COMPRESSED SENSING FOR WIDEBAND COGNITIVE RADIOS

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1 COMPRESSED SENSING FOR WIDEBAND COGNITIVE RADIOS Zhi Tian Dept. o Electrical & Computer Engineering Michigan Technological University Houghton, MI USA Georgios B. Giannakis Dept. o Electrical & Computer Engineering University o Minnesota Minneapolis, MN USA ABSTRACT In the emerging paradigm o open spectrum access, cognitive radios dynamically sense the radio-spectrum environment and must rapidly tune their transmitter parameters to e- ciently utilize the available spectrum. The unprecedented radio agility envisioned, calls or ast and accurate spectrum sensing over a wide bandwidth, which challenges traditional spectral estimation methods typically operating at or above Nyquist rates. Capitalizing on the sparseness o the signal spectrum in open-access networks, this paper develops compressed sensing techniques tailored or the coarse sensing task o spectrum hole identi cation. Sub-Nyquist rate samples are utilized to detect and classiy requency bands via a waveletbased edge detector. Because spectrum location estimation takes priority over ne-scale signal reconstruction, the proposed novel sensing algorithms are robust to noise and can aord reduced sampling rates. Index Terms spectrum estimation, compressed sensing, sub-nyquist sampling, wavelet transorm, cognitive radio 1. INTRODUCTION The emerging paradigm o Dynamic Spectrum Access shows promise to alleviate today s spectrum scarcity problem by ushering in new orms o spectrum agile wireless networks [1]. Key to this new paradigm are cognitive radios (CRs) that are aware o and can sense the environments, and perorm unctions to best serve their users without causing harmul intererence to other authorized users [2]. As such, the rst cognitive task preceding any orm o dynamic spectrum management is to develop wireless spectral detection and estimation techniques or sensing and identi cation o available spectrum. Spectrum sensing in the wideband regime aces considerable technical challenges. The radio ront-end can employ a bank o tunable narrowband bandpass lters to search one narrow requency band at a time. In each narrowband, existing spectrum sensing techniques perorm either energy detection [3] or eature detection [2]. It requires an unavorably large number o RF components and the tuning range o each lter is preset. Alternatively, a wideband circuit utilizes a single RF chain ollowed by high-speed DSP to exibly search over multiple requency bands concurrently [4]. A major implementation challenge lies in the very high sampling rates required by conventional spectral estimation methods which have to operate at or above the Nyquist rate. Meanwhile, due to the timing requirements or rapid sensing, only a limited number o measurements can be acquired rom the received signal, which may not provide su cient statistic when traditional linear signal reconstruction methods are employed. This paper aims at ast spectrum sensing at aordable complexity. A couple o key premises are capitalized to alleviate the stringent sampling requirements in the wideband regime. First, we take a multi-resolution approach to decompose the cognitive sensing task into two stages. The rst stage is coarse sensing to detect non-overlappingspectrum bands and classiy them into black, gray or white spaces, depending on whether the power spectral density (PSD) levels are high, medium or low [2]. Based on the spectrum sharing mechanism adopted [1], the second stage o ne-scale spectral shape estimation is perormed only when needed, and mostly con ned within the available (narrowband) white spaces to alleviate the sampling requirements. Second, we recognize that the wireless signals in open-spectrum networks are typically sparse in the requency domain. This is due to the low percentage o spectrum occupancy by active radios a act motivating dynamic spectrum management. For sparse signals, recent advances in compressed sensing have demonstrated the principle o sub- Nyquist-rate sampling and reliable signal recovery via computationally easible algorithms [5, 6, 7, 8]. Tailored to the above distinct nature o CR sensing, this paper derives novel compressed sensing algorithms or the coarse sensing task o spectrum band classi cation. Random sub-nyquist-rate samples are employed to ormulate an optimal signal reconstruction problem, which incorporates the wavelet-based edge detector we recently developed in [1] to recover the locations o requency bands. Because spectrum location estimation takes priority over ne-scale signal reconstruction, the novel sensing algorithms are robust to noise and can aord reduced sampling rates /7/$2. 27 IEEE IV 1357 ICASSP 27

2 Report Documentation Page Form Approved OMB No Public reporting burden or the collection o inormation is estimated to average 1 hour per response, including the time or reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection o inormation. Send comments regarding this burden estimate or any other aspect o this collection o inormation, including suggestions or reducing this burden, to Washington Headquarters Services, Directorate or Inormation Operations and Reports, 1215 Jeerson Davis Highway, Suite 124, Arlington VA Respondents should be aware that notwithstanding any other provision o law, no person shall be subject to a penalty or ailing to comply with a collection o inormation i it does not display a currently valid OMB control number. 1. REPORT DATE REPORT TYPE 3. DATES COVERED --27 to TITLE AND SUBTITLE Compressed Sensing or Wideband Cognitive Radios 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Dept. o Electrical & Computer Engineering,Michigan Technological University,Houghton,MI, PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 1. SPONSOR/MONITOR S ACRONYM(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved or public release; distribution unlimited 11. SPONSOR/MONITOR S REPORT NUMBER(S) 13. SUPPLEMENTARY NOTES See also ADM213. Proceedings o the 27 IEEE International Conerence on Acoustics, Speech, and Signal Processing (ICASSP), Held in Honolulu, Hawaii on April 15-2, 27. Government or Federal Rights 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT unclassiied b. ABSTRACT unclassiied c. THIS PAGE unclassiied Same as Report (SAR) 18. NUMBER OF PAGES 4 19a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

3 2. SIGNAL MODEL AND PROBLEM STATEMENT Suppose that a total o B Hz in the requency range [, N ] is available or a wideband wireless network. A CR receives the signal r(t) that occupiesn consecutivespectrum bands, with their requency boundaries located at < 1 < N.The requency response o r(t) is illustrated in Fig. 1. Depending on whether the PSD level is high, medium or low, each requency segment can be categorized into black, gray or white spectrum spaces [2]. White holes, and sometimes gray spaces, can be picked by the CR or opportunistic transmission, while the black holes are to be avoided or intererence control. PSD B n 1 2 n-1 c,n n wide band o interest Fig. 1. N requency bands with piecewise smooth PSD. Suppose that the time window or sensing is t [,MT ], where T is the Nyquist sampling rate. Using Nyquist sampling theory, M samples are needed to recover r(t) without aliasing. A digital receiver converts the continuous-domain signal r(t) to a discrete sequence x t C K o length K. The sampling process can be expressed in discrete-time domain in the ollowing general orm: x t = S T r t (1) where S is an M K projection matrix and r t represents the M 1 vector with elements r t [n] = r(t) t=nt, n = 1,...,M. Columns {s k } K k=1 o S can be viewed as a set o basis signals or matched lters, while the measurements {x t [k]} K k=1 are in essence the projection o r(t) onto the basis. The model in (1) subsumes all sampling schemes yielding linear measurements. For example, S = I M represents Nyquist-rate uniorm sampling, where I M is the size-m identity matrix; S = F M amounts to requency-domain sampling, where F M is the M-point unitary discrete Fourier transorm (FT) matrix. When K < M, reduced-rate sampling arises. We ocus on non-adaptive measurements where S is preset. The goal o CR sensing is to classiy and estimate the spectrum o r(t) given the sample set x t, where K < M is possible. Spectrum classi cation reers to identiying the number o subbands N and their locations {[ i, i+1 ]} N 1 i=, and classiying them into black, gray or white spaces. Spectrum estimation, on the other hand, can have dierent objectives: either to estimate the requency response o r(t) within the entire wideband, or con ne the estimation to be within the identi ed (narrowband) white spaces only. This paper primarily concerns the coarse sensing task o spectrum classi cation. N 3. MULTI-STEP COMPRESSED SENSING Our rst approach to reduced-complexity spectrum sensing takes the ollowing our steps: i) compressed random sampling to generate measurements x t rom r(t); ii) reconstruction o the requency response r = F M r t rom x t ; iii) estimation o requency band number N and locations { i } N 1 i=1 based on ˆr ; and, iv) estimation o the average amplitude o r within each identi ed band or spectrum classi cation. It is worth emphasizing that Step ii) recovers the accurate neresolution signal spectrum r represented by M requency samples at the Nyquist rate, while the available measurement set x t is o a reduced size o K(< M) elements Sub-Nyquist-rate Sampling Let F denote the non-zero requency-domain support o r(t) in the noise-ree case. In open-spectrum networks, it generally holds that F B [4], indicating the sparseness nature o r(t). Equivalently speaking, the M 1 requency response vector r contains on average K b := F M/B non-zero elements when noise ree, and K b M. The key results in compressed sensing stated that the sparse vector r can be recovered asymptotically rom K ( M) samples o r(t), as long as K K b. These samples x t can be generated rom (1) via universal non-uniorm sampling [5] or random sampling [6, 7], both o which can enable perect recovery o r when ree o noise. To distinct, we denote a reduced-rate sampling matrix as S c o dimension M K, where K b K M. A simple example o S c is a selection matrix that randomly retains K columns o the size-m identity matrix, which means that K M time instants on the sampling grid are skipped Spectrum Reconstruction With the K measurements x t = S T c r t, we now estimate the requency response o r(t) in the orm o r = F M r. For a given linear sampler S c : C M C K, we seek a nonlinear reconstruction unction R( ) :C K C M that oers an approximate reconstruction o r C M rom x t C K based on the linear transormation equality x t =(S T c F 1 M )r ; c.. (1). This is a linear inverse problem with sparseness constraint, which is NP-hard. A conceptually intuitive approach to signal reconstruction is the Basis Pursuit (BP) technique [9], which transorms the sparseness constraint on r into a convex optimization problem solvable by linear programming: ˆr = arg min r r 1, s.t. (S T c F 1 M )r = x t. (2) Besides BP, a number o e cient reconstruction methods exist, including orthogonal matching pursuit (OMP) algorithm and tree-based OMP (TOMP) algorithm [8]. Since the measurements can be complex-valued, we nd it convenient to use TOMP in our simulations, but or illustration, ormulate our signal reconstruction problem based on BP, as in (2). IV 1358

4 3.3. Band Location Estimation, which are to be identi ed. The wavelet approach is well motivated or edge detection [11]. Given ˆr, we re-cast the edge detector in [1] in discrete orm. Let φ() be a wavelet smoothing unction with a compact support. The dilation o φ() by a scale actor s is given by Having estimated r, we turn to the wavelet-based edge detector in [1] or detecting the number and requency locations o spectrum spaces. The basic idea is to view the entire wideband under scrutiny as a train o consecutive requency subbands, where the PSD is smooth within each subband, but exhibits a discontinuous change between adjacent subbands. These irregularities are in act edges in PSD, which carry key inormation on the locations and intensities o spectrum holes. To urther simpliy and expedite the coarse sensing stage, we approximately treat the spectral amplitudes within each subband to be almost at, at an unknown level α n over the n-th band. These modeling approximations are invoked to reduce the overall wideband sensing complexity. I needed, the sensing quality can be re ned ater spectrum holes are identi ed. Based on these modeling assumptions and with reerence to Fig. 1, wideband sensing can be viewed as an edge detection problem in an image depicted by ˆr in requency. Edges in this image correspond to the locations o requency discontinuities { i } N 1 i=1 φ s () = 1 s φ ( s ). (3) For dyadic scales, s takes values rom powers o 2, i.e., s = 2 j, j =1, 2,...,J. Let Φ s (τ) :=F 1 {φ s ()} =Φ(sτ) represent the inverse FT o the wavelet unction. The continuous wavelet transorm o R()( r ) is given by [1] W s R() =F{W s r(τ)} = F{r(τ) Φ(sτ)}. (4) Mapping W s R(), r(τ) and Φ(sτ) to their length-m discrete counterparts y s, r t and Φ s respectively, (4) is equivalent to y s = F M diag{φ s }r t. (5) Replacing r t in (5) by its estimate ˆr t = F 1 M ˆr, we reach the estimated wavelet transorm ŷ s = F M diag{φ s }F 1 M ˆr. (6) The derivative wavelet o r at scale s is given by z s with elements {z s [n]} M n=1 in the orm s d d (W sr()) z s : z s [n] =y s [n] y s [n 1]. (7) The boundaries { n } N 1 n=1 can thus be acquired by picking the local maxima o the wavelet modulus z s, while the band number N is determined by the number o local peaks [1, 11] Frequency Response Amplitude Estimation The estimated boundaries { n } N 1 n=1 correspond to N 1 selected indices {I n : n = + I n Δ, Δ = B/M} in the requency response vector r. Elements o r between a pair o adjacent indices belong to the same requency band. The average requency response amplitude α n o the n-th band, can thus be computed as ˆα n 1 I n I n 1 +1 I n i=i n 1 ˆr [i], n =1,...,N. (8) This simple and coarse estimator in (8) allows us to categorize the detected requency bands into black, gray, or white spaces [2], depending on whether {ˆα n } are high, medium or low. 4. ONE-STEP COMPRESSED SENSING To urther reduce the implementation complexity o coarse spectrum sensing, we now ask: can we directly detect and estimate the requency band locations rom the compressed measurements x t in (1), without having to recover the detailed requency response r? We address this question by deriving signal recovery ormulation or wavelet-based edge detection. Recall rom Section 3.3 that the band locations can be recovered rom the (N 1) peaks o the derivative wavelet modulus z s C M.WhenN M (which is generally the case), z s can be treated as a sparse vector, with only a ew nontrivial elements located at requency band boundaries; c.., see Fig. 3 or graphical validation. As such, z s can be recovered under the sparseness constraint, provided that we can nd a linear transormation equality linking z s to the compressed measurement vector x t. To this end, we rewrite (7) in matrix-vector orm as z s = Γy s, where Γ is the dierentiation matrix given by 1. Γ= (9) Putting (9) and (5) together, we obtain: M M r t = ( F 1 M diag{φ s} ) 1 ys = ( F 1 M diag{φ s} ) 1 Γ 1 z s. }{{} :=G (1) Noting that x t = S T c r t, and that z s is sparse, we reach the ollowing BP-based optimization ormulation: ẑ s = arg min z s 1, s.t. x t = ( S T c G ) z s. (11) z s Subsequently, the band boundaries { n } can be acquired rom the locations o those non-zero elements in z s, obviating the involved step o requency response estimation on r. IV 1359

5 5. SIMULATIONS We consider a wide band o interest in the range o + [5, 15]Δ Hz, where Δ is the requency resolution. Fig. 2 illustrates the spectral amplitude R() observed by a CR. During the observed burst o transmissions in the network, there are a total o N =6bands, with requency boundaries at { n } 6 n=1 = +[6, 68, 83, 119, 123, 15]Δ Hz. Among these bands (marked in Fig. 2), B 1, B 3 and B 5 have relatively high signal amplitude at levels 16, 2, and 24, respectively, while B 2 has low amplitude at a level o 2. The rest two bands, B 4 and B 6 are not occupied and are thus white spectrum holes. The sampling lower bound is thus K b /M 4%. For compressed sensing, the compression ratio K/M is set to vary rom 5% to 1% with reerence to the Nyquist rate. The noise level is n 2 w =8dB. The sampler S = S c used in (1) is uniormly random. Fig. 2 indicates that the signal recovery quality (via TOMP) improves as K/M increases. In the wavelet-based edge detector, Gaussian wavelets are used at our dyadic scales s =2 j, j =1, 2, 3, 4. Fig.3depicts the multiscale wavelet products computed rom (7) [1]. Edges in the R() are clearly captured by the wavelet transorm in all curves. As the scale actor s j increases, the wavelet transorm becomes smoother within each requency band, retaining the lower-variation contour o the noisy PSD. For requency band location estimation, Fig. 4 depicts the normalized root mean-square estimation errors (RMSE) N 1 B 1 n=1 n ˆ n 2 with respect to both the compression ratio K/M and the inverse noise level n 2 w. When either the number o samples is very small or the noise is very strong, there exhibits an estimation error oor. Nevertheless, the attained degree o estimation accuracy is bene cial to eecting CR agility at aordable sampling cost. Robustness to sample quantization errors is also illustrated. 6. REFERENCES [1] Facilitating Opportunities or Flexible, E cient, and Reliable Spectrum Use Employing Cognitive Radio Technologies, FCC Report and Order, FCC-5-57A1, March 25. [2] S. Haykin, Cognitive radio: brain-empowered wireless communications, IEEE JSAC, vol. 23(2), pp , Feb. 25. [3] H. Urkowitz, Energy detection o unknown deterministic signals, Proc. o the IEEE, vol. 55(4), pp , April [4] A. Sahai, D. Cabric, Spectrum Sensing Fundamental Limits and Practical Challenges, A tutorial presented at IEEE DySpan Conerence, Baltimore, Nov. 25. [5] P. Feng, and Y. Bresler, Spectrum-blind mimimum rate sampling and reconstruction o multiband signals, Proc. IEEE Intl. Con. on ASSP, Atlanta, pp , [6] E. J. Candes, J. Romberg and T. Tao, Robust Uncertainty Principles: Exact Signal Reconstruction rom Highly Incomplete Frequency Inormation, IEEE Trans. on Inormation Theory, vol. 52, pp , Feb. 26. [7] D. L. Donoho, Compressed Sensing, IEEE Trans. on Inormation Theory, vol. 52, pp , April 26. [8] C. La, and M. N. Do, Signal reconstruction using sparse tree representation. Proc. SPIE Con. on Wavelet Apllications in Signal and Image Processing, San Diego, Aug. 25. [9] S. S. Chen, D. L. Donoho, and M. A. Saunders, Atomic decomposition by basis pursuit, SIAM J. Sci. Comput., vol. 2, no. 1, pp , [1] Z. Tian, and G. B. Giannakis, A Wavelet Approach to Wideband Spectrum Sensing or Cognitive Radios, Proc. o Intl. Con. on CROWNCOM, Mykonos, Greece, June 26. [11] S. Mallat, W. Hwang, Singularity detection with wavelets, IEEE Trans. Ino. Theory, vol.38, pp , % 75% 1% B 1 B 2 B 3 B 4 B 5 B Fig. 2. signal requency response: (top) noise-ree X ; (rest) recovered ˆX at compressing ratios K/M = 5%, 75%, 1. s=2 1 s= x Fig. 3. wavelet modulus or edge detection at scales s =2 j. RMSE K/M = 33% K/M = 5% K/M = 75% K/M = 1% K/M=5%, 4 bits K/M=5%, 8 bits SNR (db) RMSE SNR = db SNR = 5 db SNR = 1 db SNR = 2 db Compression ratio (K/M) Fig. 4. requency band location estimation errors RMSE. Strong compression o K/M = 33% is demonstrated. Impact o quantization (bits/sample) is shown or K/M = 5%. IV 136