Optimal Quantization of TV White Space Regions for a Broadcast Based Geolocation Database
OOptimal Quantization of TV White Space Regionsfor a Broadcast Based Geolocation Database
Garima Maheshwari and Animesh Kumar
Department of Electrical EngineeringIndian Institute of Technology, BombayMumbai, India – 400076Email: [email protected], [email protected]
Abstract —In the current paradigm, TV white space databasescommunicate the available channels over a reliable Internetconnection to the secondary devices. For places where an Internetconnection is not available, such as in developing countries, abroadcast based geolocation database can be considered. Thisgeolocation database will broadcast the TV white space (or theprimary services protection regions) on rate-constrained digitalchannel.In this work, the quantization or digital representation ofprotection regions is considered for rate-constrained broadcastgeolocation database. Protection regions should not be declaredas white space regions due to the quantization error. In thiswork, circular and basis based approximations are presentedfor quantizing the protection regions. In circular approximation,quantization design algorithms are presented to protect theprimary from quantization error while minimizing the whitespace area declared as protected region. An efficient quantizerdesign algorithm is presented in this case. For basis basedapproximations, an efficient method to represent the protectionregions by an ‘envelope’ is developed. By design this envelopeis a sparse approximation, i.e., it has lesser number of non-zerocoefficients in the basis when compared to the original protectionregion. The approximation methods presented in this work aretested using three experimental data-sets.
I. I
NTRODUCTION
The wireless spectrum is a limited and valuable resource.The demand for spectrum is increasing due to the increase inthe number of wireless devices and this demand has led toresearch for efficient utilization techniques of the spectrum.The usage of TV white space by unlicensed secondary usersis an example of efficient utilization of spectrum. The spec-trum licensing agencies, Federal Communications Commis-sion (FCC) in the United States and Office of Communication(Ofcom) in the United Kingdom, have permitted access of TVwhite space by an unlicensed secondary device [1], [2].According to the existing regulations of FCC and Ofcom,TV white space can be accessed by a secondary or white spacedevice (unlicensed user) via TV white space (geolocation)database access. A certified TV white space database serviceis queried before operation by the secondary device. Thisquery includes the location of secondary device, and databaseregisters the secondary (white space) device if it is allocated a‘white’ TV channel. The TV transmitter protection regions arecalculated by the TV white space database service providersto avoid harmful interference to the primary devices of thelicensed broadcasting services. The available TV white space changes with time and space, and it is mandatory for thesecondary device to know the availability at the location andtime of current operation. The access of TV white spacedatabase takes place over the Internet [1]. Fig. 1 depicts thescheme for accessing TV white space database over the Inter-net. By design, TV white space database access is inaccessiblefor secondary devices in areas where there is unreliable orno Internet connection. The lack of internet connection isespecially prevalent in many developing or under-developedcountries where internet services are limited. In such areas, an alternate scheme for communication of the protection regionsof TV transmitters are needed.
Fig. 1. Three step explanation of database operation is explained. A whitespace (WS) device queries the database with geolocation information r anddevice id M . The database assigns an available channel and registers thedevice M . In this work, a broadcast based database service for commu-nicating TV white space availability is proposed. A broadcastbased database, such as one using a satellite, can transmit theTV transmitters’ protection regions, or simply the protectionregions . In this broadcast based database, the white spacedevice will not query and only receive the broadcast messagefrom the database. The proposed broadcast based databasewill communicate the protection regions to all the white spacedevices in a region. This broadcast based approach is assumedto make use of a wireless or digital channel, and the transfer ofinformation to a secondary device will be rate constrained (seeFig. 2). Accordingly, quantization of protection regions is ofinterest for a broadcast-style TV white space database. Whenthe protection regions are quantized, some TV white space a r X i v : . [ c s . I T ] M a y ig. 2. A broadcast based geolocation database is illustrated. It is assumedthat white space devices can receive the broadcast over a rate constrainedchannel. Thus, the database should quantize the protection regions, whileensuring the protection of primary from quantization error. area will be ‘lost’ due to quantization error. This quantizerhas to be designed to minimize the TV white space area lostdue to quantization. In summary, protection region quantizer’s design and performance are the key problems addressed inthe current work. To the best of our knowledge, this is the firstwork of its kind.
Fig. 3. This graph is obtained from the website of a certified TV whitespace service provider iconectiv in the United States [3]. The contours areprotection regions for Channel 22 near New York. Protection region labeled is almost circular while protection regions labeled and are non-circularin shape. Examples of the protection regions are illustrated in Fig. 3,which are obtained from the iconectiv website [3] for Chan-nel 22 in the New York region. Protection regions such as are nearly circular while protection regions such as and are non-circular (see Fig. 3). Motivated by this, two types ofdigital representation or quantization of the protection regionswill be considered: (i) circular protection regions with quanti-zation of radius; and (ii) non-circular protection regions withsuper-set or envelope approximations. These approximationsare illustrated in Fig. 4. The circular protection region model“marks” some portion of spatial TV white space as protected region, and it is “wasteful” compared to the non-circularprotection region model. Fig. 4. The circular approximation and basis based approximation ap-proaches are illustrated. In the circular approximation approach (left figure),the protection region is enlarged and made circular by inscribing it in acircle with maximum radius. In the basis based approximation approach, anapproximation r ap ( t ) is obtained such that the protection region is a subsetof this approximation. In both circular and non-circular models, the protectionregion has to be quantized or represented to a larger orenvelope-style region to protect the incumbent. That is, TVwhite space can be labeled as protected (due to quantization)but protection region must not be labeled as unprotected (dueto quantization). For this important regulatory reason, a tradi-tionally well-studied mean-squared error optimal quantizationmethod cannot be employed [4].
Key contributions:
For the circular model, efficient quantizerdesign algorithm will be discussed in this work to ensureprimary’s protection even after quantization! By efficient, it ismeant that for a given quantizer precision, the TV white spacearea mislabeled (lost) as protection region is minimum. For thenon-circular model, a basis (such as Fourier) based approachis detailed to obtain an envelope-approximation to the non-circular protection region. As explained in Fig. 4, a basis basedapproximation will minimize or reduce the loss in TV whitespace area beyond the circular (radius based) approximation.The main goal of our developed techniques is to obtain thequalitative graph illustrated in Fig. 5. There is a fundamentallimit of how much TV white space area is present. With largertransmission rate, the database can facilitate a more accuraterecovery of the TV white space region or the protection region.While using the circular approximation, the recovery of TVwhite space region will have a gap from the actual TV whitespace area.To clarify further, the chart shown in Fig. 6 will be usefulto identify our contributions.
Related work:
Geolocation database are well known in theliterature [1], [2], [5]. Circular protection regions for TVwhite space regions are well known (see [6]). Primary servicecontours are available as databases for countries such as USand UK [1], [2].To the best of our knowledge, a broadcast style geolocationdatabase has not been studied in the literature. Quantizationof real values where signal is always overestimated as well ig. 5. The graph qualitatively illustrates the methods described in this paper.For circular approximations, the white space area recovered by bounding theprotection region with circles has a gap when compared to the actual whitespace area. This gap can be reduced by basis based approximation for theprotection region.Fig. 6. The described problem statement in this paper can be dividedinto four segments as shown in the figure. We have addressed the issues ofquantization of circular protection regions and describing the shape of the non-circular regions using a basis function (without quantization of the coefficientscorresponding to the basis functions). as envelope style approximations are also not known in theliterature to the best of our knowledge.
Organization:
Section II formulates and solves the quan-tization of circular protection regions. Optimal algorithm ispresented to quantize the protection region, while ensuringprotection for the primary. Section III presents the basis basedenvelope-approximation for non-circular protection regions.Finally, conclusions are presented in Section IV.
Assumptions:
The quantization of transmitters’ center isnot considered. In the basis approximation case, this justcorresponds to one extra coefficient being sent and does notaffect our main results. In the circular approximation case,having quantized radius makes the problem more difficult.Nevertheless, since TV transmitters are at fixed locations, thisdata can be exchanged first before repeated broadcast of TVwhite space takes place. It is assumed that the secondarydevices will coexist by some media access control mecha-nism, since the broadcast database facilitates only ‘one-way’communication. The broadcast database will not register thesecondary devices. Distributed coexistence techniques such as collision sense multiple access can be used by the secondarydevices.II. C
IRCULAR PROTECTION REGIONS AND THEIR OPTIMALQUANTIZATION
This section deals with the quantization of (circular) pro-tection region radius. The quantization has to be designedto minimize the white space area lost due to quantizationacross all transmitters. It is assumed that the centers of theseprotection regions are already available at the receiver, so thatonly radius of protection regions have to be quantized andcommunicated. In case the protection region is not circular, thena¨ıve radius based approximation scheme depicted in Fig. 4can be used to obtain a radius.Let R := { r , r , . . . , r n } be the radius of protectionregions. For simple exposition in this paper, it is assumed thatthis set R is fixed. These radius have to be quantized in theset Q = { q , q , . . . , q m } , where log m will be the number ofbits being spent to communicate each circular region. These log m bits will index various quantization levels in the set Q .Without loss of generality, it is assumed that q , q , . . . , q m and r , r , . . . , r n are both in an increasing order. The quantizedradius set Q is known to the broadcast-database transmitter aswell as all the secondary white space devices. The protectionregion radius set R , on the other hand, is only known to thetransmitter.To ensure protection for the primary, Q ( r i ) ≥ r i for allprotection radius r i ∈ R . Unlike in traditional mean-squaredor minimax optimal quantization, where Q ( r ) is mapped to theclosest quantization level [4], quantization level for protectionregion’s radius is always the nearest larger level . Thus, thequantizer design for protection region radius is different thantraditional mean-squared optimal quantizers.The actual area of a circular protection region with radius r is πr . After quantization, the circular protection regionwill have a radius of πQ ( r ) . Since Q ( r ) ≥ r by designrequirements, so a part of TV white space region will be lost ormislabeled as protection region. This motivates the followingcost function C ( R , Q ) = π (cid:88) r ∈R (cid:8) Q ( r ) − r (cid:9) (1)which signifies the white space area lost due to quantization.This is explained with an example having four protectionradius and two quantization levels in Fig. 7. The radius r , r get mapped to q . Even though q is nearer to r , still r getsmapped to q . For any given number of quantization levels m ,an optimal quantization map Q : R → Q has to be designedto minimize the lost TV white space area, subject to a primaryprotection condition—the actual (unquantized) protection re-gion should be a subset of the quantized protection region.The cost function in (1) can be rewritten as C ( R , Q ) = π (cid:40) n (cid:88) i =1 Q ( r i ) − r i (cid:41) (2) ig. 7. Radius r , r get translated to the quantization level q and r , r get translated to the quantization level q as shown in the image. This is forthe purpose of protecting the primary transmitting device from any harmfulinterference. where Q ( r i ) maps r i to the next largest quantization levelin the set Q = { q , q , . . . , q m } . The levels in Q haveto be chosen to minimize the lost TV white space area toquantization, or C ( R , Q ) in (2). The usual technique forminimization will be to evaluate ∂C ( R , Q ) ∂q j = 0 and ∂ C ( R , Q ) ∂q j > (3)for all q j ∈ Q . Let r max = r n be the radius of the largestprotection region in R . Then q m = r max . However, thederivative does not exist since the set R is discrete andtherefore a change in q j ≥ r i to q j < r i causes a discontinuouschange in C ( R , Q ) . To gain further insights, we assume that R is obtained based on i.i.d. trials from a (continuous) probabilitydistribution p ( r ) . Then the expected value of C ( R , Q ) will beminimized. Similar to (2), this expected cost function is givenby C ( p ( r ) , Q ) = π (cid:90) r max r min (cid:8) Q ( r ) − r (cid:9) p ( r ) d r (4)where r min and r max are the minimum and maximum pro-tection region radius (over which p ( r ) is non-zero). Usingthe quantization levels in Q , the cost function in (4) can berewritten as C ( p ( r ) , Q ) = π (cid:90) q r min (cid:8) q − r (cid:9) p ( r ) d r + . . . + π (cid:90) q m q m − (cid:8) q m − r (cid:9) p ( r ) d r (5)From (5), upon taking derivative with respect to q j in accor-dance with the Leibniz integral rule [7], we get ∂C ( p ( r ) , Q ) ∂q j = 2 q j (cid:90) q j q j − p ( r ) d r − p ( q j )[ q j +1 − q j ] (6)Recall that Q is assumed to be in increasing order, that is, q < q < . . . < q m . In the above equation q j − ≤ q j ≤ q j +1 .If q j = q j − , then ∂C ( p ( r ) , Q ) ∂q j = − p ( q j )[ q j +1 − q j ] < (7)while if q j = q j +1 , then ∂C ( p ( r ) , Q ) ∂q j = 2 q j +1 (cid:90) q j +1 q j − p ( r ) d r > . (8) Using the Intermediate value theorem, with the assumptionthat p ( r ) is continuous so that ∂C ( p ( r ) , Q ) ∂q j in (6) is continuous,it can be deduced that there is some value q ∈ ( q j − , q j +1 ) such that ∂C ( p ( r ) , Q ) ∂q j (cid:12)(cid:12)(cid:12)(cid:12) q j = q = 0 . (9)With the assumption that p ( r ) is continuous, an extrema of C ( p ( r ) , Q ) with respect to q j lies between q j − and q j +1 . Toargue that the extrema is a point of minimum, the Hessian in(4) should be positive definite, which requires more assump-tions on p ( r ) . At this point it isn’t clear what assumptionsshould be made on p ( r ) , so an empirical approach is taken.An iterative algorithm will be obtained to find the set Q whichminimizes C ( r, Q ) or C ( R , Q ) . This iterative algorithm isexplained next for minimizing C ( R , Q ) , i.e., assuming that R is given.At first, note that the largest quantization level must be equalto the largest protection radius, i.e., q m = r n (10)since the largest values of Q ( r i ) need not exceed r n (themaximum protection radius) in (2). Apart from q m , each q j will be chosen from the discrete-set R only. This can beunderstood using Fig. 7. If q is chosen between r and r ,then the cost contribution of q is q − r + q − r , whichgets minimized when q = r .At a high-level, one notes that q has ( n − m ) choices in R , subsequently q has ( n − m − choices in R , and soon. So the total number of choices for the entire set Q is ( n − m )( n − m − . . . (1) = ( n − m )! , which is quite large.A fast algorithm will be developed next to solve the selectionof Q .From (6), for minimum (or an extrema) point the followingequation should be satisfied q j (cid:90) q j q j − p ( r ) d r − p ( q j )[ q j +1 − q j ] = 0 . (11)From the above, it is noted that q j depends only on q j − and q j +1 . That is, if the odd elements in the set Q are fixed, thenthe even elements can be found separately. Similarly, if theeven elements in the set Q are fixed, then the odd elements canbe found separately. This results in a separable optimizationalgorithm as detailed next. The counterpart of (11) for thediscrete case will be highlighted first. The cost function in (2)can be rewritten as π C ( R , Q )= (cid:88) r ∈R Q ( r ) − (cid:88) r ∈R r = (cid:88) r ∈ ( q j − ,q j ] Q ( r ) + (cid:88) r ∈ ( q j ,q j +1 ] Q ( r ) + E j − (cid:88) r ∈R r = (cid:88) r ∈ ( q j − ,q j ] q j + (cid:88) r ∈ ( q j ,q j +1 ] q j +1 + E j − (cid:88) r ∈R r (12)here the term E j is positive, depends on q , . . . , q j − , q j +1 , . . . , q m , and is independent of q j . In(12), it is also noted that r ∈ ( q j − , q j ] will get quantizedto q j and r ∈ ( q j , q j +1 ] will get quantized to q j +1 . Let thenumber of protection region radius r ∈ ( q j − , q j +1 ] be n j and number of radius r ∈ ( q j − , q j ] be k j . Then (12) can berewritten as π C ( R , Q ) = k j q j + ( n j − k j ) q j +1 + E j − (cid:88) r ∈R r (13)The last term is independent of Q , while the second last term E j is independent of q j and can be ignored during optimiza-tion. Since q j − and q j +1 are fixed, so is n j . Therefore, theonly choice variables are q j , which subsequently determines k j as well. The minimization of k j q j + ( n j − k j ) q j +1 andsubsequently the expression in (13) (for fixed q j − and q j +1 )can be performed by an exhaustive search over various valuesof q j in between q j − and q j +1 . In summary, for givenfixed values of q j − and q j +1 , the value of q j that (locally)minimizes C ( R , Q ) can be found out by an exhaustive search.This motivates the following Even-Odd algorithm for theminimization of cost function in (2), subject to the conditionthat quantized protection radius is always larger than the actualprotection radius:1) A random initialization for the quantization levels in theset Q is assumed. It must be noted that the quantizationlevels belong in the set R .2) The largest quantization level q m is fixed to the largestprotection radius r n .3) After a random initialization, the even quantizationlevels are fixed and the odd quantization levels areexhaustively searched according to the process outlinedin (13). It is restated that the exhaustive search for eachquantization level is separate. This results in optimalvalues for odd quantization levels with respect to thecost function in (2).4) The (locally) optimized values of the odd quantizationlevels, as obtained in the previous step, are fixed. Theeven quantization levels are now exhaustively searchedaccording to the process outlined in (13). This resultsin optimal values for the even quantization levels withrespect to the (2).5) The steps 3 and 4 above are used in an iterative manner,till the quantization levels do not change. The resultantquantization levels minimize the desired cost functionstated in (2).The simulation results are presented next. The data used inour simulation experiments is outlined first. Data used for experiments:
For circular protection regions,two sets of data were available to us. The first data-set containsthe protected service contours’ bounding radius calculatedusing protection and pollution viewpoints [6]. They make useof the latitude, longitude, EIRP and HAAT to calculate this ra-dius. This set of radius is available for all the channels between to ( channels) of United States, where transmission by awhite space device is permitted by the FCC. There were protection regions, and hence radius, in total. The second data-set contains the protected service contours’ bounding radiusobtained for India [8]. There are no white space regulations inIndia as of now. The protection radius are available only for channels in the UHF Band-III ( - MHz). There were protection regions, and hence radius, in total for India.These datasets are used to analyse the recovered TV whitespace area and test our optimal quantization Even-Odd algo-rithm. As India has extensive rural areas with negligible broad-band services, Internet connection is extremely unreliable. It isonce again emphasized that a broadcast based TV white spacegeolocation database will be quite useful for such scenarios.If b bits are used to index each radius in the set R , then m = 2 b . It is recalled that the radius set R is only known tothe database, points in R will be mapped into m quantizationlevels, and Q is agreed upon between the broadcasting geolo-cation database and the secondary devices. For comparison ofour Even-Odd algorithm discussed in Section II, a uniformscalar quantizer is used. For uniform scalar quantizer, thequantization levels are Q = { r min + ∆ , r min + 2∆ , . . . , r max } where ∆ = r max − r min m . (14)All the values of radius ( r ) greater than i − th level and lessthan or equal to i th level are translated to the i th level in orderto protect the primary.The results obtained by applying our algorithm on the data-set from United States are illustrated in Fig. 8. Increasingthe number of bits decreases the area of lost white spaceregion to quantization, and with bits (per protection radius)or quantization levels most of the white space area canbe recovered. Since there are about protection radius,they can be communicated in a lossless manner with bitsper protection radius. The results obtained by applying ouralgorithm on the data-set from India are illustrated in Fig. 9.With bits (per protection radius) or quantization levelsrecover most of the white space area. Since there are about protection radius, they can be communicated in a losslessmanner with - bits per protection radius.The evolution of quantization levels in our Even-Oddalgorithm is illustrated in Fig. 10. The initial quantizationlevels are obtained by using a scalar quantizer for b = 3 .The quantization levels obtained using the algorithm, help inrecovering more white space area as compared to uniformquantization for every bit that is sent as illustrated in Fig. 8and Fig. 9.III. E NVELOPE APPROXIMATION FOR NON - CIRCULARPROTECTION REGIONS
For some non-circular protection regions, its circular ap-proximation area can be as large as % when compared tothe protection region’s area. This area can be scavenged bythe secondary if a better approximation method is used. In thissection, protection region representation will be considered be-yond the circular approximation explained in the Introduction. ig. 8. The recovered white space area for various values of bits/protectionregion is plotted for the data-set from United States [6]. The solid linerepresents the actual real valued white space area. For smaller values ofbits/protection region, the Even-Odd algorithm based quantizer outperformsthe uniform quantizer.Fig. 9. The recovered white space area for various values of bits/protectionregion is plotted for the data-set from India [8]. The solid line represents theactual real valued white space area. As expected, the Even-Odd algorithmbased quantizer outperforms the uniform quantizer. As explained in Fig. 6, for non-circular regions quantizationis not considered in this work. This section will focus onobtaining basis-based envelope approximation as depicted inFig. 4. To the best of our knowledge, such approximationsare not present in the quantization literature. Consider a non-circular but smooth shape as depicted in Fig. 11. Let itscentroid be at the origin. Then, the shape can be parametrizedby a waveform r ( t ) as shown in the figure. With the knowledgeof origin (the center), the waveform r ( t ) is equivalent to thenon-circular protection region. The waveform r ( t ) is periodic(or has finite support) and it is expected to be smooth; so, itcan be represented via any basis function suitable for smooth Fig. 10. The evolution of quantization levels in the Even-Odd algorithmis illustrated. The initial points are obtained from a uniform quantizer. Thelargest level q is always equal to r max . In the odd steps ( q , q , q , q ) are calculated while ( q , q , q ) are held fixed. The roles are reversed inthe odd steps. The movement is illustrated by solid line, the fixed behavioris illustrated by a dashed line, and convergence of the quantization level isshown with a dotted line. signals on a compact support [9]. That is, r ( t ) = ∞ (cid:88) k = −∞ a k φ k ( t ) (15)where φ k ( t ) , k ∈ Z form the set of basis functions suitable forrepresenting smooth waveforms with support in [0 , π ] . Theset Z denotes the set of integers. Fourier series happens tobe one choice of basis functions, which will be used in thiscurrent work for simplicity of exposition. For Fourier basis,the expansion in (15) reduces to r ( t ) = ∞ (cid:88) k = −∞ a k exp ( jkt ) (16)The Fourier series coefficients are given by Fig. 11. A non-circular protection region can be viewed as a one-dimensionalwaveform r ( t ) with respect to the angle t as shown. a k = 12 π (cid:90) π r ( t ) exp ( − jkt ) d t (17)Since r ( t ) is real-valued the coefficients a k and a − k will berelated by conjugate symmetry. That is a k = ¯ a − k . So, only a k , k ≥ needs to be communicated.ur main innovation in this section is an envelope approx-imation for the waveform r ( t ) . In other words, r ( t ) has to beapproximated to r ap ( t ) such that r ap ( t ) is always larger than r ( t ) . To this end, the following approach is used. Let r K ( t ) be the K + 1 coefficient based Fourier series approximation,that is, r K ( t ) = K (cid:88) k = − K a k exp ( jkt ) . (18)Define e K as e K = max ≤ t ≤ π r ( t ) − r K ( t ) (19)It is noted that e K is the maximum error between r ( t ) and r K ( t ) . So if r ap ( t ) = r K ( t ) + e K (20)then r ap ( t ) ≥ r ( t ) , and r ap ( t ) has K + 1 Fourier coefficientsgiven by a − K , . . . , a − , e K + a , a , . . . , a K . Of these, dueto conjugate symmetry, only K + 1 coefficients need tobe sent. The coefficients that need to be sent are: [ a + e K , a , a , ..., a K ] to communicate an envelope approximation r ap ( t ) for r ( t ) . Experimental evaluation with this scheme isexplained next.There is no convenient data-set available for the shapes r ( t ) for the protected regions in the United States. Totest the Fourier series based envelope approximation method,we created a small data-set for Channel using the TVwhite spaces US Interactive Map of Spectrum Bridge [10].Across United States, there are protected service contours(excluding Alaska). These contours were hand-picked intoimages and the coverage region was segmented (using imageprocessing techniques) to obtain various shapes r ( t ) for these transmitters in Channel . Then, the Fourier basis basedenvelope approximation technique was applied. An exampleof Fourier basis based envelope is shown in Fig. 12. Observethat r ap ( t ) for various values of K always includes the actualprotection region as a subset.In the absence of real-valued data (images obtained havepixels!), a Riemann approximation [7] is used to calculate theFourier series coefficients as well as the protection region’sarea of the extracted service contours. As there is a limitationon the resolution of the images that can be obtained fromthe Interactive Map, we did observe some issues due topixelization (see Fig. 13). With a higher-resolution data-set,we will obtain better results. But such data-set is not available.Due to pixelization, in Fig. 13, the dashed line signifying whitespace recovered with basis approximation does not convergesto the solid line depicting total white space area. Based on thedata-set we could scavenge, the each pixel width is 0.412km.The results obtained by sending Fourier coefficients can beseen in Fig. 13. This graph is for the white space regionin Channel 2 of the TV spectrum in United States (exceptAlaska). The circular approximation scheme sends one coef-ficient K = 1 , but there is a gap between total white spacearea available and the white space area that can be recovered Fig. 13. This graph is applicable for protection regions in Channel 2in United States (except Alaska). The dotted line is the white space areathat can be obtained using circular approximations, which is less than theactual white space area (in solid line). If more Fourier series coefficients foreach protection region is sent, then the extra white space recovered above thecircular approximation is illustrated. by circular approximation. As the number of coefficients sentis increased, K = 2 and higher, the Fourier basis schemefills in the gap between circular approximation and actualwhite space area. It can be observed that the actual protectionregion is (7 . − . × = 2 . × sq. km.Circular approximation (with high bit-rate quantization) labels (7 . − . × = 3 . × sq. km. as protection region.So, the basis based approximation method reduces the size ofprotection region by about (3 . − . / .
49 = 49 . .IV. C ONCLUSIONS
A geolocation database that broadcasts the TV white spaceor the primary services protection regions on rate-constraineddigital channel was considered. The key issue addressed wasquantization or digital representation of the protection regions.It was observed that the protection regions can be circular ornon-circular. For circular protection regions, a fast algorithmfor optimal quantizer design was developed. The algorithmminimizes the white space area identified as protection region,while ensuring that protection region is not labeled as whitespace region due to quantization. For non-circular protectionregions, a basis based approximation was developed. A proce-dure for obtaining envelope-type approximation for protectionregion was developed, which depends on very few numberof basis coefficients. The approximation methods were testedusing three experimental data-sets. These data-sets includedcircular protection regions across all TV channels in theUnited States, circular protection regions in the UHF bandTV channels in India, and non-circular protection regions inChannel 2 in the United States.A
CKNOWLEDGMENTS
The authors benefited tremendously from stimulating dis-cussions with Prof. Anant Sahai, EECS, University of Cali- ig. 12. The approximation in (20) is illustrated. Consider the shape given by solid line in the right graph. The r ( t ) corresponding to it is given by thesolid line in the left graph. Using K = 2 and K = 11 , two different r ap ( t ) are plotted. Observe that for both values of K , the approximation is larger than r ( t ) for each value of t . This property is expected due to design of the approximation in (20). This property also ensures that the approximate protectionregion includes (or, is a superset of) the actual protection region. fornia, Berkeley, CA on this topic.This research is supported by the Ford Foundation.R EFERENCES[1] FCC 08-260, “Second report and order and memorandum opinion andorder,” US Federal Communication Commission, Tech. Rep., Nov. 2008.[2] Ofcom, “Digital dividend: Cognitive access. consultation on license-exempting cognitive devices using interleaved spectrum,” Ofcom, UK,Tech. Rep., Feb. 2009. [Online]. Available: http://stakeholders.ofcom.org.uk/binaries/consultations/cognitive/state-ment/statement.pdf[3] https://spectrum.iconectiv.com/main/home/contour vis.shtml.[4] A. Gersho and R. M. Gray,
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