Comparative Analysis of Equalization Methods for SC-FDMA

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1 Comparative Analysis of Equalization Methods for SC-MA Anton ogadaev, Alexander Kozlov SUA Russia {dak, Ann Ukhanova U otonik enmark annuk@fotonik.dtu.dk Abstract n this paper we introduce comparative analysis for different types of equalization schemes, based on the minimum mean square error (MMSE) optimization. he following types of equalizers were compared: linear equalization, decision feedback equalization (E) and turbo equalization. Performance and complexity of these schemes were tested for Single Carrier requency ivision Multiple Access (SC-MA) system with Single nput Single Output (SSO) antenna configuration. SC-MA is a common technique, which is used in the URA LE Uplink, so the results of complexity and performance analysis could be applied to find the appropriate equalization algorithm to be used in the Uplink channel of the LE the famous standard in 4G telecommunications. Simulation results in the end in this paper show bit error ratio (BER) and modulation error ratio (MER) for compared schemes.. NROUCON 3GPP Long erm Evolution (LE) is the telecommunication network of the next generation, following 3G. he main advantages of this new technology are high data rate, low latency and packet optimized radio access technology. he LE specification provides uplink peak rates of at least 50 Mbps in 0 MHz system bandwidth. or the realization of LE Uplink Single Carrier requency ivision Multiple Access (SC-MA) transmission is used. he reason of choosing this technology is that SC-MA has sufficiently low Peak-to-Average Power ratio (PAPR) of signals in comparison with Orthogonal requency ivision Multiple Access (OMA) transmission. t results in significantly lower power consumption in the user equipment (UE). One of the actual problems in this area is to provide the reliable transmission over the LE, and for this reason it is necessary to choose the equalization methos for the received signal. he main goal of this paper is to analyze and compare the performance for the existing equalization methods for SC-MA. We focus mainly on turboequlization as a primary solution, and the structure of investigated turbo equalizer bases on the idea introduced in [1]. n this work we also utilize improved adaptive coefficients solution based on the technique suggested in []. he paper is organized as follows. n Section we provide the mathematical derivation of the equivalent channel impulse response for the SC-MA. Section describes the equalization schemes. n Section V the simulation resuls are represented and discussed.. SYSEM MOEL n this section we derive a mathematical equation of the equivalent channel impulse response for the SC-MA.

2 CP OM-symbol CP OM-symbol igure 1. CP usage in SC-MA symbol One slot of information data may consist from one to several Resource Blocks (RB) and each RB has 1 orthogonal subcarriers. he channel resource allocation may be performed dynamically for each subframe of data depending on individual channel features. SC-MA may be considered as OMA, which is appended with block for each subcarrier. his conception provides power efficient signal transmission, but, on the other hand, it brings the interference component to the transmitted signal in frequency domain from neighboring subcarriers. Orthogonality between the subcarriers is maintained by use of Cyclic Prefix (CP) (see ig. 1). t prevents nter-symbol nterference (S) between SC- MA information blocks and transforms the linear convolution of the multipath channel into a circular convolution, enabling the receiver to equalize the channel. Signal transmission in SC-MA can be written as a block diagram in matrix form (see ig. ). On the transmitter side the signal is converted to the frequency domain by of the size M. After that frequency domain symbols are mapped on the localized or distributed subcarriers. After conversion to the time domain and CP insertion signal is transmitted through the channel, which is simulated as the Rayleigh fading channel. nverse process is performed on the receiver side. Lets define SC-MA symbol transmitted to the channel as a column vector s = [s P,..., s 0,..., s N 1 ], where P is CP length, N is the size of the block at the transmitter (or the total amount of available subcarriers). SC-MA symbol received from the channel is denoted as r = [r P,..., r 0,..., r N 1 ]. Lets define multipath channel impulse response by the vector of coefficients {h l } l=0,...,l, then each component of the vector r is written as L r i = h l s i l + w i, i = P,..., N 1, (1) or in the matrix form as l=0 r = Hs + w, () where w Gaussian noise vector. n () H denotes the (P + N) (P + N) matrix of channel response linear convolution: h h H = h L h L.... (3) h L h 0

3 S / P P / S S / P Sub Map Sub emap P / S x 1 1 nsert CP s Channel y 1 1 Y R Remove CP + w r igure. Block diagram of SC-MA ata symbols at the input of the transmitter are denoted as the vector x = [x 0,..., x M 1 ], where M is the number of transmitted data symbols in one SC-MA block (it is similar to the number of subcarriers allocated for one user and defines the size of the block at the transmitter side). 1 denotes the M M matrix, i.e. 1 (p+1,q+1) = 1 M e j π M pq, p, q = 0,..., M 1 (4) 1 1 denotes N N matrix. t should be mentioned that scaling factor M must be 1 used instead of N to maintain the same output signal power. is the N M matrix that maps mth frequency domain data symbol to the nth available subcarrier, where m = 0,..., M 1 and n = 0,..., N 1. he mapping matrix for localized subcarrier distribution is defined as follows: { 1, n = RAU M + m (n+1,m+1) = (5) 0, otherwise where RAU = 0,..., N 1. M t means that the equation of transmitted to the channel data block may be written as follows: s = 1 1 x (6) On the receiver side inverse process is used. So, the equation of the output signal is y = 1 1 Rr, (7) where and R are the matrices of insertion and removing of CP, which may be defined as [ ] CP =, (8) N R = [ O N P N ], (9) where N is N N identity matrix, C P is P N matrix that copies the last P rows of N, O N P is the N P null matrix.

4 requency offset fs f s igure 3. requency offset of SC-MA data blocks to the half of subcarrier band gap After substituting the (), (6) into (7), we will get the equation of the SC-MA equivalent channel: y = 1 1 (RH ) 1 1 x + w, (10) where w is the effective noise at the output of the receiver: w = 1 1 Rw (11) n practice the spectrum of the SC-MA data block should be zero symmetrical. or this purpose the frequency offset (see ig. 3) to the half of subcarrier band gap f is used. When the sampling rate is f s the frequency offset is: φ n f jπ = e t = e jπ f n fs n jπ fs = e N fs = e jπ n N (1) he use of CP in the SC-MA data blocks transforms the matrix H of linear convolution to circular one. Lets define matrix H 0 as h h L... h 1.. h h L h L H 0 = RH =. h L h L (13). 0 h L h L h L 1... h 0 H 0 is the circulant matrix and may be diagonalized by the following property: H 0 = H 0 1 = diag { h0, h 1,..., h } N 1 Matrix H 0 corresponds to the equivalent channel impulse response in the frequency domain, i.e. L h k = h l e j π N kl (15) l=0 After subcarrier mapping/demapping procedure the channel impulse response may be written in the following form: H = H0 (16) and H keeps to be diagonal and is set by the elements from H 0 as: (14) H (m+1,m+1) = H 0 (n+1,n+1), (17)

5 nput data bits Convolutional encoder nterleaver M-PSK/M-QAM mapping Sub Map nsert CP SC-MA symbol SC-MA igure 4. SC-MA transmitter with convolutional coding and interleaving where the values of m and n corresponds to (5). Matrix H defines the equivalent impulse channel response for each user subcarrier. As H is diagonal matrix, it could be written as composition of circulant matrix and / matrices: H = 1 H 1 1 (18) rom this follows that (10) may be rewritten as ( ) y = H x + w = H x + w, (19) where H is the matrix of the equivalent channel response in the URA LE Uplink with the following coefficients: h l = 1 M M 1 k=0 h ke j π N kl, l = 0,..., L, (0) where L is the length of the equivalent channel response in the Uplink and may be estimated, according to [4], as: L = (L + 1) M 1 (1) N. ESCRPON O HE USE EQUALZAON SCHEMES o provide the correct comparison of equalization schemes, we need to use the same coding in all of them. n the turbo equalization it is necessary to use error-correcting code. n this work convolutional coder with rate 1 from [5] is used. n this case the structure of the SC-MA transmitter should be changed according to ig. 4. As SC-MA may be interpreted as OMA proceeded by a block, the primitive equalization schemes may be deployed there without significant difficulties. or example, in this paper we use linear (LE) [6] frequency domain MMSE [6] equalization (see ig. 5), that is used as a starting point to more advanced schemes. MMSE criteria gives the following coefficients for LE: Hk P LE,k =, () H k + SNR 1 where H is the estimation of the channel impulse responce in frequency domain (it is calculated according to [3]), SNR is Signal-to-Noise ratio.

6 P LE,1 y P LEM, Sub emap Remove CP r igure 5. Linear equalization for SC-MA system r Sub emap P E1,1 P E,1 y Y y ~ + ŷ P E 1, M P E, M g fb CP igure 6. E structure for SC-MA system ecision feedback equalization (E) [7] is another solution that performs better than LE due to its ability to cancel nter-symbol nterference (S) component of the received signal with the help of previously received data symbols [6]. n this paper we use approach from [4], which introduces E scheme for SC-MA (see fig. 6) with frequency-domain feedforward () filter and a time-domain feedback (B) filter. he derivation of the coefficients is done in [7] and here we provide only the results for reading simplicity. he B coefficients may be obtained by solving the equation A MMSE g B = b MMSE, where matrix A MMSE and vector b MMSE are given by he coefficients are [A MMSE ] i,l = M 1 k=0 e j π M k(l i) H k + SNR 1, 1 i, l L (3) M 1 e j π M ki [b MMSE ] i = H k + SNR, 1 i 1 L (4) P E,k = k=0 H k H k + SNR 1 (1 + G B,k), (5) where G B = (g B ) and P E1,k is the first term in the (5), P E,k is the second one. Based on advanced iterative equalization and error-correcting decoding technique turbo equalization [8], [9] allows to significantly increase performance of the data transmission

7 y Y Sub emap P y~ P M Q 1 Q M ~ y L( b i,j ) s ˆ = E( i k s k ) einterleaver i L (bk,j) Pa ( i s k ) nterleaver L ( b l ) SSO decoder L A ( b l ) ecoded bits igure 7. urbo equalization for SC-MA system over a frequency selective fading channel. he approach (see ig. 7), described in this paper, is based on the turbo equalizer for SC-MA for Single nput Multiple Output (SMO) antenna configuration, which is introduced in [1] and uses adaptive coefficients. or testing purposes SSO antenna configuration is used. On the each iteration of the scheme, algorithm calculates log-likelihood ratios (LLR) of all the coded bits as follows: L(b k,j ) min ɛ i k :b i,j=0 ɛ i k min ɛ i k :b i,j=1 ɛ i k, (6) σw where σw is the power of the noise in channel, b i,j is the j-th bit of the i-th constellation point, k is the index of the transmitted symbol and ɛ i k denotes the squared Euclidean distance of the equalizer output to constellation point. hen algorithm calculates improved LLRs L (b l ) with the help of SSO decoder, based on the BCJR algorithm [5]. With the help of those improved LLRs L ](b l ) is possible to calculate the (N M) apriori symbol symbol probability matrix [P a (s ik ) [1] and compute the estimation of the transmitted symbol [1]: ŝ k = E ( ) s i k = M s i P a (s i k) (7) i=1

8 r n ~ P ( ν ) s n ~ Q( ν) q 0 0 = dn igure 8. Structure of infinite length equalizer he theoretical transfer function of the infinite length linear MMSE equalizer with a priori information could be used to derive the adaptive coefficients for turbo equalizer. he calculation is made in [] and here we introduce only the results. he equalizer consists of two filters, which are used for signal r n received from the Uplink channel and its soft estimation d n (see ig. 8). he theoretical transfer function for these filters are derived as follows: Q(ν) = P (ν) H(ν) λβ (8) where P (ν) = β = γ σ d σ d ( H (ν) ) 1 σ d H(ν) σ + σ d w σd, (9) H(ν) ( σ d σ d) dν (30) H(ν) + σ w γ = σ d λ = 1 + βσ d (31) H(ν) ( ) 1 σ d dν, H(ν) σ + σ d w σd (3) where σ d is the power of the signal estimation on the each iteration of the turbo equalizer, σ d is the power of the transmitted signal in the frequency domain, σ w is the power of the Gaussian noise in the channel and is the sampling time. V. SMULAON RESULS or equalization process simulation in the URA Uplink LE channel the typical parameters (able ) were chosen according to [3]. or performance evaluation of the equalization schemes, that were described in this paper, BER and MER values were calculated for different values of SNR. Also, for the purpose of the experiment s accuracy, different lengths of the channel impulse response values were used.

9 able PARAMEERS USE N HE SMULAON PROCESS. Carrier bandwidth 5 MHz size 51 Sampling rate 7,68 MHz Number of subcarriers 300 Number of RBs 5 Bandwidth efficiency 90% Modulation QAM-16 Lets define the SNR formula: SNR = 10 log 10 E s σw, (33) where E s is the average power of the signal, which is transmitted in the channel, σw is the power of the Gaussian noise. MER may be defined as follows: i MER = 10 log x i 10 j x j y j, (34) where x i is the transmitted constellation point, y i is its estimation at the output of equalizer. or the sake of making the correct comparison of the simulation results of turbo equalizer with linear and E schemes, convolutional encoder [5] and maximum a posteriori (MAP) decoding [5] were added to the last ones. rom the results of MER and BER performance comparison (see ig. 9) could be seen, that E scheme provides better performance than linear equalization, when the length of the channel impulse response grows up. Moreover, the length of the feedback (B) filter N B may be set less than the length of the channel impulse response without significant losses in the efficiency of the E scheme. Results from the ig. 9 shows that deployment of the turbo equalization scheme in the SC- MA allows to increase the efficiency of the transmission up to 3 db for the error probability 10 3 in comparison with E and linear schemes. rom the series of the simulations that were done during the experiment we conclude that the gain from turbo equalization does not increase significantly after 3 iterations. Moreover, it may decrease due to the the disadvantage of error propagation existence in turbo equalization. V. CONCLUSON his paper demonstrates the comparison of linear, E and turbo equalizers. or these purposes the simulation model of the SC-MA transmission was developed and tested for presented equalization schemes. he coefficients were estimated based on the MMSE technique. A E gives better performance due to its ability to remove inter symbol interference (S) component from the output signal, but suffers from error propagation as well as turbo equalizer does. he best performance is introduced by turbo equalization due to the improvement of the equalized signal estimation on the each iteration and better cancelation of the S component. he results of experiments show that usage of -3 iterations is enough to get significant increase in the performance of the SC-MA system. he use of turbo equalization in the

10 igure 9. BER and MER comparison for linear, E and turbo equalizers in SC-MA Uplink channel of the URA LE network will result in extra receiver complexity, but isolated from the base station, which does not have strong power constraints. Since this is viable, SC- MA with turbo equalization can be considered for use in the Uplink channel of the URA LE network to increase system performance and efficiency. REERENCES [1] G. Berardinelli, B. E. Priyanto,. B. Sorensen, P. Mogensen, mproving SC-MA Performance by urbo Equalization in URA LE Uplink, EEE VS Vehicular echnology Conference. Proceeding, pp , 008. [] C. Laot, R. Le Bidan,. Leroux, Low Complexity MMSE urbo Equalization: A Possible Solution for EGE, EEE ransactions on Wireless Communications, vol. 4, no. 3, May 005. [3] S. Sesia,. oufik, M. Baker, LE he UMS Long erm Evolution: rom heory to Practice, John Wiley & Sons, 009.

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