Convolutional Multiplexing for Multicarrier Transmission: System Performance and Code Design Wei Jiang and Daoben Li School of Information Engineering, Beijing University of Posts and Telecommunications, Beijing 100876, China. E-mail: [email protected] Abstract. The paper presents a new multiplexing scheme, called convolutional multiplexing (CM), to achieve high diversity gain and high spectrum efficiency for OFDM-based systems. In this scheme, data symbols are spread onto several subcarriers by the convolutional spreader. Spectrum efficiency can be improved by two approaches: high order modulation and multi-code multiplexing. With the best spreading codes searched out, the multi-code multiplexing OFDM-CM system performs better in AWGN channel, but may lose diversity gain in frequency selective fading channels. On the other hand, the single-code spreading OFDM-CM system with high order modulation can achieve the maximum diversity order. Simulation results show that the multi-code convolutional multiplexing perform better than code-division multiplexing (CDM) for OFDM-based systems. Keywords: Convolutional multiplexing, multi-code multiplexing, diversity gain, OFDM.

1. Introduction OFDM (Orthogonal Frequency Division Multiplexing) [20, 3] has been thought of as a promising technology in the future wireless communications for its advantage of relatively high spectrum efficiency, immunity to the inter-symbol interference (ISI), etc. Several standards have adopted it for the physical layer transmission, such as DAB [6], DVB [7], WLAN [19], and E-UTRA [1]. However, in wireless channel, since the data symbols are parallelized and transmitted on narrow-band subcarriers, the OFDM system suffers from frequency selective fading. Several improved schemes have been proposed to achieve diversity gain in frequency selective fading channels, such as Coded OFDM (COFDM) [15], Multicarrier Code Division Multiple Access (MC-CDMA) [21, 9, 4], or OFDM Code Division Multiplexing (OFDM-CDM) [14]. COFDM can achieve diversity gain and coding gain at the same time, but there is always a loss in spectrum efficiency. The MC-CDMA system spreads the data symbols onto multiple subcarriers according to the user-specific codes, and the signal of the different users can be multiplexed and transmitted on the same cluster of subcarriers. Though, the MC-CDMA c 2008 Kluwer Academic Publishers. Printed in the Netherlands.

CM_WPC.tex; 20/08/2008; 21:30; p.1

2 Symbol Mapper

Convolutional Spreader

IFFT

CP

D/A

De-Interleaver

FFT

Remove CP

A/D

Channel

Interleaver

De-Mapper

Detector

Figure 1. The model of the OFDM-CM system.

system suffers from multiple access interference (MAI) and the diversity gain will be reduced when the system load is high [12]. In this paper, we present a new scheme to multiplex the signal on a group of orthogonal subcarriers, by which the data symbols are spread onto several subcarriers in convolutional manner. We called it Orthogonal Frequency Division Multiplexing-Convolutional Multiplexing (OFDM-CM) system. In frequency selective fading channels, the proposed scheme can achieve significant diversity gain. To improve the spectrum efficiency, two methods can be adopted. One is high order modulation; the other is multi-code multiplexing. The former outperforms the latter in fading channels because the maximum diversity order can always be obtained. However, compared to the uncoded OFDM system with the same spectrum efficiency, the OFDM-CM system with multi-code multiplexing can achieve notable coding gain both in AWGN (Additive White Gaussian Noise) channel and in fading channels. The rest of the paper is organized as follows. Section 2 introduces the system model and describes the convolutional multiplexing scheme in detail. Section 3 discusses the performance of the proposed system in AWGN channel and in frequency selective fading channels. Section 4 introduces a free distance searching algorithm and presents several good-performance codes for single-code scheme and multi-code scheme. The numerical simulation results are given in the following section. Finally, a conclusion is drawn in the section 6.

2. OFDM-CM system model The model of the OFDM-CM system is illustrated in Fig. 1. At the transmitter, the data stream is mapped into symbols and then passed through a convolutional spreader, which introduces inter-symbol constraint in the stream. Then the symbol sequence is interleaved and transmitted by the IFFT-based OFDM transmitter. Correspondingly, at the OFDM-CM receiver, after the FFT operation, the symbols are deinterleaved and detected.

CM_WPC.tex; 20/08/2008; 21:30; p.2

3 S(l-1 )

S(l) D C(1)

S(l-N+2 )

D

C(2)

..

S(l-N+1 )

..

D

C(N-1 )

C(N)

X (l)

Figure 2. The structure of the convolutional spreader.

The convolutional spreader can be implemented by a transversal filter, as shown in Fig. 2. The complex valued coefficients on the taps compose a spreading code. The output of the spreader is the summation of the delayed symbols multiplied by the spreading code. Assume the data bits within a block are mapped into a symbol sequence of length L, which is denoted as S = [S(1), S(2), · · · , S(L)]. The spreading code of length N is denoted as C = [C(1), C(2), · · · , C(N )]. Then, the output of the convolutional spreader can be expressed as X(l) =

N X

n=1

C(n)S(l − n + 1), 0 < l ≤ L,

(1)

where S(l) = 0 for l ≤ 0. For the simplicity of the analysis, we normalize the spreading codes, i.e., N X

n=1

|C(n)|2 = 1.

(2)

To improve the spectrum efficiency of the system, high order modulation, such as M-PSK, M-QAM, can be utilized. Another approach is to use multi-code multiplexing. In this case, if there are K parallel streams, where K ≤ N , the k-th data symbol stream and the kth spreading code are denoted as S k = [Sk (1), Sk (2), · · · , Sk (L)] and C k = [Ck (1), Ck (2), · · · , Ck (N )], respectively. Then the output of the convolutional spreader is X(l) =

K X N X

k=1 n=1

Ck (n)Sk (l − n + 1), 0 < l ≤ L.

(3)

Let Y = [Y (1), Y (2), · · · , Y (L)] denote the received symbol vector after the deinterleaver. In AWGN channel, it can be expressed as Y (l) = X(l) + n(l), 0 < l ≤ L,

(4)

where n(l) is the additive complex Gaussian noise with mean zero and variance N0 .

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4 As for the frequency selective Rayleigh fading channel, if the cyclic prefix (CP) is longer than the maximum delay spread of the channel, the subcarriers undergo flat fading. We denote the fading coefficient on the l-th subcarrier as h(l), and the received symbols are Y (l) = h(l)X(l) + n(l), 0 < l ≤ L.

(5)

We also assume h(l) is independent amongst the subcarriers, which is valid when the subcarrier interval is larger than the coherence bandwidth, or ideal frequency interleaver is used. The system performance derived from this assumption can serve as a benchmark for the system design. As shown in (1) and (3), the convolutional spreader introduces intersymbol constraint, just like the convolutional code. So it can also be represented by a trellis diagram and a symbol block can be represented by a path in the trellis which starts from and ends in the all-zero state. The maximum likelihood sequence estimation (MLSE) [10, 18], which is implemented by the well-known Viterbi algorithm (VA) [11], can be used to detect the symbol sequence.

3. Performance of the OFDM-CM System 3.1. AWGN Channel In AWGN channel, the pairwise error probability [16] between the symbol vectors X i and X t for the ML detection is s

Pe (X i → X t ) = Q 



D2 (X i , X t )  , 2N0

(6)

where D2 (X i , X t ) is the squared Euclidean distance between X i and X t: D2 (X i , X t ) =

L X l=1

|Xi (l) − Xt (l)|2 .

(7)

Let Df2ree be the free squared Euclidean distance in the trellis, i.e., Df2ree = min D2 (X i , X t ), i,j

(8)

where X i and X t are two symbol sequences whose paths diverge from any state and remerge at the same, or another, state. Based on the

CM_WPC.tex; 20/08/2008; 21:30; p.4

5 Table I. Minimum Euclidean distance of modulations. Modulation

BPSK

QPSK

8PSK

16QAM

2 Dmin (Eb )

4

4

1.7573

1.6

principle of the union bound [16], the event error probability at high signal-to-noise ratio (SNR) is well approximated by s

Pe ≈ Pf Nf Q 

Df2ree 2N0



,

(9)

where Pf is the probability that a sequence with distance-Df2ree sequences is transmitted, and Nf is the average number of sequences with distance Df2ree to the transmitted sequence. Since Df2ree is the dominating factor in (9), we ignore Pf and Nf when evaluating the high-SNR performance of the OFDM-CM system in AWGN channel. It’s reasonable to compare the OFDM-CM system with the conventional OFDM system using high order modulation with the same spectrum efficiency. If BPSK modulation is used, the spectrum efficiency of the multi-code OFDM-CM system can be approximated by η = K bit/s/Hz without regarding to the cost of the cyclic prefix and 2 the pilot signal in practical systems. Let Dmin be the minimum squared Euclidean distance in the constellation for modulation of order M = 2η . Then the gain of the OFDM-CM system over the conventional OFDM system is Df2ree G = 10 × log10 2 dB, (10) Dmin 2 which can be thought of as a kind of coding gain. The parameter Dmin of some common modulations is listed in Table I, in which Eb represents the average bit energy. However, as regard to the free squared Euclidean distance of the OFDM-CM system, there is a proposition as follows.

PROPOSITION 1. The free squared Euclidean distance of the OFDMCM system is no larger than the minimum squared Euclidean distance 2 of the base modulation Dmin,B , i.e., 2 Df2ree ≤ Dmin,B . (11) Proof. We consider the more general multi-code OFDM-CM system. In fact, we only need to proof that given any transmitted symbol se2 quence, there is always another sequence that has distance Dmin,B to

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6 the correct one. Suppose S k = [Sk (1), Sk (2), · · · , Sk (L)] , 1 ≤ k ≤ K, is ′ transmitted, and let S k0 be different from S k0 only in one symbol, say ′ Sk0 (l0 ), and S k = S k for all k 6= k0 . Denoting the output sequences ′ ′ of the convolutional spreader generated by S and S as X and X respectively, then according to (3) we have ′

′

X(l) − X (l) = Ck0 (l − l0 + 1)(Sk0 (l0 ) − Sk0 (l0 ))

(12)

′

for l0 ≤ l < l0 + N , and X(l)− X (l) = 0 for other case. By substituting (2) and (12) into (7), we get

′

Since

′

2

D2 (X, X ) = Sk0 (l0 ) − Sk0 (l0 ) .

2 Dmin,B =

min′

Sk0 (l0 ),Sk (l0 ) 0

2 ′ Sk0 (l0 ) − Sk0 (l0 ) ,

(13) (14)

according to (8), the free squared Euclidean distance is upper bounded 2 by Dmin,B . When the equality in (11) holds, the OFDM-CM system performs asymptotically the same as the corresponding OFDM system in AWGN 2 Maximum channel. We call the spreading codes having Df2ree = Dmin,B Euclidean Distance (MED) codes. Especially, when BPSK is used as the base modulation, the MED codes have Df2ree = 4Eb , which is the maximum free squared Euclidean distance achievable. It can be deduced from the proposition that the single-code OFDM-CM system with MED codes performs asymptotically the same as the OFDM system with the same modulation in AWGN channel, but the multi-code OFDM-CM system may have gain over the OFDM system with high order modulation of the same spectrum efficiency. 3.2. Frequency Selective Rayleigh Fading Channels When the subcarriers of the OFDM-CM system undergo independent and flat Rayleigh fading, the pairwise error event probability of confusing the transmitted symbol sequence X i with another sequence X t is upper bounded [5] by −1   Y |X (l) − X (l)|2  i t Pe (X i → X t ) ≤ ,   4N0

(15)

l∈Gi,t

∆

where Gi,t = {l|1 ≤ l ≤ L, Xi (l) 6= Xt (l)}. Let Li,t = |Gi,t |, which is the Hamming distance between the two sequences. By defining the product

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7 distance [17] between X i and X t as ∆



2 Dprod (X i , X t ) = 

Y

l∈Gi,t



1 Li,t

|Xi (l) − Xt (l)|2 

,

(16)

we can rewrite (15) as 2 Dprod (X i , X t ) 4N0

Pe (X i → X t ) ≤

!−Li,t

.

(17)

For high SNR, the event error probability is dominated by the pairs of symbol sequences with the minimum Hamming distance, which is defined as the free Hamming distance: Lf ree = min Li,t . Furthermore, X i ,Xt

we define ∆

Df2p =

min

(X i ,X t ):Li,t =Lf ree

2 Dprod (X i , X t )

(18)

as the minimum product distance of the pairs of sequences with Hamming distance Lf ree . Then, by the principle of the union bound, the event error probability for high SNR can be approximated by ′

′

Pe ≈ Pf Nf

Df2p 4N0

!−Lf ree

,

(19)

′

where Pf is the probability that a sequence with distance-Lf ree se′ quences is transmitted, and Nf is the average number of the distanceLf ree sequences. Similar to the AWGN situation, we mainly consider the effect of ′ ′ Lf ree and Df2p , and neglect Pf and Nf when evaluating the performance of the OFDM-CM system in frequency selective fading channels, since the former two dominate the event error probability for high SNR. As seen from (19), the event error probability varies inversely with (Eb /N0 )Lf ree , so Lf ree is the diversity order of the system. Obviously, the free Hamming distance cannot be larger than the constraint length of the convolutional spreader, i.e., Lf ree ≤ N . When the spreading codes satisfy Lf ree = N , the maximum diversity order is achieved, and we call them Maximum Hamming Distance (MHD) codes. We take the conventional OFDM system combined with diversity techniques as a baseline system for comparison, in which high-order modulation is adopted to achieve the same spectrum efficiency as the OFDM-CM system. Assume the symbols are transmitted on independent Rayleigh fading sub-channels with equal energy. Maximum ratio

CM_WPC.tex; 20/08/2008; 21:30; p.7

8 combining (MRC) [16] is used at the receiver to get the optimal per2 formance. Let Dmp be the minimum product distance of the OFDM system with P -path diversity, then we have 2 = Dmp

2 Dmin . P

(20)

To compare the OFDM-CM system with the baseline OFDM system of the same diversity order, we define the coding gain of the OFDM-CM system as Df2p ′ (21) G = 10 × log10 2 dB. Dmp Note that for the baseline OFDM system, other resources, such as additional antennas, have to be used in order to achieve the diversity gain without loss in spectrum efficiency, which increases the implementation cost. We will show that the OFDM-CM system can nearly achieve the same performance as the MRC diversity of order N by MHD codes.

4. Design of the Spreading Codes As discussed in the last section, in order to get the optimal performance for OFDM-CM system, we should maximize the free Euclidean distance Df2ree in AWGN channel, and the diversity order Lf ree , as well as the minimum product distance Df2p , in frequency selective fading channels. Since the elements of the codes are complex valued, it’s quite difficult to find the optimal spreading codes. On the other hand, given a finite set of spreading codes, we can search for the best-performance codes according to their parameters of Df2ree , Lf ree , and Df2p . This can be a scheme to design the spreading codes. 4.1. Free Distance Searching Algorithm Due to the fact that the symbol sequences do not form a linear space, the distance property of the codes cannot be decided by assuming a specific sequence, like all-one sequence, is transmitted. Exhaustive search of the free distance by testing all of the possible transmitting sequences, and all state transitions, is quite complicated. As the first step, a simple algorithm is proposed to check the distance property of the spreading codes. The proposed searching algorithm is based on the well-known Viterbi algorithm [11]. It’s known that VA is an ML algorithm that tests the

CM_WPC.tex; 20/08/2008; 21:30; p.8

9 metrices of all candidate sequences. As the path corresponding to a random symbol sequence long enough can almost definitely undergo all the possible state transitions, if we detect the random sequence that is not impaired by noise, we will find out the free distance, which is just the minimum metric of the incorrect competing branches at the nodes. The free distance searching algorithm is summarized as follows. Free Distance Searching Algorithm: − Given a spreading code C, generate a sufficiently long, e.g., 1000, random symbol sequence. Get the output sequence X of the convolutional spreader according to (1) or (3). − Pass X to the sequence detector. Initialize a distance register R with +∞. − Detect the sequence with the Viterbi algorithm. When choosing the survivor path at each stage, compare the metric of the incorrect path merging into the correct state, denoted by M , with the value of the distance register. If M < R, set R = M . − When the detection is completed, R is the free distance. This algorithm is suitable for both the free squared Euclidean Distance and the free Hamming distance. Its complexity is almost the same as the conventional Viterbi algorithm, except for the need of a distance register. Note that when searching for the free Hamming distance, the Euclidean distance is used for detection, while the Hamming distance is the metric registered. When the algorithm is used for searching the product distance, the Hamming distance should first be examined, and the minimum product distance for the corresponding Hamming distance is recorded. Of cause, we only care about the minimum product distance corresponding to the free Hamming distance. 4.2. Spreading Codes for Single-code OFDM-CM System We first design the spreading codes for the single-code OFDM-CM system. In this scheme, there is only one code used for the convolutional spreading of the symbols. High-order modulation is used to increase the bit rate. It’s unrealistic to search the whole set of the possible spreading codes, because the set is infinite. In the following search, we confine the √ code symbols to be C(n) ∈ {1, j, −1, −j} for 1 ≤ n ≤ N , where j = −1, and the normalization factor √1N to satisfy the condition (2) is neglected for briefness.

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10 Table II. Spreading codes (N = 4) for single-code OFDM-CM system and their free squared Euclidean distances Df2 ree (Eb )

Spreading Code

BPSK

QPSK

8PSK

C 1a :

[1, 1, 1, 1]

2

2

0.8787

C 1b :

[1, −1, −1, −1]

4

4

1.7574

C 1c :

[1, −1, j, j]

4

4

1.7574

C 1d :

[1, j, −1, −j]

4

2

0.8787

Table III. Free Hamming distance and product distance of the spreading codes for single-code OFDM-CM system Spreading

Df2 p (Eb )

Lf ree

Code

BPSK

QPSK

8PSK

BPSK

QPSK

8PSK

C 1a

2

2

2

1

1

0.4392

C 1b

3

3

3

1.5874

1.5874

0.6975

C 1c

4

4

4

1

1

0.4392

C 1d

4

2

2

1

1

0.4392

Take N = 4 as the example. After applying the proposed free distance searching algorithm to the codes with N = 4, we list some representative codes with their parameters Df2ree in Table II, and parameters Lf ree and Df2p in Table III. We first examine the performance of the codes in AWGN channel, where the free squared Euclidean distance is the dominating factor. Remember that Df2ree is upper bounded by 2 Dmin,B . Comparing Table II with Table I, we find that the codes except C 1a are MED codes for BPSK, and C 1b and C 1c are MED codes for QPSK and 8PSK. In searching for the codes out of the all 64 codes with N = 4 and the first code symbols being 1 for BPSK modulation, 2 . They are C we find that only two codes have Df2ree < Dmin 1a and 2 [1, −1, 1, −1], both with Df ree = 2Eb . Though all the other 62 codes have the maximum Df2ree , they differ from each other in performance, because of the difference in parameters Pf and Nf in (9).

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11 Table II shows that an MED code for BPSK modulation is not necessarily an MED code for high order modulation, such as QPSK, 8PSK. For example, C 1d only has Df2ree = 2Eb for QPSK, and Df2ree = 2 0.8787Eb for 8PSK. So, compared to the Dmin shown in Table I, it has 3dB loss for QPSK and 8PSK. On the other hand, C 1b and C 1c have no loss in Df2ree . They are MED codes for all of the modulations, thus are preferred in AWGN channel. As derived in the last section, the free Hamming distance and the product distance are the two most important factors for the performance of the OFDM-CM system in frequency selective fading channels. These parameters are provided in Table III. Obviously, the best codes in AWGN channel are not necessarily the best codes in fading channel. For example, the MED code C 1b only has Lf ree = 3. On the other hand, the code C 1c has both the maximum free Euclidean distance and the maximum free Hamming distance for any modulation used. Also note that the code C 1d is an MHD code for BPSK, but not for QPSK and 8PSK, where it only has second-order diversity in frequency selective fading channels. In summary, the code C 1c is the best code for all modulation types and both kinds of channels. According to the minimum product distances of the codes given in Table III, we found that the OFDM-CM system can have gain over the conventional OFDM system with the same spectrum efficiency and the same order of diversity. For example, the OFDM-CM system with spreading code C 1b can gain about 0.8dB relative to the OFDM system with the same diversity order. However, being an MHD code, C 1c has no gain over the fourth-order diversity OFDM system. 4.3. Spreading Codes for the Multi-code OFDM-CM System The multi-code OFDM-CM system use multiple codes to spread multiple symbol sequences, and the output of the spreaders are summed up for transmission, as indicated by (3). We confine the based modulation to be BPSK. So with K spreading codes, the spectrum efficiency of the system is K bit/s/Hz. Again, we assume N = 4 and Ck (n) ∈ {1, j, −1, −j}. The search process is similar to the single-code case, and some resulting codes are listed in Table IV. C 4a is an exception that its elements are not confined in the set {1, j, −1, −j}. After applying the free distance searching algorithm to the spreading codes listed in Table IV, we have summarized their parameters Df2ree in Table V. Remember that Df2ree is upper bounded by 4Eb , so the codes listed in Table IV, except C 4a , are all the MED codes.

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12 Table IV. Spreading codes (N = 4) for BPSK modulated multi-code OFDM-CM system. C 2a : C 2c :

C 3a :





1, −1, j, j 1, 1, j, −j   1, −j, 1, j −j, −j, −1, 1



1,

j,

−j,

 1, −j, −1, 1,

j,

j,



C 2b : C 2d :



1 j  −1



1, j, −1, −j j, 1, −j, −1   1, −j, 1, j  1, −1, −j, −j 1, j, −j, 1  1, −j, −1, j   j, j, −1  1,

C 4a :

−j, e

j5π 4

, e

j3π 4

, e

j5π 4

    

Table V. Distance parameter of the spreading codes for BPSK modulated multi-code OFDM-CM system. ′

Code

Df2 ree (Eb )

G(dB)

Lf ree

Df2 p (Eb )

G (dB)

C 2a

4

0

1

16

6.0

C 2b

4

0

3

1.5874

0.8

C 2c

4

0

3

2.5198

2.8

C 2d

4

0

2

4.4721

3.5

C 3a

4

3.6

2

2

3.6

C 4a

2.3431

1.7

2

2

4.0

For the multi-code multiplexing system, searching for good codes is complicated, especially when K is large. There are plenty of codes with their Df2ree = 0, thus cannot be used. The codes listed in Table IV are some of the best codes known. Note that we have not found MED codes for K = 4. C 4a with its element not constrained in {1, j, −1, −j}, is an example that enlarging the code symbol alphabet can provide more powerful codes. In contrast, if the code symbols are confined by Ck (n) ∈ {1, −1}, we cannot find any code usable when K > 1. Comparing the OFDM-CM system to the uncoded OFDM system with the same spectrum efficiency, we can see the coding gain of the proposed system, which is also given in Table V. For spectrum efficiency η = 2 bit/s/Hz, when the MED code is used, i.e., Df2ree = 4Eb , there is no coding gain. However, for η = 3 bit/s/Hz, the OFDM-CM system can have about 3.6dB gain compared to the OFDM system with 8PSK.

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13 Similarly, for 4bit/s/Hz system, OFDM-CM using spreading code C 4a has about 1.7dB gain compared to the 16QAM OFDM system. The free Hamming distances and the minimum product distances are also provided in Table V. It can be seen that the code C 2a only has Lf ree = 1 though has Df2ree = 4Eb . We have found C 1c to be an MHD code for K = 1 case, i.e., the single-code OFDM-CM system, but for K > 1, no MHD code has been found, though most of the codes listed in the table have diversity gain. According to (20) and (21), we can calculate the gain of the OFDM-CM over the conventional OFDM system with the same spectrum efficiency and the same diversity order. For example, with diversity order being 1, OFDM-CM system with spreading code C 2a can gain about 6dB relative to the OFDM system with QPSK. Similarly, C 3a has coding gain about 3.6dB gain relative to 8PSK OFDM when achieving second-order diversity; and C 4a can gain about 4dB over 16QAM OFDM with second-order diversity. Note that C 3a and C 4a perform almost the same in frequency selective fading channels because they have equal Lf ree and Df2p , as will be verified by the simulation results given in the next section.

5. Numerical Results The performance of the proposed OFDM-CM system is evaluated by numerical simulations, both in AWGN channel and frequency selective Rayleigh fading channels. We compare the OFDM-CM system with the conventional OFDM system of the same spectrum efficiency by their bit error rate (BER). The power loss due to the cyclic prefix of both systems is not taken into account. 5.1. AWGN Channel The comparison of the OFDM-CM system and the conventional OFDM in AWGN channel is shown in Fig. 3-5. As predicted by the previous analysis, the MED codes perform asymptotically the same as the OFDM system with the same modulation. Some of the codes, such as C 1c and C 2a , have their BER-SNR curves overlapped with base modulation curve in the whole SNR range. They are the best codes in AWGN channel. Note that since C 1a has Df2ree = 2Eb for BPSK modulated OFDM-CM system, it loses 3dB relative to the MED codes, which is also the case for QPSK OFDM-CM system with C 1d . When the spectrum efficiency of the systems is high, the multi-code OFDM-CM system shows its superiority to the conventional OFDM system, as can be seen from Fig. 5. The OFDM-CM using C 3a gains

CM_WPC.tex; 20/08/2008; 21:30; p.13

14 0

10

−1

10

−2

BER

10

−3

10

−4

10

OFDM, BPSK OFDM−CM, BPSK, C

OFDM−CM, BPSK, C −6

10

1a

OFDM−CM, BPSK, C1b OFDM−CM, BPSK, C1c

−5

10

0

2

4

1d

6 8 Eb / N0 (dB)

10

12

14

Figure 3. Performance of the single-code OFDM-CM system with BPSK in AWGN channel. 0

10

−1

10

−2

BER

10

−3

10

OFDM, QPSK OFDM−CM, QPSK, C

−4

10

1a

OFDM−CM, QPSK, C1c OFDM−CM, QPSK, C1d

−5

10

OFDM−CM, BPSK, C2a OFDM−CM, BPSK, C 2d

−6

10

0

2

4

6 8 Eb / N0 (dB)

10

12

14

Figure 4. Performance of the single-code and multi-code OFDM-CM system with spectrum efficiency η = 2 bit/s/Hz in AWGN channel.

about 3.6dB over the OFDM using 8PSK at high SNR, and using C 4a gains about 2dB over OFDM using 16QAM, which can be explained by their parameters Df2ree given in Table V. Note that the gain of C 4a given in the table is a little conservative, which is because that the multipliers in (9) are not considered. To achieve the same spectrum efficiency, the single-code OFDM-CM using high-order modulation is inferior to the multi-code system using BPSK. The best-performance single-code system, which is for the code C 1c , can only perform as good as the corresponding OFDM system. This is because the free distance of the OFDM-CM system is bounded

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Figure 5. Performance of the single-code and multi-code OFDM-CM system with spectrum efficiency η = 3, 4 bit/s/Hz in AWGN channel.

by the base modulation. With this concern, the multi-code OFDMCM system with low-order modulation is a better choice for AWGN channel. 5.2. Frequency Selective Fading Channels Fig. 6 and Fig. 7 show the results for the ideal frequency selective fading channels, where the flat fading on each subcarrier is assumed to be independent and the receiver has perfect channel estimation. The performance of the conventional modulation in independent Rayleigh fading channels with MRC is also shown in the figures. It can be seen that the diversity gain of the OFDM-CM is obvious. The diversity orders are interpreted as the slope of the BER-SNR curves, which are in accordance with the parameters Lf ree given in Table III and V. It’s shown in Fig. 6 that the code C 1c performs the best, and its curve is very close to the theoretical result of the MRC diversity with order four. So C 1c is the optimum code with spectrum efficiency 1 bit/s/Hz both in AWGN channel and in frequency selective fading channels. On the other hand, other codes simulated have a loss in diversity order. Especially, the code C 2a only has first-order diversity, though it performs the best in AWGN channel. For the double-code OFDM-CM system with BPSK modulation, the best code is C 2c , as proved by Fig. 6. For higher spectrum efficiency, the results are shown in Fig. 7. It verifies that the multi-code OFDM-CM system with code C 3a or C 4a can not get fourth-order diversity, but can only have order two. Never-

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Figure 7. Performance of the OFDM-CM system with spectrum efficiency η = 3, 4 bit/s/Hz, the OFDM-CDM system with Walsh-Hadamard code, and the OFDM system with convolutional code (CC) in ideal frequency selective fading channels. The curves for 8PSK and 16QAM with independent P -path MRC diversity are also shown for comparison.

theless, they can obtain obvious coding gain relative to the second-order MRC diversity. Note that C 3a and C 4a perform almost the same at high SNR, as predicted in the last section. On the other hand, the single-code 8PSK modulated OFDM-CM system with code C 1c can achieve fourthorder diversity gain. In contrast to the AWGN case, the single-code system may perform better than the multi-code counterpart.

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17 Fig. 7 also shows the ML-detection performance of the OFDM-CDM system with 8PSK modulation, in which the Walsh-Hardamard code of length four is used. It can only achieve first-order diversity and is inferior to the multi-code OFDM-CM system with the same spectrum efficiency (using C 3a ) at high SNR. Its gain over the conventional firstorder diversity transmission is due to the increase of the Df2p . The performance of coded OFDM system in frequency selective fading channel is also simulated. A rate-1/2 recursive systematic convolutional code with generator (g1 , g2 ) = (37, 21)8 is used in the system, where g1 is the feedback generator. The spectrum efficiency of 3 bit/s/Hz is achieved by 64QAM symbol mapping, which is comparable to the single-code OFDM-CM system with 8PSK modulation and three-code OFDM-CM system with BPSK modulation. Random bit interleaving is utilized before symbol mapping. The results are also shown in Fig. 7. It’s seen that the single-code OFDM-CM system outperforms the coded OFDM system by 5 dB when BER equals to 10−4 , and the multi-code OFDM-CM also have about 3 dB gain over the coded OFDM system. However, the MLSE receiver complexity for the OFDM-CM systems is higher than the coded OFDM system, because they both have 512 states in the trellis, comparing to 16 states of the convolutional code. As regard to the coding gain in the fading channel, we find the values given in Table V are not very accurate. This is because that the gain defined by (21) is based on the approximation of the performance of ′ ′ the OFDM-CM system, where the values of Pf and Nf in (19) are neglected. However, they can be used to evaluate the performance of the codes in general. For example, according to Table V, the code C 2c outperforms C 2b by 2 dB while both of them achieve third-order diversity. This is verified by Fig. 6. We also apply the CM method to the E-UTRA OFDM system [1], and compare its performance with the specified 64-state convolutional code scheme in practical multi-path fading channel for vehicular environment [13]. In the simulations, the FFT length is set 512 and 300 subcarriers occupying the bandwidth of 5 MHz are used for transmission. Single antenna is equipped at the transmitter and the receiver. The results are given in Fig. 8, which shows that the uncoded OFDMCM systems outperforms the corresponding convolutional code OFDM system with the same spectrum efficiency by 2-3 dB for BER at the level of 10−4 .

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Figure 8. Performance comparison of the OFDM-CM and the E-UTRA OFDM system with rate-1/2 convolutional code (CC) in ITU Vehicular A channel.

6. Conclusion In this paper, we propose a new multiplexing scheme used in the OFDM-based systems, which is called OFDM-CM system. In this scheme, data symbols are spread onto multiple subcarriers by convolutional spreader. To improve the spectrum efficiency, high order modulation can be used in the single-code OFDM-CM system, or multiple loworder-modulated data streams with different spreading codes can be transmitted simultaneously in the OFDM-CM system. Base on the examination of the good codes searched out, we have found the multicode scheme performs better in AWGN channel due to its large free Euclidean distance, but it may have relatively lower diversity order in frequency selective fading channels. On the other hand, the single-code system can have the largest diversity order with well-designed codes. Numerical results show that the multi-code multiplexing OFDM-CM system is superior to the OFDM-CDM system. Moreover, to achieve the same high spectrum efficiency, the OFDM-CM system also performs better than the OFDM system with simple convolutional code. The spreading codes of the OFDM-CM system are designed by a free-distance searching algorithm. The best length-four codes ever known are presented in the paper. Only the MLSE algorithm is used in this paper for detection, which is quite complicated, especially when the spreading code is long and the number of the multiplexing codes is large. Some other reduced complexity algorithms [2, 8], which have been studied in the decoding of the convolutional code and in the equalization to the ISI channel, may be applied in practical use.

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19 Acknowledgements The research was supported by National Natural Science Foundation of China (NSFC) under grant no. 90604035. The authors would like to thank the anonymous reviewers for helpful comments that have improved the manuscript.

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