A Wavelet Approach to Wideband Spectrum Sensing or Cognitive Radios Zhi Tian Department o Electrical & Computer Engineering Michigan Technological University Houghton, MI 4993 USA ztian@mtu.edu Georgios B. Giannakis Department o Electrical & Computer Engineering University o Minnesota Minneapolis, MN 55455 USA georgios@ece.umn.edu Abstract In cognitive radio networks, the irst cognitive task preceding any orm o dynamic spectrum management is the sensing and identiication o spectrum holes in wireless environments. This paper develops a wavelet approach to eicient spectrum sensing o wideband channels. The signal spectrum over a wide requency band is decomposed into elementary building blocks o subbands that are well characterized by local irregularities in requency. As a powerul mathematical tool or analyzing singularities and edges, the wavelet transorm is employed to detect and estimate the local spectral irregular structure, which carries important inormation on the requency locations and power spectral densities o the subbands. Along this line, a couple o wideband spectrum sensing techniques are developed based on the local maxima o the wavelet transorm modulus and the multi-scale wavelet products. The proposed sensing techniques provide an eective radio sensing architecture to identiy and locate spectrum holes in the signal spectrum. I. INTRODUCTION Current wireless systems are characterized by wasteul static spectrum allocation, ixed radio unctions, and limited network coordination between mobile devices, resulting in a surprisingly large portion o the radio spectrum goes unused. The emerging paradigm o Dynamic Spectrum Access shows promise o alleviating today s spectrum scarcity problem by ushering in new orms o spectrum agile networks []. Key to this new paradigm are cognitive radios (CRs) that are aware o and can sense the environments, learn rom the environments, and perorm unctions to best serve their users without causing harmul intererence to other authorized users []. The cognitive process starts with the passive sensing o RF stimuli [3]. As such, the irst cognitive task preceding any orm o dynamic spectrum management is to develop wireless spectral detection and estimation techniques or sensing and identiication o spectrum holes. Depending on the regimes o spectrum utilization, the rontend architecture o CRs can be quite dierent [4]. In early stage o CR network deployment, the spectrum utilization is expected to be low (around 5%) and there is little spectrum scarcity. In this case, the radio ront-end starts with a tunable narrowband bandpass ilter (BPF) to search one narrow requency band at a time. Focusing on each narrow band, existing spectrum sensing techniques are largely categorized into energy detection [5] and eature detection [6]. When the spectrum utilization is medium (below %) resulting in medium spectrum scarcity, the radio ront-tend should adopt a wideband architecture to search over multiple requency bands at a time. Multiple narrowband BPFs can be employed to orm a ilterbank or wideband sensing [4], but this architecture requires an increased number o components and the ilter range o each BPF is preset. In uture networks where spectrum utilization is high (above %), the signiicant spectrum scarcity would call or dierent spectrum sharing mechanisms such as ultra-wideband cognitive radios [7], which in turn entail dierent sensing tasks or spectrum overlay. In this paper, we ocus on the wideband spectrum sensing task, without resorting to multiple narrowband BPFs. Our goal is to identiy the requency locations o non-overlapping spectrum bands and categorize these bands into black, gray or white spaces, corresponding to the power spectral density (PSD) levels being high, medium or low [3]. In a peer-topeer network adopting the equal-sharing spectrum allocation paradigm, white spaces are treated as spectrum holes that can be picked by the CR or opportunistic use. Evidently, the cognitive network o interest concerns spectrum identiication more than the detailed spectral shape over the entire wideband. Thus, many traditional spectral estimation techniques become irrelevant or unnecessarily complicated [8]. Recognizing the distinct nature o CR sensing, we model the entire wideband under scrutiny as a train o consecutive requency subbands, where the power spectral characteristic is smooth within each subband but exhibits a discontinuous change between adjacent subbands. Such changes are in act irregularities in PSD, which carry key inormation on the locations and intensities o spectrum holes. An attractive mathematical tool or analyzing singularities and irregular structures is the wavelet transorm, which can characterize the local regularity o signals [9]. Thus, it is well motivated to investigate the wavelet transorm approach to wideband spectrum sensing or CRs. There has been considerable research on wavelet analysis or time series and images [9]-[3]. Singularity detection and processing with wavelets have been applied to iltering and denoising [9], [3], compression [], and applications in image processing and elsewhere. Targeting the CR sensing task, this paper derives wavelet-based techniques or detecting irregular edges in the signal PSD as opposed to irregularities in time series. A couple o dynamic sensing solutions are ormulated based on the maxima o waveorm transorm modulus [9] and the peaks o multiscale products [3], which result in detection and estimation o the locations o spectral irregularities. We also estimate the average PSD level within each identiied subband, which carries critical inormation on spectrum holes available or opportunistic sharing. -444-38-/6/$. 6 IEEE
II. PROBLEM FORMULATION FOR SPECTRUM SENSING Suppose that a total o B Hz in the requency range [, N ] is available or a wideband wireless network. Being cognitive, this network supports heterogeneous wireless devices that may adopt dierent wireless technologies or transmissions over dierent bands in the requency range. A CR at a particular place and time needs to sense the wireless environment in order to identiy spectrum holes or opportunistic use. Suppose that the radio signal received by the CR occupies N spectrum bands, whose requency locations and PSD levels are to be detected and identiied. These spectrum bands lie within [, N ] consecutively, with their requency boundaries located at < < N.Then-th band is thus deined by B n : { B n : n < n }, n =,,...,N.The PSD structure o a wideband signal is illustrated in Fig.. The ollowing basic assumptions are adopted. a) The requency boundaries and N = + B are known to the CR. Even though the actual received signal may occupy a larger band, this CR regards [, N ] as the wide band o interest and seeks white spaces only within this spectrum range. a) The number o bands N and the locations,..., N are unknown to the CR. They remain unchanged within a time burst, but may vary rom burst to burst in the presence o slow ading. a3) The PSD within each band B n is smooth and almost lat, but exhibits discontinuities rom its neighboring bands B n and B n+. As such, irregularities in PSD appear at and only at the edges o the N bands. a4) The ambient noise is additive and white, with zero mean and two-sided PSD S w () =N /,. In the absence o noise, the normalized (unknown) power spectral shape within each band B n is denoted by S n (), which satisies the ollowing conditions: S n () =, / B n ; n S n ()d = n n. n According to a3), we may approximate S n () as: {, Bn. S n () =, / B n. With a3) anda4), the PSD o the observed signal r(t) at the CR ront-end can be written as N S r () = α ns n ()+S w (), [, N ] () n= Dynamic spectrum sharing not only concerns the identiication o spectrum holes, but also the detection o primary license-holders when a primarysecondary network (as oppose to a peer-to-peer network) is o interest. The latter task is a binary-hypothesis signal detection problem, while this paper ocuses on the ormer sensing task relevant to both network paradigms. Spectral spikes may arise in communication signals (e.g., due to signal cyclostationarity), but are not treated as PSD discontinuities. The treatment on this issue will be discussed in Section III.E. () where α n indicates the signal power density within the n-th band. The corresponding time-domain signal is r(t) = N α n p n (t)+w(t) (3) n= where S n () is the signal spectrum o p n (t) and w(t) denotes the additive noise with PSD S w (). For example, the signal component occupying B n can be a pulse train in the orm p n (t) = k= b kh(t kt s )e jπc,nt,where{b k } are digitally modulated symbols, h(t) is a pulse shaper o bandwidth ( n n ),and c,n =( n + n )/ is the center requency o this band. The spectral shape S n () is thus proportional to F{h(t)}, with F{ } denoting Fourier Transorm. The wideband spectrum sensing problem o our interest is ormulated as ollows: For a CR that receives r(t) with PSD S r () as in (), how to estimate the ollowing parameters characterizing the wideband spectral environment: N, { n } N n= and {α n }N n=? We seek answers to this problem without resorting to multiple narrowband BPFs. The use o N BPFs not only causes increased number o receiver components, but also aces challenges in tuning the local oscillator o each BPF in the absence o knowledge on N as well as the intended passband range [ n, n ], n =,...,N [4]. PSD Fig.. B n n- c,n n wide band o interest N requency bands with piecewise smooth PSD. III. WAVELET APPROACH TO SPECTRUM SENSING Based on a3) and with reerence to Fig., wideband spectrum sensing can be viewed as an edge detection problem in an image depicted by the PSD S r () in requency. Edges in this image correspond to the locations o requency discontinuities { i } i= N, which are to be identiied. This section shows that the wavelet transorm can eectively characterize the edges exhibited in the local singular structure o the PSD. In adopting the wavelet approach to spectrum sensing, we note at the outset that the wavelet transorm in existing applications is applied in lieu o Fourier transorm (FT) to characterize a time series such as r(t), or a spatial pixel graph such as in imaging [9]. In our problem, the domain o interest is requency, which is in act the duality o time t ater FT. As such, the noise component dealt in our problem has dierent characteristics rom that in conventional wavelet analysis. N
A. Wavelet Transorm o Signal PSD Let φ() be a wavelet smoothing unction with a compact support, m vanishing moments and m times continuously dierentiable. The strictly positive integer m is selected depending on Lipschitz exponent, which is a measure or the local regularity o the signal o interest [9]. Widely-used examples or φ() include the Gaussian unction and the perect reconstruction ilter bank (PRFB) []. The dilation o φ() by a scale actor s is given by φ s () = ( ) s φ. (4) s For dyadic scales, s takes values rom powers o, i.e., s = j, j =,,...,J. Letting denote convolution, the continuous wavelet transorm (CWT) o S r () is given by W s S r () =S r φ s (). (5) We note that the CWT in (5) is carried out in the requency domain, while the unction o interest S r () itsel relates to the received time-domain unction r(t) via the FT. A direct way to compute W s S r () is to irst perorm the FT on the autocorrelation unction R r (τ) :=E{r(t)r(t + τ)} such that S r () =F{R r (τ)}, ollowed by the convolution operation in (5). Equivalently, W s S r () can be computed rom r(t) in an alternative way. Let Φ s (τ) :=F{φ s ( )} = F {φ s ()} = Φ(sτ) represent the inverse FT o the wavelet unction. The inverse FT o W s S r () is given by W s S r (τ) := F {W s S r ()}, which is related to Φ s (τ) and R r (τ) via W s S r (τ) =R r (τ) Φ(sτ). (6) Thereore, an alternative to (5) is given by W s S r () =F{W s S r (τ)} = F{R r (τ) Φ(sτ)}. (7) Once the wavelet φ s () and its FT pair Φ(sτ) are determined, the computation o the CWT W s S r () involves either convolution and FT (on R r (τ)) operations as in (5), or product and FT (on the product) operations as in (7). B. Spectrum Sensing via Wavelet Modulus Maxima For the PSD S r () o interest, edges and irregularities at the scale s are deined as local sharp variation points o S r () smoothed by φ s (). As we know, the edges o a unction are oten signiied in the shapes o its derivatives. With the CWT, the irst-order and second-order derivatives o S r () smoothed by the scaled wavelet φ s () can be expressed respectively by W ss r () =s d d (S r φ s )() (8) = S r (s dφ s d )() = sf{τr r(τ)φ s (sτ)}; W s S r() =s d d (S r φ s )() (9) = S r (s d φ s d )() =s F{τ R r (τ)φ s (sτ)}. Similar to (5) and (7), the computation o W ss r () and W s S r() each has two equivalent expressions. It is shown in [9] that the local extrema o the irst derivative and the zero-crossings o the second derivative characterize the signal irregularities. In particular, the local maxima o the wavelet modulus are sharp variation points, which tend to be more accurate than local minima points (corresponding to slow variation points) or spectrum sensing purposes. The identiication o { n } N n= can thus be realized based on the ollowing proposition. Proposition. Boundaries { n } o consecutive requency bands {B n } with piecewise smooth PSD can be acquired rom r(t) by picking the local maxima o the wavelet modulus W ss r () in (8) with respect to as ˆ n = maxima { W s S r() }, (, N ) () or rom the zero-crossing points o W s S r () in (9) as ˆ n = zeros {W s S r ()} s.t. W s S r ( ˆ n )=. () Detecting the zero-crossings o W s S r () or the local extrema o W s S r() are similar procedures. When searching or ˆ n via either procedure, the scale actor s can be set to the dyadic scales s = j, j =,...,J. Only those modulus maxima or zero crossings that propagate to coarser (i.e., larger) scales are retained, while others are removed as noise [9]. C. Spectrum Sensing via Multiscale Wavelet Products In Proposition, the desired local maxima o wavelet modulus are tracked by their propagation to multiple coarser scales, with the goal o denoising. Such an idea o exploiting the multiscale correlation can be carried out in a direct (albeit nonlinear) way, giving rise to multiscale analysis techniques. In [3], edge detection and estimation is analyzed based on orming multiscale point-wise products o smoothed gradient estimators. This approach is intended to enhance multiscale peaks due to edges, while suppressing noise. Adopting this technique to our spectrum sensing problem and restricting to dyadic scales, we construct the multiscale product o J CWT gradients as J U J S r () = W s= S r() () j j= where the derivative o the smoothed PSD S r () is given by (8). Based on Proposition, it is evident that the requency edges { n } o interest (which are local maxima o W s S r() or all s J ) show up as the local maxima o U J S r (). On the other hand, noise-induced spurious local maxima o W ss r () are random at each scale and tend not to propagate through all J scales; hence, they do not show up as the local maxima o the product U J S r (). Summing up, we reach the ollowing proposition as an alternative means o spectrum sensing.
Proposition. Boundaries { n } o consecutive requency bands {B n } with piecewise smooth PSD can be acquired rom r(t) by picking the local maxima o the multiscale product U J S r () in () with respect to as ˆ n = maxima { U J S r () }, (, N ). (3) D. Spectral Density Estimation Ater { n } n= N have been detected and estimated via Proposition or, the remaining task o spectrum sensing is to estimate the PSD levels {α n } N n=. To this end, we compute the average PSD within the band B n, n =,...,N,inthe orm n β n = S r ()d. (4) n n n Based on the assumptions a3) a4) and the approximation () on the PSD shape, it is evident that β n is related to the unknown α n by β n α n + N /. The noise PSD N / can be measured oline, or deduced rom an empty band, say the n -th one, that satisies α n =and β n = N / or B n. Such an empty band almost always exists, since the spectrum utilization in current wireless systems is rather low (below %). Apparently, β n = N / is the smallest possible value or all {β n }. Summing up, we present a simple estimator or α n as ollows. Proposition 3. For each requency band B n with piecewiselat PSD as in (), its spectral density α n can be estimated rom S r () as ˆα n = β n min n β n, n =,...,N (5) where { n }, used or computing {β n } in (4), can be replaced by their estimates obtained rom the wavelet approach. The spectral density estimator in Proposition 3 is quite simple, while more elaborated methods are possible to estimate both {α n } and {S n()}, even when the signal PSD is not piecewise lat. Such solutions can take advantage o the attractive properties o the wavelet transorm in providing complete reconstructions o unctions with local structures [9]. Details are omitted here or space limit. Albeit simple, the estimator in (5) is adequate or solving the sensing problem o our interest. The primary goal o our sensing problem is to identiy the requency locations o bands {B n } and categorize them into black, gray, or white spaces [3], corresponding to PSD levels α n being high, medium or low. Thereore, coarse estimation o α n suices or requency space categorization. E. Noise Characteristic in the Wavelet Approach In our wavelet approach, local maxima o the wavelet transorm modulus may arise not only due to the requency edges { n } o interest, but also due to additional sources: isolated impulses, spikes, very-narrow-band intererence (vnbi), and additive white noise. It is o interest to investigate the degrading eect o these sources on spectrum sharing. The ollowing remarks are in due. Isolated impulses/spikes and vnbis appear as narrow peaks in an otherwise white or gray space. For wideband receivers with built-in capability to handle vnbi, it is preerred not to identiy these peaks during spectrum sensing, such that the entire white/gray space is treated as being opportunistically available or sharing. In this case, results based on multiscale products are preerred or the inherent ability to suppress isolated impulses, depending on the amount o smoothing utilized [3]. On the other hand, or narrowband receivers that rely on channelized spectrum allocation, the inormation on vnbis is useul to acquire during sensing. Regarding the ambience noise, it is interesting to observe that the noise eect in our wavelet approach to the spectrum sensing problem is not as harmul as in conventional wavelet applications. In the latter case, the wavelet transorm is imposed on a time series or an image whose noise component is random (e.g., the Gaussian noise w(t) in (3)), causing a large number o spurious local extrema at iner scales. In contrast, in our problem the CWT is applied to the PSD S r () in (), whose additive noise component S w () is white/lat. Thus, there is ew spurious edges incurred by S w (). IV. SIMULATIONS We consider a wide band o interest in the range o [5, 5] MHz. Fig. (a) illustrates the PSD S r () observed by a CR. The noise loor in the PSD is quite large at S w () =. During the observed burst o transmissions in the network, there are a total o N = 6 bands {B n }, with requency boundaries at { n } 6 n= = [5,, 7,,, 4, 5] MHz. Among these bands (marked on Fig. (a)), B, B 3 and B 5 have relatively high signal PSD at levels 4, 3, and 36, respectively, while B has low signal PSD at a level o 3, all with reerence to S w () =. The rest two bands, B 4 and B 6 are not occupied and are thus spectrum holes. In all tests, we use the Gaussian wavelet along with our dyadic scales s = j, j =,, 3, 4. Fig. (b) depicts the wavelet modulus computed rom (8), while Fig. (c) plots the multiscale products o wavelets expressed in (). Edges in the PSD S 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. In particular, the multiscale product method in (c) is very eective in suppressing the spurious local extrema caused by noise, resulting in better detection and estimation perormance. The simple spectral density estimation scheme in Proposition 3 is used to estimate the noise and signal PSD levels. The estimated values are {ˆα n} =[4.3566, 4.66, 9.4695,, 38.368,.768] corresponding to the true signal PSD values [4, 3, 3,, 36, ] respectively, and
Ŝ w () = 99.433 corresponding to the true noise PSD value. Such estimation accuracy is adequate in classiying the corresponding requency band into the coarse categories o white, gray and black spaces. The sensing capacity o the wavelet approach is evident even when the signal to noise ratio is quite low. V. SUMMARY This paper ormulates the cognitive spectrum identiication task as an spectral edge detection problem and exploits the wavelet approach or spectrum sensing o wideband channels. Solutions based on the local maxima o both gradient wavelet modulus and multiscale wavelet products are derived and tested. The proposed schemes are able to scan over a wide bandwidth to simultaneously identiy all piecewise smooth subbands, without prior knowledge on the number o subbands within the requency range o interest. The wavelet approach oers evident advantages over the conventional use o multiple narrowband BPFs, in terms o both implementation costs and lexibility in adapting to dynamic PSD structures. Since the wavelet approach targets wideband spectrum sensing, it may require high sampling rates in order to characterize the entire wide bandwidth. Nevertheless, the requirements on sampling rates can be reduced when the sensing task primarily concerns a relaxed spectral estimation problem o identiication o band types and spectrum holes, or when guard bands are inserted during CR transmissions such that it suices to obtain rough location estimates o spectrum holes. REFERENCES [] Facilitating Opportunities or Flexible, Eicient, and Reliable Spectrum Use Employing Cognitive Radio Technologies, FCC Report and Order, FCC-5-57A, March 5. [] I. J. Mitola and G. Q. Maguire, Cognitive radio: making sotware radios more personal, IEEE Personal Communi., vol. 6, pp. 3-8, Aug. 999. [3] S. Haykin, Cognitive radio: brain-empowered wireless communications, IEEE JSAC, vol. 3(), pp. -, Feb. 5. [4] A. Sahai, D. Cabric, Spectrum Sensing Fundamental Limits and Practical Challenges, A tutorial presented at IEEE DySpan Conerence, Baltimore, Nov. 5. [5] H. Urkowitz, Energy detection o unknown deterministic signals, Proceedings o the IEEE, vol. 55(4), pp. 53-53, April 967. [6] S. Shankar, C. Cordeiro, K. Challapali, Spectrum Agile Radios: Utilization and Sensing Architecture, Proc. IEEE DySpan, pp. 6-69, Nov. 5. [7] F. Granelli and H. Zhang, Cognitive UWB Radio: A Research Vision and Its Open Challenges, Intl. Workshops on Networking with UWB (NEUWB), Roma, pp. 55-59, July 5. [8] D. J. Thomson, Spectrum Estimation and Harmonic Analysis, Proceedings o the IEEE, vol. 7, no. 9, pp. 55-96, Sept. 98. [9] S. Mallat, W. Hwang, Singularity detection & processing with wavelets, IEEE Trans. Ino. Theory, vol.38, pp. 67-643, 99. [] S. Mallat, S. Zhong, Characterization o signals rom multiscale edges, IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 4, pp. 7-73, July 99. [] M. Tsatsanis, G. B. Giannakis, Time-Varying System Identiication and Model Validation Using Wavelets, IEEE Trans. Signal Processing, vol. 4, no., pp. 35-353, Dec. 993. [] Z. Cvetkovic and M. Vetterli, Discrete-time wavelet extreme representation: Design and consistent reconstruction, IEEE Trans. Signal Processing, vol. 43, pp. 68-693, Mar. 995. [3] B. M. Sadler, A. Swami, Analysis o Multiscale Products or Step Detection and Estimation, IEEE Trans. Inormation Theory, vol. 45, no. 3, pp. 43-5, April 999. S r () 5 4 3 9 B B 8.5..5 3 4 (a) x 3 4 x 4 3 x 5.5..5 3 4 (b) 3 x 5 6 x 9 4 x 3 (c) Fig.. (a) Original signal PSD; (b) wavelet transorm modulus at scales j,j =:4; (c) multiscale wavelet products. B 3 B 4 B 5 B 6