulsan, Korea 680-749Email: {tanlh,hkong,baovnq}@mail.ulsan.ac.kr

Abstract—Cognitive radio allows unlicensed (cognitive) users touse licensed frequency bands by exploiting spectrum sensing tech-niques to detect whether or not the licensed (primary) users arepresent. In this paper, we present a compressed sensing appliedto spectrum-occupancy detection in wide-band applications. Thecollected analog signals from each cognitive radio (CR) receiverat a fusion center are transformed to discrete-time signals byusing analog-to-information converter (AIC) and then employedto calculate the autocorrelation. For signal reconstruction, weexploit a novel approach to solve the optimization problemconsisting of minimizing both a quadratic (l2) error term and anl1-regularization term. Specifically, we propose the Basic gradientprojection (GP) and projected Barzilai-Borwein (PBB) algorithmto offer a better performance in terms of the mean squarederror of the power spectrum density estimate and the detectionprobability of licensed signal occupancy.

I. INTRODUCTION

The explosive growth of wireless communication has madethe problem of spectrum utilization more critical. On the onehand, the increasing diversity and demand of high quality-of-service applications are the main reasons of the crowdednature of spectrum allocation. On the other hand, the spectrumscarcity is not the result of heavy usage of spectrum, butis due to the inefficiency of the static frequency allocation.For example, the typical spectrum utilization of around fivepercent or even less is reported in [1] . To solve this problem,the Federal Communications Commission (FCC) proposed theopening of licensed bands to unlicensed users and cognitiveradio (CR) was born to improve the utilization of spectrumresource. In addition, the IEEE has established an IEEE 802.22workgroup to build the standards of WRAN based on CRtechniques [2].

Spectrum sensing for wide-band CR applications associateswith considerable technical challenges. The radio terminals arerequired to employ a bank of tunable narrowband bandpassfilters and search one narrow frequency band at a time. Thelarge bandwidth operation requires an unfavorably large num-ber of RF components. Furthermore, high speed processingunits (DSPs or FPGAs) are needed to flexibly search overmultiple frequency bands concurrently. In particular, inputanalog signals must be converted to discrete-time signals atNyquist sampling rate or higher by using analog-to-digitalconverter (ADC). However, the extremely high sampling ratesof wide-band ADCs are a main challenge with this model.Meanwhile, due to the timing requirements for rapid sensing,

only a limited number of measurements can be acquiredfrom the received signal, which may not provide sufficientstatistic when traditional linear signal reconstruction methodsare employed.

Compressed sensing (CS) [3], [4] builds on the surprisingrevelation that a signal having a sparse representation in onebasis can be recovered from a small number of projectionsonto a second basis that is incoherent with the first. Especially,compressed sampling approach can get the sparse signal atthe rates lower than the Nyquist sampling method; signalreconstruction which is a solution to a convex optimizationproblem called min-l1 with equality constraints makes useof the basic pursuit (BP) or some modified methods such asorthogonal matching pursuit (OMP), tree-based OMP (TOMP)[5], [6]. In [5], authors present a model of a single wide-bandCR that uses CS based spectrum sensing schemes.

This paper considers the problem of determining spectrumoccupancy of a wide-band system. Our proposed CR systemhas a number of CRs and a centralized fusion center thatcollects data from individual CRs with different signal-to-noiseratios (SNRs) and decides these spectra to be available or not.Specifically, the same wide-band analog signal is transformedto discrete-time signal while preserving its salient informationby using a so-called analog-to-information converter (AIC).For that kind of sparse input signal, an AIC can be acquiredat sub-Nyquist rates (matching the information rate of thesignal), and it also offers the capability of extracting the featureof data; therefore, the performance of the AIC system iscomparable to that of ADC system in term of SNR or effectivenumber of bits (ENOB) [7], [8], [9]. Then output signals ofAIC are used to generate autocorrelation vectors that will becollected at the fusion. To obtain an estimate of the signalspectrum, we propose a novel version of distributed CS algo-rithm based on [10] by using a standard approach to minimizean objective function includes a quadratic (squared l2) errorterm combined with a sparseness-inducing (l1) regularizationterm. It can be easily observed that basic gradient projec-tion (GP) can perform the high quality reconstruction [11];however, the time-consuming property makes this approachnot appropriate for spectrum sensing scheme which requirestime-constraint. Thus, the novel approach, which exploits theprojected Barzilai-Borwein (PBB) technique embedded in acontinuation heuristic to recover the efficient performance[12], is introduced to improve both time-reduction and the

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE DySPAN 2010 proceedings

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performance of signal reconstruction.We also compare the performance of the distributed com-

pressive spectrum sensing scheme with that of the scheme of[5] for a single CR to show the gains accrued from spatialdiversity and exploiting the joint sparsity structure. We use(i) the mean squared error (MSE) between the reconstructedpower spectrum density (PSD) estimate and the PSD basedon Nyquist rate sampling, and (ii) the probability of detectingspectrum occupancy over the channels as performance mea-sures.

The rest of this paper is organized as follows. Section IIcontains the model of the wide-band analog signal compressedsensing. A compressive spectrum sensing scheme for singleCR is presented in Section III. And an extention to collabo-rative compressed spectrum sensing for multiple CR is shownin Section IV. Section V demonstrates simulation results.Finally, concluding remarks are given in Section VI.

II. SYSTEM MODEL OF WIDEBAND ANALOGSIGNAL COMPSESSED SENSING

A. Signal model

We consider the frequency range of interest to be comprisedof maxI non-overlapping consecutive spectrum bands, a CRnetwork consisting of J CRs and a centralized fusion center.Sensing is performed periodically at each CR and the resultsare sent to the fusion center, where a decision is made onwhether or not there is a licensed signal present in eachchannel.

B. Overview of compressed sensing

According to Donoho [3], in CS theories, an N×1 vector ofdiscrete-time signal x = Ψs, where Ψ is the N ×N sparsitybasis matrix and s is the N ×1 vector with K � N non-zero(and large enough) entries si, can be used to reconstruct thesignal from M measurements; especially, M depends on thereconstruction algorithm and is usually much less than N . Thismeasurement can be done by projecting x on to an M × Nbasis matrix Φ that is incoherent with Ψ [13]

y = Φx = ΦΨs. (1)

The reconstruction is done by solving the following l1-normoptimization problem as

s= arg mins

‖s‖1 s.t. y = ΦΨs. (2)

Linear programming techniques, e.g., basis pursuit [14], oriterative greedy algorithms [15] can be used to solve (2).

C. Compressed sensing of analog signals

Because CS was proposed for discrete-time signal process-ing, we must use ADC sampling at Nyquist rate to discreterizethe analog signal before applying the CS. After that, thecompressed sensed data are sent to DSP blocks for furthermanipulation. While it is true that the data volume to beprocessed by DSP blocks is reduced due to the CS, a high-speed ADC sampling at Nyquist rate is still required when

the received signal is wideband. It is natural to think aboutways to avoid the high-speed ADC by applying CS to theanalog signal directly. A related idea was first described in[8], where the analog signal was first demodulated with apseudo-random chipping sequence p(t), then passed throughan analog filter h(t), and the measurements were obtained inserial by sampling the filtered signal at sub-Nyquist rate. Theserial sampling structure is appropriate for real-time process-ing. However, to achieve a satisfactory signal reconstructionquality, the order of the filter is usually higher than 10. Inaddition, because the measurements are obtained by samplingthe output of the analog filter sequentially, they are no longerindependent due to the convolution in the filter, which bringssome redundancy in the measurements.

Specifically, suppose that we have an analog signal x(t)which is K − sparse over some basis Ψ for t ∈ [0, T ] as inthe following expression:

x(t) =N∑

i=1

siψi(t), (3)

where x is the N × 1 vector x =Ψs, Ψ is the N ×N sparsitybasis matrix Ψ= [ψ0(t), ψ1(t), . . . , ψN (t)] and s an N × 1vector with K � N non-zero elements si. It has been shownthat x can be recovered using M = KO(logN) non-adaptivelinear projection measurements on to an M ×N basis matrixΦ that is incoherent with Ψ [13]. The received signal y canbe viewed as the transmitted signal plus some additive noise

y = Φx + n = ΦΨs + n. (4)

There are several choices for the distribution of Φ such asGaussian, Bernoulli.

Reconstruction is achieved by solving the l1-norm opti-mization problem as in (2). In this paper, the reconstructionproblem, that has been highly interested in solving the convexunconstrained optimization problem, is a standard approachconsisting in minimizing an objective function which includesa quadratic (squared l2-norm) error term combined witha sparseness-inducing (l1-norm) regularization term. So theproblem can be given by

mins

12‖y − ΦΨs‖2

2 + τ‖s‖1. (5)

Basic GP is able to solve a sequence of problems (5) effi-ciently for a sequence of values of τ . The gradient projectionalgorithms for solving a quadratic programming reformulationof a class of convex nonsmooth unconstrained optimizationproblems are significantly faster (in some cases by ordersof magnitude), especially in large-scale settings. Instances ofpoor performance have been observed when the regularizationparameter is small, but in such cases the gradient projectionmethods can be embedded in a simple continuation heuristicto recover their efficient practical performance. The newalgorithms are easy to implement, work well across a largerange of applications, and do not appear to require application-specific tuning.


Analog filterh(t) Autocorrelation

x(t) yk ry

pc(t)

k/Mquantizer

AIC block

Fig. 1. CS acquisition at individual CR sensing receiver.

III. COMPRESSIVE SPECTRUM SENSING ATSINGLE CR

We begin by describing the CS acquisition and recoveryscheme for a single CR (J = 1) case. Fig. 1 depicts theacquisition at a single CR sensing receiver. The analog base-band signal x(t) is sampled using an AIC. Following thecircuit implementation of the AIC system in series of previousworks [7], [8], [9], an AIC may be conceptually viewed as anADC operating at the Nyquist rate, followed by compressivesampling. Denote the N × 1 stacked vector at the input of theADC by

xk =[xkN xkN+1 ... xkN+N+1

]T

k = 0, 1, 2 ...,(6)

and the M × N compressive sampling matrix by ΦA. Theoutput of the AIC denoted by the M × 1 vector

yk =[ykM ykM+1 ... ykM+M+1

]T

k = 0, 1, 2 ...,(7)

is given byyk = ΦAxk. (8)

The respective N ×N and M ×M autocorrelation matricesof the compressed signal and the input signal vectors in (6)and (7) are related as follows:

Ry = E[ykyH

k

]= ΦARxΦH

A , (9)

where subscript H denotes the Hermitian. The elements of thematrices in (9) are given by: [Ry]ij = ry (i− j) = r∗y (j − i),[Rx]ij = rx (i− j) = r∗x (j − i).

The respective 2N × 1 and 2M × 1 autocorrelation vectorscorresponding to (6) and (7) can be expressed as follows:

rx =[

0 rx(−N + 1) ... rx(0) ... rx(N − 1)]T

,

(10)

ry =[

0 ry(−M + 1) ... ry(0) ... ry(M − 1)]T

,

(11)here the first zero values are artificially inserted. And theseabove vectors represent the first column and row of therespective autocorrelation matrices. To obtain the CS recoverylike the formula (5), we must to make the relation betweenthe autocorrelation vectors in (10) and (11). Using operationsin matrix algebra, we can derive as

ry = Φrx, (12)

note that

Φ =

[ΦAΦ1 ΦAΦ2

ΦAΦ3 ΦAΦ4

]. (13)

Denote that φ∗i,j is the (i, j)-th element of ΦA, the M × Nmatrix ΦA has its (i, j)-th element given by

[ΦA

]i,j

=

{0

φM+2−i,j

i = 1, j = 1, ..., N,i �= 1, j = 1, ..., N, (14)

and the N × N matrices Φ1, Φ2, Φ3, Φ4 are definedas Φ1 = hankel

([0N×1] ,

[0 φ∗1,1 ... φ∗1,N−1

]),

Φ2 = hankel([

φ∗1,1 ... φ∗1,N

],[φ∗1,N 01×(N−1)

]),

Φ3 = toeplitz([0N×1] ,

[0 φ1, N ... φ1, 2

]),

Φ4 = toeplitz([

φ1,1 ... φ1,N

],[φ1,1 01×(N−1)

]),

where hankel(c, r) is a hankel matrix (i.e., symmetric andconstant across the anti-diagonals), note that c is the firstcolumn and r is the last row of this matrix. toeplitz(c, r)is a toeplitz matrix (i.e., symetric and constant across thediagonals), note that c is the first column, and r is the firstrow of this matrix. And 0a×b is the a× b zero matrix.

We also know that using the wavelet-based edge detectionin [16-17], the band boundaries (locations) can be recoveredfrom 2N -1 local maxima of the wavelet modulus zs and theband number is determined by the number of local peaks; asan experiment when N � M in [5], zs can be recoveredunder the sparseness constraint, and therefore there is alinear transformation equality linking zs to the compressedmeasurement vector ry . And rx has a sparse representation inthe edge spectrum domain [5], that is

rx = Gzs, (15)

where zs is the discrete 2N × 1 vector, and G = (ΓFW)−1.The 2N × 2N matrices W and F represent respectively awavelet-based smoothing and a Fourier transform. The 2N ×2N matrix Γ is a derivative operation given by

Γ =

⎡⎢⎢⎢⎢⎢⎢⎣

1 0 · · · 0

−1. . . · · · 0

0. . .

. . .. . .

0 · · · −1 1

⎤⎥⎥⎥⎥⎥⎥⎦.

Combining (12) and (15), we can formulate the CS recon-struction of the edge spectrum as a convex unconstrainedoptimization problem:

minzs

12‖ry − ΦGzs‖2

2 + τ‖zs‖1 (16)

To solve the above problem, we use the GP approach whichis described in the Section IV for an individual CR case.The spectrum estimate can be evaluated as a cumulative sum

of terms zs =[zs (1) zs (2) · · · zs (2N)

]T

. Thediscrete components of the PSD estimate are given by

Sx (n) =n∑

k=1

zs (k) (17)


1 k,1 y,1

J k,J y,J

.

Fig. 2. Distributed compressive spectrum sensing scheme for multiple CRs.

IV. COLLABORATIVE COMPRESSED SPECTRUMSENSING

Let xj(t) be the wide-band analog baseband signal receivedat the j-th CR sensing receiver. Each CR sensing receiver pro-cesses the received signal to obtain an 2M×1 autocorrelationvector ry, j of the compressed signal, as in the CS acquisitionstep described in Section III, these vectors are then sent tothe fusion center. The fusion center applies a GP algorithmto jointly reconstruct J received PSDs Sx,j , j = 1, . . . , J andthen obtains an average PSD. The average PSD is then usedto determine the spectrum occupancy.

A. Overview of GP approach

We now describe the GP algorithm used for reconstructionof J PSDs. We write A = ΦG in terms of its columns

A =[

a1 a2 · · · a2N

]. (18)

At the j-th CR, we introduce vectors uj and vj and make thesubstitution zs,j = uj − vj , uj ≥ 0, vj ≥ 0, j = 1, · · · , J .Here uj (i) = (zs,j (i))+ = max {0, zs,j (i)} and vj (i) =(−zs,j (i))+ = max {0,−zs,j (i)} for all i = 1, . . . , 2N . Notethat (a)+ = max {0, a}. Therefore, we have ‖zs,j‖ = 1T

2N uj+1T2N vj , where 12N = [1, 1, . . . , 1]T is the vector consisting

of 2N ones. The problem (16) can be modified as

minu,v

12 ‖ry,j−A (uj − vj)‖2

2+τ1T2N uj +τ1T

2N vj

s.t. uj ≥ 0vj ≥ 0

(19)

Problem (19) can be written in more standard bound-constrained quadratic programming (BCQP) form as

minp

cT pj + 12pT

j Bpj ≡ F(pj

),

s.t. pj ≥ 0(20)

where pj =

[uj

vj

], c = τ1T

4N +

[ −b

b

], b = AT ry,j

and B =

[AT A −AT A

−AT A AT A

]. The next step is solving the

problem (20) by using a GP technique. From iterate p(k)j to

iterate p(k+1)j , we must follow the below steps,

• Step 1. Choose the scalar parameters α(k)j > 0.

• Step 2. Set: w(k)j =

[p(k)

j − α(k)j ∇F

(p(k)

j

)]+.

• Step 3. Choose the second scalar λ(k)j ∈ [0, 1].

• Step 4. Set: p(k+1)j = p(k)

j + λ(k)j

[w(k)

j − p(k)j

].

The following subsections represent two algorithms to solvethe above problem coresponding to two different ways ofchoosing α(k)

j and λ(k)j .

B. Basic Gradient Projection:The GP - Basic Algorithm

In this algorithm, we search from each iterate p(k)j along the

negative gradient −∇F(

p(k)j

), projecting onto the nonnega-

tive orthant, and performing a backtracking line search untilsufficient decrease is attained in F . We define the vector g(k)

as

g(k)j,i =

{ (∇F (pj

(k)))

i, if p(k)

j,i > 0 or(∇F (

pj(k)

))i< 0

0, otherwise.

(21)where i = 1, . . . , 2N . The procedure of this algorithm isdescribed as follows:

1) Input:

a) An initial p(0) =[

p(0)1 p(0)

2 · · · p(0)J

].

b) A 2M × J data matrix R =[ry,1 ry,2 · · · ry,J

]received from J

CR sensing receivers.c) Choose parameters β ∈ (0, 1) and μ ∈ (0, 1/2).d) Set k = 0.

2) Output: A 2N × J reconstruction matrix Zs =[zs,1 zs,2 · · · zs,J

], the average of J PSD esti-

mate vectors S(J)x .

3) Procedure:

a) Step 1. Compute α0,j as the following expression[12]:

α0,j =

(g(k)

j

)T

g(k)j(

g(k)j

)T

Bg(k)j

. (22)

Note that α0,j is solved from the expression:

α0,j = arg minαj

F[p(k)

j − αjg(k)j

]. (23)

Then to guarantee that α0,j is not too small or toolarge, we replace α0,j by mid (αmin, α0,j , αmax).Here mid (α1, α2, α3) is defined to be the middlevalue of three scalar values.

b) Step 2. Backtracking line search: chooseα

(k)j to be the first number in the sequenceα0,j , βα0,j , β

2α0,j , . . . and satisfy the followinginequality

F

((p(k)

j −α(k)j ∇F

(p(k)

j

))+)

≤ F(

p(k)j

)−

μ∇F(

p(k)j

)T(

p(k)j −

(p(k)

j −α(k)j ∇F

(p(k)

j

))+)

(24)


and update the new set of values

p(k+1)j =

(p(k)

j − α(k)j ∇F

(p(k)

j

))+

.c) Step 3. Termination test:

• Condition: We now convert p(k)j =[(

u(k)j

)T

,(

v(k)j

)T]T

to an approximate

solution z(k)s,j,GP = u(k)

j − v(k)j . And scanning

through the entire J CR to check the terminationcondition, i.e. the convergence gradient (CG)iteration is terminated when satisfying

‖ry,j − Azs,j‖22 ≤ εD ‖ry,j − Azs,j,GP ‖2

2 ,(25)

where εD is a small positive parameter. How-ever, the termination iteration is also performedwhen the number of CG steps reaches tomaxiterD.

• We perform the convergence test and terminatewith approximation solution p(k+1)

j if it is satis-fied the above conditions; otherwise we increasek to k + 1 and go back to step 1.

d) Step 4. Store the results: Store pj =[(uj)

T, (vj)

T]T

, and calculate the reconstructionvector zs,j = uj − vj . The j-th PSD estimate

vector is Sx,j (n) =n∑

k=1

zs,j (k). And the average

of J PSD estimate vectors is

S(J)x =

1J

J∑j=1

Sx,j . (26)

C. Projected Barzilai-Borwein Reconstruction Algorithm

The improvement of this algorithm is updating the step bythe following formula [18]

γ(k) = −H−1k ∇F

(p(k)

), (27)

where Hk is an approximation to the Hessian of F(p(k)

).

The procedure of this algorithm is similar to the basic GPalgorithm except the following steps:

1) Step 1. Compute step γ(k)j for the j-th CR as the

following expression:

γ(k)j =

(p(k)

j − α(k)j ∇F

(p(k)

j

))+

− p(k)j . (28)

2) Step 2. Line search: The scalar λ(k)j , (λ(k)

j ∈[0, 1]) will be found to minimize F

(p(k)

j + λ(k)j γ

(k)j

)and update the new set of values p(k+1)

j =(p(k)

j − α(k)j ∇F

(p(k)

j

))+

. Because F is quadratic, the

line search parameter λ(k)j can be evaluated by the

following closed-form expression:

λ(k)j = mid

⎧⎪⎨⎪⎩0,

(γ

(k)j

)T

∇F(

p(k)j

)(γ

(k)j

)T

Bγ(k)j

, 1

⎫⎪⎬⎪⎭ .

Note that if(γ

(k)j

)T

Bγ(k)j = 0, we choose λ(k)

j = 1.

3) Step 3. Update α(k)j : Denote

ξ(k)j =

(γ

(k)j

)T

Bγ(k)j . (29)

If ξ(k)j = 0, let α(k+1)

j = αmax, otherwise

α(k+1)j = mid

⎧⎪⎨⎪⎩αmin,

∥∥∥γ(k)j

∥∥∥2

2

ξ(k)j

, αmax

⎫⎪⎬⎪⎭ .

D. Performances

1) MSE Performance: The normalized MSE of estimatedPSD is computed by

MSE(J) = E

⎧⎪⎨⎪⎩∥∥∥S(J)

x − Sx(J)

∥∥∥2

2∥∥∥Sx(J)

∥∥∥2

2

⎫⎪⎬⎪⎭ , (30)

where S(J)x and Sx

(J) denote the average of the J PSDestimate vectors based on our compressed sensing approachand the periodogram using the signals sampled at the Nyquistrate, respectively.

2) Detection performances: We evaluate the probabilityof detection Pd based on the averaged PSD estimate S(J)

x .The detection analysis to follow, strictly speaking, holds onlyfor samples collected at Nyquist rate. We however use thisas a simple way to analyze the detection performance inthe compressive sampling case as well. The decision of thepresence of licensed transmission signals in the certain channelis made by an energy detector using the estimated frequencyresponse over that channel, i.e., the test statistic is

E(J)I =

IK∑i=(I−1)K+1

S(J)x (i), I = 1, 2, . . . ,maxI, (31)

where I is the channel index, maxI is the number of channels,and K is the number of PSD samples of each channel. ThePSD estimate of the j-th CR node can be evaluated as

Sx,j(i) =1H

H∑h=1

|Xh,j(i)|2, (32)

whereXh,j(i) is the Fourier transform of the h-th block of thereceived signal xh,j(n), j representing the CR node index,n representing the time sample index, each block containing2N time samples, and H denoting the number of blocks.Substituting (26) and (32) to (31), the test static can beobtained by

E(J)I =

1JH

IK∑i=(I−1)K+1

J∑j=1

H∑h=1

|Xh,j(i)|2. (33)

The decision rule is chosen as

E(J)I

H1><H0

μ, I = 1, 2, . . . ,maxI, (34)


0.1 0.2 0.3 0.4 0.50.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Compression rate [M/N]

Rec

onst

ruct

ion

erro

r

Basic GP, 5CRsPBB, 5 CRsBasic GP, 1 CRPBB, 1 CRRef. [5], 1 CR

Fig. 3. Reconstruction error (MSE) for Basic GP and PBB approaches versuscompression rate M/N for various number of collaborating CRs (SNRs ofactive channels varying from 8dB to 10dB).

0.1 0.2 0.3 0.4 0.50

0.050.1

0.4

0.5

0.6

0.7

0.8

0.9

1

Compression rate [M/N]

Per

form

ance

Pd, 5 CRs

Pd, 1 CR

Pd, 1CR, Ref. [5]

Pfa

, 5 CRs

Pfa

, 1 CR

Pfa

, 1 CR, Ref. [5]

Fig. 4. Probability of detection Pd and probability of a false alarm Pfa forPBB versus compression rate M/N for various number of collaborating CRs(SNRs of active channels varying from 8dB to 10dB).

where H0, H1 represent the hypotheses of the absence andpresence of primary signals, respectively, and μ is the decisionthreshold. Under H0, E(J)

I /(σ2

n/ (JH)) ∼ χ2

2JKH has acentral χ2 distribution with 2JKH degrees of freedom. Theprobability of a false alarm P

(J)fa can be obtained by

P(J)fa = 1 − Γ

(JH, μ

JH

)Γ (JH)

, (35)

where Γ(., .) is the upper incomplete gamma function [19, Sec.(8.350)], Γ(.) is the gamma function [19, Sec. (13.10)]. UnderH1, the probability of detection P (J)

d is evaluated by

P(J)d =

1a

Ia∑I=I1

Pr{E

(J)I > μ,

}(36)

where Ii, i = 1, . . . , a denote the indices of a active channels.

Parameters 2k mode

Elementary period T 7/64µsNumber of carriers K 1,705

Value of carrier number Kmin 0Value of carrier number Kmax 1,704

Duration of symbol part TU 2,048× T 224µsCarrier spacing 1/TU 4,464 Hz

Spacing between carriers Kmin and Kmax 7.61 MHz

TABLE ITHE OFDM PARAMETERS FOR THE 2K MODE.

V. SIMULATION RESULTS

The model for simulation can be briefly described in thissection. We consider at baseband, a wide frequency band ofinterest ranging from -38.05 to 38.05 MHz, containing maxI= 10 non-overlapping channels of equal bandwidth of 7.61MHz. Our simulations will focus in the 2k mode of the DVB-Tstandard. This particular mode is intended for mobile receptionof standard definition DTV. The structure of signal is followedan OFDM frame. Each frame has a duration of TF , andconsists of 68 OFDM symbols. Four frames constitute onesuper-frame. Each symbol is constituted by a set of C = 1,705carriers in the 2k mode and transmitted with a duration TS .A useful part with duration TU and a guard interval with aduration Δ (choosen to 0) compose TS . The over-samplingfactor is 2. The occupancy ratio of the total 76.1 MHz bandis 50%. The received signal is damaged by additive whiteGaussian noise (AWGN) with a variance of σ2

n = 1. Thereceived SNRs on the a = 5 active channels are randomlyvarying from 8dB to 10dB. A Gaussian wavelet function isused for smoothing. For compressive sampling, 2N is 4096,the compressed rate M/N is varying from 5% to 50% and H =160 is the number of blocks. The compressive sampling matrixΦA has a Gaussian distributed function with zero mean andvariance 1/M . The number of PSD samples of each channelis K = 25. We set αmin = 10−30, αmax = 1030 for PBBalgorithm, and use β = 0.5, μ = 0.1, and τ = 0.1

∥∥∥AT ry,i

∥∥∥∞

for both Basic and PBB algorithms.Fig. 3 illustrates MSE performance for Basic GP and PBB

algorithms and compares with the performance result in [5].In comparison with [5], our proposed approach in case of1 CR slightly decreases the MSE performance because ofthe reduced mutual incoherent of Φ in (12), however, ourapproach can reduce the hardware cost due to AIC acquisitionat the lower sampling rate. The results show that in comparisonwith Basic GP version, the PBB algorithm achieves the sameperformances while the Basic GP version takes a lot of timeto get convergence [12]. So the following results are imple-mented by using the novel PBB algorithm. This figure alsoshows the performances of signal recovery quality in whichMSE decreases when the value of compression rate M/Nincreases. However, as considering the effects of multiple CRsin spectrum sensing scheme, it is easily to observe that MSEalso decreases as the number of CRs J increases; therefore,we can obtain the lower compression rate but not degrade the


recovery performance by using more CRs in networks. Forexample, to receive MSE at 0.37, we must compress widebandsignals at the rate 0.4 in the 1 CR case, while networks with5 CRs can be implemented at the lower compression rate 0.3.

For detection performance, Fig. 4 depicts the probabilityof detection P

(J)d with respect to both the compression ratio

M/N and the number of CRs J =1, 5, under a fixed P(J)fa

of 0.01. This figure demonstrates that in order to obtainreliable performances, the joint collaboration and compressionis necessary. Especially, collaboration among CRs can avoidthe hardware cost of each CR by reducing the compression rateM/N while remaining the high detection performance. Forinstance, the probability of detection in case of 1 CR is ≈ 1at the compression rate M/N over 0.2, while the collaborationamong 5 CRs requires the compression rate M/N from thelower value 0.15.

Especially, analyzing the results reveals the interesting con-clusion, i.e., in Fig. 4, the detection performances under bothour method and the approach in [5] over the examined rangeof compression rates are similar while in Fig.3, the MSEperformances of these approaches have a bit differences.

VI. CONCLUSION

In this paper, we presented a distributed compressive spec-trum sensing scheme for CR networks. To avoid the high speedADC systems, the alternative converters called AICs are ex-ploited to acquire the salient information of received signals atsub-Nyquist rates. Moreover, the GP approach is used for jointcooperation and compressive sensing. The major barrier of GPmethod, which takes a lot of time to reach the convergence,can be solved by modifying the backtracking line search forupdating parameters. Among new fast CS techniques, PBBalgorithm, which is used to update the step of the iterationsin the recovery stage, demonstrates its outperformance, i.e.,it not only achieves high quality of signal recovery but alsoincreases the speed to quickly reach convergence.

ACKNOWLEDGMENT

This research was supported by Basic Science ResearchProgram through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science andTechnology (No. 2009-0073895)

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Projected Barzilai-Borwein Methods Applied to Distributed Compressive Spectrum Sensing

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