PROCEEDINGS OF THE 4th INTERNATIONAL CONFERENCE ON CROWNCOM 2009 Performance Comparison of Time Delay Estimation for Whole and Dispersed Spectrum Utilization in Cognitive Radio Systems Hasari Celebi and Khalid A. Qaraqe Department of Electrical and Computer Engineering Texas A&M University at Qatar PO Box 23874 Texas A&M Engineering Building Education City, Doha, Qatar {hasari.celebi, khalid.qaraqe} Abstract—In cognitive radio systems, information is transmitted and received over
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  Performance Comparison of Time Delay Estimationfor Whole and Dispersed Spectrum Utilization inCognitive Radio Systems Hasari Celebi and Khalid A. QaraqeDepartment of Electrical and Computer EngineeringTexas A&M University at QatarPO Box 23874 Texas A&M Engineering BuildingEducation City, Doha, Qatar { hasari.celebi, khalid.qaraqe } Huseyin ArslanElectrical Engineering DepartmentUniversity of South Florida4202 E. Fowler Avenue, ENB-118Tampa, FL, 33620  Abstract —In cognitive radio systems, information is transmit-ted and received over multiple dispersed bands using dispersedspectrum utilization approach whereas information is transmittedand received over a single band using whole spectrum utiliza-tion approach. In this paper, performance of both approachesare compared theoretically considering CRLB and maximumlikelihood (ML) method for time delay estimation. The resultsshow that dispersed spectrum utilization methods have a greatpotential to exploit the efficiency of spectrum utilization andsupport goal driven and autonomous cognitive radio systems. I. I NTRODUCTION Cognitive radio (CR) is a novel approach for developingintelligent and sophisticated wireless systems [1], [2], whichcan require utilization of spectrum resources dynamically [3],[4]. CR has a capability to exploit the spectrum utilizationdue to its spectrum agility. The available spectrum can bemainly in two forms;  single band  , i.e.  whole spectrum , and multiband  , i.e.  dispersed spectrum  [5]-[8] 1 . In the wholespectrum utilization approach, the transmit signal for a singleCR user occupies a single band as illustrated in Fig. 1a,whereas, in the dispersed spectrum utilization approach, itoccupies multiple bands (e.g.  K   number of adjacent or non-adjacent bands) simultaneously as shown in Fig. 1b. Notethat dispersed spectrum utilization is a method that realizesfrequency diversity in full extent in wireless systems.Since whole spectrum utilization is the conventional wayof transmitting signal numerous technologies and transceiverarchitectures have been developed for such approach. Forinstance, cognitive positioning systems (CPSs) along withfundamental limits of time delay estimation are studied con-sidering whole spectrum utilization approach in [4], [13].The dispersed spectrum utilization approach is first dis-cussed briefly in [5] and investigated with details in [6].In [6], a CR receiver architecture for the dispersed spectrumutilization based on the idea of processing the receive signal 1 Note that another way to implement dispersed spectrum utilization ap-proach is to employ orthogonal frequency division multiplexing (OFDM)technology, where the sub-carriers corresponding to the used and unused bandsare activated and nulled, respectively [6], [9]-[12]. However, such approachis not considered in this study. at multiple branches is proposed. Each branch considers oneavailable band, and down-converts the signal according tothe center frequency of that band. By this way, signals withnarrower bandwidths can be processed at each branch. Asa result, investigation on performance comparison of wholeand dispersed spectrum utilization is of considerable interestfor development of goal driven and autonomous feature of CR systems. Hence, in this paper, the performance of wholeand dispersed spectrum utilization methods are comparedtheoretically considering time delay estimation problem inCR systems. Exact and approximate Cramer-Rao lower bound(CRLB) over AWGN channel for both approaches are pre-sented. Then, performance comparison of both approachesis carried out based on exact and approximate CRLBs andoptimal Maximum Likelihood (ML) method for time delayestimation through computer simulations.The paper is organized as follows. In Section II, the systemmodel for whole and spectrum utilization approaches are pro-vided. In Section III, ML, and exact and approximate CRLBsof time delay estimate for both approaches are presented. InSection IV, results and discussions are provided. Finally, theconclusions are presented in Section V.II. S YSTEM  M ODEL To the best of author’s knowledge, there is not any solidstudy on channel behavior of dispersed spectrum utilizationmethod in the literature. In practice, the observed channelat each band can be different depending on how dispersedbands are located in the spectrum. As a result, for the sakeof simplifying the performance comparison analysis and ex-ploring the fundamental considerations, we perform analysisof both approaches considering AWGN channel in this study.In addition, we assume that all the dispersed bands experiencedifferent AWGN channels. In the following sections, systemmodels for both approaches are presented.  A. Whole Spectrum Utilization The system model shown in Fig. 2 is considered for wholespectrum utilization in this paper. The baseband transmit signal 978-1-4244-3424-4/09/$25.00 ©2009 IEEEPROCEEDINGS OF THE 4th INTERNATIONAL CONFERENCE ON CROWNCOM 2009  fPSD f c1 f c2 f cK  fPSD f c B f … PSD B  2 B  1 B  K  B  2 B  1 B  K  0  … 0 0 B  a)WholeSpectrumUtilizationb)DispersedSpectrumUtilization Fig. 1. Illustration of whole and dispersed spectrum utilization in cognitive radio systems. s ( t )  with absolute bandwidth of   B  ([-B/2,B/2]) that occupiesa whole band shown in Fig. 1a is given by s ( t ) =  l d l  p ( t − lT  s )  ,  (1)where  d l  is the real data for  l th symbol,  p ( t )  is the pulsesignal with energy  E   p  and duration  T   p , i.e.,  p ( t ) = 0  for t  ∈  [0 ,T   p ]  and  T  s  is the symbol duration. The baseband signal s ( t )  is transmitted over AWGN channel and the correspondingbaseband representation of receive signal  r ( t )  is given by r ( t ) =  αs ( t − τ  ) + n ( t )  ,  (2)where  α  and  τ   are the path coefficient and delay, respectively,and  n ( t )  is independent white Gaussian noise with spectraldensity of   σ 2 . At the receiver side,  r ( t )  is used to performML time delay estimation. MLTime DelayEstimation  +  n(t)r(t)h(t)s(t) Fig. 2. System model for whole spectrum utilization.  B. Dispersed Spectrum Utilization The system model shown in Fig. 3 is considered fordispersed spectrum utilization approach in this paper. Thebaseband transmit signals  s i ( t )  at  i th branch ( i  = 1 ,...,K  )with corresponding absolute bandwidth of   B i  that occupiesdispersed band shown in Fig. 1b is given by s i ( t ) =  l d i,l  p i ( t − lT  s )  ,  (3)where  d i,l  is the real data for the  l th symbol of signal  i ,and  p i ( t )  represents a pulse with energy  E   pi  and duration T   pi , i.e.,  p i ( t ) = 0  for  t  ∈  [0 ,T   pi ] . The baseband signal s i ( t )  is transmitted and assuming that each signal experiencedifferent AWGN channel and the corresponding basebandrepresentation of receive signal at  i th branch  r i ( t )  is givenby r i ( t ) =  α i s i ( t − τ  ) + n i ( t )  ,  (4)where  α i  and  τ   are the path coefficient and delay for  i thbranch, respectively, and  n i ( t )  is independent white Gaussiannoise with spectral density of   σ 2 i  . At the receiver side,  r i ( t ) at  i th branch is used to perform ML time delay estimation.For the fair comparison, we assume  d i,l  =  d l , which impliesthat  d l  data in case of whole band is transmitted over eachdispersed band.III. P ERFORMANCE  A NALYSIS In this section, ML, and exact and approximate CRLB ex-pressions of time delay estimate for both whole and dispersedspectrum utilization are presented.  A. Whole Spectrum Utilization Let  θ  = [ τ  ]  represent the vector of unknown signal parame-ters, where  α  is assumed to be known. The observation interval [0 ,T  ]  is considered and it can be expressed as  T   =  NT  s ,where  N   is the number of observation symbol. Then, the ML PROCEEDINGS OF THE 4th INTERNATIONAL CONFERENCE ON CROWNCOM 2009  MLTime DelayEstimation  +  n  1 (t)r 1 (t)h  1 (t)s  1 (t) +  n  2 (t)r 2 (t)h  2 (t)s  2 (t) +  n  K  (t)r K  (t)h  K  (t)s  K  (t) ... Fig. 3. System model for the dispersed spectrum utilization. estimate for  θ  is given by [14] ˆ θ ML  ≈  argmax θ   1 σ 2    T  0 αr ( t ) s ( t − τ  )d t   ,  (5)where  E   =   T  0  [ s ( t  −  τ  )] 2 d t  is the signal energy. Then, theCRLB for unbiased time delay estimators is given by [14]CRLB  = 1 γ   ˜ E .  (6)where  γ   =  α 2 /σ 2 and  ˜ E   =   T  0  [ s  ( t − τ  )] 2 d t . Assuming thatthe spectral density of   p ( t )  is constant over the  B , then theapproximate CRLB expression is given by [14]CRLB wh  = 1 4 π 2 3  SNR B 2 ,  (7)where SNR is defined asSNR  =  α 2 Nd 2 l E   p σ 2  .  (8)  B. Dispersed Spectrum Utilization Similarly, let  θ  = [ τ  ]  represents the vector of unknownsignal parameters, where  α i  are assumed to be known. Theobservation interval  T  i  at  i th branch can be expressed as T  i  =  N  i T  s , where  N  i  is considered for an integer  N  i  for i  = 1 ,...,K  . Then, the ML estimate for  θ  is given by ˆ θ ML  ≈  argmax θ   K   i =1 1 σ 2 i    T  0 α i r i ( t ) s i ( t − τ  )d t   .  (9)The exact CRLB for dispersed spectrum utilization systems isderived in [6],which has the following form,CRLB disp  = 1  K i =1 γ  i  ˜ E  i .  (10)where  γ  i  =  α 2 i /σ 2 i  and  ˜ E  i  =   T  0  [ s  i ( t − τ  )] 2 d t . Assumingthat the spectral density of   p ( t )  is constant over the  B , thenthe approximate CRLB expression is given by [14]CRLB disp  = 1 4 π 2 3  K i =1  SNR i B 2 i ,  (11)where SNR i  is defined asSNR i  =  α 2 i N  i d 2 l E   p i σ 2 i .  (12)By examining (7) and (11), the performance comparison of whole and dispersed spectrum utilization in terms of CRLBdepends on the values of SNR  B , SNR i , and  B i . The followingare three possible cases for the CRLB performance comparisonof both approaches,  CRLB disp  >  CRLB wh ,  If   K i =1  SNR i B 2 i  <  SNR B 2 , CRLB disp  <  CRLB wh ,  If   K i =1  SNR i B 2 i  >  SNR B 2 , CRLB disp  =  CRLB wh ,  If   K i =1  SNR i B 2 i  =  SNR B 2 . (13)From the third condition in (13), it can be observed thatthe same CRLB can be achieved theoretically by selecting theappropriate SNR levels and absolute bandwidths  B  for bothtechnique. Some representative applications of (13) are givenas follows. This set of equation is useful for the selection of the  K  ,  SNR i ,  B i  parameters in dispersed spectrum utilizationtechniques that can provide the same performance as wholespectrum utilization techniques with  SNR  and  B  parameters.It also can be useful to quantify the equivalent of   K   numberof dispersed bandwidth  B i  as a whole bandwidth  B  forgiven  SNR i  and  SNR , respectively. Finally, the conditionsin (13) is useful for the optimization mechanism in rangeaccuracy adaptation [5], [6], [13] which is a feature of theCPSs. During the optimization of spectrum parameters, rangeaccuracy adaptation algorithm can select the optimal spectrumparameters (e.g.  K  ,  B ,  SNR ) using the conditions in (13).IV. R ESULTS In this section, performance of both exact and approximateCRLBs are compared considering whole spectrum utilizationsystems. This is followed by performance comparison of both whole and dispersed spectrum utilization approachesconsidering the approximate CRLB and ML method.The following simulation parameters are used for comparingthe exact and approximate CRLBs. Note that these bounds arealso compared to the performance of ML. For the pulse shape,the following Gaussian second order derivative pulse shape isemployed  p ( t ) =  A  1 −  4 πt 2 ζ  2  e − 2 πt 2 /ζ  2 ,  (14)where  A  and  ζ   are parameters that are used to adjust thepulse energy and the pulse width, respectively.  A  is selectedin order to generate pulses with unit energy. For the givenpulse shape, pulse width is defined as  T   p  = 2 . 5 ζ   [6], where ζ   = 1 /B  and  B  = 1 MHz. Moreover, the symbol durationof   T  sym  = 10 ζ   is employed. Uniformly distributed delay T  a  = 0 . 8 ζ   is considered, i.e.  U  [0 ,T  a ] . The number of training PROCEEDINGS OF THE 4th INTERNATIONAL CONFERENCE ON CROWNCOM 2009  symbol  N   that is considered is  1  and  d l  = [1] . The results areobtained over  3000  channel realizations and plotted in Fig. 4.According to the results, the exact CRLB performs better thanthe approximate CRLB by approximately  3 . 7 dB. This is dueto the flat spectrum assumption made during the derivation of approximate CRLB.ML method can be implemented by cross-correlating thereceived signal with template signal and computing the min-imum square error (MSE) of time delay estimates. The timedelay estimates are subject to ambiguity errors caused by theoscillatory nature of the signal correlation function [15]. Theresults in Fig. 4 as well as the results in the literature [15]indicates that MSE is a function of SNR exhibiting thresholdphenomenon. The SNR threshold for the results in Fig. 4is  16 dB and it divides the SNR region into two distinctregimes [15], which are low and medium SNR regions.Note that the SNR point where the transition from lowto medium SNR regimes occurs is referred as SNR thresh-old. MSE demonstrates a nonlinear behavior in the lowSNR regime, whereas this behavior becomes linear after theSNR threshold. This behavior is formulated by Barankinbound [15]. An alternative definition of SNR threshold isdeveloped from the results in Fig. 4, which is the SNRpoint where the noise effects become negligible. As a result,bandwidth effects also begin to contribute to the overall MSEperformance. In other words, SNR is the dominant parameterin the low SNR region and both SNR and  B  contributesto the overall MSE performance after the SNR threshold.This definition is also useful to explain the performancegap between ML and CRLB in the low SNR (or nonlinear)region. This is because of CRLB takes both SNR and  B  intoaccount during time delay estimate over the entire SNR region.Therefore, CRLB performs better than ML in the low SNRregion and ML converges to the approximate CRLB afterthe SNR threshold. Moreover, ML performs better than theapproximate CRLB by  0 . 5 dB according to the results in Fig. 4.Performance of whole and dispersed spectrum utilizationsystems are compared considering approximate CRLB andML. Performance comparison of both approaches are con-ducted for three cases. Note that the signals from all branchesare combined in dipersed spectrum utilization method. Thenumber of available dispersed bandwidth for all three casesis considered to be  2 , i.e.  K   = 2 . In  case 1 ,  SNR 1  + SNR 2  =  SNR ,  B i  = 1 MHz and  B  = 2 MHz. For  case2 ,  SNR i  =  SNR ,  B i  =  B  = 2 MHz. Finally, the parametersfor  case 3  are given as follows;  SNR 1  +  SNR 2  =  SNR , B i  =  B  = 2 MHz. The remaining system parameters forall three cases are common and they are given as follows.Training data  d i,l  =  d l  = 1  is considered, where the numberof observation symbols for both whole and dispersed spectrumis  1 , i.e.  N  i  =  N   = 1 . In addition, it is assumed that thespectral density of the noise is the same for the whole spectrumand all the  K   branches of the dispersed spectrum techniques;i.e.,  σ i  =  σ  for  i  = 1 ,...,K  . The Gaussian second orderderivative pulse shape is used. Therefore, the pulse shape forthe whole and dispersed spectrum methods are generated using 0 5 10 15 20 25 3010 0 10 1 10 2 10 3 SNR [dB]    R   M   S   E   [  m   ] MLExact CRLB Approx. CRLB Fig. 4. Comparison of exact and approximate CRLB for whole spectrumutilization systems. T   p  = 2 . 5 ζ   where  ζ   = 1 /B , and  T   pi  = 2 . 5 ζ  i  where  ζ  i  = 1 /B i ,respectively. Furthermore, the root mean square error (RMSE)metric is considered to measure the performance of ML timedelay estimate. Note that all the results in this section areobtained over  10000  channel realizations. For the theoreticalcase, (7) is denoted as CRLB-Whole and (11) is denoted asCRLB-Dispersed. Similarly, ML time delay estimator for thewhole (5) and dispersed spectrum (9) cases are denoted asML-Whole and ML-Dispersed, respectively. The results for case 1  are plotted in Fig. 5. CRLB-Whole performs betterthan CRLB-Dispersed by  6 dB for the entire SNR region.Since the total SNR for dispersed case is equal to  SNR , 0 5 10 15 20 25 3010 −1 10 0 10 1 10 2 10 3 10 4 SNR [dB]    R   M   S   E   [  m   ] CRLB−DispersedCRLB−WholeML−DispersedML−Whole Fig. 5. ML and √  CRLB versus SNR for dispersed and whole spectrumutilization. PROCEEDINGS OF THE 4th INTERNATIONAL CONFERENCE ON CROWNCOM 2009

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