CoherentDetectionof OFDM

Transcription

1 Telematics Lab IITK p. 1/50 CoherentDetectionof OFDM Indo-UK Advanced Technology Centre Supported by DST-EPSRC K Vasudevan Associate Professor vasu@iitk.ac.in Telematics Lab Department of EE Indian Institute of Technology Kanpur

2 Telematics Lab IITK p. 2/50 Motivation forcoherentofdm For a given bit-error-rate, coherent OFDM requires least signal power This translates to longer battery life Energy efficient (green) communication technology The full system design has not been considered earlier in the literature

3 Telematics Lab IITK p. 3/50 Notation Complex quantities denoted by tilde (e.g. s n ) Subscript n denotes time index s 1, n denotes a known preamble of length L p h k, n channel response for the k th frame C N (0, σf 2 ), independent over n channel length L h w k, n AWGN for k th frame C N (0, σw), 2 independent over n

4 Telematics Lab IITK p. 4/50 SystemModel Frequency selective Rayleigh fading channel having a uniform power delay profile is assumed Data is divided into frames, QPSK modulation Channel is static over one frame, span is L h Channel span assumed by the receiver is L hr (> L h )

5 Telematics Lab IITK p. 5/50 SystemModel contd S 1, 0 Preamble QPSK symbols S 1, Lp 1 S k, 3, 0 Data QPSK symbols IFFT IFFT s 1, 0 s 1, Lp 1 s k, 3, 0 s k, 3, Ld 1 Parallel to serial and add cyclic prefix S k, 3, Ld 1 To ỹ k, n Channel s k, n receiver h k, n w k, n (AWGN) e j(ω kn+θ k )

6 Telematics Lab IITK p. 6/50 SystemModel contd s 1, n s k, 2, n s k, 3, n Preamble L p Cyclic prefix L cp Data L d s k, n Observe that Preamble length is L p Length of cyclic prefix is L cp = L hr 1 Length of data is L d Total length of the frame is L = L p + L cp + L d

7 Telematics Lab IITK p. 7/50 SystemModel contd Note that (1) s 1, n = 1 Lp 1 L p i=0 s k,3, n = 1 Ld 1 L d i=0 S 1, i e j 2πni/L p S k,3, i e j 2πni/L d.

8 Telematics Lab IITK p. 8/50 SystemModel contd We assume S k,3, i ±1 ± j. Since we require: [ E s 1, n 2] [ = E s k,3, n 2] = 2/L d = σ 2 (2) s We must have S 1, i L p /L d (±1 ± j).

9 Telematics Lab IITK p. 9/50 SystemModel contd The received signal for the k th frame can be written as (for 0 n L + L h 2): r k, n = ( s k, n h ) k, n e j(ω kn+θ k ) + w k, n (3) = ỹ k, n e j(ω kn+θ k ) + w k, n where denotes convolution and (4) ỹ k, n = s k, n h k, n. ω k [ 0.04, 0.04] radians, θ k [0, 2π) radians

10 Telematics Lab IITK p. 10/50 SystemModel contd The set of received samples can be denoted by the vector: [ ] (5) r k = r k,0... r k, L+Lh 2. For the purpose of SoF and coarse freq offset estimation for the k th frame, we assume that the channel h k, n is known at the receiver

11 Telematics Lab IITK p. 11/50 SoF and CoarseFreqOffEst Define the m th (0 m L cp + L d + L h + L hr 2) received vector as: [ ] (6) r k, m = r k, m... r k, m+lp L hr. The steady-state preamble part of the transmitted signal appearing at the channel output can be represented by a vector: [ ] (7) ỹ k,1 = ỹ k, Lhr 1... ỹ k, Lp 1.

12 Telematics Lab IITK p. 12/50 SoF and CoarseFreqOffEst contd The non-coherent maximum likelihood (ML) rule for frame detection can be stated as: Choose that time as the start of frame and that frequency ˆω k, which jointly maximize the conditional pdf: (8) max m, ˆω k 2π θ k =0 p ( r k, m ỹ k,1, ˆω k, θ k )p(θ k )dθ k.

13 Telematics Lab IITK p. 13/50 SoF and CoarseFreqOffEst contd The final detection rule is: L 1 1 max r m+i ỹk, L m, ˆω k hr 1+i e j ˆω ki (9) i=0 where L 1 = L p L hr + 1 The ideal outcome of (9) is: (10) m = L hr 1 ˆω k = ω k.

14 Telematics Lab IITK p. 14/50 SoF and CoarseFreqOffEst contd In practice, the receiver has only the estimate of the channel (ĥk, n), hence ỹ k, n must be replaced by ŷ k, n, where (11) ŷ k, n = s 1, n ĥk, n. When ĥk, n is not available, we propose a heuristic method of frame detection: L p 1 (12) max m, ˆω k r m+i s 1, i e j ˆω ki. i=0

15 Telematics Lab IITK p. 15/50 SoF and CoarseFreqOffEst contd The ideal outcome of (12) is: (13) 0 m L h 1 ˆω k = ω k depending on which channel coefficient has the maximum magnitude. When m lies outside the range in (13), the frame is declared as erased (lost). Coarse freq off est ˆω k : obtained by dividing [ 0.04, 0.04] rad into B 1 freq bins and selecting that bin which maximizes (12).

16 Telematics Lab IITK p. 16/50 SoF and CoarseFreqOffEst contd 1.0e+00 Probability of frame erasure 1.0e e e e e e-06 Lp=64 Lp=128 Lp= SNR per bit (db) For L p = 512, prob of erasure is < 10 6

17 Telematics Lab IITK p. 17/50 Channel Estimation ML channel estimation is considered Assumptions: Freq offset has been canceled SoF has been detected with outcome m 0 Define (14) m 1 = m 0 + L h 1. The steady-state, preamble part of the received signal for the k th frame is: (15) r k, m1 = s 1 hk + w k, m1

18 Telematics Lab IITK p. 18/50 Channel Estimation contd where r k, m1 = w k, m1 = h k = (16) [ r k, m1... r k, m1 +L p L hr ] T [(L p L hr + 1) 1] vector [ ] T... w k, m1 +L p L hr w k, m1 [(L p L hr + 1) 1] [ hk,0... hk, Lhr 1 [L hr 1] vector ] T vector

19 Telematics Lab IITK p. 19/50 Channel Estimation contd and s 1 = s 1, Lhr 1... s 1, (17) s 1, Lp 1... s 1, Lp L hr 2 [(L p L hr + 1) L hr ] matrix

20 Telematics Lab IITK p. 20/50 Channel Estimation contd The ML channel estimate is: find ĥk (the estimate of h k ) such that: ( r k, m1 s 1 ĥ k ) H ( r k, m1 s 1 ĥ k ) (18) is minimized.

21 Telematics Lab IITK p. 21/50 Channel Estimation contd Differentiating with respect to ĥ k the result to zero yields: and setting (19) ĥ k = ( s H 1 s 1) 1 s H 1 r k, m1. To see the effect of noise on the channel estimate in (19), consider (20) ũ = ( s H 1 s 1) 1 s H 1 w k, m1. Clearly E [ũ] = 0.

22 Telematics Lab IITK p. 22/50 Channel Estimation contd It can be shown that E [ ũũ H] = 2σ2 w L 1 σs 2 (21) I Lhr = σ2 wl d L 1 I Lhr = 2σ 2 u I Lhr. Therefore, the variance of the ML channel estimate (σ 2 u) tends to zero as L 1 and L d is kept fixed.

23 Telematics Lab IITK p. 23/50 Channel Estimation contd Magnitude response of actual channel Amplitude Amplitude Magnitude response of estimated channel Subcarriers SNR per bit is 0 db, L p = 512.

24 Telematics Lab IITK p. 24/50 Channel Estimation contd Magnitude response of actual channel Amplitude Amplitude Magnitude response of estimated channel Subcarriers SNR per bit is 10 db, L p = 512.

25 Telematics Lab IITK p. 25/50 Fine FreqOffEst Estimation rule: (22) max m, ˆω k, f L 2 1 i=0 r m+i ŷk, i e j (ˆω k+ˆω k, f )i where ŷ k, i is given in (11) and: (23) L 2 = L hr + L p 1 0 m L hr 1. ˆω k, f obtained by dividing [ˆω k 0.005, ˆω k ] rad into B 2 freq bins

26 Telematics Lab IITK p. 26/50 Fine Freqoff est contd Freq offset est error (radians) 1.0e e-03 RMS coarse RMS fine Max coarse Max fine 1.0e SNR per bit (db) RMS and max freq off est error for L p = 512.

27 Telematics Lab IITK p. 27/50 Fine Freqoff est contd Amp Coarse Freq (rad) Time (samp) Amp Fine Freq (rad) Time (samp) SoF, coarse and fine freq off est for L p = 512, SNR 0 db, B 1 = B 2 = 64.

28 Telematics Lab IITK p. 28/50 Fine Freqoff est contd Complexity (Frequency bins) Single stage Two stage Complexity of two-stage approach: B 1 + B 2 = 128. Resolution of two-stage approach is /64 = radians.

29 Telematics Lab IITK p. 29/50 Fine Freqoff est contd For obtaining the same resolution, the single stage approach requires / = 512 frequency bins. Two-stage is four times more efficient than single stage.

30 Telematics Lab IITK p. 30/50 NoiseVar Est After the channel has been estimated using (19), the noise variance is estimated as follows: ˆσ w 2 = 1 ( ) H ( ) r k, m1 s 1 ĥ k r k, m1 s 1 ĥ k 2L 1 (24) where s 1 is defined in (17).

31 Telematics Lab IITK p. 31/50 TurboDecoding Input bits length L d1 Rate-1/2 RSC encoder 1 Map to QPSK Interleaver (π) Rate-1/2 RSC encoder 2 Map to QPSK Encoder 1 QPSK symbols length L d1 Encoder 2 QPSK symbols length L d1 Total length L d = 2L d1

32 Telematics Lab IITK p. 32/50 TurboDecoding contd Generating matrix for each constituent encoders: [ ] (25) G(D) = D D + D 2. Let (26) m 2 = m 1 + L p where m 1 is defined in (14).

33 Telematics Lab IITK p. 33/50 TurboDecoding contd Define (27) r k, m2 = [ r k, m2... r k, m2 +L d 1 ] as the data part of the received signal for the k th frame. Assumptions: SoF detected Frequency offset perfectly canceled Channel estimated

34 Telematics Lab IITK p. 34/50 TurboDecoding contd Ĥ k, 0 S k, 3, 0 + W k, 0 Ĥ k, Ld 1S k, 3, Ld 1 + W k, Ld 1 L d point FFT r k, m2 OFDM receiver after synchronization. FFT output (for 0 i L d 1): (28) R k, i = Ĥk, is k,3, i + W k, i.

35 Telematics Lab IITK p. 35/50 TurboDecoding contd The variance of Wk, i is (29) 1 2 E [ Wk, i 2] = L d σ 2 w Variance of Ĥk, i is (assuming perfect channel estimates, that is Ĥk, i = H k, i ): [ 1 2 E Hk, i 2] (30) = L h σf. 2

36 Telematics Lab IITK p. 36/50 TurboDecoding contd Corresponding to the transition from state m to state n, at decoder 1, for the k th frame, at time i define (for 0 i L d1 1: ( ) 2 Rk, i Ĥk, is m, n γ 1, k, i, m, n = exp (31) 2L dˆσ 2 w where S m, n denotes the QPSK symbol corresponding to the transition from state m to state n in the trellis.

37 Telematics Lab IITK p. 37/50 TurboDecoding contd The alpha values for decoder 1 can be recursively computed as follows (forward recursion): (32) α i+1, n = m C n α i, m γ 1, k, i, m, n P (S b, i, m, n ) α 0, n = 1 for 0 n S 1 ( S 1 ) α i+1, n = α i+1, n/ n=0 α i+1, n

38 Telematics Lab IITK p. 38/50 TurboDecoding contd where (33) P(S b, i, m, n ) = { F 2, i+ if S b, i, m, n = +1 F 2, i if S b, i, m, n = 1 denotes the a priori probability of the systematic bit corresponding to the transition from state m to state n, at decoder 1, at time i, obtained from the 2 nd decoder at time l, after deinterleaving (that is, i = π 1 (l) for some 0 l L d1 1.

39 Telematics Lab IITK p. 39/50 TurboDecoding contd The recursion for beta (backward recursion) at decoder 1 can be written as: (34) β i, n = m D n β i+1, m γ 1, k, i, n, m P (S b, i, n, m ) β Ld1, n = 1 for 0 n S 1 ( S 1 ) β i, n = β i, n/. n=0 β i, n

40 Telematics Lab IITK p. 40/50 TurboDecoding contd Let ρ + (n) denote the state that is reached from state n when the input symbol is +1. Similarly let ρ (n) denote the state that can be reached from state n when the input symbol is 1. Then (for 0 i L d1 1) G 1, i+ = S 1 α i, n γ 1, k, i, n, ρ+ (n)β i+1, ρ+ (n) n=0 (35) G 1, i = S 1 α i, n γ 1, k, i, n, ρ (n)β i+1, ρ (n). n=0

41 Telematics Lab IITK p. 41/50 TurboDecoding contd The extrinsic information that is to be fed as a priori probabilities to the second decoder after interleaving, is computed as: (36) F 1, i+ = G 1, i+ /(G 1, i+ + G 1, i ) F 1, i = G 1, i /(G 1, i+ + G 1, i )

42 Telematics Lab IITK p. 42/50 Simulation Results There are two coded QPSK symbols for every uncoded bit, the SNR per bit (over two dimensions) is defined as: SNR per bit = [ 2E Ĥ k, i S k,3, i 2] [ E Ŵ k, i 2] (37) = 4L hσ 2 f L d σ 2 w.

43 Telematics Lab IITK p. 43/50 Simulation Results contd Channel length L h = 10. Channel length assumed by the receiver L hr = 2L h 1 = 19. Fade variance per dimension σ 2 f = 0.5. Length of preamble L p = 512.

44 Telematics Lab IITK p. 44/50 Simulation Results contd 1.0e e-01 BER 1.0e e-03 UC, data=512, Pr TC, data=512, Pr UC, data=1024, Pr TC, data=1024, Pr UC, data=4096, Pr TC, data=4096, Pr UC, data=4096, Id TC, data=4096, Id SNR per bit (db) Performance of ideal rx independent of data length

45 Telematics Lab IITK p. 45/50 Simulation Results contd The performance can be improved by noting that H k, i is highly correlated. Interleave the coded symbols before IFFT at tx. Deinterleave the coded symbols after FFT at rx.

46 Telematics Lab IITK p. 46/50 Simulation Results contd 1.0e e-01 Bit error rate 1.0e e e e-05 DIV, Data=512, Pr DIV, Id No DIV, Id No DIV, Data=512, Pr UC, Id DIV, Data=1024, Pr DIV, Data=4096, Pr SNR per bit (db) L d1 = 512, L d = 1024.

47 Telematics Lab IITK p. 47/50 Conclusions Performance of the practical rx close to ideal rx for L d1 = L p. The data interleaver significantly enhances the performance of the rx. BER close to 10 5 for 8 db SNR per bit.

48 Telematics Lab IITK p. 48/50 FutureWork Improve the performance for larger data lengths. Improve the accuracy of the frequency offset estimate. Use a prediction filter instead of a data interleaver. A prediction filter exploits the correlation in H k, i. Increase the overall code-rate to unity using turbo trellis coded modulation.

49 Telematics Lab IITK p. 49/50 Acknowledgement This work is supported by the India-UK Advanced Technology Center (IU-ATC) of Excellence in Next Generation Networks, Systems and Services under grant SR/RCUK-DST/Next Gen(F)/2008, sponsored by DST-EPSRC.

50 Telematics Lab IITK p. 50/50 Invited Talk This work was presented at the International Federation of Nonlinear Analysts (IFNA) World Congress, Athens, Greece, 25th June-1st July, 2012.