BER-minimized space-time-frequency codes for MIMO highly frequency-selective block-fading channels

BER-Minimized Space-Time-Frequency Codes for

MIMO Highly Frequency-Selective Block-Fading

Channels

Wei-Cheng Liu and Li-Chun Wang

Department of Communication Engineering

National Chiao Tung University, Hsinchu, Taiwan

lichun@cc.nctu.edu.tw, Tel: +886-3-5712121 ext 54511

Abstract—In this paper, we present bit error rate (BER)-minimized space-time-frequency (STF) block codes for multi-input multi-output (MIMO) highly frequency-selective block-fading channels. We consider the IEEE 802.15.3a ultra-wide band (UWB) channel models (CM) 1–4. Based on a new STF block codes design criterion with the objective of minimizing BER, we develop an efficient searching algorithm for the design of the optimal STF block codes which maximize the coding gain. For 128 subcarriers with two subcarriers jointly encoding with 2–4 transmitting antennas, we find that the optimal STF block codes for all the IEEE 802.15.3a UWB channel models CM 1–4 can be found. Furthermore, the designed STF block codes outperform the recently published high-rate full-diversity STF codes [1] by 1 dB. Last, the proposed STF codes can be decoded by maximum likelihood decoding approach, which is simpler than the sphere decoding principle used in [1].

Index Terms—Bit error rate (BER), block-fading chan-nels, IEEE 802.15.3a channel model, multi-input multi-output (MIMO), space-time-frequency (STF) block codes, ultra-wideband (UWB).

I. INTRODUCTION

T

HE space-time-frequency (STF) coding is a technique which provides error control ability in input multi-output (MIMO) systems, which are usually combined with the orthogonal frequency-division multiplexing (OFDM) tech-nology. The main purpose of using the STF coding is to achieve the full diversity gain. For example, in [1], the authors proposed STF codes which achieve the diversity gain of

NtNrKL, where Nt is the number of transmit antennas,

Nr is the number of receive antennas, K is the number of independent fading blocks in one codeword, and L is the

number of taps of channel impulse response (CIR) between any pair of transmit and receive antennas. The space diversity, time diversity, and frequency diversity are NtNr, K, and L, respectively.

However, in a highly frequency-selective fading channel, the number of taps of CIR could be very large. For example, in the IEEE 802.15.3a UWB channel model [2], the number of taps of CIR is infinity theoretically and about 1000 to 2000 practically. Thus, it is difficult to achieve the full frequency

1_{This work is supported by the National Science Council, Taiwan, under}

the contract NSC95-2221-E-009-147.

diversity under the highly frequency-selective fading channel. Thus, it motivates us to turn to a more fundamental problem: How to design BER-minimized STF codes for MIMO highly frequency-selective fading channels? Here the block-fading channel is defined as follows: The channel remains the same within one fading block and is independent from one block to another one [1].

The difficulties of design BER-minimized STF block codes for the MIMO highly frequency-selective block-fading chan-nels can be discussed in three aspects. Note that we take the IEEE 802.15.3a UWB channel model as an example in this paper. 1) First, the IEEE 802.15.3a channel model has four different sets of parameters, named CM1, CM2, CM3, and CM4. For different channels, we have to design different codes to reflect the channel characteristics. One challenging issue arises: Is there a universal code which is optimal for all the four channel models CM 1–4 for given numbers of subcarriers and transmit antennas? 2) As the numbers of subcarriers and transmit antennas increase, the number of all possible codes becomes astronomical. Thus, the second challenge is how to search the optimal codes efficiently. 3) Because traditional STF coding methods focus on linear codes, it will be challenging to examine if there exist nonlinear optimal STF block codes. To our best knowledge, the design of STF block codes for the MIMO-OFDM systems under the IEEE 802.15.3a channel models considering all the three aforementioned challenges has not been seen in the literature.

Here, we introduce some related works about space-frequency (SF) codes and STF codes for the MIMO-OFDM systems. In [3], the authors analyzed the rate-diversity tradeoff for the MIMO-OFDM channels and presented two asymp-totically optimal SF code constructions. In [4], the authors investigated STF codes for MIMO-OFDM and found an equivalence between antennas and subcarriers. The authors then suggested a complexity-reduced scheme with coding across subcarriers only. In [5], the authors proposed an adap-tive STF coding scheme according to the space-frequency water-filling procedure for MIMO-OFDM systems. In [6], the authors considered STF codes over MIMO-OFDM block-fading channels and derived a sphere packing lower bound on the average word error probability and an upper bound

for pairwise word error probability, but they did not show how to design the optimal codes to achieve these bounds. In [1], authors proposed a systematic design method for high-rate full-diversity STF codes for broadband MIMO block-fading channels. In [7], authors presented rate-two STF block codes for multiband UWB-MIMO communication systems using rotated multidimensional modulation. We will show by simulation that our proposed STF codes have better BER performance than the codes in [1] and [7] do.

The objective of this paper is to design the universally optimal STF block codes for the MIMO-OFDM systems under four kinds of IEEE 802.15.3a UWB channel models, i.e., CM 1–4. The rest of this paper is organized as follows. In Section II, we introduce our system model. In Section III we describe the design criterion, an efficient searching algorithm, and the optimal codes in some examples. In Section IV, we discuss the properties of our proposed optimal STF block codes. We show the numerical results in Section V and give our concluding remarks in Section VI.

II. SYSTEMMODEL

Figure 1 shows our system block diagram. First, we divide the information bits into groups. Each group has two bits. Then we pass the bits to our STF block encoder. For example, if we want to encode across two transmit antennas and two subcarriers, then the codeword can be expressed as a matrix. Then we use an OFDM modulator to allocate every elements in the codeword to corresponding subcarriers and transmit antennas. That is, dij is allocated on the i-th subcarrier and

j-th antenna, for i = 1, 2 and j = 1, 2. The transmitted signals

pass the IEEE 802.15.3a UWB channel. The receiver recovers the original information bits via inverse operations as in the transmitter: We first use an OFDM demodulator to find the codewords. Then we use a maximum likelihood (ML) STF block decoder to find the original information bits.

III. THEUNIVERSALLYOPTIMALSTF BLOCKCODES DESIGN

In this section, we describe a criterion and a efficient searching algorithm of the universally optimal STF block codes.

A. The Optimum Criterion

Our goal is to design the STF block codes to minimize

Pe in [8, (40)]. For given SNR ρ, number of transmit antennas Nt, number of receive antennas Nr, number of

OFDM blocks jointly encoded K, and number of OFDM

subcarriers jointly encoded M , it is equivalent to maximize

the term q =
r_n=1eig_n(S ◦ RM) by designing the matrix

S = (D − ˆD)(D − ˆD)H, where D and ˆD are two distinct

STF block codes codewords,◦ denotes the Hadamard product [9], andRM is the auto-covariance matrix of which definition can be found in [8]. Similar to the rank and determinant criteria of the space-time block coding (STBC) [10], we have to maximize the minimalq along the pairs of distinct codewords.

We first consider the simplest case. LetNibe the number of input information bits for each codeword D. Let M = Nt=

Ni = 2. Let b1, b2 ∈ {0, 1} be the two input bits. We use the binary phase shift keying (BPSK) modulation. Lets1 and

s2 be the two corresponding symbols, thensi = mod(bi) for

i = 1, 2, where mod(x) =



1, if x = 1,

−1, if x = 0. The codewordD

is a 2× 2 matrix with each element being 1 or −1, i.e., D ∈

{1, −1}2×2_{. Then there are 2}2·2 _{= 16 different codewords.} Since there are two input bits, there are 22= 4 possible inputs, i.e., b1b2 ∈ {00, 01, 10, 11}. Hence, we have to choose four distinct codewords for these four different inputs.

For the convenience of expression, let us define the de-modulation function dem(x)
mod−1(x) and the mul-tiple digits version of dem(·) is defined as dem(x)
[dem(x1), dem(x2), · · · , dem(xm)], where the vector x stands for anm-digit number and the i-th digit is xi for 1≤ i ≤ m.

The following equation gives each codeword D a

unique positive integer n as its subscript: Dn =

 d11 d12 d21 d22  : bd(dem(d) + 1) = n 

, where the function

bd(x) is to transform a binary number x into its decimal form

andd = [d11, d12, d21, d22]. Now, the set that contains all the codewords isC = {D1, D2, . . . , D16}. Let B be a subset of

C and B contains four codewords. Now, our problem can be

mathematically described as finding a setB∗ such that

B∗= arg max B⊂C,|B|=4 D, ˆD∈B,D= ˆmin D r  n=1 eig_n(S ◦ RM), (1) where |B| is the number of elements of B.

B. An Efficient Searching Algorithm for the Optimal STF Block Codes

In order to simplify the representation of our problem and provide more insight, we introduce a graph representation to our code space. We represent each codeword as a vertex with number n, and between any two distinct vertices there is an

undirected edge with metricq. Then, for the M = Nt= Ni= 2 case, we can use a complete graph [11] with 16 vertices which is denoted by K16 to represent our code space. Then our problem becomes to find the optimalK4∗ inK16such that the minimal metric inK₄∗ is the largest one among that of all

K4 in K16. There are 16₄ = 1820 distinct K4 in K16. To find the minimal metric within eachK4we need to search for ₄

2

= 6 metrics. Thus, we need to do 1820· 6 = 10920 times of searching to findK₄∗.

For generalM , Nt, andNi, there are 2MNt _{vertices for the} BPSK case. The complexity of the complete search becomes

O2₂MNtNi ₂Ni

2 

. The complexity grows rapidly as M , Nt, andNi increase. Thus, it is necessary to find a more efficient algorithm to search for the optimal STF block codes.

For the case M = Nt = Ni = 2 and CM1, we find that the metricq takes only on eight different values. Sorting these

values in the decreasing order, we then haveq ∈ {64, 16.3314,

many searching steps if we search K4 subject to the largest m metrics for m = 1, 2, . . ., until we find all K₄∗ for a certain value of m. Let use take the M = Nt = Ni = 2 case as an example. For m = 1, we only consider the edges with the

largest metric 64. Obviously, it does not contain anyK4. For

m = 2, we consider the edges with the largest two metrics:

64 and 16.3314. After searching, we also find that there is no

K4 in this graph. Form = 3, we consider the edges with the largest three metrics: 64, 16.3314, and 16. We find that there are totally eightK4in this graph, they are:{1, 7, 12, 14}, {1, 8, 10, 15}, {2, 8, 9, 15}, {2, 8, 11, 13}, {3, 5, 12, 14}, {3, 6, 12, 13}, {4, 5, 11, 14}, and {4, 6, 9, 15}. Note that we do not need to search for the case m > 3, because we already

find that the max-min value ofq is 16.

We use the same method to search the optimal STF block codes for CM2, CM3, and CM4. CM2 and CM3 both have the same optimal STF block codes as CM1 does, but there are nine optimal STF block codes for CM4. These nine codes contains the eight optimal codes which are the same as that of CM1 and an additional codes{4, 6, 11, 13}. This is an interesting discovery that for different channel model, the optimal codes may be different. Thus, in order to design the optimal codes, we have to take the channel model into account.

In order to design the codes that are optimal for all channel model, we choose the eight optimal STF block codes for CM1 out of 22·2₄ = 1820 candidates. The next step is to transform these codes to a code structure. Take the code

{3, 5, 12, 14} as an example. According to the

matrix-indexing procedure we defined in Section III-A, we find these four integers correspond to the following codewords: D3 =  −1 −1 1 −1  , D5 =  −1 1 −1 −1  , D12 =  1 −1 1 1  , D14 =  1 1 −1 1 

. We assign these four codewords to the information

bits 00, 01, 10, and 11, respectively. Note that we can choose another assignment and the max-min value of q will not

change. To discover the code structure from these codewords, we first consider the element in the first row and first col-umn of them. They are Di[1, 1] = {−1, −1, 1, 1}, where

i ∈ {3, 5, 12, 14}. Since each position can take values on −1

or 1, there are totally 24 = 16 possibilities. We establish a truth table of these 16 values, as a function of s1 and s2. For some cases, we find it is more convenient to express the function in terms of b1 and b2. Use this table to check the function f (s, b) for all the elements of Di, we finally find the code structure is



s1 s2

−s2 s1 

. It is the Alamouti coding

scheme [12]. The other seven optimal code structures are  s1 s2 −s1s2 s1  ,  s1 s2 s2 −s1s2  ,  s1 s2 s2 −s1  ,  s1 s2 −s1s2 −s1  ,  s1 s2 −s2 −s1s2  ,  s1 s2 −s2 s1s2  , and  s1 s2 s1s2 −s1  .

The pseudo code of our proposed searching algorithm for the optimal STF block codes can be found in Algorithm 1. Note that in the ninth line we only consider the vertices with degree being at least three because any vertex in a K4 must

satisfy this condition.

Algorithm 1: The searching algorithm for the optimal

STF block codes. input :M , Nt,Ni, and CM. output: B∗. B∗← ∅ 1 G ← K2MNt 2 found← False 3 foreach 1≤ i, j ≤ 2MNt _do 4 S ← (Di− Dj)(Di− Dj)H 5

E(G)i,j←
rn=1eign(S ◦ RM(CM))

6

metric← list of distinct values of E(G) in

7

decreasing order

form ← 1 to Length(metric) do

8

F ← ({e : e ∈ E(G), e ≥ metric[m]}, {v : v ∈

9 V (G), deg(v) ≥ 3}) foreach B, {B ⊂ F, |V (B)| = 2Ni} do 10 ifB is K2Ni then 11 found← True 12 B∗← B∗∪ {B} 13 iffound then 14 returnB∗ 15

C. Optimal STF Block Codes for the Other Cases

We use the algorithm described in Section III-B to find the optimal STF block codes for the case M = Ni = 2 and

Nt = 3. We find that there are 54 different optimal STF block codes for all the four CM out of 23·2₄ = 635376 candidates. In order to simplify the expression of the code matrix, we define s3
s1s2, si

−s_i for i = 1, 2, 3, and

sijk
[si sj sk]. Then, among these 54 optimal STF block codes, 28 of them have the form of [sT₁₂₃
sT_a]T and the other 26 codes have the form of [sT₁₂₃sT_b]T, wherea ∈ {3’12, 23’1,

13’2’, 3’12’, 23’1’, 3’21’, 132, 231, 21’3’, 3’1’2, 231’, 3’2’1’, 21’3, 3’1’2’, 312, 2’13’, 321, 2’3’1, 312’, 2’13, 2’3’1’, 1’3’2’, 32’1, 2’31, 31’2, 1’32, 2’31’, 31’2’} and b ∈ {13’2, 3’12, 23’1, 3’21, 3’12’, 23’1’, 3’2’1, 21’3’, 3’1’2, 132’, 231’, 21’3, 3’1’2’, 2’13’, 2’3’1, 1’3’2, 312’, 2’13, 321’, 2’3’1’, 2’31, 31’2, 32’1’, 2’31’, 31’2’, 1’32’}. For the case M = N_i= 2 andNt= 4, we find 5148 different optimal STF block codes for all the four CM out of24·2₄ = 174792640 candidates. Due to the space limit, we do not list all codes here. One of the optimal STF block codes is



s1 s1 s2 −s1s2

s2 −s1s2 s1 s2 

.

For the case M = 3 and Ni = Nt = 2, there is an interesting fact. We find that there does not exist any optimal STF block codes for all the four CM out of22·3₄ = 635376 candidates. For the caseM = Nt= 3 and Ni= 2, we find that there does not exist any optimal STF block codes for all the four CM out of23·3₄ = 2829877120 candidates. For the case

TABLE I

THE CODING GAIN OF THE OPTIMAL CODES WE HAVE FOUND INSECTION III.

Coding Gain (dB) CM1 CM2 CM3 CM4

Nt= 2 0 0 0 0

Nt= 3 0.64 0.66 0.71 0.84

Nt= 4 0.49 0.50 0.54 0.61

STF block codes for all the four CM are the same. There are totally 1464 optimal codes out of 24·3₄ = 11710951848960 candidates and they are all nonlinear. For the case M = 4

and Ni = Nt = 2, we find that there does not exist any optimal STF block codes for all the four CM out of ₂2·4

4

= 174792640 candidates.

IV. PROPERTIES OF THEOPTIMALSTF BLOCKCODES

A. Coding Gain

The coding gain of a code can be computed via CG = 1

4Nt[
_r

n=1eign(S ◦ RM)]1/r. In Table I we list the coding gain of the optimal codes we have found in Section III. For the Nt = 2 case, the optimal STF block code is Alamouti code. Its coding gain is one [10]. For theNt= 3 and Nt= 4 cases, we find that the coding gain is greater than 0 dB by a little amount. Thus, we can predict that the BER performance of these three codes will be very close.

B. Diversity Order

The diversity order of a code isrKNr. The optimal codes we found above all have the same value of r for the same

values ofM , Ni,Nt, and CM and the same modulation. Thus, for the same values ofK and Nr, the optimal codes achieves the same diversity order under the same condition. The values of r for different kinds of optimal codes are listed in Table

II. From this table, we find a interesting fact. Sometimes the optimal codes achieve different diversity order for different values of CM. For example, when M = 3, Ni = 2, Nt= 2, and the modulation is BPSK, the diversity order is two for CM1 and CM2 and three for CM3 and CM4.

V. NUMERICALRESULTS

Our simulation environment is an MIMO-OFDM system. The number of total subcarriers is 128 and the sub-band bandwidth is 528 MHz. We apply the IEEE 802.15.3a UWB channel model CM 1–4 [2].

A. BER Comparison with STF Codes in [1] and [7]

Figure 2 shows the BER comparison of our code with Chusing’s code [7] and Zhang’s code [1] for theM = 4, Ni= 2, Nr= 1, Nt= 2 case in the IEEE 802.15.3a UWB channel model CM4. We can see that the diversity gains of the three codes are the same, but our code has better BER performance than Chusing’s and Zhang’s codes do. At BER = 10−4, the coding gain between our code and Chusing’s code is about 8 dB and the coding gain between our code and Zhang’s code is about 1 dB.

TABLE II

THE VALUES OFrWHICH IS THE RANK OF MATRIXS ◦ R_MFOR DIFFERENT KINDS OF OPTIMALSTFBLOCK CODES.

M Ni Nt CM modulation r 2 2 2 1–4 BPSK 2 2 2 2 1–4 QPSK 2 2 2 3 1–4 BPSK 2 2 2 4 1–4 BPSK 2 3 2 2 1,2 BPSK 2 3 2 2 3,4 BPSK 3 3 2 3 1 BPSK 2 3 2 3 2–4 BPSK 3 3 2 4 1–4 BPSK 3 4 2 2 1 BPSK 3 4 2 2 2,3 BPSK 2 4 2 2 4 BPSK 4

B. Impact of Number of Transmit Antennas Jointly Encoded (Nt) for Two Subcarriers Jointly Encoded (M = 2)

Figure 3 shows the impact of number of transmit antennas jointly encoded on the BER for CM1, CM2, CM3, and CM4 for the optimal STF block codes for the M = Ni = 2 case. Figs. 3(a), 3(b), and 3(c) are for the cases Nt= 2, 3, and 4, respectively. For each sub-figure, the BER decreases as CM increases. This phenomenon can be explained by the coding gain. In Table I, the coding gain increases as CM increases for the casesNt= 3 and 4, thus the BER decreases.

Moreover, we find a surprising fact. The BER in Figs. 3(a), 3(b), and 3(c) are almost the same for the same CM. In other words, the BER for a certain CM does not change as the number of transmit antennas increases. This result is quite different from the STBC case. In STBC, increasing the number of transmit antennas will decrease the BER performance [10]. Thus, we may conclude that in the MIMO-UWB systems, using multiple transmit antennas does not provide significant improvement to the BER performance, because the UWB channels already possess rich diversity inherently. In the uncoded UWB systems using multiple antennas, there exists the same phenomenon [13].

VI. CONCLUSIONS

In this paper, we study the BER-minimized STF block codes designed for the MIMO highly frequency-selective block fad-ing channels. We consider the IEEE 802.15.3a UWB channel model. Based on the BER analysis under the aforementioned environment in [8], we provide a BER-minimized design criterion, an efficient searching algorithm for the optimal STF block codes, and optimal BER performance curves. Compared with other space-frequency-time codes [1], [7] for MIMO-OFDM communication systems under the UWB channel, our code has about 1 and 8 dB coding gain at BER = 10−4, respectively. On the other hand, increasing the number of transmit antennas does NOT improve the BER performance for the MIMO-UWB systems when M = 2.

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Fig. 1. The system block diagram.

0 2 4 6 8 10 12 14 16 18 20 10−4 10−3 10−2 10−1 100 Eb / N0 (dB)

Bit Error Rate

Chusing‘s Code Zhang‘s Code Our Code

Fig. 2. The BER comparison of our code versus Zhang’s code [1] and Chusing’s code [7] for four subcarriers jointly encoded, two input information bits for each codeword, one receive antenna, and two transmit antennas jointly encoded in the IEEE 802.15.3a UWB channel model CM4. The modulation is BPSK. 0 2 4 6 8 10 12 14 16 18 20 10−3 10−2 10−1 100 E_b / N₀ (dB)

Bit Error Rate

CM1 CM2 CM3 CM4 (a) 0 2 4 6 8 10 12 14 16 18 20 10−3 10−2 10−1 100 E b / N0 (dB)

Bit Error Rate

CM1 CM2 CM3 CM4 (b) 0 2 4 6 8 10 12 14 16 18 20 10−3 10−2 10−1 100 E_b / N₀ (dB)

Bit Error Rate

CM1 CM2 CM3 CM4

(c)

Fig. 3. The effect of different number of transmit antennas jointly encoded (Nt) on the BER for CM1, CM2, CM3, and CM4 for the optimal STF block codes for two subcarriers jointly encoded and two input information bits for each codeword. The modulation is BPSK. (a)Nt = 2. (b) Nt = 3. (c) Nt= 4.