Key-Words: - Software defined radio, Walsh Hadamard codes, Lattice filter, Matched filter, Autoregressive model, Gauss-Markov process.

Transcription

1 G Suhitra, M L Valarmathi A Novel method of Walsh-Hadamard Code Generation using Reonfigurable Lattie filter and its appliation in DS-CDMA system GSUCHITRA, MLVALARMATHI Department of ECE, Department of CSE Government College of Engineering, Government College of Tehnology Bargur, Coimbatore India susisiv@yahoooin, ml_valarmathi@rediffmailom Abstrat: - Walsh Hadamard odes are widely used as signature odes in the urrent wireless standards suh as IS-9 CDMA, WCDMA, CDMA2000, and image transform appliations It is also used in onjuntion with error orretion algorithms for Reed-Muller odes and in dyadi invariant signal proessing In this paper, a novel method of generating Walsh Hadamard(WH) odes using lattie filter is proposed The obtained results onur with the properties of WH ode The entire WH ode set an be generated by hanging the refletion oeffiients without modifying the filter struture This feature ensures reonfigurability and enables them to be used in software radio Moreover, this reonfigurable struture an be used as a mathed filter for multiuser detetion in Diret Sequene Code division Multiple Aess System (DS-CDMA) BER performane of a DS- CDMA system employing a orrelator reeiver and the proposed mathed filter reeiver for despreading is simulated in an AWGN hannel and flat fading Rayleigh hannel Simulation results showing the generation of WH odes from the autoregressive model of first order Gauss-Markov proess is also obtained Key-Words: - Software defined radio, Walsh Hadamard odes, Lattie filter, Mathed filter, Autoregressive model, Gauss-Markov proess Introdution A reonfigurable transeiver should support multiple air interfaes suh as CDMA, OFDM, MC- CDMA with dynami reonfigurability in a single terminal [] This is ahieved in Software defined Radio (SWR) by a hange in the software without any hange in the hardware element As mentioned in [2], the size of the software to be downloaded should be small to redue the reonfiguration time This an be optimised by parameterisation method, in whih ommon aspets of different ommuniation standards an be supported by the reonfigurable transeiver, whih is identified and a ommon proessing proedure is installed in the devie The two approahes in parameterisation method are Common Funtion tehnique (CF) and Common operator approah (CO) In [], CO tehnique applied to the two operators namely Fast Fourier Transform (FFT) and Reonfigurable Linear Shift Register(R-LFSR) in digital domain is disussed In CO tehnique, ommon elements with similar strutural aspets that an be desribed by a general equation and independent of typial standards are identified From the design and implementation point of view these seleted CO funtions should be reusable and reonfigurable so that they an be applied to any standards In this paper, lattie filter module is used as a ommon element in realising WH ode generator, Mathed filter and Gauss Markov model representing the slowly varying signals Generation of WH odes from the lattie filter struture of Gauss Markov model and the Bit Error Rate (BER) performane of the mathed filter whih is realised by the proposed lattie filter struture in a diret sequene ode division multiple aess system (DS-CDMA), is also disussed In CDMA systems, all the users transmit at the same frequeny where individual users are identified by assigning distint signature odes to them Spreading odes used in CDMA systems inlude Walsh-Hadamard sequenes, Orthogonal Variable Spreading Fator odes, Gold odes, Kasami odes, m-sequenes et [6, 7] Walsh Hadamard odes are widely used as signature odes for downlink transmission [3] In general, these odes are generated using linear feedbak shift register (LFSR) LFSR an be realised by a tapped delay line or finite impulse response (FIR) filter struture with binary inputs [] In the proposed method, the LFSR used to generate WH odes is replaed by the autoregressive (AR) or infinite impulse response (IIR) model realised using the lattie filter (LF) struture Lattie filters possess E-ISSN: Issue 3, Volume 9, July 203

2 G Suhitra, M L Valarmathi several important properties suh as a modular struture, low sensitivity to parameter quantization effets, and a simple method to ensure filter stability Depending on the values of the refletion oeffiients, the LF an be used to generate different WH ode sequenes The required refletion oeffiients an be stored in a look up table or it an be software downloaded Auray an be inreased by using higher order filters with less omputational omplexity sine the modular struture of LF allows the filter order to be inreased or dereased without having to reompute the refletion oeffiients In setion 2, relation between LFSR and AR modelling using Levinson Durbin algorithm is disussed Setion 3 desribes the generation of WH odes using LF In setion, appliation of the proposed LF struture as a mathed filter in DS- CDMA system is desribed and simulation results showing the BER performane of a DS-CDMA system using the onventional and mathed filter method of despreading is presented In setion, relation between the AR model of WH odes and first order Gauss Markov proess is disussed 2 Relation between Linear Feedbak Shift Register and Autoregressive (AR) modelling In general, maximal length sequenes like pseudo noise sequenes are produed by using a LFSR [7] For example, the blok diagram to generate a pseudo noise sequene of length using an LFSR of length is shown in Fig In Fig, D and + represents a delay element (z - ) and exlusive OR operation respetively PN sequene output D D D D + Fig Blok diagram of a binary LFSR The problem of generating a given sequene [ 0, N- ], of length N, using an LFSR is stated as follows: The feedbak onnetion polynomial g(d) has to be determined The initial register ontents of the shortest LFSR that ould produe the sequene (D) has to be determined One method that provides a solution to this problem is Massey s algorithm [7] This involves solving a set of system equations of the form shown in equation () Assuming a LFSR of length, the set of equations are given by, g g 2g 3 g 2 3 = 6 7 () From equation (), it an be inferred that the matrix on the left is a Toeplitz matrix, and these equations represent the Yule-Walker equations of an AR proess in signal modelling This implies that it may be possible to generate the maximal length sequenes from the impulse response of an AR deterministi signal model [2] In stohasti signal modelling, these equations an be solved by using the Levinson-Durbin algorithm Levinson-Durbin reursion algorithm [6] is widely used to solve the Prony normal equations and autoorrelation normal equations The autoorrelation method, a modified form of Prony s normal equations, is applied to a fixed length sequene (windowed sequene) The autoorrelation normal equations in matrix form for a real proess X is a set of (p+) linear equations in (p+) unknowns a(), a(2),, a(p) and ε p+ given by, r ( 0) r( ) r( p ) r( p) ( ) r( 0) r( p 2) r( p ) r r ( p ) r( p 2) r( 0) r( p 2) r( p) r( p ) r( ) r( 0) a a In matrix notation, eqn2 an be written as, Rp + ap+ = εp+ Up+ (3) The autoorrelation sequene r(k) is given by, r N ( k) = x( n) x( n k) ;k 0 n = k () N represents the length of the sequene x(n) ( ) = εp+ 0 ( ) p 0 (2) E-ISSN: Issue 3, Volume 9, July 203

3 G Suhitra, M L Valarmathi and the minimum modelling error ε p+ is given by, k = p εp+ = r( 0) + a( k) r( k) () k = Eqn2 an be solved by using the Levinson Durbin Algorithm It represents an all-pole model of the form given by, b ( ) 0 H z = (6) p + ( ) k a p k z k = Eqn (6) an be implemented using the diret form- II digital filter struture The refletion oeffiients, one of the by-produts obtained by solving the autoorrelation normal equations an be used to implement an all-pole IIR lattie filter sensitive to quantisation effets sine these strutures are implemented by using the quantised numerator and denominator oeffiients diretly This drawbak of sensitivity beomes worse as the number of rowded poles inreases and its effet an be redued by using asade or parallel form of filter strutures However filters realised with lattie strutures possess good numerial properties sine it an be realised as a asade of regular modules Fig2 and Fig3 shows the single stage and p-th order all pole lattie filter respetively where T p is referred as p th refletion oeffiient and f, g stands for all pole and all-pass response respetively This modular struture makes it more suitable for VLSI implementation In this paper, generation of WH odes using LF obtained by solving the autoorrelation normal equations is disussed sine the signifiane of these refletion oeffiients is that they are all bounded by one in magnitude resulting in a stable all-pole model f j+ g j+ T j+ * -T j+ D f j g j 3 GENERATION OF WALSH- HADAMARD CODES USING THE RECONFIGURABLE LATTICE FILTER (RLF) In this setion, the proposed method of generation of WH odes from the impulse response of all-pole model realized using LF struture is desribed A numerial example showing the generation of WH odes of length 8 is also disussed Fig2a 3 AR modelling of WH ode generation f j+ T j+ f j WH ode generation using AR modelling is shown in Fig g j+ g j Fig2b Fig2a, 2b Single stage of an all pole lattie filter f p f p- f p-2 f f 0 T p T p- T Lattie u(n) filter f(i) + H N Sgn g p g p- g p-2 g g 0 Fig3 An order p all pole lattie filter D -f(i-) As disussed in [2], the poles and zeros of the transfer funtion in diret form strutures are very Fig Walsh Hadamard ode generation from all-pole response of Lattie filter E-ISSN: Issue 3, Volume 9, July 203

4 G Suhitra, M L Valarmathi The algorithm to obtain the AR model of WH ode is as follows: Table Refletion Coeffiients of Walsh- Hadamard ode of length 8 a) The autoorrelation values for a Walsh Hadamard ode sequene of length N are alulated b) Levinson Durbin algorithm is applied to these autoorrelation values and the refletion oeffiients T, T 2, T N/2 are obtained ) An IIR all pole lattie filter is implemented using these refletion oeffiients To get highly preise results, the model order an be inreased from (N/2) to (N/2)+,,N d) The all pole response {f k : k=0,,2,,(n- )} of this filter is obtained by giving a unit step sequene as input e) Sine the first bit of all the Walsh Hadamard ode sequenes start with one, initialise the first bit of the ode as H(,)= The remaining bits H(,i), i=2,3,,n are then given by, H N (,i)=signum{[f(i)-f(i-)]};i=2,3,,n 32 Numerial Example As an example, onsider a Walsh Hadamard ode set of size and length 8 in whih eah row represents a ode sequene Eah row in the matrix H 8 represents a ode sequene The following pattern an be observed from the ode sequenes The ode sequene 2 an be generated from ode sequene by hanging signs at bit positions 2,,6,8 In the same manner, ode sequenes, 6, 8 an be obtained from ode sequenes 3,, 7 respetively Table and Table2 shows the refletion oeffiients for a Walsh Hadamard ode of length 8 and 6 respetively generated by the proposed algorithm Code Code 2 Code 3 Code Code Code 6 Code 7 Code The following observations an be noted from Table and Table2 a) It an be observed that the numerial values of refletion oeffiients are same for odes and 2, 3 and, and 6, 7 and 8, 9 and 0, and 2, 3 and, and 6 but only with a sign hange in the odd bit positions Hene it an be inferred that two different ode sequenes an be generated from the same set of refletion oeffiients by hanging the sign in odd positions whih means that the ode generation arhiteture is reonfigurable b) To generate a ode of length 8, the number of refletion oeffiients to be stored in memory is only Hene to generate a ode of length N, the number of memory elements needed is only N/2 This implies that the memory requirements are less ) The AR model order has to be inreased to 2N to generate ode sequenes of length N This an be aomplished by adding extra lattie filter modules and using the orresponding set of refletion oeffiients Fig shows the magnitude spetrum for a sample ode sequene in 8, 6 length WH ode sets E-ISSN: Issue 3, Volume 9, July 203

5 G Suhitra, M L Valarmathi and their respetive impulse responses from the WH-AR model It is observed that the AR model and the WH ode have nearly the same magnitude spetrum distribution Hene the proposed AR model an be used to generate the WH odes Table3 shows the all pole response, the first order differene of the all pole response and the WH ode generation from this first order differene for a sample WH ode sequene of length 6 by using the algorithm proposed in setion 3 It an be observed that the WH ode sequene generated by the proposed method is the same as the original WH ode Table: 2 Refletion Coeffiients of Walsh- Hadamard ode of length 6 Code Code 2 Code 3 Code Code 9 Code 0 Code Code Code 3 Code Code Code Code Code 6 Code 7 Code E-ISSN: Issue 3, Volume 9, July 203

6 G Suhitra, M L Valarmathi Magnitude Magnitude AR model6 walsh6 Magnitude spetrum frequeny AR model8 walsh8 Magnitude spetrum frequeny Fig Magnitude spetrum of a sample sequene of WH-ode set of length 8, 6 and their respetive AR models Table3 Proposed method of WH ode generation for a sample sequene of length 6 Refletion f difof Signum WH Coeffiients (difof) ode RLF as a Mathed filter in DS- CDMA system In ommuniation systems, binary messages transmitted aross a noisy hannel are deteted by using a mathed filter at the reeiving end [6], [7] Mathed filter is an optimum filter that maximises the signal to noise ratio of a signal in the presene of additive white Gaussian noise A mathed filter operates by orrelating a known signal with an unknown signal and detets the presene of the known signal pattern This is equivalent to onvolving the inoming unknown signal with the known signal pattern In the traditional method of diret sequene spread spetrum tehnique, spreading is aomplished by multiplying the information bearing signal b(t) with the spreading odes at the transmitting end At the reeiving end, the original message is reovered by orrelating the reeived spreaded signals with a replia of the signature sequene used at the transmitting end This proess is alled as despreading[8],[9] In CDMA systems, despreading is ahieved by sliding orrelators, surfae aousti wave (SAW) mathed filters and E-ISSN: Issue 3, Volume 9, July 203

7 G Suhitra, M L Valarmathi digital mathed filters [7] As mentioned in [3], the mathed filter used in CDMA systems is tuned to math a ode sequene It is in general realized using a tapped delay line filter struture ie, FIR struture or moving average model where the input data pattern is represented by digital words and the spreading ode pattern of + or - are stored as the multipliation oeffiients As disussed in setion 3, WH ode generator an also be realized using the AR model In this paper, despreading is done at the reeiving end by the AR model of WH ode generator realized by a lattie filter instead of the digital mathed filter Spreaded data Χ Code generator Integrate and dump filter Deision devie Fig6 Conventional method of despreading Fig6 shows the onventional method of despreading DS-CDMA system employing the proposed reonfigurable mathed LF is shown in Fig7 in appendix The modified algorithm of despreading is disussed below: a) Multiplier and ode generator are replaed by the AR model WH (z) of the WH ode b) The reeived signal after demodulation is onverted into NRZ polar format represented as R(z) It is then passed through the AR lattie filter with transfer funtion WH (z) and the filter output response f(n) is obtained whih is given by, ----(7) Where F - indiates inverse Fourier transform ) This output f(n) is then passed through an integrator that integrates the lattie filter response over a period equal to the length of the spreading ode and outputs one value during eah period b e d) Output of integrator is then passed through a deision devie and an estimate b e (n) of the transmitted data is obtained It is given by, N b e = sgn f ( k) (8) k = where N is the length of the spreading ode e) The proposed MF reeiver an be reonfigured to math the users spreading odes by using their respetive refletion oeffiients stored in a look-up table Simulation Results In this setion, BER performane of a DS- CDMA system employing onventional method and the proposed MF method of despreading is investigated WH odes are used as signature odes to distinguish different users and spreading at the transmitting end is aomplished using the traditional approah BPSK modulation is used The signal at the reeiving end is given by, rx = K Hi s + W i= i (9) i=,2,,k represents the number of users, s i is the transmitted spreaded signal of user i W is the additive white Gaussian noise, H i represents the flat fading Rayleigh hannel of user i Fig8 shows the BER performanes in a DS- CDMA system employing onventional method and proposed mathed filter method of despreading for 8 and 6 lengths Walsh odes respetively in an AWGN plus Rayleigh flat fading hannel with four user senario It an be observed that the output of the proposed model and onventional method are marginally the same Fig9 shows the multi-user BER performanes employing onventional method and proposed mathed filter method of despreading of 6-length Walsh odes in an AWGN plus Rayleigh flat fading hannel for SNR=6dB and SNR=2dB as a funtion of number of users in a DS-CDMA system E-ISSN: Issue 3, Volume 9, July 203

8 G Suhitra, M L Valarmathi BER performane in an AWGN plus Rayleigh hannel for a 6-length ode 0 x 0-03 AWGN plus Rayleigh hannel for a 6 length ode TR=6dB LF=6dB BER BER LF TM SNR(dB) Number of users BER performane in an AWGN plus Rayleigh hannel for a 8-length ode LF TM 0 x 0-00 AWGN plus Rayleigh hannelfor a 6 length ode TR=2dB LF=2dB BER BER SNR(dB) Number of users Fig8 BER performanes employing onventional method and mathed filter AR signal model of 8, 6-length Walsh odes in an AWGN plus Rayleigh flat fading hannel with four user senario in a DS-CDMA system Fig9 Multi-user performanes employing onventional method and mathed filter AR signal model of 6-length Walsh odes in a DS-CDMA system E-ISSN: Issue 3, Volume 9, July 203

9 G Suhitra, M L Valarmathi Simulation results show that the BER performane of the AR lattie model is slightly better than the onventional method of despreading This slight differene is due to round off noise effets that our in lattie filter strutures Effet of round off noise an be redued if the proposed LF struture is replaed by saled normalised LF based on Shur algorithm These filter strutures provide the added advantages of pipelining, parallel proessing, Givens rotation implementation using CORDIC algorithm and low power VLSI implementation Relation between AR model of WH ode and Gauss-Markov proess In transform oding, slowly varying signals are represented by Gauss Markov model [] In this type of modelling, if the distribution of observation of the urrent random variable is dependent only on one previous observation, then it represents an autoregressive proess (AR) signal model For a wide sense stationary input random vetor X =(X, X 2,,X n ) T, of unit variane, the ovariane matrix for this single step orrelation is given by, where ρ represents the orrelation oeffiient (0) Prinipal Component analysis of this ovariane matrix (ie,) Karhunen-Loeve transform gives the eigen values and their orresponding eigen vetors respetively These eigen vetors are in general arranged in the order of inreasing eigen values The resulting eigen vetors represent unitary spreading odes In [0], KLT bases generated for low values of orrelation oeffiient from this ovariane matrix an be used in the design of spreading odes It has been shown that these varying power spreading odes, when applied in DS-CDMA ommuniations, performs omparable to or marginally better than the widely used Gold odes and outperforms Walsh odes in AWGN and Rayleigh flat fading hannel onditions By applying the same proedure as disussed in Setion to these eigen vetors, a set of binary spreading odes an be generated In this paper, the autoorrelation values of these eigen vetors are alulated and used to find the refletion oeffiients of the lattie filter struture It is observed that the binary ode set generated from the KLT bases of the Gauss Markov signal model realised by the proposed lattie filter struture are nearly the same as the WH odes Numerial Example Table gives the refletion oeffiients generated for a Gauss Markov proess with orrelation oeffiient ρ=09 Table shows the deimal equivalent of the WH ode set of size 8, and KLT odes generated from a ovariane matrix of size 8 by 8 for the orrelation oeffiients taking the values of ρ=02 and ρ=09 Table Refletion Coeffiients of KLT binary ode of length 8 Code Code 8 Code Code Code 2 Code 7 Code 3 Code The order of KLT ode sequene in the table orresponds to the ode generated from the eigen vetor that orresponds to the asending eigen value It an be noted from the table that only one ode sequene is different between the WH odes and the KLT odes It an be observed that by inreasing the values of the orrelation oeffiient from 02 to 09, the number of bits in whih they differ from the original Walsh Hadamard ode set is only or 2 bits Hene the lattie filter struture representing an AR Gauss Markov proess an be used to generate Walsh-Hadamard odes Sine the lattie filter transfer funtion represents a Gauss Markov proess, this lattie filter struture an be interpreted as mathed with a slowly varying transmitted signal or slowly varying hannels Thus this struture an be used as a mathed filter reeiver or as an equaliser at the reeiving end E-ISSN: Issue 3, Volume 9, July 203

10 G Suhitra, M L Valarmathi Table Deimal values of Walsh Hadamard Codes and KLT odes of length 8 Walsh- Hadamard ode set Proposed KLT ode set for ρ= Proposed KLT ode set for ρ=09 Conlusion In software radio (SWR), the main aspet is to use flexible omponents suitable for all types of ommuniation systems The main signifiane of the proposed Walsh Hadamard Code generation is that a new ode an be generated by just hanging the refletion oeffiients The proposed lattie filter AR signal model an also be used as a mathed filter for multiuser detetion in a DS-CDMA system Simulation results show that the BER performanes of onventional method and mathed filter method of despreading are marginally the same The AR lattie filter representing the Gauss Markov model an be used to generate the Walsh- Hadamard odes when the refletion oeffiients are hosen aording to the signature ode of the partiular user In general, interferene and hannel noise effets in wireless ommuniation reeivers are redued by using equalisers implemented using filter strutures Reonfiguring the LF to at as an equaliser for other wireless standards suh as OFDM, MC-CDMA et and integrating the equaliser and spreading operations in the LF struture by optimising the refletion oeffiients at the reeiving end in a DS-CDMA system is a further extension of this work This work an be extended to implement the generation of the various spread spetrum odes suh as Gold ode, Kasami ode and multilevel integer odes using LF Referenes: [] Lokesh BThiagaraajan, Samir Attallah, Ying- Chang Liang, Reonfigurable Transeivers for Wireless Broadband Aess Shemes, IEEE Wireless Communiations, June 2007, pp8-3 [2] LAlaus, JPaliot, CRoland, YLouet, DNoguet, Promising Tehnique of Parameterization for Reonfigurable Radio, the Common operators Tehnique: Fundamentals and Examples, Journal of Signal Proessing System, Springer, 20 [3] Ken Chapman, Paul Hardy, Andy Miller, and Maria George, CDMA Mathed Filter Implementation in Virtex Devies, Appliation Note: Virtex Series, XILINX, Marh 2000 [] Jie Chen, Guoliang Shou, Changming Zhou, High speed low power omplex mathed filter for W-CDMA:Algorithm and VLSI Arhiteture, IEICE TransFundamentals, January 2000, Vol E83-A,No [] Qinghua Shi, Yong Liang Guan, CLLaw, Channel-Mathed Spreading odes for the Downlink of MC-CDMA, ICC 2007 Proeedings, pp [6] Monson H Hayes, Statistial Digital Signal Proessing and Modelling, John Wiley & Sons, 2002 [7] Todd K Moon, Wynn CStirling, Mathematial Methods and Algorithms for Signal Proessing, Pearson Eduation [8] Simon Haykin, Digital Communiations, John Wiley & Sons [9] Proakis J G,Digital Communiations, Third Ed MGraw Hill 99 [0] Ali NAkansu, Radha Poluri, Design and Performane of Spread Spetrum Karhunen- Loeve Transform for Diret Sequene CDMA ommuniations, IEEE Signal Proessing Letters, Deember 2007, Vol, No2, pp [] Edward RDougherty, Random Proesses for Image and Signal Proessing, Prentie Hall of India Private Limited, 2003 [2] Keshab KParhi, VLSI Digital Signal Proessing Systems Design and implementation, John Wiley & Sons,999 [3] AliNAkansu,Radha Poluri, Walsh-Like Nonlinear Phase Orthogonal Codes for Diret Sequene CDMA Communiations, IEEE Transations on Signal Proessing, (7), July 2007, E-ISSN: Issue 3, Volume 9, July 203

11 G Suhitra, M L Valarmathi APPENDIX Data BPSK modulator X BPSK demodulator Polar onverter Lattie filter WH ode generator j(t) Deision devie Integrator Fig7 Proposed DS-CDMA system E-ISSN: Issue 3, Volume 9, July 203