Bounds on the Decoding Error Probability of Binary Block Codes over ...

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Bounds on the Decoding Error Probability of Binary Block Codes over Noncoherent Block AWGN and Fading Channels Xiaofu Wu, Haige Xiang, Cong Ling, Xiaohu You, and Shaoqian Li

Abstract— We derive upper bounds on the decoding error probability of binary block codes over noncoherent block additive white Gaussian noise (AWGN) and fading channels, with applications to turbo codes. By a block AWGN (or fading) channel, we mean that the carrier phase (or fading) is assumed to be constant over each block but independently varying from one block to another. The union bounds are derived for both noncoherent block AWGN and fading channels. For the block fading channel with a small number of fading blocks, we further derive an improved bound by employing Gallager’s first bounding technique. The analytical bounds are compared to the simulation results for a coded block-based differential phase shift keying (B-DPSK) system under a practical noncoherent iterative decoding scheme proposed by Chen et al [1]. We show that the proposed Gallager bound is very tight for the block fading channel with a small number of fading blocks, and the practical noncoherent receiver performs well for a wide range of block fading channels. Index Terms— Upper bounds, noncoherent decoding, block fading channels, block codes, turbo codes.

I. I NTRODUCTION

N

ONCOHERENT techniques in communications have attracted much research interest in recent years. The basic noncoherent technique may be traced back to the use of simple differential detection, where the carrier phase is assumed to be constant over two consecutive symbols and information symbols are differentially encoded before transmission. As this simple noncoherent detection suffers from a performance penalty compared to ideal coherent detection, more advanced noncoherent techniques [2], [3] have been extensively studied. Manuscript received Nov.26,2004, revised Aug.13, 2005 and Jan.22,2006. The associate editor coordinating the review of this paper and approving it for publication was Prof.Fred Daneshgaran. This work was supported in part by the National Science Foundation of China under Grants 60402026, 60302006, and 60390540. The work of X. Wu was also supported by the National High Technology Research and Development Program (863 program) of China under Grant 2001AA123052, by the Opening Research Foundation of the National Mobile Communication Research Laboratory at Southeast University under Grants N200501, and by the National Key Laboratory of Communication of China at the University of Electronics Science and Technology Xiaofu Wu, and Haige Xiang are with the Department of Electronics, Peking University, Beijing 100871, China (Email: [email protected], [email protected]) Cong Ling is with the Dept. Electronic Engineering, King’s College London, Strand, London WC2R 2LS, United Kingdom (Email: [email protected]) Xiaohu You is with the National Communication Research Laboratory, Southeast University, Nanjing 210096, China (Email: [email protected]) Shaoqian Li is with the National Key Lab. of Communication, Univ. of Electronics Science and Technology of China, Chengdu 610054, China (Email: [email protected])

In particular, Divsalar and Simon introduced a fundamental technique of multiple-symbol differential detection, which is a generalization of simple differential detection by extending the observation interval beyond two symbols [2]. The common feature of noncoherent receivers is that they do not require an explicit carrier phase reference. Therefore, they are especially suitable for wireless communication systems where carrier synchronization is difficult due to fading, severe interference, or the introduction of excessive overhead in general. Theoretically, as a time-varying channel is a priori unknown, the noncoherent paradigm is most appropriate in this setting. In practice, the pilot-based coherent detection is often employed due to its low-complexity on implementation. The pilot-based coherent approach, indeed, can be viewed as a suboptimal implementation of the optimal noncoherent receiver. An obvious disadvantage of this approach is that it requires excessive overhead in rapidly time-varying channels. For reliable communications over noncoherent channels, coding should be introduced. The problem of integrating errorcontrol coding with noncoherent detection has been studied in recent years [4]–[8]. On one hand, it was noted that optimal codes for coherent detection are not necessarily optimal for noncoherent detection; therefore various noncoherent codes have been proposed [5], [6], [8]. On the other hand, many works also reported that powerful turbo-like codes originally designed for additive white Gaussian noise (AWGN) channels still work well for some noncoherent channels with welldesigned iterative receivers [1], [9]–[11]. Essentially, this depends on the dynamics of the channel, which is often characterized by the coherence time of the channel (denoted as L). When L → ∞, a code which is optimal for coherent detection is also optimal for noncoherent detection [12]. This, however, is not necessarily true for finite values of L. In this paper, we consider a block AWGN (or fading) channel model [1], [10], which is a good model to capture the dynamics of the carrier phase (or fading) process. By a block AWGN (or fading) channel, we mean that the carrier phase (or fading) is assumed to be constant in a block but independently varying from one block to another. The block size L can be thought of as the channel coherence time, which reflects the rapidity of the change of carrier phase (or fading). In practice, the assumption of independence among blocks requires sufficient separation in time or frequency among blocks. Therefore, block AWGN (or fading) channels may provide good approximations for time-division multiple-access (TDMA), frequency-hopping, or block-interleaved channels.

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Block Encoder

Fig. 1.

Channel Interleaver

2

BPSK Modulator

Block AWGN (or Fading) Channel with a Block Size of L

B-DPSK Modulator

Block AWGN (or Fading) Channel with a Block Size of L +1

The block diagram of a coded BPSK system in comparision with a coded B-DPSK system.

So far, considerable efforts have been made in deriving upper bounds on the decoding error probability of coded systems over the block fading channel [13]–[16], under coherent detection and maximum likelihood decoding. The union bound was derived in [13]. In [15], Malkam¨aki and Leib derived a tighter bound, which is based on a ”limit-before-average” technique. The other bounds can be found in [14], [16]. Currently, there is little work on the performance analysis of coded systems over the noncoherent block fading channel. A union-Chernoff bound was only given for convolutional codes over the noncoherent block AWGN channel [5]. In this paper, we derive upper bounds on the decoding error probability of binary block codes over noncoherent block AWGN and fading channels, with applications to turbo codes. The analytical results are compared with the simulation results for a coded block-based differential phase shift keying (B-DPSK) system, where a practical noncoherent iterative decoding scheme proposed by Chen et al. [1] is employed. The paper is organized as follows. In Section II, the system model of a binary block code over a noncoherent block AWGN (or fading) channel is described, and the decoding metric is derived under noncoherent maximum-likelihood (ML) decoding. The coded B-DPSK system [1], along with its practical receiver structure, is also summarized in Section II. The exact pair-wise error probability (PEP) and its Chernoff bound are derived in Section III. In Section IV, union bounds are introduced, and the related problem of weight enumeration is addressed. Section V is dedicated to deriving a tight Gallager bound for the block fading channel with a small number of fading blocks. Numerical examples and simulations are discussed in Section VI, followed by conclusions in Section VII. II. S YSTEM MODEL AND PRELIMINARY Consider the transmission of a binary block code over a block AWGN (or fading) channel. For ease of analysis, we consider a coded binary phase shift keying (BPSK) system, which is marked by real lines in Fig. 1. At the transmitter, the input information bits are first encoded by a block encoder, and the coded bits are interleaved. Then the interleaved bits are BPSK modulated before transmission over a block AWGN (or fading) channel with a block size of L.

Since the block AWGN channel can be viewed as a special case of the block fading channel, we first present a general model for the block fading channel. We consider the received complex base-band samples at mth block, which can be represented as follows √ m rjm = α ˜ m γs sm j + zj ,

j = 1, · · · , L; m = 1, · · · , M (1)

where the jth transmitted symbol sm j in mth block takes values from {+1, −1}. Throughout this paper, we assume that the bits ”0” and ”1” are mapped by the BPSK modulation to ”+1” and ”−1”, respectively. The complex channel gain α ˜ m is assumed to be constant in a block of size L but independently varying from one block to another, zjm is the zero-mean h¯ ¯ i additive 2 complex Gaussian noise with variance E ¯z m ¯ = 1 and j

γs = Es /N0 denotes the signal-to-noise ratio (SNR). Let ˜ = (α α ˜1, α ˜2, · · · , α ˜ M ) denote the column vector of collected complex fading gains. As we only consider Rayleigh fading channels in this paper, the probability density function (pdf) ˜ takes the form of of α ³ ´ ¡ ¢ 1 1 ˜ 2 = M exp −α ˜Hα ˜ , ˜ = M exp − |α| f (α) (2) π π where the notation (·)H is used to represent the Hermitian ˜ will be used in deriving transpose of a matrix. The pdf f (α) a tight Gallager bound in Section-V. The complex channel gain can also be written as α ˜ m = αm ejθm , where αm denotes the fading envelope, and θm denotes the unknown carrier phase, which is uniformly distributed in the interval (−π, π]. A. Noncoherent ML Metric for the Block AWGN Channel Letting αm = 1 in (1), we obtain a block AWGN channel, over which the received samples at mth block are given by √ m rjm = ejθm γs sm j + zj , j = 1, · · · , L; m = 1, · · · , M. (3) ¡ ¢T Let s = s1 , s2 , · · · , sM denote the transmitted coded se¡ 1 2 ¢ m m M T quence with sm = (sm 1 , s2 , · · · , sL ), r = r , r , · · · , r denote the corresponding received samples, and θ = (θ1 , θ2 , · · · , θM ) denote the unknown carrier phase vector. The noncoherent ML decoder attempts to choose the transmitted

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+1

S1

S2 Block 1

Fig. 2.

SL

+1

S1

S2

3

SL

Block 2

+1 Block 3,

, M -1

S1

S2 Block

SL M

The block diagram of a coded BPSK system in comparision with a coded B-DPSK system.

coded sequence s to maximize the posterior probability of r Z π M Y 1 p (r|s) = Eθ [p (r|s, θ)] = p (rm |sm , θ m ) dθm 2π −π  m=1  M L ³ X 1 X ¯ m ¯2 ¯ m ¯2 ´ ¯rj ¯ + γs ¯sj ¯  = · exp − πL m=1 j=1 ¯ ¯  ¯X ¯ ¯ L m m ∗¯ √ ¯  · I0 2 γs ¯ rj (sj ) ¯¯ (4) ¯ j=1 ¯ where x∗ denotes the conjugate of x, and I0 (x) is the zeroth order modified Bessel function of the first kind. In order to simplify the metric in (4), we use the approximation I0 (x) ≈ ex , which is valid for large values of x. With this approximation, a noncoherent ML decoder chooses a coded sequence s to maximize the metric ¯ ¯ ¯ M ¯X X ¯ L m ¡ m ¢∗ ¯ ¯ rj sj ¯¯ . ¯ ¯ ¯ m=1 j=1 Unfortunately, the above metric does not lend itself to an efficient evaluation of the system performance. Thus we further introduce the following quadratic metric ¯ ¯2 ¯ M M ¯X X X ¯ L m ¡ m ¢∗ ¯ ¯ rj sj ¯¯ , η= ηm = (5) ¯ ¯ m=1 m=1 ¯ j=1 which, indeed, have also been employed in many noncoherent detection schemes [6], [8]. B. Noncoherent ML Metric for the Block Fading Channel For the block fading channel, we consider the case of no channel side information (CSI) at the receiver, in which a noncoherent approach is more natural. As demonstrated by (1), both the fading coefficients and the received samples of noise are complex Gaussian. Therefore, conditioned on the transmitted signals, the received signals are jointly complex Gaussian. Then, the posterior probability of r given s [1] is p (r|s) =

M Y m=1

p (rm |sm )

½ ¾ h i−1 H H exp −rm IL + γs (sm ) sm (rm ) i h = H π L det IL + γs (sm ) sm m=1   M X L X ¯ m ¯2 1 ¯ rj ¯  · exp − = LM π LM (1 + γs ) m=1 j=1 ¶ µ γs (6) ·exp 1 + Lγs M Y

This suggests the noncoherent ML metric ¯2 ¯ ¯ M M ¯X X X ¯ L m ¡ m ¢∗ ¯ ¯ rj sj ¯¯ . η= ηm = ¯ ¯ m=1 m=1 ¯ j=1

(7)

It is worth pointing out that the metric (7) is a true noncoherent ML metric, which is different from the case of the block AWGN channel, as we do not introduce any approximation in obtaining (6). C. Coded B-DPSK System and Its Practical Noncoherent Receiver Although the noncoherent ML decoder can achieve the best possible performance, the complexity issue often makes it impractical. Therefore, various low-complexity noncoherent iterative receivers have been proposed in recent years [1], [10], [11], [17], [18]. Many works [17], [18], which depend on M ary differential phase-shift keying (M -DPSK), often fail to exploit the full potential of powerful turbo-like codes, as the application of M -DPSK may change the distance spectrum of these codes. The B-DPSK, as a block modulation code, was proposed in [1], which relates all symbols directly to the first reference symbol in each block as illustrated in Fig. 2, i.e., sl+1 = s0 sl , l = 1, · · · , L, s0 = +1. The block diagram of the coded B-DPSK system is also given in Fig. 1, which is marked by dashed lines. Obviously, this technique requires that the carrier phase (or fading) keeps constant in a modulation block, which holds for the block AWGN (or fading) channel. Due to the insertion of the reference symbol at each channel block, the transmission rate is reduced by a factor of L/(L + 1) if the block size of the channel is L + 1. The structure of B-DPSK can be well exploited to develop a practical noncoherent iterative decoding scheme, in which a phase-quantization approach for each fading block was used and a practical linear-complexity Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm [1, Eq. (11)] was proposed. We refer readers to [1] for more details. Basically, they show that turbo-like codes such as turbo codes and repeat accumulate (RA) codes perform well in the block fading channel when combined with B-DPSK. Later in this paper, we focus on the coded BPSK system for derivation purposes. The derived bounds, however, can be straightforwardly applied to the coded B-DPSK system. III. PEP The PEP, which represents the probability of choosing the coded sequence ˆs when the coded sequence s was transmitted, is the fundamental expression for the construction of the union or other upper bounds on the average error probability performance of a system. In this section, the exact PEP is

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derived, which is of a theoretical importance but difficult to compute in general. Hence the Chernoff bound on the PEP is also pursued. A. Block AWGN Channel Letting ηm (or ηˆm ) denote the noncoherent metric (5) for the correct (or incorrect) coded sequence s (or ˆs) incident to the mth block, then the PEP of choosing the coded sequence instead of the actual transmitted coded sequence s is given by à M ¯ ! M X X ¯ P (s → ˆs) = Pr ηˆm > ηm ¯¯s . (8) m=1

m=1

Due to the independence assumption of the unknown carrier phase among blocks, the summations at the right hand side of (8) represent sums of independent random variables. Exact evaluation of (8) was once regarded as a tough task [4, Section-IV]. Nonetheless, we managed to obtain a closedform expression of the PEP by employing the residue theorem. The derivation is summarized in the Appendix, and the PEP takes a form of P X ˆ P (s → s)= Res [G(s), sp < 0] ,

G(s) = s−1

m=1

1 1 − 4s2 τ (hm ) ¾ ½ 4s(1 − sL)τ (hm ) , ·exp −γs 1 − 4s2 τ (hm )

λ

For the block fading channel, two approaches can be employed to compute the¡ PEP P ¯ (s ¢ → ˆs). One is to compute the conditional PEP P s → ˆs¯α at first, and then averaging it over α gives P (s → ˆs). Another approach is to directly compute it. We choose the later approach for its simplicity. As concluded in the Appendix, the exact PEP can be accurately evaluated as P (s → ˆs) =

P X

Res [G(s), sp < 0] ,

p=1

G(s) = s−1

M Y

1 , 1 + 4sτ (h ) (γ m s − s(1 + Lγs )) m=1

(12)

and its Chernoff upper bound is given by M Y

1 . 1 + 4λτ (h ) (γ m s − λ(1 + Lγs )) m=1 (13) From (13), it is easy to show that the optimum Chernoff parameter is γs λopt = , (14) 2(1 + Lγs ) P (s → ˆs) ≤ min λ

P (s → ˆs) ≤ (9)

where τ (h) = h(L−h), hm denotes the component Hamming distance at mth block between coded sequences ˆs and s (i.e., Hamming distance between subsequences ˆsm and sm ), sp is the pth of the P (1 ≤ P ≤ M ) distinct, negative poles of G(s), and Res[·] denotes the residue. Unfortunately, the above expression is numerically difficult to compute for large values of M . Therefore, we have to resort to some suboptimum but computationally efficient bounds. The Chernoff bound is just one of such bounds. Also shown in the Appendix, the Chernoff bound on the PEP is given by P (s → ˆs) = min

B. Block Fading Channel

which is independent of the component Hamming distance (hm ). Therefore, the Chernoff bound in this case has a closedform expression, namely,

p=1 M Y

4

M Y

1 2 τ (h ) 1 − 4λ m m=1 ½ ¾ 4λ(1 − λL)τ (hm ) ·exp −γs .(10) 1 − 4λ2 τ (hm )

M X

1 + Lγs . 1 + Lγs + γs2 hm (L − hm ) m=1

(15)

To gain insight into how the PEP is governed at high SNRs, we further express (15) as P (s → ˆs) ≤

M Y

(1 + γs hm (1 − hm /L))

m=1

≈ γs−D

D Y

1 , h (1 − hmt /L) t=1 mt

(16)

© ¯ ªD where mt ¯1 ≤ mt ≤ M t=1 is the effective index set of the fading blocks in which the component Hamming distance meets 0 < hmt < L. All of these effective fading blocks contribute to the diversity order D. In [13], the PEP under coherent detection was given as ¶−1 Z M µ γs 1 π/2 Y dθ. (17) hm 1+ Pc (s → ˆs) = π 0 sin2 θ m=1 Using sin2 θ ≤ 1, (17) is upper bounded by

The above Chernoff form has also been given in [6]. It is clear that its exponential behavior provides an upper bound on the exponential rate of the error probability. It has been proven in [7] that the exponential rate satisfies M X P (s → ˆs) ≤ min(hm , L − hm ) I = − lim γs →∞ γs m=1

(11)

The parameter I reflects the fact that due to the noncoherent metric (5), the receiver cannot resolve 180o of phase rotation.

Pc (s → ˆs) ≤

M Y

−1

(1 + γs hm )

.

(18)

m=1

Comparing (16) with (18), the noncoherent loss lies in the extra term of (1 − hm /L) , due to which, the PEP for noncoherent detection does not always decrease with the component Hamming distance hm . Except for the extreme case of hm = L, noncoherent detection can achieve the same diversity order D as coherent detection. Therefore, we can

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expect that the union-Chernoff bound of noncoherent detection approaches that of coherent detection in the high SNR region, which is demonstrated to be the truth as shown in Fig. 4. IV. UNION BOUND AND ITS APPLICATION A. Union Bound and Weight Enumeration Consider a binary linear (N, K) block code C (with a code rate of Rc = K/N ) transmitted over a noncoherent block AWGN (or fading) channel. In general, a union bound on the average word error probability can be computed as X XX Pe = P (s)P (error|s) ≤ P (s)P (s → ˆs) . (19) s

s

ˆ s

where P (s) is the priori probability of transmitting the code sequence s. Without loss of generality, we always assume that the code sequence in C is transmitted with a priori equal probability P (s). To simplify (19), we often require the so-called geometrically-uniform property in a coherent setting, which is not necessarily the truth when noncoherent detection is employed. Nonetheless, the simplification of (19) here is rather straightforward thanks to the closed-form expression of the PEP obtained in Section II. For example, in the case of the block fading channel, we have obtained the exact PEP (see Eq. (12)), which, conceptually, is only related to the Hamming distance distribution over all blocks, namely, h = (h1 , h2 , · · · , hM ). As the code is linear, (19) in this case can be simplified as X X Pe ≤ P (s0 → ˆs) = A(h)P (h), (20) ˆ s

h

where s0 denotes the all-zero codeword, A(h) denotes the multi-block weight enumerator, which is the number of codewords in C with weight distribution h, P (h) denotes the exact PEP (Eq. (9) or (12)) associated with h. As the exact PEP is numerically difficult to compute, we focus on the union-Chernoff bound in the remainder of this paper. Hence the exact PEP P (h) in (20) is replaced by its Chernoff bound P U (h)(see Eq. (10) and (15)). At this point, we should say more about the multi-block weight enumerator. Note that the value of P (h) remains invariant under permutation 4

π(h) = (hπ1 , hπ2 , · · · , hπM ) , where the sub-index {π1 , π2 , · · · , πM } represents a possible arrangement of the sub-index sequence {1, 2, · · · , M }. We introduce the equivalent class [h] by grouping all permutations of h as a single unit for any fixed h [14], [16], and further denote the permutation invariant weight enumerator by A[h], which accounts for the number of the codewords in C, whose multi-block weight distribution vector is in the set [h]. Therefore, (20) can now be rewritten as X Pe ≤ A[h]P U (h). (21) [h]

Given the sum-weight of h (i.e.,

PM

m=1 hm 4

= h), the weight

in each block must meet 0 ≤ v ≤ d = min(h, L). Let nv

5

denote the number of blocks with weight v among M blocks. Collect all nv ’s of a given [h] to form a complete weight d distribution (CWD) set {nv }v=0 . It is clear that all elements d in [h] have the same CWD set {nv }v=0 , and vice versa. The concept of weight distribution over sub-blocks was first introduced in [13]. Under the assumption of uniform interleaving, the computation of the average permutationinvariant weight enumerator A[h] was given in [13], [14]. As implicitly given in [13], the average permutation-invariant weight enumerator can be computed as µ ¶−1 d µ ¶n Y L N M! (22) A[h] = Ah n0 !n1 ! · · · nd ! v=0 v v h where Ah designates the number of codewords in C whose Hamming weight is h, N = M · L is the code length. So far, most of results are devoted to the computation of word error probability. In practice, it is also important to compute the bit error probability. To obtain a similar upper bound on the bit error probability, we require the knowledge of average input-output (permutation-invariant) weight enumerator Aw [h], which is given by µ ¶−1 d µ ¶n Y L N M! (23) Aw [h] = Aw,h n0 !n1 ! · · · nd ! v=0 v v h where Aw,h denotes the number of codewords in C with input weight w and output weight h. Define A[h] =

k X w Aw [h]. k w=1

(24)

Then the union bound on the bit error probability admits the same form of (21) with A[h] replaced by A[h] . B. Application to the Coded B-DPSK System The union bound derived above can be directly applied to the coded B-DPSK system [1]. To understand the essential idea, let us investigate the weight distribution of the code for the coded B-DPSK system in comparison to the coded BPSK system. Consider that a block code of length N = M · L is distributed over two systems as shown in Fig. 1, with the same channel interleaver for each system. For comparison purposes, the block size of the channel is assumed to be L for the coded BPSK system while it is assumed to be L + 1 for the coded B-DPSK system (due to the insertion of a reference symbol). The number of channel blocks is M for each system. If the reference symbol for the B-DPSK is assumed to ”+1”, it can be well understood that the weight distribution of the code (i.e., A[h] for all h) for the coded B-DPSK system, which depends on the employed block code, the channel interleaver and the number of channel blocks, remains unchanged in comparison to the coded BPSK system. Hence, the union bound is still applicable with Es /N0 = Rc ·Eb /N0 ·L/(L+1), and the only modification is to replace L with L + 1 in Eq. (10) and (15) due to the insertion of a pilot. It can be well understood that other upper bounds such as the Gallager bound derived in the next section is also applicable with a similar modification. We will not repeat it there.

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V. A TIGHT GALLAGER BOUND FOR THE BLOCK FADING CHANNEL WITH A SMALL NUMBER OF FADING BLOCKS In this section, we focus on the block fading channel with a small number of fading blocks (small values of M ). The union bound often diverges in this case. Hence we derive an improved bound by employing Gallager’s first bounding technique [19]. In the following derivation, we restrict our discussion on the word error probability, but the extension to bit error probability is straightforward. Basically, the word error probability can be upper-bounded as Pe

= Pr(error, α ∈ RE ) + Pr(error, α ∈ RE ) ≤ Pr(α ∈ RE ) + Pr(error, α ∈ RE ). (25)

where RE is the Gallager region, selected as a subset in the space of the fading vector, and RE denotes the complementary set of RE . To further develop a tractable bound, one has to find a suitable upper bound on the PEP conditioned on the fading vector α. A. Conditional Pairwise Error Probability Following the same procedure in the Appendix, we can show that the Chernoff bound on the conditional PEP is given by M Y ¯ ¢ P s → ˆs¯α =min

¡

1 2 τ (h ) λ 1 − 4λ m m=1 ½ ¾ 4λ(1 − λL)τ (hm ) 2 · exp −αm γs (26) 1 − 4λ2 τ (hm )

where τ (h) = h(L − h). This is formally the same as that of the block AWGN channel (see Appendix) except a change in SNR. For further development, it is more convenient to fix the Chernoff parameter in (26) for all h. A sensible choice is λ∗ = γs /2(1 + Lγs ) as indicated by (14). Hence, we obtain a suboptimum but uniform Chernoff bound on the conditional PEP as ( M ) X ¯ ¢ ¡ 2 αm χ(hm ) P s → ˆs¯α ≤ C(h)exp − m=1 H

© ª = C(h)exp −α D(h)α ,

=

χ(hm ) = D(h) =

regular geometric shape to approximate the optimum Gallager region. As investigated in [14], the simple sphere region RE defined as ( ) M X ¯ 2 2 2 ¯ RE = α k α k = |αm | < r m=1

(

M X ¯ 2 ¯ ˜ ˜ α kαk = |˜ αm |2 < r2

≡

)

m=1

is a reasonable choice. The above equivalent description of ˜ later in RE will be used for change of variables (α → α) deriving Eq. (32). The parameter r2 in RE is to be optimized for getting the tightest bound within this framework. Then it is straightforward to show Z M −1 2m 2 X r (28) Pr (α ∈ RE ) = f (α)dα = 1 − e−r m! α∈RE m=0 since the integral in (28) over RE is recognized as the cumulative distribution function of a chi-square random variable with 2M degrees of freedom. The second part of (25) can be calculated as Z ¢ ¡ Pe (α)f (α)dα (29) Pr error, α ∈ RE = α∈RE

With the traditional union bounding technique, we have X Pe (α) = P (s)P (error|s, α) s X X

≤

s

P (s)P (s → ˆs|α)

(27)

M Y

1 , 2 τ (h ) 1 − 4λ m ∗ m=1 4λ∗ (1 − λ∗ L)τ (hm ) γs , 1 − 4λ2∗ τ (hm ) diag{χ(h1 ), χ(h2 ), · · · , χ(hM )}.

B. Derivation of the Tight Gallager Bound To further develop the Gallager bound (25), the traditional union bound is often employed to compute ¢ ¡ Pr error, α ∈ RE . In this framework, it is essential to find a

(30)

ˆ s

As shown in (27), the Chernoff bound on the conditional PEP is only related to h = (h1 , h2 , · · · , hM ). In the case of a linear binary block code with BPSK modulation, (30) can be written as Pe (α)

= P (error|s0 , α) X ≤ P (s0 → s|α) s6=s0

≤

X

© ª A(h)C(h)exp −αH D(h)α

h 4

where

C(h)

6

= PeU (α)

(31)

where s0 denotes the all-zero codeword. Then (29) can be further bounded ³as in (32), ´ at the top of the next page, where f (β) = π1M exp −β H β , and the last equation is obtained by change of variables 1/2

˜ = (IM + D(h)) β = Pα

˜ α,

and by further noting © ª ˜ :α ˜Hα ˜ > r2 RE = α n o = β : (P −1 β)H (P −1 β) = β H Σβ > r2 −1

(33)

(34)

with Σ = P −H P −1 = (IM + D(h)) . With the form of (32), the remaining derivation follows straightforwardly [14], [20]. By employing the classic

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¡

¢

Z

P error, α ∈ RE ≤ Z

X

Z PeU (α)f (α)dα

=

X

© ª ˜ H D(h)α ˜ f (α)d ˜ α ˜ A(h)C(h)exp −α

h

˜ α∈R E

α∈RE

7

© ª 1 ˜ H D(h)α ˜ ˜ H α)d ˜ α ˜ A(h)C(h)exp −α exp(−α πM h ˜ α∈R E Z X © ª 1 ˜ H (IM + D(h)) α ˜ dα ˜ A(h)C(h)|IM + D(h)|−1 = exp −α M −1 π |IM + D(h)| =

h

=

X

˜ α∈R E −1

A(h)C(h)|IM + D(h)|

³ ´ · P r β H Σβ > r2

(32)

h

1

10

characteristic-function approach [21], the word error probability can be upper-bounded by ) ( M −1 2m X X r U −r 2 (35) Pe = min − A(h)C(h)F (h) + 1 − e r m! m=0

−1

10

h

F (h) =

P X

Res[G(s), sp < 0],

−2

10 Bit Error Rate

where

noncoh, noncoh, Chernoff, Chernoff, L+1=5, L+1=5, hmax=100 hmax=100 noncoh, noncoh, Chernoff, Chernoff, L+1=9, L+1=9, hmax=60 hmax=60 coh, coh, Chernoff Chernoff coh, coh, Exact Exact bound bound Sim, Sim, L+1=9 L+1=9

0

10

−3

10

−4

10

p=1 −5

sp is the pth of the P (1 ≤ P ≤ M ) distinct ,negative poles of M 2 Y −1 G(s) = s−1 esr (s + 1 + τ (hm )) . m=1

Clearly, we can show that ∀π, F (π(h)) = F (h); A(π(h)) = A(h); C(π(h)) = C(h).

10

−6

10

−7

10

1

2

3

4

5

6

7

8

Eb/N0, dB

Fig. 3. Union bounds on the bit error probablity of turbo codes for block AWGN channels under both noncoherent and coherent detections.

Hence, (35) can be finally written as VI. NUMERICAL RESULTS AND DISCUSSION ) ( M −1 2m X 2 X r Consider a rate-1/3 turbo code with generator polynomials (36) PeU = min − A[h]C[h]F [h] + 1 − e−r r m! (37, 21) in octal [22]. The coded B-DPSK system is assumed. m=0 h Unless explicitly stated otherwise, the input information bit where length is assumed to be K = 256 . d Y ¡ ¢−nv 2 Fig. 3 shows the union-Chernoff bound on the bit error 1 − 4λ∗ v(L − v) C[h] = , probability for the noncoherent block AWGN channel. The v=0 union bound was evaluated by counting the codewords whose and Hamming weight is subject to h ≤ hmax , where hmax denotes P d X the maximum allowable output weight. Thus, the numerically 2 Y −n F [h] = Res[s−1 esr (s + 1 + τ (v)) v , sp < 0] computed bound is, in fact, an approximation of the union p=1 v=0 bound. The truncation can not be avoided in many cases d as the number of summations involved in computing (21) if the CWD set {nv }v=0 representation of [h] is employed. U Letting ∂Pe /∂r = 0, we implicitly obtain the optimum increases rapidly when the value of hmax increases, which is especially pronounced when the block size L + 1 increases1 . value of r as the solution to the equation ( P ) The exact union bound and its Chernoff form under coherent d X X 2 Y detection are also depicted in Fig. 3 as a benchmark of −n A[h]C[h] Res[esr (s + 1 + τ (v)) v , sp < 0] comparison. Under coherent detection, the Chernoff bound is p=1 v=0 h looser than the exact union bound by about 1-2 dB, which, we 2 e−r r2(M −1) = . (37) anticipate, may also be the case in the noncoherent setting. We (M − 1)! run the simulations under noncoherent iterative decoding [1]. For the block fading channel with a small value of M , the In all simulations, the number of quantization levels (phase residues in (36) and (37) can be accurately evaluated without much numerical difficulty. Hence, the proposed Gallager bound in this case can be efficiently evaluated.

1 However, it should be mentioned the computation load decreases when the block size L + 1 is larger than a threshold. In fact, the computation load is minimized in the case of L + 1 = N + 1, i.e., the quasi-static fading channel.

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8

1

10

of 1 can also be observed in principle at a possibly lower WEP. In the noncoherent setting, we should further note the fact that a channel block full of bit ”1” does not contribute to diversity due to a noncoherent loss (see Eq. (16)).

noncoh, Chernoff, L+1=5, hmax=100 noncoh, Chernoff, L+1=9, hmax=60 coh, Exact bound, L+1=9, hmax=60 coh, Chernoff, L+1=9, hmax=60 Sim, L+1=9

0

10

−1

10

−2

Bit Error Rate

10

0

10 −3

10

−4

10

−1

10 −5

−6

10

−7

10

3

4

5

6

7 8 Eb/N0, dB

9

10

11

12

Frame Error Rate

10

−2

10

Bound, M=1 Bound, M=2 Bound, M=3 Out. Prob., M=1 Out. Prob., M=2 Out. Prob., M=3 Sim., M=1 Sim., M=2 Sim., M=3

−3

10

Fig. 4. Union bounds on the bit error probablity of turbo codes for block fading channels under both noncoherent and coherent detections.

−4

10

0

2

4

6

8

10 Eb/N0, dB

12

14

16

18

20

Fig. 5. Tight Gallager bounds on the word error probability of turbo codes for the noncoherent block fading channel with a small number of fading blocks.

0

10

−1

10

Slope=3 −2

10

Slope=2 Word Error Probability

quantization for each channel block) is fixed at 20, and the overall number of iterations is set to 12. As shown in Fig. 3, the noncoherent iterative receiver proposed in [1] works well even for short frame sizes. We also investigate the effect of block size (channel coherence time) on the system performance, and about 0.5 dB improvement has been observed when the block size increases from L + 1 = 5 to L + 1 = 9. Similar results are presented in Fig. 4 for the noncoherent block fading channel. The exact union bound [13] under coherent detection is also shown. It is observed that the performance gap between coherent and noncoherent detection is less than 2 dB considering the fact that a loose Chernoff bound is used for noncoherent detection. In Fig. 5, we show the proposed Gallager bound on the word error probability (WEP) for the block fading channel with a small number of fading blocks. Simulation results are also given, which seems to be surprisingly good considering the fact that only one pilot is inserted for each block (with a large value of block size). It should be noted that we assume random interleaving between coding and modulation (i.e., random multiplexing [23]). For comparison, we also depict the outage probability for BPSK inputs, as the limiting WEP performance of coherent detection [24]. In the coherent setting, the asymptotic slope (diversity) of the WEP with BPSK inputs is limited by the Singleton bound [23]. Therefore, the optimal diversity, which can be achieved by an MDS (maximum distance separable) code in the coherent setting, equals M for a rate-1/3 binary code transmitted over the block fading channel with M (M ≤ 3) fading blocks. In Fig. 6, the analytical bounds show a slope of 2 for the rate1/3 turbo code with K = 100 when the WEP is around 10−8 . Currently, it is not easy to give a conceptual justification for this phenomenon due to random multiplexing. For instance, consider the rate-1/3 binary code C over the block fading channel with M = 3. By assuming random multiplexing, the codewords of Hamming distance h in C can be managed so that all the bits ”1” of the codeword distribute over 2 fading blocks with a small but nonzero probability if h ≤ 2(L + 1), and therefore a slope of 2 can be observed. Similarly, a slope

−3

10

−4

10

−5

10

−6

10

−7

10

0

5

10

15

20 Eb/N0, dB

25

30

35

40

Fig. 6. The diversity of the tight Gallager bound at M =3 for the rate-1/3 turbo code with K=100.

VII. CONCLUSION AND FUTURE WORK In this paper, we have derived union bounds for binary block codes over noncoherent block AWGN and fading channels. A true union bound was given for the noncoherent block fading channel without introducing any approximation. Unfortunately, the exact union bound is generally difficult to compute. Hence a suboptimum but computationally efficent union-Chernoff bound was derived and compared to simulation results. It was shown that the B-DPSK scheme, along with the noncoherent iterative receiver proposed in [1], is very efficient for both noncoherent block AWGN and fading channels, especially at high SNRs. For the block fading channel with a small number of fading blocks, the union bound often diverges. Hence we derived a tight bound by employing Gallager’s first bounding

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technique. The effectiveness of the proposed bound has been demonstrated by simulations. We are somewhat surprised to observe that the B-DPSK scheme, along with the practical noncoherent receiver, also works well even for large values of (L + 1). There are several topics for future research. A prompt work is to search convolutional codes for the noncoherent block fading channel. With the Chernoff bound on the PEP (15), one can perform such a search according to the systematic approach proposed in [25] for the coherent block fading channel. The diversity of 2 was observed for a rate-1/3 turbo code over M = 3 fading blocks under random multiplexing. It remains an interesting problem to further exploit if nonrandom multiplexing approach [23] can achieve full diversity in the noncoherent setting. A more challenging work is to derive tight bounds for general noncoherent block fading channels, especially with a medium value of M . This topic was also discussed in [14]. Future work also includes the code design for the block fading channel under both coherent and noncoherent detection. Finally, it is our hope that ”good” codes designed for the block fading channel of block size L may also work well in the realistic correlated fading channel with a similar coherence time.

9

A. Block AWGN Channel

√ For an AWGN channel, rj = ejθ γs sj + zj , j = 1, 2, · · · , L. Given s and θ, it is understood that the elements of β are jointly complex Gaussian variates. Therefore, given s, the characteristic function of (A.2) can be evaluated as n H ¯ o n h H ¯ io ¯ ¯ E esβ F β ¯s = Eθ E esβ F β ¯s, θ o n  ∗ −1   exp sµH F (I − 2sR ) µ θ θ θ = Eθ ∗   det (I − 2sRθ ) n o −1 exp sµH F (I − 2sR∗θ ) µ = (A.4) det (I − 2sR∗ ) where I is the identity matrix, · n ¯ o √ ¯ µθ = E β ¯s, θ = γs ejθ and Rθ

In this appendix, we derive a closed-form expression of the PEP. The Chernoff upper bound on the PEP is also given. In essence, we aim to evaluate the probability à M ! ¯ X ¯ P (s → ˆs) = P r (ˆ ηm − ηm ) > 0¯s . (A.1) m=1

A basic step in computing (A.1) is to find the characteristic function for ηˆm − ηm given the coded sequence s. For simplicity of notation, we shall drop the subscript ”m” with the understanding that we are referring to the mth block. From the definition of ηˆ and η, we can write the difference ηˆ − η in the Hermitian quadratic form ηˆ − η

¯2 ¯ ¯2 ¯ L L ¯X ¯ ¯X ¯ ¯ ¯ ∗¯ ∗¯ rj sˆj ¯ − ¯ rj sj ¯ = ¯ ¯ ¯ ¯ ¯ j=1 j=1    L L X X 1 = rj (ˆ sj − sj )∗   rj (ˆ sj + sj )∗  2 j=1 j=1    L L X 1 X + rj (ˆ sj − sj )∗   rj (ˆ sj + sj )∗  2 j=1 j=1 = β H Fβ

(A.2)

where # " P · ¸ L sj − sj )∗ 0 1/2 j=1 rj (ˆ , F = β = PL 1/2 0 sj + sj )∗ j=1 rj (ˆ

(A.3)

4

= ejθ µ

(A.5)

¯ o 1 n ¯ E (β − µθ )(β − µθ )H ¯s, θ 2 · ¸ 2h 0 = 0 2(L − h) 4

E VALUATION OF THE PEP

¸

=

= R A PPENDIX

−2h 2(L − h)

(A.6)

As implicitly defined in (A.5-A.6), the parameters µ, R are independent of the channel phase θ. After some routine manipulations, (A.4) can be written as ½ ¾ n H ¯ o s(1 − sL)τ (h) 1 ¯ exp −γ E esβ F β ¯s = s 1 − s2 τ (h) 1 − s2 τ (h) 4

=gm (s)

(A.7)

where τ (h) = h(L − h). Hence, Z ∞ Z ²+i∞ M Y ds P (s → ˆs)= dz gm (s) exp(sz) 2πi 0 ²−i∞ m=1 Z ²+i∞ M Y ds = s−1 gm (s) 2πi ²−i∞ m=1 I ds = G(s) 2πi X = Res [G(s), sp < 0] (A.8) p

QM where G(s) = s−1 m=1 gm (s) with gm (s) defined as in (A.7), ² < 0, sp is the pth of the P (1 ≤ P ≤ M ) distince, negative poles of G(s), and Res[·] denotes the residue. In deriving (A.8), a key step is to close the contour of integration from the negative imaginary axis to the positive imaginary axis with a semicircle sweeping the entire left half complex plane in a counterclockwise direction. Since G(s) decays for large values of |s| at least as |s|−3 , the semicircle itself does not contribute to the total integral but encloses all the integrand’s left half plane poles sp < 0, p = 1, 2, · · · , P . In general, the exact PEP (A.8) is difficult to compute. Therefore, it is practically interesting to derive the easy-tocompute Chernoff bound. The Chernoff bound in this case

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was derived in [6] and given by P (s → ˆs) ≤ min λ

R EFERENCES M Y

gm (s),

(A.9)

m=1

where λ is the Chernoff parameter to be optimized.

B. Block Fading Channel √ For the block fading channel, rj = α ˜ γs sj + zj , j = 1, 2, · · · , L. Given s, the elements of β are jointly complex Gaussian variates. Therefore, Eq. (A.2) is of a Hermitian quadratic form in complex Gaussian variates if s is given. Hence, the results in [21] can be directly applied, and the characteristic function of (A.2) takes a simple form of n o n H ¯ o exp sµH F (I − 2sR∗ F)−1 µ ¯ E esβ F β ¯s = det (I − 2sR∗ F) 1 = (A.10) det (I − 2sR∗ F) as

n ¯ o ¯ µ = E β ¯s = 0,

(A.11)

and ¯ o 1 n ¯ R= E (β − µ)(β − µ)H ¯s ·2 ¸ 2h(1 + hγs ) −2h(L − h)γs = (A.12) −2h(L − h)γs 2(L − h) (1 + (L − h)γs ) Hence, P (s → ˆs) =

P X

Res [G(s), sp < 0] ,

(A.13)

p

where G(s) = s−1

M Y

1 . 1 + λτ (hm )(γs − λ(1 + Lγs )) m=1

Similarly, P (s → ˆs) ≤ min λ

10

M Y

1 , 1 + λτ (h )(γ m s − λ(1 + Lγs )) m=1 (A.14)

ACKNOWLEDGMENT The authors wish to acknowledge to the Associate Editor and anonymous reviewers for their helpful comments and constructive suggestions. The authors would like to thank one of referees for addressing the issue of diversity in Fig. 5 in the earlier manuscript of this paper.

[1] R. Chen, R. Koetter, U. Madhow, and D. Agrawal, “Joint noncoherent demodulation and decoding for the block fading channel: A practical framework for approaching shannon capacity,” IEEE Trans. Commun., vol. 51, pp. 1676–1689, Oct. 2003. [2] D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of mpsk,” IEEE Trans. Commun., vol. 38, pp. 300–308, Mar. 1990. [3] H. Leib, “Data-aided noncoherent demodulation of dpsk communications,” IEEE Trans. Commun., vol. 43, pp. 722–725, Feb./Mar./Apr. 1993. [4] D. Divsalar, M. K. Simon, and M. Shahshahani, “The performance of trellis-coded mdpsk with multiple symbol detection,” IEEE Trans. Commun., vol. 38, pp. 1391–1403, 1990. [5] R. Knopp and H. Leib, “M-ary phase coding for the noncoherent awgn channel,” IEEE Trans. Inform. Theory, vol. 40, pp. 1968–1984, Nov. 1994. [6] Y. Kofman, E. Zehavi, and S. S. (Shitz), “Nd-convolutional codes-part i: Performance analysis,” IEEE Trans. Commun., vol. 43, pp. 558–575, Mar. 1997. [7] ——, “Nd-convolutional codes-part ii: Structure analysis,” IEEE Trans. Commun., vol. 43, pp. 576–589, Mar. 1997. [8] F. W. Sun and H. Leib, “Multiple-phase codes for detection without carrier phase reference,” IEEE Trans. Inform. Theory, vol. 44, pp. 1477– 1491, July 1998. [9] G. Colavolpe, G. Ferrari, and R. Raheli, “Noncoherent iterative (turbo) detection,” IEEE Trans. Commun., vol. 48, pp. 1488–1498, Sept. 2000. [10] R. Nuriyev and A. Anastasopoulos, “Pilot-symbol-assisted coded transmission over the block-noncoherent awgn channel,” IEEE Trans. Commun., vol. 51, pp. 953–963, June 2003. [11] I. Motedayen-Aval and A. Anastasopoulos, “Polynomial-complexity noncoherent symbol-by-symbol detection with application to adaptive iterative decoding of turbo-like codes,” IEEE Trans. Commun., vol. 51, pp. 197–207, Feb. 2003. [12] D. Raphaeli, “Noncoherent coded modulation,” IEEE Trans. Commun., vol. 44, pp. 172–183, Feb. 1996. [13] S. Zummo and W. Stark, “Performance analysis of coded systems over block fading channels,” in IEEE Vehicular Technology Conference, (VTC’02/Fall), Vancovour, Canada, 2002, pp. 1129–1133. [14] X. Wu, H. Xiang, and C. Ling, “New gallager bounds in block fading channels,” IEEE Trans. Inform. Theory, submitted for publication. [15] E. Malkam¨aki and H. Leib, “Evaluating the performance of convolutional codes over block fading channels,” IEEE Trans. Inform. Theory, vol. 45, pp. 1643–1646, July 1999. [16] X. Wu, Y. Cheng, and H. Xiang, “The engdahl-zigangirov bound for binary coded systems over block fading channels,” IEEE Commun. Lett., vol. 9, pp. 726–728, Aug. 2005. [17] M. Peleg and S. Shamai, “Iterative decoding of coded and interleaved noncoherent multiple symbol detected dpsk,” Electron. Lett., vol. 33, pp. 1018–1020, June 1997. [18] I. D. Marsland and P. T. Mathiopoulos, “On the performance of iterative noncoherent detection of codedm-psk signals,” IEEE Trans. Commun., vol. 48, pp. 588–596, Apr. 2000. [19] S. S. (Shitz) and I. Sason, “Variations on the gallager bounds, connections, and applications,” IEEE Trans. Inform. Theory, vol. 48, pp. 3029–3051, Dec. 2002. [20] C. Ling, K. H. Li, and A. C. Kot, “Performance of space-time codes: Gallager bounds and weight enumeration,” IEEE Trans. Inform. Theory, submitted for publication. [21] M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques. New York: McGraw-Hill, 1966. [22] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near shannon limit errorcorrecting coding and decoding: Turbo-codes,” in Proc. ICC’93, Geneva, Switzerland, May 1993, pp. 1064–1070. [23] J. Boutros, E. C. Strinati, and A. G. i Fabregas, “Turbo code design for block fading channels,” in Proc. 42th Annu. Allerton Conf. Communication, Control and Computing, Allerton, IL, Sept.-Oct. 2004. [24] E. Malkam¨aki and H. Leib, “Coded diversity on block fading channels,” IEEE Trans. Inform. Theory, vol. 45, pp. 771–781, Mar. 1999. [25] M. Chiani, A. Conti, and V. Tralli, “Further results on convolutional code search for block fading channels,” IEEE Trans. Inform. Theory, vol. 50, pp. 1312–1318, June 2004.

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Xiaofu Wu (S’02-M’05) was born in Jan. 1975. He received the B.S. and M.S. degrees in electrical engineering from the Nanjing Institute of Communications Engineering, Nanjing, China, in 1996 and 1999, respectively, and the Ph.D. degree in electrical engineering from the Peking University, Beijing, China, in 2005. From 1999 to 2001, he was a Lecturer at the Nanjing Institute of Communications Engineering. Since Sept. 2005, he has been with the Southeast Univeristy as a Post-Doctoral research fellow at the National Mobile Communication Research Laboratory. He also held a guest research fellowship at the National Key Laboratory of Communications, Univ. of Electronics Science and Technology of China, Chengdu. His research interests are in the general area of coding and information theory, with recent emphasis on near Shannon-limit codes and Network Coding.

Haige Xiang is the Professor of the School of Electrical Engineering and Computer Science, Peking University. His research interests are in the general area of digital communication and signal processing, such as wireless and satellite communication networks, multi-user detections and interference cancellationmultiple carrier communications and OFDM systems, smart antenna and MIMO systems, channel code technique, software-radio and communication system on chip.

Cong Ling (A’01-S’02-M’05) received the B.S. and M.S. degrees in electrical engineering from the Nanjing Institute of Communications Engineering, Nanjing, China, in 1995 and 1997, respectively, and the Ph.D. degree in electrical engineering from the Nanyang Technological University, Singapore, in 2005. From 1998 to 2001, he was a Lecturer at the Nanjing Institute of Communications Engineering. He has been a Lecturer at King’s College London since 2005. His research interests are in the general area of wireless communications, with emphasis on spread-spectrum, coding and iterative processing, and MIMO communication. Dr. Ling was a recipient of the Singapore Millennium Scholarship.

Xiaohu You was born in August 25, 1962. He received his Master Degree and Ph.D. Degree from Southeast University, Nanjing, China, in Electrical Engineering in 1985 and 1988, respectively. Since 1990, he has been working with National Mobile Communications Research Laboratory at Southeast University, where he held the rank of professor. His research interests include mobile communication systems, signal processing and its applications. He has contributed over 20 IEEE journal papers and 2 books in the areas of adaptive signal processing, neural networks and their applications to communication systems. He was the Premier Foundation Investigator of the China National Science Foundation. From 1999 to 2002, he was the Principal Expert of the C3G Project, responsible for organizing China’s 3G Mobile Communications R&D Activities. Now he is the Principal Expert of the national 863 FuTURE Project.

11

Shaoqian Li received the B.S.E.E. degree from University of Electronic Science and Technology of Xi’an, China, in 1982 and the M.S.E.E. degree from University of Electronic Science and Technology of China (UESTC), Sichuan, in 1984. Since July 1984, he has been a Faculty Member with the Institute of Communication and Information Engineering, UESTC, China, where he is a Professor, since 1997, and Ph.D supervisor, since 2000. He is also the director of National Key Lab of Communication in UESTC and is a Member of Telecommunication Subject with National High Technology R & D Program of China (863 Program). His main fields of research interest include mobile communications systems (3G, Beyond 3G), wireless communication(CDMA, OFDM, MIMO and etc.), integrate circuit in communication systems and radio/spatial resource management.