A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems

Transcription

1 A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems Wei Zhang, Xiang-Gen Xia and P. C. Ching EE Dept., The Chinese University of Hong Kong ECE Dept., University of Delaware A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.1/21

2 Outline Motivation Space-frequency coding High-rate full-diversity SFC design Summary and future work A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.2/21

3 Multiple Antennas System 1 h i, j 1 M M M t M r h i,j, τ i,j : are the multipath gain and delay from the jth Tx to the ith Rx, respectively (j = 1,,M t ; i = 1,,M r ) h i,j = [ h i,j (0) h i,j (1) h i,j (L 1) ] τ i,j = [ τ i,j (0) τ i,j (1) τ i,j (L 1) ] A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.3/21

4 Advantages of MIMO System MIMO increases capacity of communication MIMO uses independent channel fading due to multipath propagation to increase capacity No extra bandwidth is required MIMO gives reliable communication Multiple independent samples of the same signal at the receiver give rise to diversity A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.4/21

5 Multiple Antennas Coding Space-time coding (STC) achieves the diversity gain M t M r is effective in flat-fading channels only ST-OFDM is used for broadband communication where MIMO channel exhibits FSK induced by multipath fading. Its diversity order is M t M r, rather than M t M r L (full). SF-OFDM It encodes the information symbols onto different Tx (space) and OFDM subchannels (frequency bins). It can achieve the potential full diversity M t M r L. A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.5/21

6 Advances in SFC Design Lee-Williams (2000), Gong-Letaief (2002), Hong-Hughes (2002) used STC directly as SFC, but without guarantees of full-diversity Bölcskei (2001) organized the columns of FFT matrix into full-diversity SFC, but the symbol rate is only 1 M t L Su-Safar-Olfat-Liu (2003) improved the rate of full-diversity SFC to 1 L by repetition mapping from STC Su-Safar-Liu (2004) achieved the full-rate (rate 1) and full-diversity SFC Our focus is on the design of high-rate (R > 1) full-diversity SFC. A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.6/21

7 SF-OFDM System Model OFDM Tx 1 1 OFDM Rx SF Encoder SF Decoder OFDM Tx OFDM Rx M t M r H l (a) Typical MIMO-OFDM system SF Encoder H(e j2 k/n ) SF Decoder (b) Equivalent MIMO system A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.7/21

8 Signal Model Assume a vector S, of which the elements are from the constellation A (QPSK, or QAM), is encoded into C via f : S C where S = [ S 0 S 1 S Ns ] T, C = c (1) 0 c (1) 1 c (1) N 1 c (2) 0 c (2) 1 c (2) N 1 c (M t) 0 c (M t) 1 c (M t) N 1. A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.8/21

9 Signal Model (cont.) At the ith receive antenna, after FFT the signal at the kth subchannel, denoted by r (i) k, is given by r (i) k = M t j=1 H i,j (k)c (j) k + n (i) k where H i,j (k) = L 1 l=0 h i,j(l)e j2πkτ i,j(l)/t s and c (j) k is the symbol transmitted from the kth subchannel at the jth Tx. The ML detection at the receiver is given by Ĉ = arg min C M r i=1 N 1 k=0 r (i) k Mt j=1 H i,j (k)c (j) k 2 A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.9/21

10 Pair-wise Error Probability Suppose that the ML detector decides in favor of E wrongly when C is in fact transmitted, the pairwise error probability can be upper bounded by [Bölcskei-Paulraj, 2000] P(C E) ( m i=1 ) Mr ( ) mmr Es λ i 4σn 2 where m is the rank of R(C,E)[R(C,E)] H and {λ 1,,λ m } are the nonzero eigenvalues of R(C,E)[R(C,E)] H. R(C, E) = [D 0 (C E) T, D 1 (C E) T,, D L 1 (C E) T ], and D l = diag{h(l)e j2πkτ(l)/t s } N 1 k=0. A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.10/21

11 SFC Design Criteria In the high-snr region, the average error probability for communication over a fading channel usually behave as: P G c SNR G d where G c is referred to as the coding advantage and G d is called the diversity order. Here, G c = ( m i=1 λ i) M r and G d = mm r. The SF code design criteria can be summarized as (rank criteria) Maximize the minimum rank m of R(C, E) (determinant criteria) Maximize the minimum product, m i=1 λ i of R(C,E)[R(C,E)] H for all distinct pairs C and E. A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.11/21

12 Full-Diversity SFC Design Assume N M t L, an SFC constructed by f : S Ns 1 C M t N achieves full diversity if and only if R(C,E) has full rank M t L for all C E, where R(C, E) = [D 0 (C E) T, D 1 (C E) T,, D L 1 (C E) T ] Equivalently, (C E) T has full rank M t B i are linearly independent with B j for i j (i,j = 0,1,,L 1), where B i = D i (C E) T A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.12/21

13 High-rate SFC Design N 1 M t S Block S 1 S 2 SFC SFC ~ G ~ G 1 2 Concatenation N M t OFDM T C 1 S J SFC ~ G J OFDM Mt (a) S: Input block of size NM t 1. S i : Sub-block of size KM t 1 for i = 1,,J, where J = N/K and K = 2 log 2 (MtL). G i : Encoded matrix of size K M t for i = 1,,J. C T : SFC given by C T = [ GT 1 G T 2 GT J ]T. A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.13/21

14 High-rate SFC Design (cont.) The rate of the proposed SFC is R = N s N = M t Why do we divide the input block in that way? It facilitates the low-complexity decoding at receiver; J is always an integer since K is power of 2; K M t L, so we move to pursue the full rank of K M t L matrix R( G i, Ẽi) rather than R(C,E) Our focus next is to make at least one matrix G i, (1 i J) so that R( G i, Ẽi) achieve full rank when S i Ŝi 0. Equivalently, full-diversity SFC is desired. A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.14/21

15 High-rate SFC Design (cont.) SFC KM t 1 KM t 1 K M t K M t S i Precoding X i Reshape G i Hadamard ~ G i (b) (precoding) X i = ΘS i, where Θ =vander( θ 1 θ 2 θ KMt ) and θ k = e j π(4k 3) 2KM t reshaping KM t 1 vector X i into K M t matrix G i (Hadamard) G i = G i ( H a Mt 1 b 1 ), where H a Mt is the first M t columns of the a a Hadamard matrix. a = 2 log 2 M t and b = K a. A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.15/21

16 Example of High-rate SFC For M t = 2, L = 2, and N = 4, the design of rate-2 SFC is shown as the following steps, get K = 4, J = 1, a = 2 and b = 2 construct the 8 8 precoding matrix Θ precoding the data vector S 8 1 with Θ into X = [ x 1 x 2 x 8 ] T after reshaping and Hadamard, we finally get x 1 x 5 C T x 2 x 6 = x 3 x 7 x 4 x 8 A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.16/21

17 Numerical Analysis An SFC denoted by C is a full-diversity if R(C,E) has full rank, i.e. (1) det (R(C,E)) H R(C,E) 0 for all distinct pairs C and E. By exhaustive search in space of (C E), we can numerically prove the full rank of (1) for BPSK, QPSK and 4-PAM, (M t = 2 and L = 2). Dimension of seach space for specific modulation BPSK QPSK 4-PAM ~ S S C E A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.17/21

18 Simulation Results 10 0 Two ray channel model, [0, 0.5] µ s, 2 bits/s/hz Rate 1, QPSK [Su, 2004] Rate 2, BPSK 10 1 Symbol Error Rate SNR (db) Figure 1: Symbol Error Rate, 2 bits/s/hz, τ = [0,0.5]µ s A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.18/21

19 Simulation Results (cont.) 10 0 Two ray channel model, [0, 0.2] µ s, 2 bits/s/hz Rate 1, QPSK [Su, 2004] Rate 2, BPSK 10 1 Symbols Error Rate SNR (db) Figure 2: Symbol Error Rate, 2 bits/s/hz, τ = [0,0.2]µ s A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.19/21

20 Summary and Future Work Proposed a design of high-rate (rate M t ) SFC The proposed SFC was validated to achieve the full-diversity by numerical and simulation results General theory of high-rate full-diversity SFC design are needed (for any number of Tx and Rx, any channel length, and any constellations) Low-complexity decoding (sphere decoding) A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.20/21

21 Thank you! A Design of High-Rate Space-Frequency Codes for MIMO-OFDM Systems p.21/21