Time-Efficient Wideband Spectrum Sensing based on Compressive Sampling Yanbo Wang, Caili Guo, Xuekang Sun, Chunyan Feng School of Information and Communication Engineering Beijing University of Posts and Telecommunications Beijing, China 100876 Email:
[email protected]
Abstract—Compressed spectrum sensing (CSS) is proposed to detect spectrum opportunities efficiently over a wideband. However, most of existing CSS approaches will cause high computation costs for signal recovery when spectrum bandwidth goes large. As a result, it prolongs time for spectrum detection, which however runs counter to the original purpose of finding out spectrum opportunities over a wideband as rapidly as possible. To reduce the time consumed in signal reconstruction and realize real-time detection, we propose a novel decomposition compressed spectrum sensing (D-CSS) scheme. In D-CSS, a sparse sampling matrix is constructed first, and then it equivalently means a decomposition of the reconstructing process into two recovery subtasks. In doing so, we can scale down the overall problem and reduce the entire time for wideband spectrum detection compared with current CSS methods for a given desired sensing accuracy. Furthermore, the sparse character of our designed sampling matrix not only facilitates the operations of signal sampling and signal recovery, but also relieves the burden on random seeds generator and memory storage, which alleviates the overall implementation cost in CR practice. Index Terms—Compressed spectrum sensing, time-efficient, signal recovery time, D-CSS, computation complexity
I. I NTRODUCTION Cognitive radio (CR) [1] is proposed in order to grab available frequency resource from licensed spectrum, which offers a way to overcome the bottleneck of frequency shortage in wireless communications by rapidly sensing and then opportunistically utilizing the unoccupied spectrum without harmful interference to any primary user (PU). To exploit spectrum opportunities as more as possible, detecting over a wide frequency range from hundreds of megahertz to several gigahertz will be required. A major implementation challenge for spectrum sensing in the wideband regime lies in the very high sampling rates required by conventional spectral estimation methods which have to operate at or above the Nyquist rate. To overcome this problem, compressive sampling (CS) technique [2, 3], as an efficient way of signal acquiring and processing, is adopted in spectrum sensing [4]. This is well motivated by the fact of sparsity that there are plenty of temporarily unused spectrum opportunities in the wideband CR scenario. This technique uses a small number of samples which are determined by the information rate rather than the band width of signals. ∗ This work is supported by Chinese National Nature Science Foundation (61372116).
Therefore, the sampling rates of CS can be considerably lower than the Nyquist sampling rate in wideband spectrum sensing. Several CS schemes [4, 5] have been developed to detect wideband PU frequency occupancy at much lower sampling rate than conventional detection methods, which is mainly determined by the sparsity S and total length N of the underutilized spectrum. Real-time spectrum sensing is of great importance in wideband CR system, since we attempt to find more spectrum opportunities in a fast way for the sake of spectrum utilization improvement. However, most existing signal recovery algorithms in CSS need a great amount of calculations to reconstruct the original signal and the computational complexity of them increases dramatically when the signal length N goes large, because the increase of N leads to larger sampling measurements M = o(S log N ) at the same time. For example, the basis pursuit (BP) algorithm uses a dense, unstructured measurement matrix consuming time about o(M 3/2 N 2 ) ; the running time of the orthogonal matching pursuit (OMP) algorithm is about o(SM N ) demonstrated in [6]. An applicable remedy to reduce the time for signal recovery is to reform the greedy searching manner of these classical recovery algorithms. Improved schemes [7, 8] based on classical greedy iterative algorithms have been developed to achieve faster signal recovery, however, they still suffer high complexity when N and M are very large. To fundamentally reduce the high computational complexity of signal recovery algorithms involved in existing CSS due to large N and M , this paper proposes a decomposition compressed spectrum sensing (D-CSS) scheme for wideband spectrum sensing. To this end, this work takes advantage of the fact that the available historical knowledge about subchannel occupancy of primary users can be used to accelerate the process of spectrum opportunity detection. Specifically we employ the probability of occupancy in each subchannel as prior knowledge, which can be easily achieved by analyzing historical statistics using technology in [9]. Given such prior knowledge, the D-CSS scheme designs a sampling matrix and decomposes a large scale problem into two smaller scale problems where S and M are halved, which can considerably reduce the amount of calculations during greedy searching to reconstruct signal and thereby reduce the entire time for wideband spectrum sensing to realize real-time detection.
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Simulation results show that the overall sensing time of our proposed D-CSS is reduced enormously in contrast with that of conventional CSS methods, while keeping the same desired sensing accuracy. In addition, the sampling matrix used in DCSS scheme possesses a sparse property of matrix structure. This matrix structure allows half of its entries being zeros, which requires less random seeds generator and less storage memory compared with dense structured sampling matrix in conventional CSS. The rest of this paper is organized as follows. In Section II, we give the system model and a brief overview of compressed sensing. Section III presents the proposed novel scheme for signal reconstruction and compares them with conventional methods. In Section IV, simulation results are presented to measure the performance of proposed scheme. And we conclude in Section V. II. S YSTEM M ODEL AND P RELIMINARIES Consider a wide frequency of W Hz that hosts both primary communication systems and secondary CR users. A slotted frequency segmentation structure is adopted in which the entire wideband spectrum is divided into N non-overlapping equal−1 bandwidth subchannels centered at {fn }N n=0 . At a particular time and geographical region, the PU network is underutilized with only S(N ) channels occupied by active PUs, whereas the rest N − S channels are idle. Those temporarily idle subchannels are termed spectral holes and are available for opportunistic spectrum access by secondary users. Given the received signal r(t) and sparsity order S, the wideband spectrum sensing task for a CR detector is to correctly detect the occupancy states of the N spectrum bands in purpose of finding frequency opportunities. We assume that the high-layer, e.g., the medium access cognitive radios keep quiet during the detection interval such that the only spectral power remaining in the air is emitted by the primary users s(t) in addition to background noises w(t), in other words, r(t) = s(t) + w(t). For simplicity and to compare various CSS techniques in an identical setup, we consider the scenarios where all the active subchannels are of similar power levels and the signal propagates via an additive white gaussian noise (AWGN) channel. A CR detector adopting CS techniques starts with an analogto-digital sampler that collects M discrete-time measurements from r(t). Suppose that the time window for sensing is t ∈ [0, N /RN q ], where RN q is the Nyquist sampling rate. The discrete form representation of the CS sampling operation on the received r(t) at the CR detector can be expressed in the following general form: xt = Φrt = ΦF −1 rf = Θrf ,
(1)
where xt is an M × 1 vector of sampled measurements, rt is an N × 1 vector of the discrete form of with elements rt [n] = r(t)|t=n/RN , n = 0, ..., N − 1, rf is the N × 1 q sparse frequency response, F −1 represents the N × N inverse discrete Fourier transform matrix, and Φ is an M × N (M < N ) sampling matrix. An example construction of Φ
is by choosing elements that are drawn independently from a random distribution, e.g., Gaussian, Bernoullim. Evidently, the actual sampling rate R = (M /N )RNq in collecting the sample vector is lower than the Nyquist rate RNq , while the recovery of rt or rf in (1) will enable spectrum sensing at full frequency resolution. When the sensing matrix Θ meets the condition of restricted isometry property (RIP), the traditional CSS solves the problem of recovering sparse rf from xt consists in solving the l0 minimization problem [3]: ∧
rf = arg min rf 0 rf
s.t. xt = Θrf ,
(2)
∧
where rf is the estimated form of rf presenting the estimation of occupancy states of wideband spectrum. Then Mean ∧ Squared Error (M SE) between rf and rf : N ∧ 2 i=1 (rf − rf ) (3) M SE = N is used to evaluate the signal recovery performance. Linear programming techniques, e.g., basis pursuit (BP), or iterative greedy algorithms, e.g., orthogonal matching pursuit (OMP), can be used to solve the NP-hard problem (2). OMP is a classical algorithm to solve problem (2) and its computation complexity is o(SM N ). Consider that if we attempt to halve S and M where sparsity S is determined by the PUs and the number of measurements M depends on S and N , we may reduce the amount of calculations for signal recovery. Our proposed scheme will achieve this purpose by decomposing a large scale problem into two smaller scale subtasks using a specially designed sampling matrix. III. D ECOMPOSITION C OMPRESSED S PECTRUM S ENSING This section develops the decomposition compressed sensing (D-CSS) scheme for time-efficient wideband spectrum sensing. The overall procedure of our D-CSS scheme is illustrated in Fig. 1. The D-CSS first designs a sparse sampling matrix and then uses this matrix to sample the received signal to get compressed measurements. Next, these collected measurements are divided into two groups to reconstruct signal individually by solving smaller scale problem rather than directly reconstructing signal which is a large scale problem. Eventually, a final decision on the spectrum occupancy is made based on the estimated wideband signal. Details of each step are elaborated next. The decomposition of a large scale problem in our proposed D-CSS scheme is achieved by using a specially designed sampling matrix. This sampling matrix is constructed based on the prior knowledge about occupancy probability of each subchannel that is able to reflect the occupancy sate in each subchannel according to [9]. To construct such sampling matrix, we first sort the occupancy probability of each subchannel in ascending order and record their index as a set: Sort(rf ) = {{k1 , ..., kN }|P (rfk1 ) < ... < P (rfkN ), ki ∈ {1, ..., N }}, (4)
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Fig. 1.
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where rfi , i = 1, 2, ..., N terms the ith subchannel and P (rfi ) terms the probability of being occupied in subchannel i. Then, we form another two sets by picking up the odd and even elements from Sort(rf ), respectively: Supp(r1f ) = {k1 , k3 , k5 , ...kN −1 }
(5)
Supp(r2f ) = {k2 , k4 , k6 , ..., kN },
(6)
and where Sort(rf ) = Supp(r1f ) ∪ Supp(r2f ). Finally, we exploit an M /2 × N Gaussian random matrix Φcon = [φ1 , φ2 , φ3 , · · ·, φN −1 , φN ] to construct our sparse sampling matrix, where φi , i = 1, 2, · · · , N are column vectors. Define Φ1 = Φcon , Φ2 = Φcon ,
if φj = 0, j ∈ Supp(r2f ) if φj = 0, j ∈ Supp(r1f )
and the sparse sampling matrix is constructed as: T ∧ , Φ = Φ1 Φ2
(7)
(8)
where Φcon = Φ1 + Φ2 . Such a decomposition of Φcon for ∧ constructing sampling matrix Φ is termed as D-method. The reason why sampling matrix is constructed in this way will be explained in the following. According to sampling operation expressed in formulation ∧ (1), we use Φ as sampling matrix and have ∧ Φ1 Θ1 −1 xt = Φ rt = (9) F rf = rf , Φ2 Θ2 where Θ1 = Φ1 F −1 , Θ2 = Φ2 F −1 and T xt = x1 x2 x3 · · · xM −1 xM T . rt = r1 r2 r3 r4 · · · rN −1 rN Apparently, equation (9) equals: rk1 φk1 + rk3 φk3 + · · · + rkN −1 φkN −1 x1t = x2t rk2 φk2 + rk4 φk4 + · · · + rkN φkN (10) where T x1t = x1 x2 · · · xM /2 T . x2t = xM /2+1 xM /2+2 · · · xM Setting r1t = rt , r2t = rt ,
if rj = 0, j ∈ Supp(r2f ) if rj = 0, j ∈ Supp(r1f )
x1t = Φ1 rt = Φ1 r1t = Φ1 F −1 r1f = Θ1 r1f x2t = Φ2 rt = Φ2 r2t = Φ2 F −1 r2f = Θ2 r2f .
(11)
where r1f and r2f are expression of r1t and r2t in frequency domain. When detecting spectrum opportunities, traditional CSS usually needs to reconstruct rf by solving problem stated in (2). But in D-CSS, we can achieve resolution by solving formulations as follows: ⎧ ∧ ⎪ r1f = arg min r1f 0 s.t. x1t = Θr1f ⎪ ⎪ ⎪ r1f ⎨ ∧ r2f = arg min r2f 0 s.t. x2t = Θr2f . (12) ⎪ ⎪ r2f ⎪ ⎪ ⎩ ∧ ∧ ∧ rf = r1f + r2f Compared (2) with (12), using M length measurements xt and M × N sampling matrix Φ to reconstruct signal rf in CSS equivalently means using M /2 length measurements x1t , x2t and M /2 × N sampling matrix Φ1 , Φ2 to reconstruct signal r1f and r2f respectively in D-CSS. That is say, the DCSS decompose a large scale problem (2) in CSS into two smaller scale subtasks (12). The underlying reason is that, given accurate prior knowledge on occupancy probability, both Supp(r1f ) and Supp(r2f ) contains S/2 subchannels being occupied. So r1f and r2f are S/2-sparse and thus M/2 length measurements x1t and x2t are enough to reconstruct them. The overall number of measurements in D-CSS is still M that is the same with CSS. It is worth pointing out that the D-CSS itself has ability to find worse-detection performance by checking up if the sparsity of the reconstructed signal is S/2. When the sparsity ∧ ∧ order of the reconstructed signal either r1f or r2f is not S/2, we need to update the prior knowledge using technology in [9] because they are not accurate enough to reflect the occupancy states in wideband. In terms of computation complexity for signal recovery, we have known that consuming time of most algorithms for compressed signal recovery are tightly related with size of K, M and N . For example, the running time of OMP algorithm is about o(SM N ). It is obvious that the complexity of algorithms based on OMP increases dramatically when the signal length N is very large leading to large sampling measurements M . However, with the D-CSS scheme, we scale the computational complexity down by decomposing a large scale problem into two smaller scale subtasks, because number of measurements and sparsity order are halved in subtasks. For convenience of comparison, we define that when OMP algorithm is used in CSS to solve the reconstruction problem (2), we still call it OMP algorithm; when OMP algorithm is used in D-CSS to solve the reconstruction problem (12), we call the algorithm DOMP. Similarly BP and DBP in Section IV are defined in the same way. A. Analysis of Computation Complexity We here take OMP and DOMP for example to further explain the reason for reduction of signal recovery time in
the proposed D-CSS method. To evaluate the computational complexity of both OMP and DOMP, we count the number of multiplications and additions for signal recovery, termed as m and a. Other operations such as loop counting, indexing etc are not counted. According to the description of OMP algorithm in [3], when the iteration times is n, number of multiplications and additions to solve (2) using OMP are calculated to be: + 3M n(n+1) momp = nM N + n(n+1)(2n+1) 3 2 . n(n+1) aomp = N M − N + (3M − 4) 2 + n(n+1)(2n+1) 3 (13) Similarly, keeping the same times of iteration as n, number of multiplications and additions using DOMP to solve (12) are: mdomp = nM2 N + n(n/2+1)(n+1) + 3M n(n/2+1) 3 4 adomp = N M − 2N + ( 32 M − 4) n2 ( n2 + 1) + n(n/2+1)(n+1) 3 (14) Compared (14) with (13), DOMP reduces the number of both multiplications and additions needed for signal recovery and the reduction is: 3
2
2
red m = nM2 N + n +n + M ( 9n8 + 2 3 2 9n2 3n red a = N + M ( 8 + 4 ) + n −n 2
3n 4 )
.
(15)
Such reduction showed in (15) demonstrates that DOMP is very time efficient compared with OMP. If we set M = N/2 and n = M/4, then (15) becomes 3
2
red m = 21N + 7N 512 128 3 5N 2 red a = 5N + 512 128 + N.
(16)
From (16), we conclude that D-CSS scheme can achieve huge time reduction for signal recovery when N goes large, which will be demonstrated via simulation. Similar conclusion can be drawn when our D-CSS scheme employs BP and other algorithms such as [7, 8] for compressed signal reconstruction. B. Decision and Sensing Performance ∧
Having obtained rf from (12), the CR detector can finally make decisions on the spectrum occupancy by comparing the signal energy within each subchannel with a decision threshold γ which is obtained according to Neyman-Pearson theorem, as follows: ∧ ∧ 2 (17) d = | rf | ≥ γ . ∧
The binary decision vector d of length N indicates the sensing outcomes on the monitored wideband spectrum. If an entry of ∧
d is 0, then the corresponding frequency band is decided to be idle, presenting a potential spectrum opportunity for CRs. If the entry is 1, then the corresponding frequency band is decided to be occupied by primary users. The signal to noise ratio (SNR) is defined to be the signal energy of wideband signal over the entire spectrum scaled by the power of the white noise. Performance metrics that we adopt are probability of missdetection Pm and probability of false-alarm Pf . The two detection errors indicate the interference to PUs and the
wastefulness of available spectrum opportunities for CRs, respectively. Given the true state vector d of the target wideband spectrum, Pm and Pf can be expressed as follows: ⎧ ⎧ ⎫ ⎫ ⎨ dT (d = ∧ ) ⎬ ⎨ (1 − d)T (d = ∧ ) ⎬ d d Pm = E , Pf = E , ⎩ 1T d ⎭ ⎩ ⎭ N − 1T d (18) where E denotes expectation, and 1 denotes the all-one vector. IV. S IMULATION This section presents simulation results to verify the sensing performance and computation costs of the proposed D-CSS scheme (DBP, DOMP), using the classical CSS (BP, OMP) approaches as benchmarks. Consider a wideband CR system . with N = 128 subchannels over the given wide frequency band. To model the sparsity in frequency utilization, only S = 10 randomly chosen subchannels are occupied during each detection period. The CSS and D-CSS schemes are respectively employed for sparse signal recovery in the presence of ambient AWGN. For compressed sensing, the compression ratio M/N is set to 50% and the entries in the sampling matrix Φ are drawn from independent Gaussian distributions. When ∧ the D-CSS scheme is employed, the sampling matrix Φ is constructed according to D-method. Fig. 2 shows the MSE between original signal and signals reconstructed by both CSS and D-CSS schemes. It is obvious that the proposed D-CSS scheme can appropriately recover the original signal just like CSS. It is worth remembering that the problem tackled here is spectrum sensing and not spectrum estimation. Thus, not necessarily the exact reconstruction of signal facilitates unveiling the active subbands, so such minor corruption showed in figure will not degrade the sensing performance. We analyze the sensing performance of the proposed wideband spectrum detection scheme D-CSS via Monte Carlo simulations and set Pf = 0.1, varying the signal to noise ratio SNR from -10dB to 20dB. The detection probability Pd = 1 − Pm of conventional CSS schemes and that of proposed D-CSS schemes are illustrated in Fig. 3. Evidently, the D-CSS scheme has similar sensing performance with traditional CSS scheme. Our contribution to reduce the signal recovery time in DCSS is demonstrated in Table I and Table II. It is worth mentioning that we achieve such time reduction while keeping desired sensing accuracy which CSS can provide. We execute each algorithm (BP, OMP, DBP and DOMP) 1000 times respectively on the same computer and then average the execution time to describe the signal recovery time. In Table I, we can easily see that D-CSS is more time-efficient in contrast with traditional CSS adapting to different SNR situations. To demonstrate the relationship between time reduction and signal length, we vary N to test the advantage of D-CSS when N goes large and simulation results are shown in Table II. As we can see, the saved time not only between OMP and DOMP but also between BP and DBP increases when signal length N grows. Such phenomenon validates our initial
TABLE I C ONSUMING TIME OF EACH SIGNAL RECOVERY SCHEME .
0.1 BP DBP OMP DOMP
0.09 0.08
SNR -10dB -10dB 0dB 0dB 10dB 10dB
0.07
MSE
0.06
CSS BP OMP BP OMP BP OMP
Time 0.0686s 0.0118s 0.0633s 0.0116s 0.0635s 0.0093s
D-CSS DBP DOMP DBP DOMP DBP DOMP
Time 0.0499s 0.0096s 0.0442s 0.0080s 0.0460s 0.0076s
0.05
Saved time 27.26% 18.59% 30.29% 26.65% 27.57% 22.30%
TABLE II T IME SAVED VERSUS SIGNAL LENGTH N.
0.04 0.03 0.02 0.01 0 −10
−5
0
5 SNR (dB)
10
15
20
Fig. 2. MSE between the original signal and signals reconstructed by BP, OMP, DBP and DOMP, with compression ratio M/N=50% and SNR varying from -10dB to 20dB.
1 0.9 0.8 0.7
Pd
0.6 0.5 0.4 0.3
BP OMP DBP DOMP
0.2 0.1 −10
−5
0
5 SNR (dB)
10
15
N 128 128 256 256 512 512
CSS BP OMP BP OMP BP OMP
Time 0.3001s 0.0118s 0.6048s 0.0284s 3.3570s 0.1045s
D-CSS DBP DOMP DBP DOMP DBP DOMP
Time 0.1013s 0.0061s 0.1585s 0.0123s 0.6487s 0.0229s
Saved time 66.23% 48.37% 73.79% 56.85% 80.68% 78.07%
sensing accuracy. Using the designed sparse matrix according to D-method, our proposed decomposition compressed spectrum sensing (D-CSS) scheme can accelerate the process of detecting by decomposing a large scale reconstructing problem into subtasks and thus reduce the undesired high computational complexity for signal recovery which dominates the entire sensing time. In terms of the implementation cost, sparse matrix employed here requires not only less random seeds to generate the matrix entries but also a small memory size for storage, because it has more zero entries compared with dense matrices in CSS. Such sparse structure also facilitates fast implementations of both data collection and signal reconstruction. These conclusions can offer guidelines on the system design that has requirements of short detection time and low implementation cost for practical CR applications. R EFERENCES
20
Fig. 3. Detection performance of BP, OMP, DBP and DOMP, with S=10, N=128, M=64 and Pf =0.1.
purpose of reducing signal recovery time when signal length N goes large. Compared with CSS, we reduce the entire time for spectrum sensing by reducing signal recovery time while keeping desired sensing performance, because in other steps, such as sampling or deciding, D-CSS spends same time with CSS scheme. Generally, the proposed scheme can also reduce the computational comlexity of other signal recovery algorithms such as in [7, 8] to reduce time for wideband spectrum sensing. This is of significant importance for the practical implementation of CR system which entails real-time spectrum detection to find more spectrum opportunities. V. C ONCLUSION In this paper, we detect spectrum opportunities more quickly compared with traditional CSS scheme for a given desired
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