Este artículo intenta explicar los criterios que sustentan la idea de que la contabilidad China ha sido conformada durante el pasado siglo con arreglo a factores culturales, económicos y políticos. Asimismo se propone estudiar los fenómenos históricos experimentados por la contabilidad china en el transcurso de los años, y evaluar comparativa y críticamente los efectos culturales de los mismos con las transformaciones políticas y económicas que han tenido influencia en el desarrollo de la contabilidad China. This paper attempts to argue the criteria which claim that Chinese accounting has been shaped by together with cultural, economical and political factors in the last century. This paper also aims to compare historical phenomena which occurred in Chinese accounting over the years, and then to assess comparatively or critically the effects of culture with politic and the economic transformations on the development of Chinese accounting.Contabilidad comparativa, contabilidad china, factores culturales, historia, sociología. Chinese accounting, comparative accounting, cultural factors, evolution, history, sociology.

Barbero, Luis

Cowan, C.F.N.

Ratnarajah, Tharmalingam

Sellathurai, Mathini

Xiao, Pei

English

Solas, Cigdem

Ayhan, Sinan

Research Papers in Economics

1st Edition, Volume II,

Lai, Alice Loui So Yee.: “Philosophical Issues

Mattessich’s Critique of Accounting,

Toohey, Peter.: “The Cultural Logic of Historical Periodization”, Handbook of Historical Sociology,

The historical evolution of accounting in China (novissima sinica): effects of culture (2nd part).

This paper presents a method to significantly reduce the preprocessing complexity of the sphere decoder (SD) in frequency-selective channels. The method consists of calculating an approximate QR Decomposition (AQRD) of the channel matrix, making use of its special Toeptliz and block-Topelitz structure in single and multiple-antenna frequency-selective channels, respectively. The AQRD obtains the QR decomposition of a small submatrix of the channel matrix and extends that result to the rest of the matrix, resulting in a considerable complexity reduction compared to the original full QR Decomposition (FQRD). Simulation results show that, despite the lower complexity of the AQRD, it causes only a small bit error rate (BER) performance degradation in the SD.</p

Barbero, Luis G.

Ratnarajah, T.

Cowan, C. F. N.

Heriot Watt Pure

A sphere decoder with approximate QR decomposition for frequency-selective channels

This paper presents a method to significantly reduce the preprocessing complexity of the sphere decoder (SD) in frequency-selective channels. The method consists of calculating an approximate QR decomposition (AQRD) of the channel matrix, making use of its special Toeptliz and block-Topelitz structure in single and multiple-antenna frequency-selective channels, respectively. The AQRD obtains the QR decomposition of a small submatrix of the channel matrix and extends that result to the rest of the matrix, resulting in a considerable complexity reduction compared to the original full QR decomposition (FQRD). Simulation results show that, despite the lower complexity of the AQRD, it causes only a small bit error rate (BER) performance degradation in the SD

Barbero, L

Xiao, P

Ratnarajah, T

Sellathurai, M

Surrey Research Insight

A Sphere Decoder with Approximate QR Decomposition for Frequency-Selective Channels

Queen's University Belfast Research Portal

Barbero, L

Xiao, P

Ratnarajah, T

Sellathurai, M

University of Surrey

A Sphere Decoder with Approximate QRDecomposition for Frequency-Selective ChannelsLuis G. Barbero, Pei Xiao, T. Ratnarajah, Mathini SellathuraiThe Institute of Electronics, Communications and Information TechnologyQueen’s University Belfast, BT3 9DT, United KingdomAbstract— This paper presents a method to significantlyreduce the preprocessing complexity of the sphere decoder(SD) in frequency-selective channels. The method consistsof calculating an approximate QR Decomposition (AQRD)of the channel matrix, making use of its special Toeptlizand block-Topelitz structure in single and multiple-antennafrequency-selective channels, respectively. The AQRD ob-tains the QR decomposition of a small submatrix of thechannel matrix and extends that result to the rest of thematrix, resulting in a considerable complexity reductioncompared to the original full QR Decomposition (FQRD).Simulation results show that, despite the lower complexityof the AQRD, it causes only a small bit error rate (BER)performance degradation in the SD.Index Terms— sphere decoder (SD), approximate QRDecomposition (AQRD), frequency-selective channel.I. INTRODUCTIONThe sphere decoder (SD) has been recently pro-posed as a means of achieving optimal maximumlikelihood (ML) performance in the detection of singleinput-single output (SISO) and multiple input-multipleoutput (MIMO) systems in frequency-selective chan-nels [1], [2]. However, for large block sizes and, es-pecially, at high signal to noise ratio (SNR), the re-quired full QR Decomposition (FQRD) of the channelmatrix has a critical effect in the overall complexityof the algorithm [2]. In this paper, an approximateQR Decomposition (AQRD) is proposed to significantlyreduce the preprocessing complexity of the SD, withoutgreatly affecting the performance. It makes use of thespecial Toeplitz structure of the channel matrix whichresults in the convergence of the elements of its FQRDfor large block sizes [3]. Although that convergenceproperty has been previously applied to the elements of aCholesky factorization in the context of joint zero forcing(ZF) detection in time division-code division multipleaccess (TD-CDMA) systems [4], [5], its application tothe preprocessing stage of the SD in frequency-selectivechannels has not been previously studied.We consider a MIMO system with M transmit andN receive antennas, denoted as M ×N , in a block-fading frequency-selective propagation environment. Thechannel between transmitter j and receiver i is mod-elled as a linear finite impulse response (FIR) fil-ter with L independent symbol-spaced taps, denotedas hi,j = [hi,j(0), hi,j(1), . . . , hi,j(L − 1)]T, wherehi,j(l) ∼ CN (0, σ2l ) for i = 1, . . . , N , j = 1, . . . , M andσl is set according to the power delay profile (PDP) of thechannel for l = 0, . . . , L− 1, such that E[hHi,jhi,j]= 1.For notation purposes, we consider the transmission ofMT symbols over a span of T symbol-spaced timeinstants, resulting in a block size T , with T > L.The MT -vector of transmitted symbols per block isdenoted as x = [x1, x2, . . . , xMT ]T, where the first Msymbols represent the symbols transmitted from all Mantennas at time instant k = 0, and subsequently fork = 1, . . . , T − 1. The symbols are taken independentlyfrom a constellation O of P points and E[|xi|2]=1/M for i = 1, . . . , MT . In order to avoid inter-blockinterference, L − 1 zeros are appended per transmitantenna at the end of each block. Correspondingly, thevector of received symbols y spans over K = T +L−1time instants, i.e. y = [y1, y2, . . . , yNK ]T, and can bewritten asy = Hx + v ,whereH =H(0)... . . .H(L − 1) · · · H(0). . . ... . . .H(L − 1) · · · H(0). . . ...H(L − 1)(1)andH(l) =h1,1(l) · · · h1,M (l)... . . ....hN,1(l) · · · hN,M (l)for l = 0, . . . , L−1. The vector v = [v1, v2, . . . , vNK ]T,contains independent complex Gaussian noise sampleswhere E[|vi|2]= N0 = σ2 for i = 1, . . . , NK. Finally,1we assume that the channel is perfectly known at thereceiver and that N ≥ M 1.A. Sphere DecoderThe main idea behind the SD is to reduce the complex-ity of the maximum likelihood detector (MLD), given byx̂ML = arg minx∈OMT‖y − Hx‖2 ,by searching over only the vectors x that satisfy‖y − Hx‖2 ≤ R2 , (2)i.e. searching over the lattice vectors Hx that lie withina hypersphere of radius R around the received vectory and achieving the same ML performance [6]. Duringthe preprocessing stage, the SD makes use of the FQRDdecomposition of H = Q̃ [RT0]T [7], where Q̃ is anNK × NK unitary matrix, R is an MT × MT uppertriangular matrix, and 0 is a 0-matrix of the appropriatesize. Partitioning Q̃ = [QQ′] into two submatrices Qand Q′, containing the first MT columns and the lastNK − MT columns of Q̃, respectively, and removingconstant terms, the metric calculation in (2) can berewritten as‖QHy − Rx‖2 ≤ R2 . (3)Due to the upper triangular matrix R, the solution to (3)can be obtained recursively by performing a constraineddepth-first tree search through a tree with MT levelswhere P branches originate from each node on thetree [6].One important problem of the SD in frequency-selective channels is that the size of H increases withthe block size, an effect that is not present in the flat-fading case. Thus, there is a dramatic increase in thecomplexity of the FQRD with the block size. That effectis more noticeable at high SNR, where the preprocessingcomplexity can become the dominant factor in the overallcomplexity of the SD [2]. In order to overcome thatproblem, an AQRD is proposed in the next section toconsiderably reduce the preprocessing complexity of theSD.II. APPROXIMATE QR DECOMPOSITIONIn order to introduce the AQRD, we take into accountthat the channel matrix H in (1) has a block-Toeplitzstructure with N ×M blocks. Thus, the matrices R and1Although the discussion in this paper concentrates on the MIMOcase, the alternative proposed here also applies to the SISO case,where H is Toeplitz instead of block-Toeplitz.Q in (3) can be written asR=R(1)(0) · · · R(L)(L − 1) 0. . . ... . . .R(L)(0) R(T )(L − 1). . . ...0 R(T )(0),(4)with M × M blocks R(n)(l) for l = 0, . . . , L − 1 andn = l + 1, . . . , T , for a given l, and asQ =Q(1)(L − 1) · · · Q(T )(T + L − 2)... . . ....Q(1)(0) Q(T )(L − 1). . . ...0 Q(T )(0), (5)with N × M blocks Q(n)(m) for n = 1, . . . , T andm = 0, . . . , n + L − 2, respectively.Using the fact that R can also be obtained through theCholesky factorization of the hermitian block-ToeplitzGram matrix G = HHH, it has been previously shownthatlimn→∞R(n)(l) = R(l) , (6)i.e. each one of the L sequences R(n)(l) in n convergesto a given value R(l) [3], [8]. Further, as a consequenceof (6), it can be directly shown thatlimn→∞Q(n)(m) = Q(m) , (7)i.e. the sequences Q(n)(m) converge in n to a givenvalue Q(m), andlimm→∞Q(n)(m) = 0 , (8)i.e. the sequences Q(n)(m) converge in m to the 0-matrix [9].The AQRD proposed in this section makes use of theabove convergence results to obtain the FQRD of H inthe following three steps:1) Initially, a FQRD is performed on the submatrix ofH resulting from selecting its first MB columns,with L < B < T . The output of that FQRDcorresponds to the first MB columns of R andQ, i.e. R(n)(l) for l = 0, . . . , L − 1 and n =l + 1, . . . , B and Q(n)(m) for n = 1, . . . , B andm = 0, . . . , n + L − 2, respectively.2) The remaining M(T − B) columns of R areobtained using (6) and settingR(n)(l) = R(B)(l)for n = B + 1, . . . , T and l = 0, . . . , L − 1.23) The M(T − B) columns of Q are obtained, firstusing (7) to setQ(n)(m) = Q(B)(m)for n = B + 1, . . . , T and m = 0, . . . , B + L− 2,and then using (8) to setQ(n)(m) = 0for n = B+1, . . . , T and m = B+L−1, . . . , T +L − 2.Thus, the FQRD of H is obtained approximating thelast T −B block-columns of R (Q) by the B-th block-column of R (Q), performing a one-block down-shiftfrom block to block and introducing 0 in the top-rightnon-computed blocks of Q. If we approximate the totalnumber of real operations required by the FQRD ofH using the modified Gram-Schmidt (MGS) method asNF ≈ 8NL(MT )2 [10], the use of the AQRD reducesthat figure to NA ≈ 8NL(MB)2, where B < T . Similarreductions in complexity can be obtained by applying theabove convergence results to other QR methods like theSchur algorithm, optimized for Toeplitz or block-Toeplitzmatrices [11], [12].The convergence result in (6) has been previouslyused to propose an approximate Cholesky factorizationof a Hermitian block-Toeplitz matrix in the context ofZF joint detection of TD-CDMA systems [4], [5]. Inthis paper, instead of considering sub-optimal detectors,the AQRD is applied to the preprocessing stage of theoptimal SD in frequency-selective MIMO channels. Inaddition, given that the QR decomposition is needed,this paper includes the convergence results of the blocksof Q, describing the extension procedure to obtain thefull matrix Q.0 10 20 30 40 5010−510−410−310−210−1100 nexpected value of  F R M = N = 2, T = 50 L = 4 L = 8 L = 12exponential PDPuniform PDPFig. 1. Expected value of FR(n) in a 2× 2 system with T = 50.III. SIMULATION RESULTSThis section shows the convergence of the FQRD of Hand the BER performance of the SD when the FQRD isreplaced by the AQRD. In order to show the convergenceof the FQRD of H, we define, as a distance measure, thestochastic processes FR(n) = {‖R(n)(0) − R(T )(0)‖F :n = 1, . . . , T} and FQ(m) = {‖Q(T )(m)‖F : m =0, . . . , T + L − 2}, where ‖ · ‖F denotes the Frobeniusnorm 2. Fig. 1 shows the expected value of FR(n) ina 2 × 2 system with block size T = 50 for a differentnumber of taps L. Both uniform and exponential PDPshave been considered. In the former, σ2l = 1/L forl = 0, . . . , L − 1, while in the latter, σ2l = e−l/σ2h forl = 0, . . . , L−1 with σ2h =∑L−1l=0 e−l. It can be observedhow the expected value decreases with n, i.e. R(n)(0)is a better approximation of R(T )(0) as n increases,especially in the exponential PDP case. Based on that,the AQRD can select the parameter B so that the average‖R(B)(0) − R(T )(0)‖F is below an arbitrary threshold.If L increases, the parameter B also needs to increaseto maintain the distance constraint. However, the effectis less noticeable in the exponential PDP case, giventhat the additional taps have a exponentially decreasingenergy. It has also been observed that if B is selectedoutside the end tail of the average FR(n), its valueremains almost unchanged with the increase of T , for agiven L and distance constraint. That indicates that thenumber of taps is the determinant factor in the choiceof B. Those last results have not been included due tospace constraints.0 10 20 30 40 50 6010−310−210−1100 M = N = 2, T = 50 mexpected value of  F Q L = 4 L = 8 L = 12uniform PDPexponential PDPFig. 2. Expected value of FQ(m) in a 2× 2 system with T = 50.Fig. 2 shows the expected value of FQ(m) in thesame scenario. For the range of selectable values of m(m = L− 1, . . . , T +L− 2), it can be observed how the2The stochastic processes FR(n) and FQ(m) have been selected,due to space constraints, as illustrative examples of the convergenceof R(n)(l) and Q(n)(m) in n and of Q(n)(m) in m, respectively.34 6 8 10 12 14 1610−510−410−310−210−1100Eb/N0 (dB)BER M = N = 2, T = 15, L = 3, uniform PDP  ASD−6ASD−9SD4−QAM16−QAMFig. 3. BER performance of the SD and the ASD as a function ofthe Eb/N0 in a 2×2 system with T = 15, L = 3 and uniform PDP.average FQ(m) decreases with m, approaching its limit0. Thus, the parameter B can be selected to keep theaverage ‖Q(T )(B+L−2)‖F below an arbitrary threshold.As in Fig. 1, B would need to increase with L to keepthe same distance constraint, with a less noticeable effectin the exponential PDP case. It is worth noting, though,that, for m ≤ L − 2 in the exponential PDP case, theaverage FQ(m) initially increases, reaches a maximumand then decreases. Therefore, in the exponential PDPcase, B needs to be selected so that B +L− 2 is withinthe region of monotonic convergence of the averageFQ(m), a condition which is always guaranteed since,by definition, B > 0. Finally, it has also been observedthat, for a given L, B remains almost unchanged with theincrease of T for the same distance constraint, providedthat B + L − 2 is outside the end tail of the averageFQ(m).The performance of the original SD has been com-pared to that of the SD using the AQRD, i.e. ASD, toshow how the latter can approximate the performanceof the former with a considerably reduction in prepro-cessing complexity. Fig. 3 shows the BER performanceof the SD and the ASD-B in a 2 × 2 system withT = 15, L = 3, uniform PDP and B = 6, 9 as afunction of the SNR per bit at each receive antenna,Eb/N0 = log−12 (P )/σ2. Both 4 and 16-quadratureamplitude modulation (QAM) constellations have beenTABLE IAVERAGE NUMBER OF REAL OPERATIONS OF THE SD AND THEASD IN THE SYSTEM OF FIG. 3 WITH 4-QAM AT Eb/N0 = 14 dB.SD ASD-9 ASD-6Preprocessing stage 41,534 19,010 11,528Tree search stage 2,239.6 2,330.2 2,768.3Total 43,773.6 21,340.2 14,296.34 6 8 10 12 14 1610−510−410−310−210−1Eb/N0 (dB)BER M = N = 3, T = 10, 4−QAM, exponential PDP  ASD−4 (L = 2)ASD−6 (L = 2) SD (L = 2)ASD−4 (L = 4)ASD−6 (L = 4)SD (L = 4)Fig. 4. BER performance of the SD and the ASD as a function ofthe Eb/N0 in a 3× 3 system with T = 10, 4-QAM and exponentialPDP.used. It can be observed how, in both cases, the ASD canapproximate the performance of the SD. However, for the16-QAM case, the higher constellation order causes theapproximation of the AQRD to have a more detrimentaleffect on the BER, thus requiring a higher value of Bto approach the performance of the SD. For the samesystem, with 4-QAM and at Eb/N0 = 14dB, Table Ishows the average number of real operations of thefixed-complexity preprocessing stage and the variable-complexity tree search stage in SD and the ASD bothusing the Schnorr-Euchner enumeration. The radius R isinitially set to ∞ and updated every time a lattice vectoris found inside the hypersphere. The clear reduction inpreprocessing complexity only marginally increases theaverage complexity of the tree search stage, resultingin an overall complexity reduction. It has also beenobserved that increasing the block size T does not havesignificant effect on the choice of B for a given systemin order to achieve the same relative performance, thusincreasing the complexity reduction.Finally, Fig. 4 shows the effect the number of taps Lhas on the performance of the ASD in a 3 × 3 systemwith T = 10, 4-QAM and exponential PDP. In this case,it can be observed how the increase in the number oftaps L requires an increase in the value of B if we wereto maintain the same relative performance. In particular,it can be observed how when L = 4, the performancedegradation of the ASD-4 causes its performance to gobeyond that of the SD and ASD when L = 2, which is alower-diversity scenario. In addition, it can be seen howthe difference in performance between the SD and theASD-6 increases. Thus, as pointed out at the beginningof this section and in Section II, the number of taps L hasa more determinant effect on the choice of B in the ASDthan the block size T . It is important to note that due tothe errors introduced in the AQRD, the performance of4any ASD is expected to have an error floor at high SNR.However, that problem can be overcome in practice byselecting a value of B that places the ASD error floorbelow the error floor fixed-point quantization error floor.IV. CONCLUSIONThis paper has presented a method to reduce the pre-processing complexity of the SD in frequency-selectivesingle or multiple-antenna channels. The method relieson the convergence of the elements of the QR decom-position of the Toeplitz or block-Toeplitz channel matrixto perform an AQRD, a QR decomposition of a smallsubmatrix of the channel matrix, as opposed to theFQRD originally required by the SD. The size of thatsubmatrix depends on the number of antennas, the blocksize, the constellation order and the number of taps, withthe latter being the most determinant factor in order toachieve a specific level of performance. The resultingASD approximates the performance of the SD whilehaving a considerably lower preprocessing complexity.In addition, the size of the submatrix can be selected suchthat the error introduced by the AQRD is lower than thatof the fixed-point quantization, effectively removing theeffect of the former. This fact and the reduced complexitymake the ASD more suited for hardware implementationthan the original SD.ACKNOWLEDGEMENTThis work was supported by the UK Engineering andPhysical Sciences Research Council under grant numberEP/C004132/1.REFERENCES[1] H. Vikalo, B. Hassibi, and U. Mitra, “Sphere-constrained MLdetection for frequency-selective channels,” IEEE Trans. Com-mun., vol. 54, no. 7, pp. 1179–1183, Jul. 2006.[2] H. Vikalo and B. Hassibi, “Maximum-likelihood sequence de-tection of multiple antenna systems over dispersive channelsvia sphere decoding,” EURASIP Journal on Applied SignalProcessing, vol. 2002, no. 5, pp. 525–531, May 2002.[3] J. Rissanen and L. Barbosa, “Properties of infinite covariancematrices and stability of optimum predictors,” InformationSciences, vol. 1, no. 3, pp. 221–236, Jul. 1969.[4] H. R. Karimi and N. W. Anderson, “A novel and efficient solu-tion to block-based joint-detection using approximate choleskyfactorization,” in Proc. 9th IEEE International Symposium onPersonal, Indoor and Mobile Radio Communications (PIMRC’98), vol. 3, Boston, MA, USA, Sep. 1998, pp. 1340–1345.[5] M. Vollmer, M. Haardt, and J. Götze, “Comparative study ofjoint-detection techniques for TD-CDMA based mobile radiosystems,” IEEE J. Sel. Areas Commun., vol. 19, no. 8, pp. 1461–1475, Aug. 2001.[6] A. D. Murugan, H. E. Gamal, M. O. Damen, and G. Caire, “Aunified framework for tree search decoding: Rediscovering thesequential decoder,” IEEE Trans. Inf. Theory, vol. 52, no. 3, pp.933–953, Mar. 2006.[7] M. O. Damen, H. E. Gamal, and G. Caire, “On maximum-likelihood detection and the search for the closest lattice point,”IEEE Trans. Inf. Theory, vol. 49, no. 10, pp. 2389–2402, Oct.2003.[8] J. Rissanen, “Algorithms for triangular decomposition of blockHankel and Toeplitz matrices with application to factoringpositive matrix polynomials,” Mathematics of Computation,vol. 27, no. 121, pp. 147–154, Jan. 1973.[9] X.-W. Chang, M. J. Gander, and S. Karaa, “Asymptotic prop-erties of the QR factorization of banded Hessenberg-Toeplitzmatrices,” Numerical Linear Algebra with Applications, vol. 12,no. 7, pp. 659–682, Sep. 2005.[10] G. H. Golub and C. F. V. Loan, Matrix Computations. London,UK: The Johns Hopkins University Press Ltd., 1996.[11] J. Chun, T. Kailath, and H. Lev-Ari, “Fast parallel algorithmsfor QR and triangular factorization,” SIAM Journal on Scientificand Statistical Computing, vol. 8, no. 6, pp. 899–913, Nov.1987.[12] M. Vollmer, M. Haardt, and J. Götze, “Schur algorithms forjoint-detection in TD-CDMA based mobile radio systems,”Annals of Telecommunications, vol. 54, no. 7, pp. 365–378, Jul.1999.5

The historical evolution of accounting in China (novissima sinica): effects of culture (2nd part).

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