Channel Estimation for OFDM In Time-Variant Multi-path Environment

Channel Estimation for OFDM In Time-Variant Multi-path Environment1 Zheng Li, Xia Lei, Wanbin Tang, Yue Xiao and Shaoqian LiNational Key Lab of CommunicationUniversity of Electronic Science and Technology of China, Chengdu, P.R.CAbstract—In this paper, we address the problem of chan n el estimation in time-variant multi-path environment in Orthogonal frequen cy division multiplexin g systems(OFDM). Based on the assumption that the channel varies in a linear fashion during an OFDM block duration, a n ovel iterative chan n el estimation arithmetic with n oise an d in terferen ce suppression usin g pilot ton es is in vestigated. Through iterative n oise an d in terferen ce suppression to determine the positions of channel taps, then we compute the time averages an d slopes of that to get the time domain chan n el matrix, so such method can improve the precision of the estimation. To in crease the spectral efficien cy and reduce the computational complexity, assuming the positions of chan n el taps are n ot chan ged in on e frame duration, a frame based on such estimation scheme is presen ted. Alon g with the channel estimation technique, we also analyze the optimum pilot ton es placemen t. Theoretical an alysis an d simulation results show that our assumption is reason able, an d the proposed channel estimation arithmetic has a good performance with low computational complexity and high spectral efficiency.In dex Terms—OFDM, channel estimation, iterative, time-variantI.I NTRODUCTIO NOrthogonal frequency division multiplexing (OFDM), due to its high data rate transmission capability with high bandwidth efficiency and robustness to multi-path delay, has recently been widely used in wireless communication, and examples are digital audio broadcasting, digital high-definition television broadcasting[1]. H owever, one of the major disadvantages of OFDM is the sensitivity of its performance to frequency offset resulting from Doppler shift due to a mobile environment, which causes intercarrier interference (ICI)[2].Channel estimation in time-vary environment is important for both intercarrier interference and coherent demodulation, and some methods have been proposed. Though blind channel estimation technique tries to estimate the channel without any knowledge of transmitted data, it requires large mount of received data and its computational complexity is unacceptable[3]. Zijian Tang[4] bases on BEM model and Yasamin Mosto[5] assumes the channel varies in a linear fashion during a block period to estimate channel state information using pilot tones. In the long channel length scenarios, however, such methods require too many pilot tones and their results are influenced severely by noise and interference, especially in the low received signal-to-noise plus interference ratio (SNIR).Based on the assumption in this paper, that the channel is approximated to variation in a linear fashion during an OFDM block, a frame is proposed based on channel estimation arithmetic with iterative noise and interference suppression, which makes the channel estimation more accurate, spectral efficiency increased and computational complexity reduced.The organization of this paper is as follows. System modelis described in Section II. Proposed time-variant channel estimation arithmetic is provided in Section III. Simulation results are discussed in Section IV, and conclusions are givenin Section V.Notation: L denotes the maximum normalized length of channel; N denotes the FFT size of an OFDM block; + denotesM-P inverse; A denotes a matrix and A H is the conjugated transpose of A;a denotes a vector , diag(a) is a diagonalizedmatrix by placing a at diagonal while off-diagonal elementsare set to be zero;T a denotes the transpose of a ; mean(a) denotes the average of the values in vector a;A[a] denotes the submatrix within A, defined by the index vector of desired columns in a and all the rows are desired.II.S YSTEM M ODELIn the OFDM system, pilot insertion block is used to addpilots into the information data sequences to form N sequences.Then through IDFT block, the sequences {()}nX m are transformed into time domain signal{()}nx k with the following equation:211()(),01j kmNNn nmx k X m e k NNπ−==≤≤−¦ (1)Where N is the FFT size, ()nx k denotes the k th sample of then th OFDM block in a time domain.Assuming that the maximum normalized channel length isL, the time domain received signal can be written as1()(,)()(),01Ln n n nly k h k l x k l z k k N−==−+≤≤−¦ (2)where ()nz k denotes the channel additive white Gaussiannoise(AWGN) of the k th sample in the n th OFDM symbol, and(,)nh k l represents the l th tap of the k th sample in n th OFDM block, andnh can be expressed as(0,0)0(0,1)(0,1)(1,1)(1,0)0(1,3)(1,2)(1,1)(1,2)(1,0)000(1,1)(1,0)n n nn n n nn n nn nh h L hh h h hh L L h L L h Lh N L h N−ªº«»«»«»=«»−−−−−«»«»«»−−−«»¬¼"""##%%##""###%##""nh(3)By taking FFT of the time domain received signal, the frequency domain received signal is obtained asn n n n n nY y X Z X Z==+=+Hn nW Wh W F (4)Where [(1),,()]Tn n nY Y Y N="and [(1),,()]Tn n nX X X N="denote the n th received and transmitted block in frequency domain respectively.nZ represents AWGN of the n th OFDM block.W is N-point FFT matrix.nh is the N N×time domainSupported by National Natural Science Foundation of China (NSFC) (No. 60496313 and 60602009)channel matrix of the n th block, and =H n n F Wh W denotes frequency domain channel matrix of the n th block. As to the k th sample of the n th OFDM block ()n Y k 10()(,)(,)()()0-1N n n n n i i k Y k k k X k i X i Z k k N −=≠=++≤≤¦n n ICIF F(5)Where the first item is the expected signal and the second represents the ICI.When the multi-path channel is time-invariant during one ODFM block, n h is a circulant matrix, and n F is a diagonal matrix, then ICI term of (5)is zero.H owever, if the Doppler frequency arises, and channel varies rapidly, then ICI appears, and n h will not be a circulant matrix any more.III.T IME -V ARIANT C HANNEL E STIMATIONWhen the normalized Doppler frequency is below 0.2, linear approximation is a good estimation of channel time-variations [5][6][7]. Therefore time variations of channel impulse response (,)n h k l can be approximated by straight lines withslopes during an OFDM block period. Due to suchapproximation, the time domain channel matrix of the n thblock (3) can be expressed as [8]=+∂n n n h v İ (6)Where,0,1,1,1,0,3,2,1,2,0,1,000000n n L n n n n n n L n L n n L n v v v v v v v v v v v v −−−−ªº«»«»«»=«»«»«»«»«»¬¼n v """##%%##""###%##"" (7) ,0,1,1,1,0,3,2,1,2,0,1,000000n n L n n n n n n L n L n n L n εεεεεεεεεεεε−−−−ªº«»«»«»=«»«»«»«»«»¬¼n İ"""##%%##""###%##"" (8)((1)/2,(3)/2,,(1)/2)diag N N N ∂=−−−−−" (9)n v and n İdenote the time averages and slopes of the channel impulse response of the n th OFDM block, and they are both circulant matrix. So if the 2L parameters: ,0,1,,n n L v v −",,0,1,,n n L εε−"are estimated, n h will bedetermined. A.Pilot –Assisted MethodAssuming 2P L ≥pilot tones are placed at subcarriers 1[]P p p p =", the received signal at pilot subcarries is 10,,10,()(,)()(,)()(,)()+z (),1kN n k k k n k k n i i p i p N n n k i i pY p p p X p p i X i p i X i p k P−=∈≠−=∉=++≤≤¦¦n n n F F F (10)Defining,,1[(),,()]T n p n n P Y Y p Y p =",,1[(),,()]T n p n n P X X p X p =",,1[(),,()]T n p n n P U U p U p =".We can obtain,,,n p n p n p Y X U =+H p n p W h W (11)Where p Wis the FFT matrix at positions of pilot subcarries.,n p U is the ICI due to data and noise, which can be expressed as10,()(,)()(),1N n k k n n k i i p U p p i X i Z p k P −=∉=+≤≤¦n F (12)Based on the former analysis, Chen [7]has given the solution of n h , however, in order to do further study , we must give more predigested expression. Substituting (6) into(11), we obtain(),__,_,__,[][][][][][][][]n p n e n e n pn e n pn e n e n p Y J v J J J U v J J J U J h U εε=++§·=+¨¸©¹=+n n n n n A B A B G(13) ,()n p X =n p A W (14) =∂H n p p n B W W A (15) _,0,1[,,,0,,0]T n e n n L v v v −="" (16)_,0,1[,,,0,,0]T n e n n L εεε−="" (17) ()[][]J J =n n n G A B (18) ___[][]n e n e n e v J h J ε§·=¨¸©¹ (19) Where, _[]n e v J and _[]n e J εdenote the subvectors within _n e v and _n e εdefined by the index vector J , where [0,1,,1]J L =−".Considering the computational complexity, the least square solution is obtained as _,n e n p h Y =+n G (20)Through equation(20), we can get _n e v and _n e ε, and then n h is obtained. But such solution has some disadvantages as follows:1) 2L pilot tones are required per block at least. If the maximum normalized channel length L is large, many pilot tones will reduce spectral efficiency severely.2) Though the channel length is L , only a small part of which are channel impulse responses (channel taps), the others are noise and interference. Such solution will be influenced severely by noise and interference, especially in the low received SNIR.B.Noise/Interference SuppressionIn order to improve the precision of estimation, a novel channel estimation arithmetic with noise and interference suppression is proposed. We can do as follows:Step ing formula(20), get time averages (_[]n e v J ) and slopes (_[]n e J ε) of the channel taps of the n th OFDM block. If no iteration, then compute the maximum value of _[]n e v J ,labeled T , and set Threshold (0)h T =T /E 1.Step pute _(()()),a n e h T mean v i T u i J =<∈, where uis the times of iteration. Set T h (u+1)=E 2Ta .Compare_()n e v i with (1)h T u +, if the former is bigger, then we label it channel tap, otherwise, labeled noise and interference. Therefore, the positions of channel taps 1,,L Q Q ′!are obtained,labeled []1L Q =Q ,Q ′".Step3. Update J with Q , go back to Step1. After iterating several times, the taps due to noise andinterference will be suppressed, the time averages and slopes of channel taps will become more accurate. Then the time domain channel matrix n h is determined.The optimum way to define E 1 and E 2is to relate them to the received SNIR. But estimation of the received SNIR may not be feasible in a high mobile environment. On one hand, choosing small E 1 and large E 2will increase the chance of losing channel taps with significant values and only improves the performance in low received SNIR, whereas large E 1 and small E 2 will reduce the risk of losing channel taps with considerable values at the price of less efficiency in high noise/interference cases. So, it is better to choose E 1 and E 2such that losing taps below Threshold doesn’t introduce considerable performance loss. C.Frame Based Channel EstimationAt the transmitter, usually several OFDM blocks form aframe, and then are transmitted. Duration of one frame isseveral milliseconds generally, so we can assume the positions of channel taps (Q ) are not changed within this duration. On this assumption, if Q of the first OFDM block is determined, that of other blocks is fixed. So we can use iterative channel estimation arithmetic with noise and interference suppressionto decide the positions Q of the first block. Once Q isdetermined, the time averages and slopes of other blocks canbe computed directly according to(20), without iteration. Therefore such method can reduce the computational complexity.Because the first block is needed to determine the positions of the channel taps, 2L pilot tones are required at least. The other blocks are needed no less than 2L ’ pilot tones just to compute the _[]n e v Q and _[]n e Q ε, where L ’ is the amount of channel taps. Generally, L ’ is much smaller than L , such method can also increases the spectral efficiency.What’s more, we can use time domain channel interpolation to reduce the amount of pilot tones [9].D.Placement of Pilot TonesIn time invariant frequency-selective channels, the equally-spaced pilot tones should be optimal [3].But in the time-variant channels, pilot tones of equispaced groups have proved to be the optimal [7][10].In our proposed arithmetic, pilot tones placement should minimize ,n p U in(11). Frequency domain channel matrix n F has main values at diagonal and nearby, so group by pilot tones can minimized ,n p U and pilot tones placed in equispaced groups is optimal. IV.S IMULATION RESULTSTo demonstrate the effectiveness of the proposed approach for time-variant multi-path channels, we compare proposed channel estimation method with conventional two methods:chen’s method [7] and DFT interpolation method [11]. Define the average normalized mean square error (ANMSE) of channel estimation as [7]11200112100ˆ(,)(,)1ANMSE (,)N L M k n N L i k n k n k n M k n −−==−−===­½−°°°°=®¾°°°°¯¿¦¦¦¦¦n n n h h h (21) Where ˆ(,)k n n h denotes the estimated value of (,)k n nh ,M denotes the amount of OFDM blocks, and N denotes the FFT-Size. In our simulations, system parameters are summarized in Table 1.Table 1 Parameters of the Simulation SystemParameters Values Bandwidth 10M Modulation 16-QAM FFT-Size(N) 512 Cyclic prefix144The parameters of the selected six-path Rayleigh fading channel are listed in Table 2. Table 2 Parameters of ChannelPathNumber Relative Time delay(us )Relative Power Ratio 1 0 02 2 -634 -12 4 6 -185 8 -246 10-30 Table 3 Spectral efficiency of different schemesSchemeηScheme1 50% Scheme2 50%Scheme3 50% Scheme4 78.75% For Table 2, the normalized maximum channel length L is101. So in Chen’s method, the pilot tones of each OFDM block are no less than 202. We get the simulations in the followingways:1)Scheme1: DFT interpolation estimation method,256pilot tones per OFDM block are equispaced on the FFT grid: {0,2.4,Ă,510}2)Scheme2: Chen’s method,256 pilot tones per OFDM block are equispaced in 64 groups and 4 tones each group: {0,1,2,3,8,9,10,11,Ă,504,505,506,507}3)Scheme3: proposed channel estimation method, 256 pilot tones per block are equispaced in 64 groups and 4tones each group: {0,1,2,3,8,9,10,11,Ă,504,505,506, 507} Fig.1 gives the ANMSE of Scheme3 under different thresholds when normalized Doppler frequency is 0.05. Using such thresholds, the performance of once iteration has greaterimprovement than no iteration, especially in low received SNR.And the performance keeps the same nearly over there are more time iterations. We can also see that Threshold 2 has the better performance than Threshold1 when no iteration, after iteration, the difference becomes small, but Threshold2 alsoperformances better. In the following simulations usingScheme3, we choose Threshold 1.Fig.2 and Fig.3 show the ANMSE and bit error rate (BER) of different channel estimation methods using zero-forcing (ZF) detection respectively. We can see that Chen’s method even has worse performance than the DFT interpolation method, because the former influences by noise and interference severely, especially in low receiver SNR. And our proposed method has the best performance.In order to increase the spectral efficiency and reduce the computational complexity, frame based channel estimation method is proposed. We define the Scheme 4 asScheme4: proposed frame based channel estimation method,10 block per frame, 512 pilot tones of first block 64 pilot tones of the other blocks are equispaced in 16 groups and 4tones each group: {12,13,14,15,44,45,46,47,Ă, 492, 493, 494,495}Let’s define spectral efficiency η the ratio of data subcarriers (total subcarriers subtracts subcarriers used for pilot tones ) and total subcarriers per frame.Fig. 4 shows that Scheme3 and Scheme4 almost have thesame performance of ANMSE, and we can conclude that ourassumption that the positions of channel taps are not changed in one frame duration is reasonable.From Table 3, we can get that Scheme4 has higher spectral efficiency than Scheme3. Considering the computational complexity, we must compute the M-P inverse of n G u times, matrix multiplication of +n G and ,n p Y u +1 times, noise and interference suppression u times per OFDM block for Scheme3;whereas for Scheme4, only the first block per frame need to do that, the other blocks only need matrix multiplication of +n G and ,n p Y once. To sum up, the proposed frame based channel estimation method has good performance with highspectral efficiency and low computational complexity.V.C ONCLUSION An novel channel estimation arithmetic with noise and interference suppression has been analyzed in multi-path timevariant environment. Through iterative method of estimation, noise and interference can be suppressed, and the positions ofchannel taps can be determined, then we compute the time variations and slopes to determine the time domain channel matrix, and such method can improve the performance ofestimation. Assuming the positions of the channel taps are notchanged in one frame duration, frame based channel estimation method has also been investigated. Using which, computational complexity will be reduced and spectral efficiency will beincreased. Simulation results have proved that the proposedmethod has a better performance.SNR(dB)A N M S E (dB )Fig. 1 Performance of ANMSE using Scheme3: normalized Dopplerfrequency equals 0.05. (Threshold 1: E 1 = 10, E 2=2; Threshold 2: E 1 = 20, E 2=4)1012141618202224262830--SNR(dB)A N M S E (dB )Fig. 2 Performance of ANMSE using different channel estimation methods:normalized Doppler frequency equals 0.0530101214161820222426281010-10-10SNR (dB)B E RFig.3 Performance of BER using different channel estimation methods: normalized Doppler frequency equals 0.05-SNR (dB)A N M S E (dB )Fig. 4 Performance of ANMSE using Scheme3 and Scheme4: normalized Doppler frequency equals 0.05R EFERENCES[1]Yun Hee Kim; Iickho Song; Hong Gil Kim; Taejoo Chang; Hyung Myung Kim, “Performance analysis of a coded OFDM system in time-varying multipath Rayleigh fading channels”, IEEE Trans. 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