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Optimal Binary Training Sequence Design for Multiple-Antenna Systems Over Dispersive
Fading Channels Shan-An Yang and Jingshown Wu
Abstract—Accurate and efficient channel estimation is important in mul-tiple-antenna communication systems in order to effectively reduce the mu-tual interference among different transmitting antennas. For a nondisper-sive channel that is modeled by a single tap for each transmitting and re-ceiving antenna pair, the well-known Hadamard sequences can be applied to estimate the channel coefficients. However, for a dispersive channel that has multipath problem and is modeled by multiple taps, the optimal se-quences must have both good autocorrelations and cross correlations. The existence of binary sequences with such good property is an open problem. In this paper, we devise an algorithm to find these sequence sets. These codes can be applied in multiple-antenna systems.
Index Terms—Channel estimation, fading, MIMO, training sequence.
I. INTRODUCTION
Efficient channel estimation is important for multiple-antenna sys-tems especially when the number of antennas increases. To avoid the degradation of estimation accuracy due to interference, an intuitive way is to transmit training sequences for each transmitting antenna in turn [1]. For a system withM antennas, this scheme requires M times band-width compared with a single antenna transmitter system. However, orthogonal training sequences can be simultaneously applied for each transmitter antenna to estimate the channel efficiently [2], [3]. For a single tap coefficient discrete channel model, it is well known that or-thogonal sequences are the optimal training sequences that minimize the estimation errors if the additive noises are identical independent Gaussian random processes. In this case, a Hadamard matrix can be ap-plied. However, in the case of multipath channel, the channel for each pair of transmitting and receiving antennas should be modeled by sev-eral taps. It can be proven that the training sequences should have both good autocorrelation and cross correlation. Existence of such training sequence sets is still an open problem. In this paper, we discuss the existence of such optimal binary training sequence sets and propose a search algorithm.
Manuscript received October 9, 2000; revised July 10, 2001. This work was supported in part by the National Science Council and the Ministry of Education, Taiwan, R.O.C., under Grants NSC89-2213-E002-061 and 89-E-FA06-2-4.
The authors are with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan (e-mail: wujsh@cc.ee.ntu.edu.tw).
Digital Object Identifier 10.1109/TVT.2002.800638 0018-9545/02$17.00 © 2002 IEEE
Fig. 1. The burst data structure forM-antenna transmission.
II. PROBLEMDESCRIPTION
The multiple-antenna system under consideration hasM transmit-ting antennas andN receiving antennas. The burst data structure for each transmitting antenna is shown in Fig. 1, wherex (1 : L); =
1; 2 . . . M; denotes the training sequence to be transmitted from the th antenna. L is the length of each training sequence. The training sequences are embedded in each burst. Data bursts from different an-tennas will have different training sequences that are designed together so that the coexistence of the training sequences does not affect the channel estimation accuracy. Provided that the burst is short and the channel is quasi-static within a burst, the output of discrete equivalent channel can be expressed as
y (k) = M =1 V i=0 h (i)x (k 0 i) + n (k) (1) whereV is the order of channel memory and h (i) denotes the
re-sponse of the th receiving antenna of the receiver to a discrete unit sample applied in the th transmitting antenna. n (k) is assumed to be
identical independently distributed Gaussian random noise. With good synchronization, small value ofV is enough to well approximate the channel.
With simple manipulations, we can prove that the training sequence set is optimal if the training sequence in each antenna is not only or-thogonal to its shifts withinV taps but also orthogonal to the training sequences in other antennas and their shifts withinV taps (see Ap-pendix). In other words, the optimal training sequences should satisfy the following equation:
L0V k=1
x (k)x (k + s) = 0
where ; = 1 M
s = 0 V; when 6= 1 V; when = . (2) III. PROPERTIES AND ASEARCHALGORITHM OF THEOPTIMAL
TRAININGSEQUENCES FORBPSK
In this section, we will discuss existence of the optimal training se-quence selected from the binary phase-shift keying (BPSK) constella-tion. In other words, we will discuss the binaryf1; 01g-sequence sets satisfying (2). First, we prove thatx (k) = x (k + P ) for k V .
For convenience, we denoteP = (L0V ). Consider s = 1 and = in (2); we have
P k=1
x (k)x (k + 1) = 0: (3)
Lety(k) = x (k)x (k + 1). Thus, y(k) = 01 implies x (k) =
0x (k + 1). Since Pk=1y(k) = 0, half of y(k); k = 1 P; are
01. Since P=2 is an even number (see Property 1 in Table I), we have x (P + 1) = (01)P=2x (1) = x (1): As a consequence, finding
optimal sequencesx (1 : L) is equivalent to obtain x (1 : P ), which
satisfies the following equation:
p k=1
x (k)x (k + s mod P ) = 0
where ; = 1 M
s = 0 V; when 6= 1 V; when = . (4) Hereafter, we will call a sequence set that satisfies (4) as a(P; V; M) code. Some properties of this code are listed in Table I. ForM 2 andV 1, we require that x1(1 : P ); x1(2 : P + 1); x2(1 : P );
andx2(2 : P + 1) are mutually orthogonal. This leads to result that
P must be a multiple of four. (This is the same reason that the order of a Hadamard matrix is multiples of four.) Thus, we have Property 1. Property 2 states that multiplying01 to any sequence in a (P; V; M) code can result another(P; V; M) code. Properties 3, 4, and 5 are quite trivial and can be easily verified. Property 3 implies that we can find all the(P; V; M 0 1) codes and then find all the (P; V; M) codes based on all the(P; V; M 01) codes. Property 4 says that if we reverse every sequence simultaneously, the result is still a(P; V; M) code. Property 5 implies that if we shift every sequence by the same amount, the result is still a(P; V; M) code. Property 6 gives an upper bound of M when the valuesP and V are given.
Based on the above properties, we can restrict the first entry of each seqeunce to 01 (or 1) and restrict the sequences permutation in a (P; V; M) code to the order in the (P; V; 1) code table without loss of generality. We describe the search algorithm as follows.
Step 4) Find all of the(P; V; 1) codes for the given values of P and V . Construct the (P; V; 1) code table; each code in the code table is presented by an unique number. As in Fig. 2, for example,(P; V; 1) #2 represents the second (P; V; 1) code in the(P; V; 1) code table.
Step 5) Find all the(P; V; 2) codes from all the pairs of (P; V; 1) by checking the orthogonality conditions. At the same time, we construct an index table in order to reduce the complexity forM = 3.
Step 6) Letm = 3.
Step 7) Construct the(P; V; m) code table and index table by ap-plying the(P; V; m 0 1) code table and index table. Note that a(P; V; m) code is composed of two (P; V; m 0 1) codes, where the lastm 0 2(P; V; 1) codes of a (P; V; m 0 1) code is identical to the first m 0 2(P; V; 1) codes of the other(P; V; m01) code. As a result, we only have to check the orthogonality of the first(P; V; 1) code in the former (P; V; m 0 1) code and the last (P; V; 1) code in the latter (P; V; m 0 1) code to determine whether these two codes constitute a(P; V; m) code when m 3.
Step 8) Ifm > M or the newly constructed code table is empty, then the process is finished. Otherwise, increasem by one and go back to Step 4).
In Fig. 2, we illustrate how the code table and index table are con-structed. As illustated in the first three columns, code(P; V; 2) #1 is composed of code(P; V; 1) #1 and #3. We build the code table for M = 2 by testing all possible pairs of (P; V; 1) and list them in the third column. At the same time, an index table is constructed with its contents pointing to the starting positions of the corresponding codes in the code table. In order to find the(P; V; 3) codes, we start from the
TABLE I
SOMEPROPERTIES OF A(P; V; M)CODE
Fig. 2. The construction of the code table and the index table.
first element in the code table forM = 2. Since the second element of (P; V; 2) #1 is (P; V; 1) #3, we refer to the third and fourth elements in the index table which are denoted byk3 and k4. So, we search from k3th through (k4 0 1)th codes in the code table to see if any code can pair with(P; V; 2) #1 to constitute a (P; V; 3) code. We only have to check the orthogonality between the first(P; V; 1) code in (P; V; 2) #1 and the last(P; V; 1) code in the other (P; V; 2) codes between k3th through(k4 0 1)th. After finding all the (P; V; 2) codes which can be
paired with(P; V; 2) #1, we continue with (P; V; 2) #2 and then #3 until reaching the last element of the code table.
A. Computational Complexity Analysis
For(P; V; M) codes, restricton on the first entry of the sequence has a reduction factor of2M times. In addition, restriction on the permuta-tion reduces the computapermuta-tion complexity byM factorial times. It can be proven that the total number of(P; V = 1; M = 1) sequences isnCn=2. For greater values ofM, the computational complexity de-pends on the total number of codes that exist and is hard to analyze. Thus, we take(P; V ) = (16; 1) as an example and give the numerical results in Table II. Columns (A) and (B) demonstrate the total number of codes found by the algorithm and the total number of codes without any restriction, respectively. Column (C) shows the reduction factor defined as a ratio between Column (B) and Column (A). Column (D) gives the probability of finding such(P; V; M) code by random guess. It is observed that the values in Column (A) increase at first, reaches its maximum value atM = 3, and decreases thereafter. On the con-trary, the values in Column (B) increase all the way untilM = 8. To determine that two sequences satisfy (4), we have to test orthogonality (2V + 1) times. Each time it requires an operation of bit-wiseXOR between twoP -bits words and an operation to count the total number of ones in aP -bit word. In Columns (E) and (F), we show the total numbers of orthogonality tests required in each round for the proposed algorithm and for a method by directly searching all combinations of f01; 1g in each bit. This implies that our algorithm provides a feasible approach for a personal computer to determine existence or nonexis-tence of(P; V; M) codes for a small value of P . Although the pro-posed algorithm will become computationally impractical for a large value ofP , it suffices for the purpose of designing training sequences in multiple-antenna systems since they are often short.
B. An Alternative Way of Construction
If we do not intend to find the maximum value of M, we can construct such codes by applying shifts of a binary al-most perfect sequence (BAPS) [4], [5]. Take M = 2 as an example; we can choosex1(k) = ak andx2(k) = ak+p=4, where
TABLE II
COMPARISONBETWEEN THEALGORITHM AND THEDIRECTEXHAUSIVE SEARCHWITHOUTANYRESTRCTION
TABLE III
THEMAXIMUMVALUES OFMGIVEN(P; V )
sequence f01; 01; 1; 01; 01; 01; 01; 01; 1; 1; 01; 1; 01; 1; 1; 1g is a BAPS withp = 16. Thus,
x1= f01; 01; 1; 01; 01; 01; 01; 01; 1; 1; 01; 1; 01; 1; 1; 1g
and
x2= f01; 01; 01; 01; 1; 1; 01; 1; 01; 1; 1; 1 0 1; 01; 1; 01g
satisfy the conditions of the optimal training sequences for (P = 16; V = 3; M = 2). However, we see in Table III that the maximal number of achievableV is 4, not 3. Therefore, although applying BAPS helps to construct such code, it does not achieve the maximum value ofM or V .
IV. SEARCHRESULT ANDNUMERICALSIMULATION In Table III, we list the maximal numbers of achievableM given P andV . In the table, N means that no such (P; V; M) code exists. One can observe that theM value for some (P; V ) combinations achieve the upper bound given by Property 6, while others do not. For example, whenP = 16 and V = 1, the maximum achievable M is 8, which is just the upper bound given by Property 6. However, when(P; V ) = (12; 1), the maximal achievable M is only 5, not 6.
In Table IV, we list at least one example for all existing codes withP 16. Although we have proved the optimal property of the proposed training seqeunces under the assumption of quasi-coherent channel, we are also interested in their performance in an environment with Doppler frequency shift. We perform numerical simulation to compare three different training sequence sets. The sequences under test are listed in Table V. The first one is the optimal sequence set with (P; V; M) = (8; 1; 4) as proposed. The second sequence set
TABLE IV
EXAMPLES OF THE(P; V; M)CODES
TABLE V
THECODESUNDERTEST IN THENUMERICALSIMULATION
is constructed with a well-known pseudorandom binary sequence (PRBS) with different shifts in different transmitter antennas. Since we want to keep the cross correlation low between shifts, the maximal number of transmitting antennas is three with a PRBS of length seven. The transmitted power is increased to compensate the shorter length for a fair comparison. The third one is an arbitrarily chosen sequence set. The channel tap coefficients are assumed to be indepen-dent complex Gaussian random variable with uniformly distributed phaseand Rayleigh distributed amplitude. The transmitted power from each transmitting antenna is assumed to be the same. Here, we use the well-known Jakes’ model to perform the simulation. The results are shown in Fig. 3. We see that when the Doppler frequency shift
Fig. 3. Performance comparison of three sequence sets with Doppler frequency shift.
is not severe, the advantage of the proposed sequence set remains unchanged.
V. CONCLUSION
In this paper, we study the design of optimal trianing sequence sets for multiple-antenna communication systems in a dispersive fading en-vironment. The conditions of the optimal training sequences for the multiple-antenna systems are proposed and proven. We also propose an algorithm to search the optimal training sequences and analyze the complexity of the algorithm. Existence of such codes is shown by an exhaustive search for code length less than or equal to 16. Examples of the search result are listed in a table. Numerical tests are performed to test their performance in a nonideal environment. We believe that these sequences can be used for channel estimation in multiple-antenna com-munication systems.
APPENDIX
PROOF OF THECONDITION FOROPTIMALTRAININGSEQUENCES The firstV sampled values are discarded to avoid the interference from symbols prior to the first training symbol. Thus, we can rewrite (1) in matrix form as
y = XH + N (A.1)
wherey = [y (V + 1) y (V + 2) 1 1 1 y (L)]T (see the equation at the bottom of the page). IfXHX is invertible, the least squares es-timation for the channel matrixH is given by
^
H = (XHX)01(XHy ): (A.2) The above estimation is unbiased. Thus, the estimation error and vari-ance are given by
e = ^H 0 H (A.3)
and
E[ke k2] = E tr e eH
= tr (XHX)01XHE N N H X(XHX)01 : (A.4)
In the case of independent white discrete Gaussian noises, we have E N N H = n2IL0V: (A.5)
Therefore, (A.4) is simplified as
E ke k2 = n2tr[(XHX)01] = n2 (MV +M) k=1 1 k (A.6) X = x1(1 : V + 1) x2(1 : V + 1) 1 1 1 xM(1 : V + 1) x1(2 : V + 2) x2(2 : V + 2) 1 1 1 xM(2 : V + 2) .. . ... ... ... x1(L 0 V : L) x2(L 0 V : L) 1 1 1 xM(L 0 V : L) H = [h1 h2 1 1 1 hM ]T; N = [n (V + 1) n (V + 2) 1 1 1 n (L)]T x (i : j)^=[x (i) x (i + 1) 1 1 1 x (j)]; h = [h (V ) h (V 0 1) 1 1 1 h (0)] i = 1 L 0 V; j = i + V; = 1; 2; . . . ; M and = 1; 2; . . . ; N:
wherekdenotes the eigenvalues ofXHX. By applying Cauchy in-equality, we have (MV +M) k=1 1 k (MV +M) k=1 k M2(V + 1)2: (A.7) Thus, minimization of (A.7) requires thatkis constant for allk = 1 (MV + M). This implies that XHX is a diagonal matrix with
all the diagonal entries equal to a constant. ACKNOWLEDGMENT
The authors would like to thank the reviewers for their valuable com-ments and suggestion.
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