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tx094超宽带（UWB）系统的抗多径性能分析,机械毕业设计

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ON DESIGNING THE OPTIMAL TEMPLATEWAVEFORM FOR UWB IMPULSE RADIO IN THEPRESENCE OF MULTIPATHAli Taha , Keith M. ChuggCommunication Sciences InstituteUniversity of SouthernCaliforniaLos Angeles, CA 90089-2565taha , chuggABSTRACTUsing an appropriate template waveform matched tothe received signalallows extracting the energy of thereceived signal eciently. This eciency becomes vi-tal for Ultra Wide Bandwidth (UWB) Impulse Ra-dio in the presence of multipath, where each pathundergoes a dierentchannel causing distortion inthe received pulse shape due to a varietyoffactorssuch as dierentamounts of attenuation for dier-ent frequencies 1, 2. In such a situation, using aclean ideal line of sight path signal as a template maydegrade the performance due to the mismatches be-tween the template waveformand the received signal.Furthermore, because of inherent ltering in the RFprocessing (i.e., antennas, ampliers, etc.), it is oftendicult to determine even such a clean line of sightpulse. In this paper, algorithms for designing opti-mal template waveforms for UWB Impulse Radio aredeveloped and the improvementover a more tradi-tional template waveform used for this kind of radiois illustrated.1. INTRODUCTIONDue to sending a sub-nanosecond pulse in each frameperiod, impulse radio enjoysavery high multipathresolution capabilityandavery low duty cycle sig-nal with huge spread spectrum processing gain 3,4, 5. On the other hand ultra-wide bandwidthsuggests that the higher frequencies attenuate morethan the lower frequencies 1, 2, causing distortionin the shape of the received pulse. The delay spreadof the impulse radio received signal is many manypulse durations even for indoor applications. Thesephenomena motivateus to design an algorithmwhichderives an optimal template waveformat the receiverthat captures the most amount of energy with theleast number of correlations. Since the eects of thechannel are somehowembedded in the received sig-nal, we can compute the optimal template waveformbased on the received signal online. This makes ouralgorithm adaptive, since with changes in the chan-nel, the received signal changes, and so does our opti-mal template based on the received signal. Weshowthe improvementachieved by this optimal templatewaveform compared to more traditional second orderderivative of Gaussian waveform 6, by applying ouriterative algorithm to real data obtained from mea-surement experiments taken in the Wireless RadioLab of the University of Southern California. Usingour template waveform algorithm helps us adapt ourtemplate to dierentenvironments based on the re-ceived signal whichembodies all the channel charac-teristics, including those of the antennas on the wave-forms which are sometimes not well-understood. Thealgorithm developed in this section is not limited toUWB systems only, and can be applied to any kindof communication system.In Section II, optimal long-tailed template wave-form design is presented, which then leads us to de-sign optimal single path template waveform in Sec-tion III. Conclusion remarks are made in section IV.2. LONG-TAILED TEMPLATEUsing the digital sampling oscilloscope, 9 measure-ments have been taken in the Wireless Radio Lab ofthe University of Southern California. These mea-surements are shown in Fig. 1. These are the re-ceived signals from a pulser that generates monocy-cles. It is worth mentioning that each measurementis the average of 256 received proles at the samelocation to get a more stable measurement and ne-glect some transient eects. We sample each mea-surement at a rate greater than Nyquist rate andnormalize them to have unit energy. After these pro-cedures, we representeach measurementbyavector,nts0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.4Time (nanoseconds)Amplitude (Volts)Figure 1: Measurements from the UWB radio re-ceived waveform taken at the Wireless Radio Lab atthe University of Southern Cly, ri=ri1ri2:rint,fori =1;2;:;9: Nowwend the vector w =w1w2:wntfor which the func-tion,F =NXi=1j j2(1)is maximum. In this case, we nd the nearest vectorto all the measurementvectors in the sense that itcaptures the most energy out of the measurements ifwe just want to do a single correlation at the receiver1.We set w to be of unit energy for normalizationpurposes. This is a constrained optimizationproblem(e.g., see appendix C in 7) with kwk =1: Solvingthis optimization problem, weget(A + I)w =0where A = M Mtand M is a matrix whose ithcolumn is ri. Therefore, w is simply the normal-ized eigen vector of matrix A corresponding to itslargest eigen value since F = wtAw = wt(;w)=;kwk2= ;. This normalized eigen vector is sim-ply the optimal template waveform when wewanttodo only one correlation against the received signal atthe receiver. This problem can be generalized in astraightforward manner to the case when wewanttodesign two or more orthogonal template waveformsthat capture the energy of the received signal opti-mally. The solution is the eigen vectors correspond-ing to the largest eigen values of matrix A. Sincethe matrixA is symmetric, these template waveformscan be selected orthogonal. Fig. 2 shows 9 orthonor-mal template waveforms corresponding to the ninenonzero eigen values of matrix A. The rst template1This is equivalent to a LS criterion when kwk is con-strained to be constant.0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.2Time (nanoseconds)Amplitude (Volts)0 10 200.2Time (nanoseconds)Amplitude (Volts)0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.4Time (nanoseconds)Amplitude (Volts)0 10 200.2Time (nanoseconds)Amplitude (Volts)0 10 200.500.51Time (nanoseconds)Amplitude (Volts)Figure 2: Orthonormal templates as the output ofthe algorithm.waveform at the upper left corner of the gure cap-tures 58:39% of the total energy contained in all themeasurements by just one correlation with each mea-surement. The second template in the upper middleof the gure, captures 23% of the total energy out ofall the measurements with just one correlation witheach measurement. The rest of the templates cap-ture 8:38%;3:61%;2:21%;1:71%;1:20%;0:83%;0:67%respectively.For nine measurements, we should beable to capture the whole energy with at most ninesingle long-tailed orthonormal template waveforms.By looking at matrix A,we see that each elementsomehow computes the average of the correlationbetween two specied components of each measure-ment, over all the measurements.3. SINGLE PATH TEMPLATEAs Fig. 1 shows, there are a lot of paths in the re-ceived signal. Wewant to resolve the paths and ndthe optimal template waveform based on these in-dividual paths. In that case we desire a short-tailedtemplate waveform(i.e., with support much less thanthe delay spread) and wemay use it to do a selectivemultiple combining 8 for the most dominant paths.Dene the new objective function F asF = jr;LXj=1cjw(n ;nj)j2(2)where risatypical received waveform,njis the delayassociated to its jth path and cjis the correspondingamplitude. As the above formulasuggests, we assumeonly the rst L dominant paths in our model. In or-der to minimize F with respect to w, cjs, and njs,ntswe rst minimize F conditioned on a given waveformw as an initial estimation. A good initial estima-tion can be the truncated version of the long-tailedtemplate waveform obtained in the last section. Af-ter nding the optimized values of cjs and njs forj =1;2;:;L based on this initial waveform, weusethese values of the coecients cjs and delays njs tond the optimized waveform w of length m. Nowwe use this new optimized waveform w to computethe new values of the coecients and delays and werepeat this procedure again and again until conver-gence occurs for the waveform w. The length of w,m, is a design parameter. If we assign a very smalllength for the template waveform, then it will not beeective, since it requires more correlations againstthe received signal to capture the same amountofenergy. Therefore, we can choose an initial value form, and then obtain the template waveform and com-pute the number of correlations to capture a speciedamount of energy out of the received signal. Then weincrease m, and repeat the same procedure again. Ifthe reduction in the number of correlations to cap-ture energy is signicant, we increase m again up tothe point where the reduction is not worth choosing alonger template waveform or we get negligible valuesfor the template waveform after some point. For thecase when we know the width of one received pulse,we can simply choose m such that it meets the widthof the pulse. So F = j(r ;PLj=1cjwI(n ; nj)j2where subscript I means the initial estimation forw. Assume the received signal before sampling asr(t)=s(t)+n(t) where n(t) is the additive whiteGaussian noise with power spectral level ofN02: Thereceived signal r(t) consists of several paths at spe-cic delays ni= in;i =1;2;:;L, and amplitudescis, for i =1;2;:;L: Assuming selectivecombiningfor the rst L dominant paths, we ignore the rest ofthe paths:r(t)=LXi=1ciw(t ;ni)+n(t) (3)Finding the maximum likelihood (ML) estima-tor is equivalent to nding the Minimum MeanSquared Estimates (MMSE) of cis and nis, be-cause n(t)isAWGN. Dening c =c1c2:cLtandn =n1n2:nLtand ignoring the irrelevanttermofthe aboveintegral in calculating the MMSE of c andn we get the following estimations 6:n = argmax(X+(n)R;1X(n) (4)andc = R;1X(n) (5)whereX(n)=ZT0r(t)0BBBw(t; n1)w(t; n2).w(t; nL)1CCCAdt (6)and the correlation matrix R isR =0BBBR(n1; n1) R(n1;n2) R(n1; nL)R(n2; n1) R(n2;n2) R(n2; nL).R(nL; n1) R(nL;n2) R(nL; nL)1CCCA(7)where R(ni;nj)=RT0w(t; ni)w(t ;nj)dt.Using the values obtained for delays and ampli-tudes in (4) and (5) to compute the optimal w,werepeat the whole procedure with our new w insteadof wIuntil the optimal template waveform convergesto its nal format. In order to compute the new w ,weneedtominimize F = jr ;PLj=1cjw(n ; nj)j2.In this equation, r is an n 1vector, and w is anm 1vector, and in order to write the above equa-tion correctly,we need to add zeros to each w(n; nj)suchthatitbecomesavector of order n 1too,i.e., w(n ; nj) = 00:0w1w2w3:wm000:0twherewehave added njzeros at the beginning of thevector, and then wehave the unknown coecients,wjs, which are to be determined, and nally weaddn ; m ; njzeros to complete the dimension as ann1vector.In order to minimize F with respect to wjs forj =1;2;:;m,we need to take the derivatives of Fwith respect to each wjand equate them to zero.Fwp=2LXk=1c2kwp+2LXk=1Xlm,or j0. The above linear system of equations canbe solved for the given values of cis and njs usingany standard algorithm for solving a linear systemof equations available in anynumerical computationbook.ntsThe last step is to normalize the template wave-form obtained from all the above steps in order tomake it of unit energy as the constraint of our op-timization, so that we can compare its performancewith any other unit energy template in terms of theamount of the captured energy after correlation.wopt=wkwk(9)where woptis the optimal template waveform as theoutput of our algorithm.In order to determine whether the algorithm hasconverged to woptor not, we can use the followingcriterion: If kw(k+1)opt;w(k)optk for some positive ,then stop running the algorithm, otherwise continuefrom step two. The output of the algorithm after thekth iteration has been denoted by w(k)opt. Here, depends on the accuracy needed. The smaller the ,the better the approximation. Fig. 3 demonstratesthe output of the algorithm, wopt, along with thesecond-order derivative of Gaussian waveform. Fig.4 shows the owchart of the algorithm.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.30.20.6Time (ns)Amplitude (V)Dash: SecondOrder Derivative of Gaussian WaveformSolid: Optimal Template WaveformFigure 3: Optimal template along with the second-order derivativeofGaussianwaveformAs (4) suggests, there is a nonlinear complexity as-sociated to the exhaustive search for nding the op-timal values of the dominant pathss arrival times.However, we can simply use a suboptimal linearsearch when we assume a negligible overlap betweenadjacent paths. This can be explained by looking at(7). In this case, we can see that matrix R becomesstrongly diagonal, so does its inverse in (4). There-fore, (4) suggests that we need to search for thosevalues of n where kXk2becomes maximum. Sincewe can search for each dominant path independentlyin this case, this simply means to nd those values ofInitialEstimationfor w(t)Non-LinearExhaustiveSearch for nLinear Algorithmfor Finding cLinearAlgorithm forw(t)Figure 4: Flowchart of the algorithmn for which the magnitude of each componentofX ismaximum. This is a linear complex searchintermsofthe number ofcomponents ofX.Itisworth mention-ing that this suboptimal algorithm becomes optimalfor the case when there is no overlap between theadjacent paths at all. Because of the excellentmulti-path resolution capability of impulse radio due to itsultra wide bandwidth, we can employthe suboptimalalgorithmwith some condence. Since the results ob-tained by the suboptimal algorithm match those ofoptimal algorithm with a high precision, the resultspresented here reect those obtained by running thefast linear suboptimal algorithm. Running the sub-optimal algorithm on a various generated data usingcomputer simulation has shown that the algorithmresolves the paths successfully under dierentmul-tipath scenarios where two dierent paths can evenoverlap with each other.For anygiven w(t)ateach iteration of the algo-rithm, cis and nis are the maximumlikelihood esti-mates of the amplitudes and delays of dierent paths.Specically, nis are obtained through an exhaustivesearch to minimize the magnitude of the dierencebetweenPLi=1ciw(t;ni) and the received waveformr(t). Also at the sametime,due to the AWGN natureof the problem, cis are mean squared estimationsntswhich minimize the mean squared error. By theseexplanations, we see that after estimating the ampli-tudes and delays of dierent paths, the mean-squarederror becomes smaller during any iteration. For thesecond part of each iteration, given the estimates ofamplitudes and delays, we compute the shape of thetemplate waveform by taking derivatives to minimizethe mean-squared error; therefore, we get a smallermean-squared error after the second half of each iter-ation. Since the sequence of mean-squared errors is adecreasing sequence bounded from belowby zero, weconclude that this sequence is convergent.Running the algorithm when L = 3 for the mea-surementshown in the upper left of Fig. 1 and withthe initial estimationas the input to the algorithmtobe the second-order derivative of Gaussian waveform,demonstrates about 0.93 dB improvementin terms ofthe captured energy out of the three most dominantpaths with respect to that of second order derivativeof Gaussian. Similar results are obtained by runningthe algorithmon the rest of the measurements. Also,the algorithm converges very fast, and in fact afterthe second iteration, there is no more improvement.This veries the optimalityof our template waveformshown in Fig. 3 over the second order derivativeofGaussian.To demonstrate the robustness of the algorithm,we consider a very bad initial estimation of the tem-plate waveform, which is just a unit energy rectangu-lar pulse (Flat Template Waveform)over the interval0 t 2 nanoseconds. The captured energy us-ing this template waveform is only 7 percentofthetotal energy. Fig. 3 shows the template waveformafter only two iterations of the algorithm. It cap-tures 97 percent of the energy and it is identical tothe template waveform obtained using the second-order derivative of Gaussian as the initial estimation.The 0.9 dB improvement in extracting the energyout of the received signal using the optimal templatewaveform compared to the second order derivativeofGaussian, can even further mitigate the already lowfading margin in UWB impulse radio.4. CONCLUSIONTwo algorithms to design optimal template wave-forms were introduced: One for optimal one-timecorrelation long-tailed templates and the other formulti-correlation short-tailed templates. Weshowedhowwe can design optimal templates in the sense ofcapturing the most amount of energy with the leastnumber of correlations against the received signal inthe presence of multipath. Simulation results veri-es the robustness and accuracy of our iterativeal-gorithm. Applying our algorithm to real data ob-tained from measurements, we notice the capabilityof the algorithm in resolving the dominant paths ofthe multipathalongwith estimatingthe optimaltem-plate waveform