Parametrization of the Local Scattering Function Estimator for Vehicular-to-Vehicular Channels

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1 Parametrization o the Local Scattering Function Estimator or Vehicular-to-Vehicular Channels Laura Bernadó, Thomas Zemen, Alexander Paier, ohan Karedal and Bernard H Fleury Forschungszentrum Teleommuniation Wien (tw), Vienna, Austria nstitut ür Nachrichtentechni und Hochrequenztechni, Technische Universität Wien, Vienna, Austria Department o Electrical and normation Technology, Lund University, Sweden Department o Electronics Systems, Aalborg University, Denmar contact: bernado@twat Abstract Non wide-sense stationary (WSS) uncorrelatedscatterering (US) ading processes are observed in vehicular communications To estimate such a process under additive white Gaussian noise we use the local scattering unction (LSF) n this paper we present an optimal parametrization o the multitaperbased LSF estimator We do this by quantizing the mean square error (MSE) For that purpose we use the structure o a twodimensional Wiener ilter and optimize the parameters o the estimator to obtain the minimum MSE (MMSE) We split the observed ading process in WSS regions and analyze the inluence o the estimator parameters and the length o the stationarity regions on the MMSE The analysis is perormed considering three dierent scenarios representing dierent scattering properties We show that there is an optimal combination o estimator parameters and length o stationarity region which provides a minimum MMSE NTRODUCTON n vehicular communications the ading process does not ulill the wide-sense stationary (WSS) uncorrelated-scattering (US) property Methods have been proposed to deal with non- WSSUS processes by repeating the experiment and perorming the second-order statistics calculation in the sample domain instead o the temporal domain [] n reality it is very diicult to obtain dierent realizations o the same experiment, while eeping the same experimental conditions Thereore we use the idea o splitting an observed process in small regions, in which we can assume that the WSSUS property holds [] n [] the length o these stationarity regions is assessed by means o the collinearity measure without optimizing the estimator parameters n [] an analysis o the estimator parameters is done or pedestrian mobility n this paper we compute the optimal parametrization o the LSF estimator or vehicular scenarios We minimize the mean square error (MSE) taing into account the length o the WSSUS region Contributions o the Paper The objective o the paper is to characterize the properties o the local scattering unction (LSF) estimator introduced in [] The estimator which is used to calculate the scattering unction This research was supported by the EC under the FP Networ o Excellence projects NEWCOM, as well as the project COCOMNT unded by Vienna Science and Technology Fund (WWTF) The Telecommunications Research Center Vienna (tw) is supported by the Austrian Government and the City o Vienna within the competence center program COMET o non-stationary processes n order to assess the perormance o the estimator, we deine the MSE based on the structure o a two-dimensional time-requency Wiener ilter [] We show that the Wiener ilter coeicients can be calculated rom the LSF t is noteworthy to mention that we do not use the Wiener ilter or iltering a signal, but or characterizing the perormance o an estimator Furthermore, we assess the length o the WSSUS region by means o the minimum MSE (MMSE) Notation Time t, requency, delay τ and Doppler shit ν are discrete variables Organization o the Paper n Section we introduce the two-dimensional (D) Wiener Filter concept to calculate the MSE o the estimator n Section the relation between the LSF and the ilter coeicients is deined The calculation o the ilter coeicients and their optimization, ie the choice o the estimator parameters which minimizes the MSE, are analyzed in Sections V and V respectively Conclusions are presented in Section V D WENER FLTER We want to optimize the parameters o the LSF estimator n order to do that, we use a D time-requency Wiener ilter described in Fig [] We consider a signal R(t, ) to be iltered which consists o the true channel transer unction H(t, ) corrupted with additive noise N(t, ) The optimum LSF parameter set minimizes the MSE between the output o the ilter Ĥ(t, ) and H(t, ) R(t, ) = H(t, )N(t, ) Fig D WENER FLTER H (t, ) D Wiener ilter scheme H(t, ) The estimated transer unction at the ilter output reads Ĥ(t, ) = M i= N j= - e(t, ) a i,j R(t i, j) ()

2 where a i,j are the ilter coeicients and M and N denote the length o the sequence R(t, ) in time and requency respectively The MSE is given by where E = E{ e(t, ) } e(t, ) =H(t, ) Ĥ(t, ) and E{} denotes expectation n order to ind the coeicients o this D ilter, we use the orthogonality principle E{e(t, )Ĥ (t, )} = to obtain the Wiener-Hop equation E{H(t, )R (t, )} = a(t,, t, ) t, E{R(t, )R (t, )} () We assume that the noise N(t, ) is a zero-mean process statistically independent o the channel requency response H(t, ) Under this assumption we can rewrite () as E{H(t, )H (t, )} = t, a(t,, t, ) E{R(t, )R (t, )} () From () we see that E{H(t, )H (t, )} is the timerequency (TF) auto-correlation unction (ACF) o H(t, ) Similarly, E{R(t, )R (t, )} is the TF-ACF o R(t, ) For short we rename these two TF-ACFs as R H and R R n order to deine the Wiener ilter, we must assume to have stationary processes, thereore the TF-ACFs will depend only on the time and requency lags and can be expressed as R H ( t, ) and R R ( t, ) For the sae o simplicity, we use the compact notation or (): R H = AR R From this identity we obtain the coeicients o the D Wiener ilter: A T = R T HR R () with R R = R R ( t, ) R R ( t, N ) R H R ( t, N ) R R( t, ), where R R (, ) R R (M, ) R R ( t, ) = RR (M, ) R R(, ) n the above expressions, () H denotes Hermitian transposition and () complex conjugation The matrix R H ollows the same structure, whereas A stems rom rewriting () in matrix notation, ie ĥ = A T r, where the vectors ĥ and r are the channel transer unction and its noisy version respectively: ĥ() Ĥ(, ) ĥ = with ĥ() = ĥ(m ) Ĥ(,N ) and r = r() r(m ) with r() = R(, ) R(,N ) FLTER MPLEMENTATON The TF ilter in () can be written as the D TF-convolution Ĥ(t, ) =a(t, ) R(t, ) () with a(t, ) denoting the ilter coeicients deined in () Perorming the Fourier transorm in both dimensions, we can express () as F t F {Ĥ(t, ) } = F t F {a(t, ) R(t, )} () SĤ(ν, τ) = A(ν, τ)s R (ν, τ), where SĤ(ν, τ) and S R (ν, τ) are the Doppler-delay spreading unctions o Ĥ(t, ) and R(t, ) n () F and F denote the Fourier transorm and its inverse respectively The ilter coeicients A(ν, τ) in () are given by A(ν, τ) =F t F { RH ( t, ) R R ( t, ) } = S H(ν, τ) S R (ν, τ) = C H(ν, τ) C R (ν, τ) with C H (ν, τ) and C R (ν, τ) denoting the Doppler-delay scattering unctions o H(t, ) and R(t, ) respectively n order to obtain Ĥ(t, ) we only need to reverse the sequence o Fourier transorms perormed in () at the output o the ilter V CALCULATON OF THE FLTER COEFFCENTS The ilter coeicients determined in () depend on the scattering unctions C R (ν, τ) and C H (ν, τ) We calculate them using a multitaper estimator in which the windows are chosen to be the discrete prolate spheroidal sequences (DPSS) [] Since we are dealing with non-wss processes, the observed process at the input o the ilter is divided in time regions over which we can assume that the WSS property holds This allows us to calculate the second-order moments o the process and, consequently, the ilter coeicients The estimate o C H (ν, τ) reads Ĉ H(ν, τ) = () L H (Gl) (ν, τ) () l= The ranges o the variables τ and ν in () are {,, N } and { M/, M/ } respectively Moreover, and

3 are the number o tapers in time and requency respectively and H(t## t, ## ) t = M/ = N/ τ ) C R (ν,!) R (t,) LSF ESTMATOR H (t,) Calculation Coeicients C H (ν,!) Fig (b) S-b (Highway- m/h) (c) S-a (Rural- m/h) (d) S-b (Rural- m/h) e(t,) A (ν,!) WSS region (e) S-a (Urban- m/h) () S-b (Urban- m/h) Fig Normalized squared envelope o the channel response measured in the three investigated scenarios (a) S-a (Highway- m/h) H(t,) C H (ν, τ ) C R (ν, τ ) H (t,) D WENER FLTER () is the tapered version o H(t, ) and Gl (t##, ## )e jπ(νt τ ) are the DPSS time-requency tapers n () t { M } or each region Since everything taes place locally, the estimate o the scattering unction in () is named local scattering unction [], [] The ilter coeicients deined in () will be also local They are ed bac to the ilter ater the estimation procedure (Fig ) n order to show explicitly this locality, we introduce the superscript in the notation Note that the ilter coeicients are calculated rom estimated unctions, thereore we will use estimates o the ilter coeicients We use the sampled version o the LSF deined in [] The WSS region is inite with dimensions M in time and N in requency The local ilter coeicients are deined as R(t,) = H(t,)N(t,) Gl (t##, ## )e jπ(νt A (ν, τ ) =!! H (Gl ) (ν, τ ) = N/ M/ Calculation o the ilter coeicients or each WSS region Even though this solution is considered optimal or the Wiener-iltering problem, our estimation o the LSF orces us to perorm another optimization step n the estimator () there is an optimal number o windows in time and requency that reduces the overall MSE The number o windows is determined by the time-bandwidth product rom () Besides that, we can also optimize the length o the stationarity regions M V O PTMZATON OF THE PARAMETERS OF THE ESTMATOR Recent measurements have reported non-stationary channel responses or vehicle-to-vehicle propagation [], and we thereore mae use o such measurements to demonstrate a solution or the above optimization problem Further details regarding the measurements are ound in [] As observed in [], communications between vehicle with opposite driving directions experience the largest temporal variations We consider three such scenarios (transmitter and receiver move at the same speed): Scenario (S): Highway environment There is a strong line o sight (LOS) component with diuse components and some relecting paths Considered speeds: (a) m/h and (b) m/h Scenario (S): Rural environment There is a strong LOS component with diuse components but no additional paths Considered speeds: (a) m/h and (b) m/h Scenario (S): Urban environment There is a strong LOS component with large diuse contributions and additional paths created by relections coming rom near objects Considered speeds: (a) m/h and (b) m/h n scenario (highway) the delay spread is short and some contributions arrive at late delays n scenario (rural) the delay spread is longer with no urther contributions n scenario (urban) the largest delay spread is observed with lots o paths coming rom close relecting objects Figure shows the squared magnitude o the time-varying impulse responses measured in the three scenarios Since the measured channel transer unctions have a signal-to-noise ratio (SNR) o roughly, we consider these as the true

4 transer unctions H(t, ) Noise-corrupted transer unctions R(t, ) are created by adding random noise to H(t, ) such that an SNR o is achieved We use the sampled channel transer unction H(t, ) = H(t T rep, B/Q), where T rep = µs is the repetition time o the channel sounder, B = MHz is the measurement bandwidth and Q = is the number o requency bins The variables t and are continuous We consider a measurement run o s with { Q } and t { S }, where S = samples A Number o Time-Frequency Tapers (Choice o (, )): n a irst step we select a given stationarity length M and optimize the parameters and For this irst analysis we set M = The choice o the time-bandwidth product determines the number o tapers used in the estimation process []: <N t W t in the time domain and <N W in the requency domain, where N t W t = i/ t and N W = j/ τ (N t W t = i and N W = j when using normalized t and τ) Our optimization problem is thus to solve ( ) E argmin, K The term to be optimized is the mean o the MSE per region, computed over a total number o K WSS regions The MSE per region is deined as E = MN M t= = N H (t, ) Ĥ (t, ) with < i, < j, and M, N denoting the extent o the WSS region We evaluate the MSE or all possible combinations o and rom to The resulting MSEs or all scenarios are plotted in Fig versus and using the gray-scale code reported on the right We highlight the minimum MSE with a red square at the pair (, ) which achieves it n each red square the value o the minimum scaled by is written As we can see in Fig, while increasing the number o tapers in both time and requency domains the MSE decreases rapidly and beyond a certain point starts to grow slowly This is due to the act that, by increasing the number o tapers, on one hand we reduce the variance, but on the other hand we increase the bias There is a trade-o between these two eects which corresponds to the best combination o the number o tapers in time and requency to minimize the MSE B Length o the Stationarity Region: The stationarity region is deined by two parameters: M (or the time domain) and N (or the requency domain) M has to be selected in order to ensure that the WSS property and N the US property When choosing M properly, we can also ensure that contributions at dierent delays are not correlated (they come rom dierent scatterers) We deine dierent regions with lengths M =,, and samples and calculate the MSE The length o this region is short enough to ensure (a) S-a (Highway- m/h) (c) S-a (Rural- m/h) (e) S-a (Urban- m/h) Fig x x x (b) S-b (Highway- m/h) (d) S-b (Rural- m/h) () S-b (Urban- m/h) MSE versus (, ) or the three dierent scenarios x x the US property to hold within one WSS region Although the optimization o the number o tapers was done previously in Sec V-A, here we analyze the dependency o the length o the stationarity region with the optimal (, ) pair Table reports the results plotted in Fig TABLE MMSE FOR THE OPTMAL (, ) COMBNATON FOR DFFERENT SCENAROS AND DFFERENT WSS REGON LENGTHS THE VALUES OF MMSE ARE SCALED BY S S S WSS region length M a (,)- (,)- (,)- (,)- b (,)- (,)- (,)- (,)- a (,)- (,)- (,)- (,)- b (,)- (,)- (,)- (,)- a (,)- (,)- (,)- (,)- b (,)- (,)- (,)- (,)- The MMSE has the same variation depending on M regardless o the scenario n Sec V-A we showed that there is an optimal parametrization which results in the minimum MSE (Fig ) n that case, the MSE gets smaller and then larger again while increasing the number o considered tapers in our estimator We observe a similar eect with the length o the stationarity region: when increasing the length o this region, the MMSE gets smaller and then grows again Figure depicts this result and shows that the increase is aster or x

5 higher speeds (S) Furthermore, when we increase the length o the WSS region, we observe a longer period o time and consequently more variations o the process Thus, we need to consider more windows in our multitaper estimator (Tab ) MMSE ( ) [linear scale] Fig Scenario Scenario Scenario a version b version M MMSE versus the WSS-region length or all scenarios n Fig, or a given scenario (we compare curves with the same color/marer), the achieved MMSE is smaller or the case (b), ie when the cars drive slower This shows the nonstationarity o the process and its relation to the MSE The less time-varying the channel is (the slower the cars drive), the smaller the achieved MMSE is We have to highlight the dierence between scenarios, even though all o them consider a strong LOS component Due to the speed o the cars it is clear that there are more time luctuations in scenario and results in the largest MMSE Even though the cars in scenario drive slower than in scenario it has a larger MMSE This is due to the large delay spread and numerous path contributions in scenario where we observe a more time-varying process So ar the presented results have been calculated with a SNR o n Fig we compare or scenario -a the MMSE or dierent SNRs o, and As expected, the pair (, ) achieving the MMSE depends on the length o the stationarity region This combination is written next to their corresponding MMSE value in Fig The MMSE decreases when SNR increases The shape o the curve is the same but shited upwards or downwards depending on the SNR The higher the SNR is, the lower the MMSE is V CONCLUSONS We assessed the perormance o the local scattering unction (LSF) by means o the mean square error (MSE) For that purpose we used a two-dimensional Wiener ilter We showed that there is a best parameters combination or the LSF which results in the minimum MSE (MMSE) and that the MSE decreases until it reaches a minimum and increases again when the number o tapers in the LSF increases The number o tapers achieving the MMSE increases when considering longer stationarity regions There is a length or which the MMSE is the minimum We chose the vehicular communications MMSE ( ) [linear scale] (,) (,) (,) (,) (,) (,) (,) (,) (,) SNR = SNR = SNR = (,) (,) (,) M Fig MMSE or S-a versus M or dierent SNRs environment to quantiy it and showed that, or a given scenario, the slower the cars drive, the smaller the MMSE is When comparing dierent scenarios, we showed that the MMSE depends on the non-stationarity o the environment The MMSE is smaller or more stationary scenarios n this sense, the highway scenario, due to the high speeds o the cars, is the most time-varying and the rural scenario is the least, given that it presents a short delay spread and no urther contributions Although the speeds o the cars in the urban scenario are the lowest, the scattering environment contributes to creating a long delay spread and thereore a more timevarying environment REFERENCES [] S Bendat and A G Piersol, Engineering Applications o Correlation and Spectral Analysis ohn Wiley & Sons, nc, [] G Matz, Doubly underspread non-wssus channels: Analysis and estimation o channel statistics, in Proc EEE nt Worshop on Signal Processing Advance Wireless Communications (SPAWC), Rome, taly, une, pp [] A Paier, T Zemen, L Bernadó, G Matz, Karedal, N Czin, C Dumard, F Tuvesson, A F Molisch, and C F Meclenbräuer, Non-WSSUS vehicular channel characterization in highway and urban scenarios at GHz using the local scattering unction, in Worshop on Smart Antennas (WSA), Darmstadt, Germany, February [] S Kaiser, Multi-carrier CDMA mobile radio systems-analysis and optimization o detection, decoding and channel estimation, PhD dissertation, German Aerospace Center (DLR) nstitute o Communications and Navigation, Germany, [] D Slepian, Prolate spheroidal wave unctions, Fourier analysis, and uncertainty - V: The discrete case, The Bell System Technical ournal, vol, no, pp, May-une [] G Matz, On non-wssus wireless ading channels, EEE Trans Wireless Commun, vol, no, pp, September [] L Bernadó, T Zemen, A Paier, G Matz, Karedal, N Czin, C Dumard, F Tuvesson, M Hagenauer, A F Molisch, and C F Meclenbräuer, Non-WSSUS vehicular channel characterization at GHz - spectral divergence and time-variant coherence parameters, in Proceedings o the XXXth URS General Assembly, Chicago, USA, August [] A Paier, Karedal, N Czin, H Hostetter, C Dumard, T Zemen, F Tuvesson, C F Meclenbräuer, and A F Molisch, First results rom car-to-car and car-to-inrastructure radio channel measurements at GHz, in nternational Symposium on Personal, ndoor and Mobile Radio Communications (PMRC ), September, pp [] D B Percival and A T Walden, Spectral Analysis or Physical Applications Cambridge University Press,