On the Coding Scheme for Joint Channel Estimation and Error Correction over Block Fading Channels

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(1)On the Coding Scheme for Joint Channel Estimation and Error Correction over Block Fading Channels Chia-Lung Wu∗ , Po-Ning Chen∗ and Yunghsiang S. Han†‡ , Yan-Xiu Zheng§ ∗ Dept.. of Comm. Eng., National Chiao-Tung Univ., Taiwan, R.O.C. Email: clwu@banyan.cm.nctu.edu.tw, qponing@mail.nctu.edu.tw † Graduate Institute of Comm. Eng., National Taipei Univ., Taiwan, R.O.C. Email: yshan@mail.ntpu.edu.tw ‡ Dept. of Computer Science and Inform. Eng., National Chi Nan Univ., Taiwan, R.O.C. § Inform. & Commun. Research Lab., Industrial Technology Research Institute of Taiwan, R.O.C. Email: zhengyanxiu@itri.org.tw Abstract—In this work, we propose a novel systematic code construction scheme for joint channel estimation and error correction for channels with independently varying fading subblocks. Unlike the existing noncoherent codes that are designed with the help of computer search, a code of desired code length and code rate can be directly generated with our coding scheme. We then compare our codes with the three-times-repetitive (12, 6) code proposed by Xu et al. for use of channel quality indicator (CQI) in uplink control for IEEE 802.16m. Simulations show that our constructed (36, 6) code has comparable performance to Xu’s code when channel coefficients changes randomly in every 12 symbols. If the channel taps remain constant in the entire coding block of length 36, our code outperforms Xu’s code by 0.7 dB. This indicates that the new constructed code adapts more robustly to the two simulated scenarios. For frequency selective channels of unit memory order, our simulation results suggest that our code that takes in consideration the varying characteristic of channels can achieve better performance at median-to-high signal-to-noise ratio over the computer-searched, union-bound-minimized code of length less than the varying subblock size. A side advantage of our code construction scheme is that its systematic structure makes it maximum-likelihoodly decodable by the priority-first search algorithm. The decoding complexity is therefore significantly decreased in contrast to that of exhaustive decoder for the structureless computer-searched codes.. I. INTRODUCTION The coding technique that combines channel estimation and error correction has received general attention recently. Several previous works [1]–[5], [9], [11], [12] have substantiated that such a joint noncoherent design can improve the system performance over existing separate designs. Theoretical evidence that the coherent channel capacity and noncoherent channel capacity almost coincide to each other at median SNR range such as 30 dB further suggests the potential of such technique [15]. The error correcting code design that jointly considers channel estimation is especially useful in situation when either the fading is rapid enough to preclude a good estimate of channel taps or the cost of implementing the channel estimators is high. One example is the reliable delivery of often short-inlength control signal such as channel quality indicator (CQI). 978-1-4244-5213-4/09/ $26.00 ©2009 Crown. in a highly mobile environment. At this background, Xu et al. proposed a novel nonlinear coding scheme suitable for blind noncoherent detection of the transmitted control signal to the 802.16m standard body [13]. In the proposal, the uplink CQI information is encoded using a (12, 6) code. The codeword will then be repeatedly transmitted three times (perhaps through different OFDM channels) in order to further benefit from diversity gain (which can be equivalently regarded as a (36, 6) coding scheme). Since most of the existing blind-detectable noncoherent codes are designed with the help of computer search, they exhibit no apparent structure for efficient decoding. The operation-intensive exhaustive search therefore becomes the only decoding option, of which the dramatically increasing decoding complexity prevents its practical use for codes of long codeword length or high code rate. In this work, we take a different approach in such code design. Based on self-orthogonality framework, we propose a systematic (N, K) coding scheme that can deal with any given N and K for channels with possibly varying channel coefficients in a coding block. It is an extension of our previous work that targets the blind detection over channels with static (i.e., constant) channel coefficients during the transceiving of a codeword [14]. Simulations show that our constructed (36, 6) code has almost the same performance as Xu’s three-timesrepetitive (12, 6) code when the channel independently varies its coefficients three times in a coding block. In case the channel remains constant during the entire coding block, our constructed code has 0.7 dB performance improvement over Xu’s code. Xu’s code is specifically designed for a frequencynonselective OFDM system, while our systematic code construction scheme can also be applied in a frequency selective environment. Our simulation results indicate that with a proper design, a blind-detectable noncohrent code can be made robust for channels whose taps may vary more often than a coding block. A side advantage of our code construction scheme is that its systematic structure makes it maximum-likelihoodly decodable by the priority-first search algorithm. Thus, when being. 1272.

(2) compared with the operation-intensive exhaustive decoder, the III. C ODE D ESIGN decoding complexity is greatly reduced especially when codes We summarize the proposed code construction scheme [14] of longer code length is adopted. Throughout this work, superscripts “H” and “T ” are specifi- for P = 0 (frequency nonselective) and P = 1 (frequency cally reserved for the matrix operations of Hermitian transpose selective) in the following algorithm. and transpose, respectively [8]. Step 1. Fix b1 = −1,2 and choose a certain integer ∆ defined later. Find 2K codewords of the (N, K) code by repeating Steps 2–4 for 0 ≤ i ≤ 2K − 1. II. S YSTEM M ODEL Step 2. Let ρmin = 0 and ρ = i · ∆. Suppose that a codeword b = [b1 · · · bN ]T is transmitted Step 3. For = 2 to N , assign the -th bit of the i-th codeword, b , according to that if ρ < ρmin + γ , then b = −1; else, over a block fading channel of memory order P , of which b = 1 and ρmin = ρmin + γ , where channel coefficients may vary in every Q symbols, where bi ∈ {±1} and Q > P . By letting L N + P and M L/Q, γ = |AP (b1 , . . . , b−1 , b = −1)|, the system can be modelled by: which will be defined shortly. y = Bh + n, Step 4. Store the ith codeword b, and goto Step 2 for the next codeword until all 2K codewords are selected. where n is zero-mean white Gaussian distributed, H Now, as far as the code design for frequency nonselective h hH hH · · · hH 1 2 M channels is concerned, A0 (b1 , . . . , b ) is simply the set of all binary ±1-sequences of length N , whose first bits are with hk [h0,k h1,k · · · hP,k ]T , and assigned as the arguments indicate, and which at the same time satisfy that B B1 ⊕ B2 ⊕ · · · ⊕ BM T Bk Bk = Q for 1 ≤ k < M ˜ P bk ]. Here, 0Q×P ˜ k ··· E with Bk [0Q×P IQ ][bk Eb (2) T B M BM = N − (M − 1)Q. represents a Q × P all-zero matrix, IQ is a Q × Q identity matrix, bk [b(k−1)Q−P +1 · · · b(k−1)Q+1 · · · bkQ ]T is a For channels of memory order P = 1, the conditions to define portion of the transmitted codeword b, A1 (b1 , . . . , b ) are the same as those to define A0 (b1 , . . . , b ) ⎡ ⎤ except that condition (2) is replaced with 0 0··· 0 0 ⎧

(3). ⎢ ⎥ Q c1 ⎪ T ⎪ ⎢1 0 . . . 0 0⎥ B B = ⎪ 1 1 ⎪ ˜ ⎢ ⎥ −1 ⎪ E ⎪ c1 Q

(4) ⎢ ⎥ ⎨ . . Q ck ⎣0 1 . 0 0⎦ T Bk Bk = for 2 ≤ k ≤ M − 1 c k Q ⎪ 0 0 · · · 1 0 (Q+P )×(Q+P ) ⎪

(5) ⎪ ⎪ N − (M − 1)Q cM ⎪ T ⎪ ⎩ BM BM = equates the logical left-shift operator, and “⊕” is the direct sum cM N − [(M − 1)Q − 1]+ 1 (3) operator for two matrices. Also, for notational convenience, we take nj = 0 for j > L, and bj = 0 for j ≤ 0 and j > N . where c1 = −((Q − 1) mod 2), c2 = −(Q mod 2), cM = + + Under such system setting, y is an M Q × 1 received vector −((N −[(M −1)Q−1] −1) mod 2), and [x] max{x, 0}. Note that conditions (2) and (3) are devised based on the selfwith yj = 0 for j > L. orthogonal codeword property that guarantees to maximize the It can be derived that the joint maximum-likelihood decoder system signal-to-noise ratio regardless of the statistics of h [3], [12] upon the reception of y is given by: [14]. M. 2 It remains to determine the integer ∆. In order to have ˆ = arg max. y k y H. , b (1) k − PBk adequate number of codewords selected, ∆ must satisfy b∈C k=1. where y k [y(k−1)Q+1 y(k−1)Q+2 · · · ykQ ] is the output portion affected by bk , and PBk Bk (BTk Bk )−1 BTk . In the above derivation, we assume that the receiver, although it knows nothing about h, has perfect knowledge about the values of P and Q. 1 For. two matrices A and B, the direct sum of A and B is defined as.

(6) A 0 A⊕B= . 0 B. |AP (b1 = −1)| . (4) 2K − 1 We however found that letting ∆ be the largest integer satisfying (4) as we did in [14] may not generate the alphabetically uniform-pick code with the best error performance. In certain cases, the second largest integer satisfying (4) is indeed a better choice. Further investigation that follows along this direction suggests that a better choice of ∆ will yield a ∆≤. 2 Codeword b and {P M Bk }k=1 in (1) is not one-to-one corresponding unless the first element of b, namely b1 , is fixed. We thus fix b1 = −1 in our code design.. 1273.

(7) code with minimum pairwise distance in the sense of larger M ¯k }M and {Bk }M (1), i.e., k=1 PB¯k − PBk 2 , where {B k=1 k=1 ¯ and b. respectively correspond to codewords b It may not be practical to examine the minimum pairwise distance for all 2K codewords for the determination of the best ∆. Instead, we choose K codewords as representatives. These representative codewords correspond to ρ = 2j ∆ for 0 ≤ j ≤ K − 1. Subject to (4), we then adopt the ∆ that minimizes the pairwise distance among these K codewords. When N > K + 4 and P = 0, the proposed process of determining ∆ is indeed equivalent to that the ∆-th codeword must be of the form. on-the-fly, and will therefore cause much less delay in the decoding. For the evaluation of the second metric f2 , however, one needs to know all received symbols, but its computational complexity is much less than that of f1 . Continuing the derivation from (1) based on BTk Bk = Gk for 1 ≤ k ≤ M , we establish that: ˆ = arg min 1 b b∈C 2. wm,n,k =. K+1. where u is a maximum-length shift-register sequence. In other words, the first K + 3 bits are fixed as [−1 · · · − 1 1 1], and the last bit is always equal to 1. This is because under P = 0, all binary ±1-sequences satisfy (2), which results in that (2j+1 ∆)-th codeword is exactly the logical left-shift of (2j ∆)-th codeword. We close this section by pointing out that the size of set AP (b1 , . . . , b ) has explicit formula for P = 0 and P = 1. It is given below for which the derivation is omitted. AP (b1 , . . . , b ) = ⎧ N − 2 , for P = 0; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Q − ( mod Q) ⎪ ⎪ 1 {|cτ − m | ≤ Q − ( mod Q)} ⎪ ⎪ ⎪ Q−( mod 2Q)+cτ −m ⎪ ⎪ ⎪ M −1 ⎪ ⎪ N − [(M − 1)Q − 1]+ − 1 Q ⎪ ⎪ , + ⎨ × Q+ck+1 N −[(M −1)Q−1] −1+cM 2. k=1 m=1 n=1. 2. ⎪ ⎪ for 1 ≤ τ < M and P = 1; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ N − [(M − 1)Q − 1]+ − 1 ⎪ ⎪ ⎪ ⎪ N −[(M −1)Q−1]+ −1+cM −m ⎪ ⎪ ⎪ 2 ⎪ ⎪ ×1 {|cM − m | ≤ N − [(M − 1)Q − 1]+ − 1} , ⎪ ⎪ ⎩ for τ = M and P = 1,. . −wm,n,k b(k−1)Q−P +m b(k−1)Q−P +n. . where for 1 ≤ m, n ≤ Q + P ,. [−1 · · · − 1 1 1 u 1], . k=τ +1. M Q+P. Q+P. P P . ∗ δi,j,k Re{˜ ym+i,k y˜n+j,k },. i=0 j=0 H ˜ k [01×P y H y1,k · · · y˜Q+2P,k ]T , y k 01×P ] = [˜. and δi,j,k is the (i, j)-th entry3 of matrix Dk G−1 k . M Q+P Q+P 1 By adding a constant 2 k=1 m=1 n=1 |wm,n,k | to the decoding criterion, the on-the-fly metric f1 that suits for the recursive computation of the priority-first search is given by: f1 (b1 , . . . , b ) = f1 (b1 , . . . , b−1 )+ ⎧ P P. ⎪ ⎪ ⎪ α − b δi,j,k Re{˜ ys+i,k · uj,k (b1 , . . . , b )}, s,k ⎪ ⎪ ⎪ ⎪ i=0 j=0 ⎪ ⎪ ⎪ for P < s ≤ Q; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ P P ⎪. ⎨ αr,k − b δi,j,k Re{˜ yr+i,k · uj,k (b1 , . . . , b )} ⎪ ⎪ i=0 j=0 ⎪ ⎪ ⎪ +αs,k+1 ⎪ ⎪ ⎪ P P ⎪. ⎪ ⎪ ⎪ −b δi,j,k+1 Re{˜ ys+i,k+1 · uj,k+1 (b1 , . . . , b )}, ⎪ ⎪ ⎪ ⎪ i=0 j=0 ⎪ ⎩ otherwise, where s [( + P − 1) mod Q] + 1, r s + Q, k max{/Q, 1}, αs,k . where τ = /Q + 1, and. s−1. n=1. |ws,n,k | +. 1 |ws,s,k | , 2. m = and ⎧ 0, ⎪ ⎪ ⎪ uj,k (b1 , . . . , b+1 ) = uj,k (b1 , . . . , b ) ⎪ for = 1 or ( = [(τ − 1)Q − 1]+ + 1 and 2 ≤ τ ≤ M ); ⎪ ⎪ ⎪ 1 ∗ ⎨ ∗ b y˜s+j,k + b+1 y˜s+j+1,k + for 1 < < Q; b1 b2 + · · · + b−1 b , 2 ⎪ ⎪ ⎪ with initial values f1 (b1 , . . . , b ) = 0 for = 0, and ⎪ ⎪ b[(τ −1)Q−1]+ +1 b[(τ −1)Q−1]+ +2 + · · · + b−1 b , ⎪ ⎪ uj,k (b1 , . . . , b(k−1)Q−P +1 ) = 0 for 0 ≤ j ≤ P and 1 ≤ ⎩ + for [(τ − 1)Q − 1] + 1 < < τ Q and 2 ≤ τ ≤ M. k ≤ M . The low-complexity decoding metric f2 is given by f2 (b1 , . . . , b ) = f1 (b1 , . . . , b ) + h(b1 , . . . , b ),. IV. O PTIMAL P RIORITY-F IRST S EARCH D ECODING In this section, we will derive two decoding metrics that can be used by the priority first search algorithm [6], [7]. Both metrics will lead to the optimal maximum-likelihood decoding. The difference is that the first metric f1 can be computed. 3 When N − (M − 1)Q = 0, the designated BT B M M in (3) has no inverse. In such case, we redefine

(8). 0 0 . DM 0 1. 1274. N −[(M −1)Q−1]+ −1.

(9) 0. 10. where. Proposed (36,6) Xu et al. (12,6). h(b1 , . . . , b ) ⎧ Q+P Q+P. ⎪ ⎪ ⎪ α − |vm,k (b1 , . . . , b )| − βs,k ⎪ m,k ⎪ ⎪ ⎪ m=s+1 m=s+1 ⎪ ⎪ ⎪ ⎪ for P < s ≤ Q; ⎪ ⎪ ⎪ ⎪ ⎨ Q+P Q+P. αm,k+1 − |vm,k+1 (b1 , . . . , b )| − βs,k+1 ⎪ ⎪ ⎪ ⎪ m=s+1 m=s+1 ⎪ ⎪ Q+P Q+P ⎪. ⎪ ⎪ ⎪ ⎪ + αm,k − |vm,k (b1 , . . . , b )| − βr,k ⎪ ⎪ ⎪ m=r+1 m=r+1 ⎪ ⎩ otherwise,. −1. WER. 10. −2. 10. −3. 10. −4. 10. 0. 1. 2. 3. 4. 5. 6. 7 8 E /N (dB) b. where s, r and k are defined the same as for f1 (·),. 9. 10. 11. 12. 13. 14. 15. 0. Fig. 1. Word error rates (WERs) for the constructed (36, 6) code and Xu’s three-times-repetitive (12, 6) code over flat fading channel with coefficients varying independently in every 12 symbols.. vm,k (b1 , . . . , b ) = vm,k (b1 , . . . , b−1 ) + ws,m,k b , and βs,k. = βs−1,k −. Q+P. |ws,n,k | −. n=s+1. 1 |ws,s,k | 2. with initial values vm,k (b1 , . . . , b(k−1)Q−P +1 ) = 0 and β0,k = Q+P m=1 αm,k . V. S IMULATION R ESULTS In our simulations, the channel parameters follow those in [12], where h is zero-mean complex-Gaussian distributed with E[hhH ] = (1/(P + 1))IP +1 . We first compare our constructed (36, 6) code with Xu’s three-times-repetitive (12, 6) code over frequency nonselective channels. As shown in Figure 1, the two codes has comparable performance when channel coefficients vary independently in every 12 symbols. In case the channel coefficients remain constant over the entire coding block, the proposed (36, 6) code performs 0.7 dB better than Xu’s code as shown in Figure 2. It should be emphasized that when P = 0, AP (b1 , . . . , b ) is irrelevant to the design parameter Q; hence, the (36, 6) code in Figure 1 is identical to the one used in Figure 2. This indicates that the proposed (36, 6) code can adapt more robustly to the two simulated scenarios than Xu’s code. Figures 3 simulates three half-rate codes over frequency selective channels of memory order 1, in which the channel coefficients vary independently in every 15 symbols. The three codes are identified by (28, 14)(Q = 29), (28, 14)(Q = 15) and CS(14, 7), which respectively denote the constructed (28, 14) code with design parameter Q = 29 (i.e., assuming at the design stage, the channel coefficients remain constant during the entire decoding block L = N + P = 28 + 1 = 29), the constructed (28, 14) code with design parameter Q = 15 (i.e., assuming the channel coefficients vary in every 15 symbols at the design stage), and the computersearched (hence, structureless) (14, 7) code that minimizes the union bound derived based on the assumption that the channel taps remains constant during the decoding block (i.e.,. Q = L = N + P = 14 + 1 = 15, which is exactly the simulated channel). As anticipated, (28, 14)(Q = 29) seriously degrades since its assumption at the design stage does not match the characteristic of the true simulated channel. This suggests that the assumption that the channel coefficients remain constant in a coding block is very critical in the code design, and should be made with caution. A striking result from Figure 3 is that the constructed (28, 14)(Q = 15) code performs markedly better than the CS(14, 7) code at medium-to-high signal-to-noise ratios, despite that the CS(14, 7) code is the computer-optimized code specifically for the simulated channel. This suggests that when the channel memory order and varying characteristic are prior known (i.e., P and Q), performance gain can be obtained by enhancing the inter-Q-block correlation, and the system favors a longer code design. In Table I, we summarize the decoding complexity for the (28, 14)(Q = 15) code simulated in Figure 3, measured by the average number of node expansions per information bit. It shows, as previously mentioned, that the decoding metric f2 requires less decoding efforts than the on-the-fly decoding metric f1 . The performance of our constructed code can be further (slightly) improved if the codewords are selected uniformly from all feasible (c1 , c2 , · · · , cM ) ∈ {−1, 0, 1}M . For example, select only half (i.e., 213 ) of the codewords according to c1 = 0 and c2 = −1 for the (28, 14)(Q = 15) code, and pick the remaining half of the codewords from those binary sequences satisfying (3) with c1 = 0 and c2 = 1. This however will slightly increase the decoding complexity. The trade-off between selecting codewords from fixed (c1 , . . . , cM ) or multiple (c1 , . . . , cM )’s is thus evident.. 1275.

(10) TABLE I AVERAGE NUMBERS OF NODE EXPANSIONS PER INFORMATION BIT FOR THE (28, 14)(Q = 15) CODE SIMULATED IN F IGURE 3. SNR f1 f2 ratio of f1 /f2. 3dB 1658 766 2.2. 4dB 1367 625 2.2. 5dB 1074 482 2.2. 6dB 899 392 2.3. 7dB 701 321 2.2. 8dB 593 254 2.3. 0. 10dB 448 177 2.5. 11dB 356 149 2.4. 12dB 309 133 2.3. 13dB 277 121 2.3. 14dB 244 104 2.3. 15dB 232 92 2.5. that the channel coefficients h vary nonstationarily as the periods Q1 , Q2 , . . ., QM are not equal is straightforward. Such design may be suitable for, e.g., the frequency-hopping scheme of Global System for Mobile communications (GSM) and Universal Mobile Telecommunications System (UMTS), or the time-hopping scheme in IS-54, in which cases the channel coefficients change (or hop) at protocol-aware scheduled time [10].. 10. Proposed (36,6) Xu et al. (12,6). WER. 9dB 488 219 2.2. −1. 10. R EFERENCES. −2. 10. 0. 1. 2. 3. 4. 5. 6. 7 8 E /N (dB) b. 9. 10. 11. 12. 13. 14. 15. 0. Fig. 2. Word error rates (WERs) for the constructed (36, 6) code and Xu’s three-times-repetitive (12, 6) code over flat fading channel with coefficients unchanged during the transmission of a codeword. 0. 10. −1. WER. 10. −2. 10. −3. 10. (28,14)(Q=29) (28,14)(Q=15) CS(14,7) −4. 10. 3. 4. 5. 6. 7. 8. 9. 10 11 Eb/N0 (dB). 12. 13. 14. 15. 16. 17. 18. Fig. 3. Word error rates (WERs) for the (28, 14)(Q = 29) code, the (28, 14)(Q = 15) code and the CS(14, 7) code over channels of memory order 1, whose coefficients varying independently in every 15 symbols.. VI. C ONCLUSION An extension of the code design for combined channel estimation and error correction to channels with independently varying fading subblocks is established in this work. This design can directly construct a code of any desired code length and code rate, of which the performance is shown to be comparable to the best computer-searched code for the channels simulated. Although we only derive the coding scheme and its decoding metric for a fixed Q, further extension to the situation. [1] M. Beko, J. Xavier and V. A. N. Barroso, “Noncoherent communication in multiple-antenna system: Receiver design and codebook construction,” IEEE Trans. Signal Process., vol. 55, no. 12, pp. 5703-5715, Dec. 2007. [2] M. J. Borran, A. Sabharwal and B. Aazhang, “On design criteria and construction of noncoherent space-time constellations,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2332-2351, Oct. 2003. [3] O. Coskun and K. M. Chugg, “Combined coding and training for unknown ISI channels,” IEEE Trans. Commun., vol. 53, no. 8, pp. 13101322, Aug. 2005. [4] J. Giese and M. Skoglund, “Space-time code design for unknown frequency-selective channels,” Proc. IEEE. Int. Conf. Acoust., Speech, Signal Process., Orlando, FL, USA, May 2002. [5] J. Giese and M. Skoglund, “Single- and multiple-antenna constellations for communication over unknown frequency-selective fading channels,” IEEE Trans. Inform. Theory, vol. 53, no. 4, pp. 1584-1594, Apr. 2007. [6] Y. S. Han and P.-N. Chen, “Sequential decoding of convolutional codes,” The Wiley Encyclopedia of Telecommunications, edited J. Proakis, John Wiley and Sons, Inc., 2002. [7] Y. S. Han, C. R. P. Hartmann and C.-C. Chen, “Efficient priority-first search maximum-likelihood soft-decision decoding of linear block codes,” IEEE Trans. Inform. Theory, vol. 39, no. 5, pp. 1514–1523, Sep. 1993. [8] D. Harville, Matrix Algebra From a Statistician’s Perspective, 1st edition, Springer, 2000. [9] B. Hochwald, T. L. Marzetta, T. J. Richardson, W. Sweldens and R. Urbanke, “Systematic design of unitary space-time constellations,” IEEE Trans. Inform. Theory, vol. 46, no. 6, pp. 1962-1973, Sep. 2000. [10] R. Knopp and P. A. Humblet “On coding for block fading channels,” IEEE Trans. Inform. Theory , vol. 46, no. 1, pp. 189-205, Jan. 2000. [11] T. L. Marzetta and B. M. Hochwald, “Capacity of a mobile multipleantenna communication link in Rayleigh flat-fading,” IEEE Trans. Inform. Theory, vol. 45, no. 1, pp. 139-157, Jan. 1999. [12] M. Skoglund, J. Giese and S. Parkvall, “Code design for combined channel estimation and error protection,” IEEE Trans. Inform. Theory, vol. 48, no. 5, pp. 1162-1171, May 2002. [13] C. Xu, H. Sun, J.-K. Fwu, Q. Li, Y. Zhu, J. Wang, S. Zhang, Y. Gao, H. Yin, R. Vannithamby and S. Ahmadi, IEEE 802.16m UL PHY Control: CQI Feedback Channel Design, C80216m-UL PHY Ctrl-08 065r1, Intel Corporation, IEEE 802.16 UL PHY Control RG, 11th, Nov. 2008. [14] C.-L. Wu, P.-N. Chen, Y. S. Han and S.-W. Wang, “A self-orthogonal code and its maximum-likelihood decoder for combined channel estimation and error protection,” Proc. IEEE International Symposium on Information Theory and its Applications, Auckland, New Zealand, Dec. 2008. [15] S.-H. Wu, “Effects of estimation errors on the noncoherent capacity over time-varying channels,” Proc. IEEE Vehicular Technology Conference, Montreal, Canada, Sep. 2006.. 1276.

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