UMTS addresses future packet services over channels

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1 6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 7 Low-Cost Approximate Equalizer Based on Krylov Subspace Methods for HSDPA Charlotte Dumard, Florian Kaltenberger, and Klemens Freudenthaler Abstract The throughput of the High Speed Downlink Packet Access (HSDPA) sub-system of UMTS suffers significantly from multiple access interference in the wireless channel. A linear minimum mean square error () equalizer at the receiver achieves higher throughput than a conventional RAKE receiver, at the cost of higher complexity. We introduce an iterative algorithm based on Krylov subspace projections, approximating the equalizer with negligible loss of performance for the receiver. Slow variations of the channel can be exploited to allow further acceleration of the algorithm. Computational complexity as well as storage requirements are strongly reduced. Index Terms UMTS, HSDPA,, Krylov subspace methods, low complexity equalizer. Similar investigations were carried out in [4] for an iterative multi-user receiver. However, due to interference cancelation, the Krylov based algorithm had to be applied for every user independently. Thus only parallelization of the computations was made possible but no computational complexity reduction. Notation is specified in Section II and Krylov subspace methods are described in Section III. The system and signal models are presented in Section IV. Section V deals with the application of Krylov subspace methods to HSDPA. Simulations are presented and results on throughput performance discussed in Section VI. Conclusions are drawn in Section VII. I. INTRODUCTION UMTS addresses future packet services over channels with fluctuating quality via its HSDPA sub-system. A simulation environment specifically tailored to HSDPA for investigating HSDPA receiver requirements was developed in [], []. Multiple access interference as well as the interference from signaling channels like the Synchronization Channel (SCH) and the Common Pilot Channel (CPICH) significantly degrade the performance of an HSDPA system [] when a RAKE receiver is used. We showed recently that an equalizer for User Equipment (UE) capability classes 6 achieves the required throughput [], but at the cost of high complexity. In this paper we describe the implementation of Krylov subspace methods [3] to approximate the equalizer, trading accuracy for efficiency. We investigate the throughput of HSDPA employing such an equalizer by simulations. To this aim, we have carried out numerical experiments using the frequency selective channel models specified by the International Telecommunication Union (ITU). We obtained results for the Pedestrian A and B and Vehicular A channel models with 6QAM modulation. The Krylov equalizer allows computational complexity and storage requirements reduction with almost no loss of performance. Manuscript received August 4, 5; revised February 7, 6 and May 6, 6; accepted June 9, 6. The associate editor coordinating the review of this paper and approving it for publication was G. Vitetta. This work was partly funded by Kplus, Infineon Technologies and the ARC Seibersdorf research GmbH through the ftw. project C9 and by the Wiener Wissenschafts- Forschungs- und Technologiefonds (WWTF) through the ftw. project I Future Mobile Communications Systems. The authors would like to thank Christoph Mecklenbräuker, Joachim Wehinger, and Steffen Paul for their support. C. Dumard is with the Forschungszentrum Telekommunikation Wien (ftw.), Donau-City Str. /3, A- Vienna, Austria ( dumard@ftw.at). F. Kaltenberger is with ARC Seibersdorf research GmbH. K. Freudenthaler is with Johannes Kepler University of Linz. Digital Object Identifier.9/TWC /7$5. c 7 IEEE II. NOTATION We denote a column vector with elements a[i] by a and a matrix with elements [A] i,l by A. Their transpose and conjugate transpose are given by. T and. H respectively. The l and A norms of a are denoted through a and a A = ah Aa respectively. The Q Q identity matrix is denoted as I Q. The complex conjugate of b C is b. ( ; ) denotes an open interval. III. KRYLOV SUBSPACE METHODS We consider the linear system Ax = b,wherea is a known invertible matrix of size Q Q and b a known vector of length Q. Krylov subspace based algorithms [3] iteratively compute an approximation of x starting from an initial guess x and using projections on Krylov subspaces. We describe in this section the special case when A is hermitian. A. Krylov Subspace Projection Method The Cayley-Hamilton theorem states that there is a minimum polynomial R of degree R Q such that R(A) = and I Q, A,...,A R are linearly independent, which we can rewrite as A = R r= a ra r. The coefficients a,,a R C R are defined by R. Our aim is to approximate A as a linear combination of the first s terms, where s R s A a r A r. () r= We consider an initial guess x for x and define the initial error r = x x and b = b Ax = Ar. Thus x = x + A b and using () we write s x x + a r A r b = x s. () r= x s is the approximation of x at the step s and y s = x s x is the residual vector. We also define the error at step s by

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 7 6 r s = x x s = r y s. Note that if s = R the approximation () becomes an equality and r s = and y s = r. The residual vector y s is computed as element of the Krylov subspace of A and b } with dimension s defined by K s =span { b, A b,...,a s b. We constrain the error r s to be uncorrelated to K s. In other words, we project x onto K s such that Ar s K s. B. The Krylov Subspace Based Algorithm We write y s as element of K s as y s = V s z s, where z s C s and V s =[v,...,v s ] is an orthonormal basis of K s. V s is iteratively obtained by applying the Gram-Schmidt orthonormalization to the basis B s =[V s, A s b]. The condition Ar s K s becomes V H s Ar s = V H s b = V H s AV sz s. (3) Furthermore, the vectors v i for i {,...,s} are such that Av i K i+. Thus v H l Av i =if l>i+ and T s =V s H AV s is an upper Hessenberg matrix. A being symmetric, T s is consequently tridiagonal symmetric. We denote its elements on the main diagonal as α i R and on the secondary diagonals as β i (; + ). We also know that b = b v. Inserting these results into (3), we obtain z s = T s b e, where e = [,,...,] T has length s. We see that z s is proportional to c (s) first, first column of T s. To compute c(s) first, we apply the matrix inversion lemma for partitioned matrices [5] to the iterative relation T s β s ẽ s T s = β s ẽ T, s α s where ẽ s =[,...,, ] T has length s. Thisgivesthe following set of iterative equations c (s) (s ) c first = first + γs c (s ) [] βs c(s ) [ β s c (s) = βs c (s ) ] (4) γ s, where c (s) is the column of T s, and γ s = α s βs c (s ) [s ] is a scalar. Finally, we obtain our approximation at step s with x x s = x + b V s c (s) first.thefinal step S is referred as the number of iterations in the algorithm or as the dimension of the Krylov subspace where we project x. The corresponding algorithm is summarized in Table I. IV. HSDPA SYSTEM DESCRIPTION HSDPA introduces three new physical channels into UMTS Release 5. The High Speed Physical Downlink Shared Channel (HS-PDSCH) is the data channel shared by all HSDPA users of a single cell in time and code domain. It consists of up to 5 subchannels corresponding to 5 Walsh-Hadamard channelization codes with Spreading Factor (SF) 6. The Transmission Time Interval (TTI) is ms, which is also called a subframe. The High Speed Shared Control Channels (HS- SCCH) inform the selected users on the used Modulation and Coding Scheme, the current Hybrid Automatic Repeat Request (HARQ) process, and the Redundancy and Constellation Version (RV) of the retransmission. An uplink signalling channel, TABLE I KRYLOV SUBSPACE BASED ALGORITHM FOR AN HERMITIAN MATRIX. input A, b, x, S b = b Ax 3 v = b/ b 4 u=av 5 α=v H u 6 c first =c =/α 7 w=u αv 8 for s =,...,S 9 β = w v s =w/β u=av s α=v H s u 3 γ =α β c [s ] 4 c first, c using eq. (4) 5 w=u αv s βv s 6 end 7 V S =[v,...,v S ] 8 output x S = b V S c first + x the so-called High Speed Dedicated Physical Control Channel (HS-DPCCH), carries positive and negative acknowledgements (ACK/NACK), as well as the Channel Quality Indicator (CQI). Three closely coupled procedures govern the performance of the HSDPA [6]: Adaptive Modulation and Coding (the data rate is adjusted depending on measured channel quality in each subframe by puncturing and repetition of the rate /3 turbo-coded data stream), Fast HARQ (the data from previous transmissions are stored to enable joint decoding of retransmissions with incremental redundancy; retransmissions are requested until correct decoding or a maximum number of attempts is exceeded) and Fast Packet Scheduling (for each TTI the allocation of channelization codes on the HS-PDSCH to the users is controlled). For each subframe the HSDPA transmitter generates the HS-PDSCHs for the scheduled users, the SCH, the Primary Common Control Physical Channel (P- CCPCH) and the Primary-CPICH (P-CPICH) according to [7]. To achieve total transmit power density I or =db, further interfering channels are generated by the so-called Orthogonal Channel Noise Simulator (OCNS) [8] and added to the chip stream. They have a fixed SF 8 and simulate the users or control signals on the other orthogonal channels of the downlink. The HS-SCCHs are not transmitted but signalled error-free to the receiver. All channels except for the SCH are modulated, spread by orthogonal Walsh-Hadamard sequences and subsequently scrambled by the cell-specific Gold sequence. They are weighted and added to a single chip stream s[i] according to [9]. The weighting factors are calculated from the relative power ratios of the channels to the total transmit power spectral density (E c /I or ). We assume a frequency selective, time-variant channel model where each tap of the impulse response has a Jakes Doppler spectrum with a maximum Doppler frequency given by the UE speed. The channel is implemented as a time-variant finite impulse response filter with sample spaced taps. We use oversampling factor and root raised cosine filters at the transmitter and receiver. The sample spaced filter coefficients are generated from the ITU channel models by a sinc interpolation. The received signal after downsampling is r[i] = L h l= h l [i]s[i l]+n[i], where

3 6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 7 s[i l] is the transmitted signal delayed with l, h l [i] are the downsampled and filtered channel coefficients, L h denotes the delay spread in chips and n[i] is i.i.d. zero-mean additive white Gaussian noise with variance σ z. V. KRYLOV EQUALIZER FOR HSDPA It has been shown that a equalizer with filter length L f allows considerable amelioration of the bit error rate []. However, the inversion of a matrix of size L f L f is required, which is not implementable in real time. We present in this section the equalizer and its low-complexity approximation using Krylov subspace methods. Computational Complexity (CM) 4 Krylov A. The Equalizer Every subframe, the equalizer minimizes the mean square error between the equalized received signal f H r i and the transmitted signal s[i d]: f =argmine{ f H r i s[i d] },whered is the delay caused by the system and r i = [r[i],...,r[i L f +]] T contains the L f received samples. A solution is given by f =(HH H + σ I Lf ) H d, (5) where σ denotes the ratio σs/σ z between transmitted signal power and noise variance and H d is the d-th column of H. H is the channel matrix with Toeplitz structure and size L f (L h + L f ), denoting the convolution with the channel impulse response H = h h Lh h h Lh. B. Complexity Comparison The solution (5) can be approximated by the Krylov subspace method. We compare the computational complexity of the and the approximate Krylov solutions. Results are given in approximated number of complex multiplications (CM). Using the exact filter (5), we have to compute the matrix A = HH H + σ I Lf. Given the structure of H, this corresponds to about.5l f (L f +) CM. Its inverse using [] and the product A b cost about.5l f +7.5L f CM and L f CM respectively. This leads to a total computational complexity C.5L h (L h +)+L f (.5L f +7.5) CM. Using the Krylov based algorithm, two matrix-vector products in H(H H v s )+σ v s (L h (L h + L f ) CM) and two inner products to compute α and β (L f CM) are required at every step s. The total computationalcomplexityafter S iterations is then C Krylov S(L f L h +L f +L h ) CM. Furthermore, using the Krylov subspace method allows storage savings: instead of storing A, only v s for s {,...,S} is stored. The computational complexity as well as the equalizer performance increase with L f, thus a tradeoff needs to be found. A reasonable choice is L f 3L h [], where the delay spread L h is given by the channel model used. A comparison of the computational complexity is shown in Fig Krylov Subspace Dimension Fig.. Computational Complexity for the exact equalizer and the Krylov based equalizer, function of the Krylov subspace dimension S {,...,8}. forl f =48and L h =5(Vehicular A channels in our simulations). The approximate number of CM is shown versus the Krylov subspace dimension (or iteration number) S. C. Choice of the Parameters The error r S resulting from the Krylov subspace method is bounded by [] r S A r A ( ka ka +) S, where k A > is the condition number of A (ratio of largest and smallest eigenvalues). Convergence is thus assured, but the convergence speed depends strongly on the matrix A and on the initial guess x. It is necessary to appropriately choose the parameters S and x. If no information on f is available at the receiver, we choose x =[,...,] T. However, the equalizer (5) depends only on the channel estimate H. WhenH is varying slowly, the equalizers from one subframe to another are assumed to be strongly correlated. Thus a more suitable choice for initial guess is x (sub+) = f (sub),where (sub) denotes the subframe index. We consider these two approaches, called Standard and Adaptive Krylov respectively. VI. SIMULATION SETUP AND RESULTS Throughput simulations for an HSDPA receiver with UE capability 6 [3] were carried out for 6QAM modulation. At the transmitter, the fixed reference channel H-Set 3 as defined in [8] is generated. The relative power ratios of the simulated physical channels to the total transmit power spectral density (E c /I or ) are set in compliance to the HSDPA test cases [8]. The E c /I or of the HS-PDSCH is varied. No pathloss is assumed (Îor = I or ). The interference from other cells and the noise is modelled as AWGN with variance σz = I oc. Table II summarizes the simulation setup. The simulations include retransmissions as required by the H-Set 3 testing.

4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 7 63 TABLE II SIMULATION SETUP FIXED REFERENCED CHANNEL H-SET 3 Modulation QPSK 6QAM Nominal Avg. Inf. Bit Rate [kbps] 6 33 Inter-TTI Distance No. of HARQ Processes 6 6 Coding Rate.67.6 No. of Physical Channel Codes 5 4 SIMULATION PARAMETERS RV coding sequence {,, 5, 6} {6,,, 5}. x UE capability class 6 Combining P-CPICH E c/i or SCH E c/i or P-CCPCH E c/i or OCNS soft db db db on Î or/i oc db Equalizer length L f 48 Channel coefficient estimation PA3 Turbo decoding Krylov adaptive Krylov adaptive Krylov standard3 Krylov standard PA3 min.req. PB3 min. req. least squares max-log-map - 8 iterations. PB HS PDSCH E c /I or [db] Fig.. Throughput for ITU-PA3 and PB3 models with 6QAM modulation and OCNS, using Standard or Adaptive Krylov subspace method with S {,, 3} iterations. We show also the minimum requirements for the two models. Throughput simulations are carried out for the frequencyselective Rayleigh fading channels ITU Pedestrian A (PA3), B (PB3) and Vehicular A (VA3 and VA) [8]. The UE speed is 3 kmh, 3 kmh and kmh for PB3/PA3, VA3 and VA respectively. The symbols in Fig., 3 and 4 at -3dB and -6dB show the minimum requirements given by the UMTS standard [8]. Fig. and 3 show the throughput for the frequency selective channels PA3 and PB3 and for VA3 and VA respectively. We compare the throughput for the exact equalizer, and the Standard and Adaptive Krylov equalizers with varying subspace dimension. For a slow varying channel (PA3 and PB3), the Adaptive equalizer converges faster to the. x Krylov adaptive3 Krylov standard3 Krylov adaptive Krylov standard VA3 min. req. VA min. req. VA VA HS PDSCH E c /I or [db] Fig. 3. Throughput for ITU-VA3 and VA models with 6QAM modulation and OCNS, using Standard or Adaptive Krylov subspace method with S {,, 3} iterations. We show also the minimum requirements for the two models. performance than the Standard one. While increasing the UE speed, the Standard equalizer performance gets closer to the Adaptive one (VA3) until it outperforms it (VA). These results show that, when the channel changes slowly, the equalizer will show small variations and thus time coherence from one subframe to another can be exploited. However, when the UE is moving fast, no such information is beneficial. For the investigated channels, a Krylov subspace dimension S 3 is sufficient to attain the throughput. Referring to Fig., we see that the computational complexity is reduced by about a half order of magnitude. In Fig. 4 we show for comparison the throughput obtained using a Least Mean Square (LMS) equalizer as implemented in []. The interference caused by the OCNS degrades the performance of the LMS equalizer that performs worse than the exact or the Krylov approximate : a loss up to 9 db can be observed. VII. CONCLUSION A low-cost approximation of an equalizer for HSDPA has been introduced. This equalizer was tested with frequency selective fading channels (ITU-PA3, PB3, VA3, VA) and turned out to perform as well as the exact equalizer, while the computational complexity and the storage requirements of the algorithm are strongly reduced. Exploiting the temporal coherence of the channel in two consecutive subframes allows further computational complexity reduction. REFERENCES [] F. Kaltenberger, K. Freudenthaler, S. Paul, J. Wehinger, C. F. Mecklenbräuker, and A. Springer, Throughput enhancement by cancellation of synchronization and pilot channel for UMTS high speed downlink packet access, in Proc. IEEE Workshop on Signal Processing Advances in Wireless Commun. (SPAWC) 5. [] K. Freudenthaler, F. Kaltenberger, S. Geirhofer, S. Paul, J. Wehinger, C. F. Mecklenbräuker, A. Springer, and J. Berkmann, Throughput simulations for a UMTS high speed downlink packet access equalizer, in Proc. IST Mobile Summit 5.

5 64 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 7. x PA3 PB3 VA3 VA no OCNS OCNS HS PDSCH E /I [db] c or Fig. 4. Throughput for the ITU-PA3, PB3, VA3 and VA models with 6-QAM modulation, using the LMS equalizer. We show both results with and without OCNS channel. [3] Y. Saad, Iterative Methods for Sparse Linear Systems, nd ed. SIAM, 3. [4] C. Dumard and T. Zemen, Double Krylov subspace approximation for low-complexity iterative multi-user decoding and time-variant channel estimation, in Proc. IEEE Workshop on Signal Processing Advances in Wireless Commun. (SPAWC) 5. [5] T.K.MoonandW.Stirling,Mathematical Methods and Algorithms. Prentice Hall,. [6] I. Forkel and H. Klenner, High speed downlink packet access (HS- DPA): a means of increasing downlink capacity in WCDMA cellular networks? in Proc. 5th European Wireless Conference 4. [7] Members of 3GPP, Technical specification group radio access network; multiplexing and channel coding (FDD) (3GPP TS 5. version 6.5.), 3GPP, Tech. Rep., June 5. [8], Technical specification group radio access network; User Equipment (UE) radio transmission and reception (FDD) (3GPP TS 5. version 6.4.), 3GPP, Tech. Rep., Mar. 4. [9], Technical specification group radio access network; spreading and modulation (FDD) (3GPP TS 5.3 version 6.3.), 3GPP, Tech. Rep., June 5. [] S. Geirhofer, C. Mehlführer, and M. Rupp, Design and real-time measurement of HSDPA equalizers, in Proc. IEEE Workshop on Signal Processing Advances in Wireless Commun. (SPAWC) 5. [] H. Krishna and S. D. Morgera, The Levinson recurrence and fast algorithms for solving Toeplitz systems of linear equations, IEEE Trans. Acoust., Speech, Signal Processing, vol. 35, no. 6, pp , June 987. [] T. Kailath and A. H. Sayed, Fast Reliable Algorithms for Matrices with Structure. SIAM, 999. [3] Members of 3GPP, Technical specification group radio access network; UE radio access capabilities definition (3GPP TS 5.36 version 6.3.), 3GPP, Tech. Rep., Mar. 4.