Convolutional Multi-code Multiplexing for OFDM ... - Google Groups

Convolutional Multi-code Multiplexing for OFDM Systems Wei Jiang and Daoben Li School of Information Engineering, Beijing University of Posts and Telecommunications Beijing 100876, China Email: [email protected] d1

d2

Sy mb o l M ap p er

Sy mb o l M ap p er

S1

Co n v o lu tio n al Sp read er 1

S2

X

Co n v o lu tio n al Sp read er 2

Interleaver

IFFT

CP

D/A

...

Abstract— The paper proposes a new multiplexing scheme for the OFDM-based systems with significant coding gain and diversity gain, which is called Convolutional Multi-code Multiplexing (CMM). In this scheme, the data symbols are spread onto orthogonal subcarriers by convolutional spreaders, and multiple data streams spread by different codes can be transmitted on the same frequency band simultaneously. The free Euclidean distance and the free Hamming distance are found to dominate the performance of the OFDM-CMM system in AWGN channel and frequency selective fading channel, respectively. Some codes with good performance are found by computer search. Compared to the conventional uncoded OFDM system with the same spectrum efficiency, the OFDM-CMM system can achieve significant diversity gain in frequency selective fading channel, and notable coding gain both in AWGN channel and fading channel.

dK

Sy mb o l M ap p er

SK

Co n v o lu tio n al Sp read er K

Fig. 1.

A/D

The structure of the OFDM-CMM transmitter

Remo v e CP

Fig. 2.

Y FFT

D e-In t erleav er

Detec tor

Sˆ

dˆ De-M ap p er

The structure of the OFDM-CMM receiver

I. I NTRODUCTION The OFDM (Orthogonal Frequency Division Multiplexing) system [1], [2] suffers from frequency selective fading, as the data symbols are parallelized and transmitted on narrow subcarriers. Several improved schemes have been proposed to achieve diversity gain in frequency selective fading channels, such as Coded OFDM (COFDM) [3], Multicarrier Code Division Multiple Access (MC-CDMA) [4]–[6], or OFDM Code-Division Multiplexing (OFDM-CDM) [7]. COFDM can achieve diversity gain and coding gain at the same time, but there is always a loss in spectral efficiency. The MC-CDMA system spreads the data symbols onto multiple tones 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 system suffers from multiple access interference (MAI) and the diversity gain will be reduced when the system load is high [8]. In [9], the authors proposed the method of convolutional spreader to achieve diversity gain without loss of spectrum efficiency for multi-carrier systems. But there is no coding gain with this method. In this paper, we propose a new scheme using multiple convolutional spreaders to multiplex the signal, which is called Orthogonal Frequency Division MultiplexingConvolutional Multi-code Multiplexing (OFDM-CMM) system. The data symbols are spread onto several subcarriers by convolutional spreaders and the parallel data streams are added together for simultaneous transmission. Compared to the uncoded OFDM system with the same spectral efficiency, the proposed method can achieve notable coding gain in AWGN (Additive White Gaussian Noise) channel. In frequency selec-

tive fading channel, significant diversity gain with order equal to the length of the spreading codes can be achieved. The rest of the paper is organized as follows. Section II introduces the system model and describes the new multiplexing scheme in detail. Section III discusses the performance of the proposed system in AWGN channel and in frequency selective fading channel. Section IV presents several spreading codes found by computer search and analyses their performances. The numerical simulation results are presented in the following section. Finally, a conclusion is drawn in the last section. II. OFDM-CMM S YSTEM M ODEL The transmitter of the OFDM-CMM system is illustrated in Fig. 1. The parallel data streams are mapped into symbols and then passed through different convolution spreaders, which introduce inter-symbol interference in the streams. The spread symbols of all the streams are summed up and interleaved. Then the symbols are modulated onto orthogonal subcarriers by employing inverse fast Fourier transform (IFFT), and padded with cyclic prefix (CP) to resist the inter-symbol interference, just as the conventional OFDM system does. Correspondingly, Fig. 2 shows the structure of the OFDMCMM receiver. After the FFT operation, the symbols are deinterleaved and detected. The convolutional spreader can be implemented by a transversal filter, as is shown in Fig. 3. The complex valued coefficients on the taps form a spreading code. The output of the spreader is the summation of the delayed symbols multiplied by the spreading codes.

S k (l)

S k (l-1) D

C k (1)

S k (l-N+2)

D

C k (2)

...

connecting each state is 2K . So the complexity of the MLSE algorithm is O(2KN ), which is prohibitive when the spreading code is long and the number of the multiplexing codes is large. Some other reduced complexity algorithms, which have been studied in the decoding of the convolutional code and in the equalization to the ISI channel, such as the Fano algorithm, the Stack algorithm [12], may be applied in the OFDM-CMM system for practical use.

S k (l-N+1)

...

D

C k (N-1)

C k (N)

X (k ,l) Fig. 3.

III. P ERFORMANCE OF THE OFDM-CMM SYSTEM WITH M AXIMUM -L IKELIHOOD D ETECTION

The structure of the convolutional spreader

Assume the length of the spreading code is N , and there are K parallel streams, where K ≤ N . Denote the k-th data bit stream as dk = [dk (1), dk (2), · · · , dk (L)], where 1 ≤ k ≤ K, and L is the block length. We assume Sk = BPSK modulation is used, and dk is mapped √ into √ [Sk (1), Sk (2), · · · , Sk (L)], where Sk (l) ∈ { Eb , − Eb }, and Eb is the energy per bit on average. The k-th spreading code is denoted as Ck = [Ck (1), Ck (2), · · · , Ck (N )], and the set of multi-code is {C1 , C2 , · · · , CK }. So, the summation of the output of the convolutional spreaders can be expressed as: X(l) =

K
N


Ck (n)Sk (l − n + 1), 0 < l ≤ L,

(1)

A. AWGN channel When Y is received, the ML detection of the symbols can be expressed as ˆ = arg max P (Y|X) . X

In AWGN channel, it’s well known that the pairwise error probability [13] between the symbol vectors Xi and Xt is   2 (X , X ) D i t  , (6) Pe (Xi → Xt ) = Q  2N0 where D2 (Xi , Xt ) is the squared Euclidean distance between Xi and Xt , i.e.,

k=1 n=1

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


2

|Ck (n)| = 1, 1 ≤ k ≤ K.

(2)

n=1

(3)

where n(l) is the additive complex Gaussian noise with mean zero and variance N0 . As for the frequency selective Rayleigh fading channel, as long as the CP is larger than the delay spread of the channel, and the interleaving block length is large enough, the fading on the subcarriers can be assumed to be flat and independent. 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.

D2 (Xi , Xt ) =

L


2

|Xi (l) − Xt (l)| .

(7)

l=1

Let Df2 ree be the free squared Euclidean distance in the trellis: (8) Df2 ree = min D2 (Xi , Xt ), i,t

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

(5)

X

(4)

As is shown in Fig. 3 and Eq. (1), the convolutional spreader introduces inter-symbol constraint, just as 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], [11] algorithm can be used to detect the symbol sequence. With BPSK modulation, the number of the states in the trellis diagram equals to 2K(N −1) , and the number of branches

where Xi and Xt are two symbol sequences whose paths diverge from any state and remerge at the same, or another state after one or more transitions. Due to the lack of the linearity, we cannot assume a certain sequence, such as the all-one sequence, is transmitted to find the free distance, just as we do for the convolutional codes. Based on the principle of the union bound [13], the event error probability at high signal-to-noise ratio (SNR) is well approximate by   Df2 ree , (9) P e ≈ P f Nf Q  2N0 where Pf is the probability that a sequence with distanceDf2 ree sequences is transmitted, and Nf is the average number of sequences with distance Df2 ree to the transmitted sequence. When evaluating the performance of the OFDM-CMM system in AWGN channel, we neglect the effect of the Pf and Nf and regard Df2 ree as the dominating factor. It’s reasonable to compare the OFDM-CMM system with the conventional OFDM system using high order modulation with the same spectrum efficiency. Since BPSK modulation is assumed to be used, the spectrum efficiency of the OFDMCMM system can be approximated by η = K without regarding to the cost of the cyclic prefix and the pilot tone

∆

TABLE I M INIMUM E UCLIDEAN D ISTANCE OF M ODULATION Modulation

BPSK

QPSK

8PSK

16QAM

2 Dmin (Eb )

4

4

1.7573

1.6

where Gi,t = {l|1 ≤ l ≤ L, Xi (l) = Xt (l)}. Let Li,t = |Gi,t |, and  L1  i,t ∆ 2 2  Dprod (Xi , Xt ) = |Xi (l) − Xt (l)| . (15) l∈Gi,t

2 in practical systems. Let Dmin be the minimum squared Euclidean distance in the constellation with modulation order M = 2η . Then the gain of the OFDM-CMM system over the conventional OFDM system is

G = 10 × log10

Df2 ree dB, 2 Dmin

(10)

Li,t is the Hamming distance between Xi and Xt , and 2 (Xi , Xt ) can be called the product distance [15] beDprod tween the two sequences. Then (14) can be rewritten as  −Li,t 2 (Xi , Xt ) Dprod . (16) Pe (Xi → Xt ) ≤ 4N0 For high SNR, the event error probability is dominated by the pairs of symbol sequences with the minimum Hamming distance in the event error, which is defined as the free Hamming distance:

which can be thought of as a kind of coding gain. The 2 of some common modulations is listed in parameter Dmin Table I. As regard to the free squared Euclidean distance of the (17) Lf ree = min Li,t . i,t OFDM-CMM system, there is a proposition as follows: Proposition 1: The free squared Euclidean distance of the Furthermore, we define OFDM-CMM system is no larger than the minimum squared ∆ 2 Df2 p = min Dprod (Xi , Xt ) (18) Euclidean distance of the base modulation. Especially, when (Xi ,Xt ):Li,t =Lf ree BPSK is used, (11) as the minimum product distance of the pairs of sequences Df2 ree ≤ 4Eb . Proof: Without loss of generality, we only consider with Hamming distance Lf ree . Then, by the principle of the the BPSK case. In fact, we only need to proof that given union bound, the event error probability for high SNR can be any transmitted symbol sequence, there is always another approximated by sequence that have distance 4Eb to the correct one. Suppose  −Lf ree Df2 p

Sk = [Sk (1), Sk (2), · · · , Sk (L)] , 1 ≤ k ≤ K, is transmitted,
, (19) P e ≈ P f Nf and let Sk0 be different from Sk0 only in one symbol, say 4N0
Sk0 (l0 ), and Sk = Sk for all k = k0 . Denoting the output
sequences of the convolutional spreader generated by S and where Pf is the probability that a sequence with Hamming


S as X and X respectively, then according to (1) we have distance Lf ree is transmitted, and Nf is the average number √    of the distance-Lf ree sequences.
2|Ck0 (l − l0 + 1)| Eb , l0 ≤ l < l0 + N   X(l) − X (l) = As in the AWGN channel, we mainly consider the param0, others.

eters Lf ree and Df2 p , and neglect the effect of Pf and Nf (12) when evaluating the performance of the OFDM-CMM system By substituting (2) and (12) into (7), we get in frequency selective fading channels, since the former ones
(13) dominate the event error probability for high SNR. As is D2 (X, X ) = 4Eb . the event error probability varies inversely So, the free squared Euclidean distance is upper bounded by seen from (19), Lf ree /N ) , so Lf ree is the diversity order of the with (E b 0 4Eb . system. Obviously, the free Hamming distance cannot be larger When the equality in (11) holds, the OFDM-CMM system than the register length of the convolutional spreader, i.e., performs the best. We call the spreading codes having Df2 ree = ≤ N . When the spreading codes satisfy Lf ree = N , we L f ree 4Eb the Maximum Euclidean Distance (MED) codes. Note call them the Maximum Hamming Distance (MHD) codes. that for the MED codes, Pf in (9) equals to one. Moreover, for the conventional OFDM system with diver2 B. Rayleigh fading channel be the minimum product distance of the OFDM sity, let Dmp system that has the same diversity order and the same spectrum When the subcarriers of the OFDM-CMM system undergo efficiency with the OFDM-CMM system. Then the coding gain independent and flat Rayleigh fading, as is described in of the OFDM-CMM system relative to the convention OFDM section II, the pairwise error event probability of confusing system is the transmitted symbol sequence Xi with another sequence Df2 p
Xt is upper bounded [14] by dB. (20) G = 10 × log10 2  −1 Dmp  |X (l) − X (l)|2  i t Note that in order to achieve the diversity gain without loss , (14) Pe (Xi → Xt ) ≤   4N0 in spectrum efficiency for the conventional OFDM system, l∈Gi,t

TABLE II S PREADING CODES WITH N = 4 C1a : C1c : C2a :



[1, 1, 1, 1] [1, −1, j, j]   1, −1, j, j 1, 1, j, −j

C2c :

1, −j, 1, j −j, −j, −1, 1

C1b :

[1, −1, −1, −1] 

C2b :





C2d :

 1, j, −j, 1    1, −j, −1, j  1, j, j, −1



 C3a :

    

C4a :



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

−j, e

j5π 4

,e

j3π 4

,e

 

j5π 4

    

TABLE III D ISTANCE PARAMETER OF THE S PREADING C ODES


Code

Df2 ree (Eb )

G(dB)

Lf ree

Df2 p (Eb )

G (dB)

C1a C1b C1c

2 4 4

−3.0 0 0

2 3 4

1 1.5874 1

−3.0 0.8 0

C2a C2b C2c C2d

4 4 4 4

0 0 0 0

1 3 3 2

16 1.5874 2.5198 4.4721

6.0 0.8 2.8 3.5

C3a

4

3.6

2

2

3.6

C4a

2.3431

1.7

2

2

4.0

other technologies must be employed, such as antenna diversity. When the energy is identically spread onto P independent fading channels to achieve diversity gain, we have 2 Dmin . (21) P IV. S PREADING C ODES FOR THE OFDM-CMM SYSTEM 2 = Dmp

As is stated in the last section, in order to get the optimal OFDM-CMM system performance, we should maximize the coding gain, or Df2 ree , in the AWGN channel, and the diversity order Lf ree , as well as the product distance Df2 p , in the frequency selective fading channel. 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 Df2 ree , Lf ree , and Df2 p . This can be a scheme to design the spreading codes. Take N = 4 as the example. Using the free distance searching algorithm proposed in [9], we have found some representative codes by computer search, which are listed in Table II. In the search, we confine the code symbols to be Ck (n) ∈ {1, j, −1, −j} for √ 1 ≤ n ≤ N and 1 ≤ k ≤ K (except C4a ), where j = −1, and the normalization factor √1 to satisfy the condition (2) is neglected. N Let’s examine the codes for the AWGN channel first, where the free squared Euclidean distance is the dominating factor.

After applying the free distance searching algorithm to the spreading codes listed in Table II, we have summarized their parameters Df2 ree in Table III. Remember that Df2 ree is upper bounded by 4Eb , so the codes listed in Table III, except C1a and C4a , are all the MED codes. In searching for the codes with the first symbols being 1 and K = 1, i.e., there is no multi-code multiplexing, out of the all 64 codes, we find that only two codes have Df2 ree < 4Eb . They are C1a and [1, −1, 1, −1], both with Df2 ree = 2Eb . Though all the other 62 codes have the maximum Df2 ree , they also differ from each other in performance, because of the difference in parameters Pf and Nf in (9). We find eight codes, C1c being one of them, outperform the others, as will be seen in the next section. As to the code with K > 1, i.e., there is multi-code multiplexing, searching for the good codes for the AWGN channel is more complicated, especially when K is large. There are plenty of codes with their Df2 ree = 0, thus cannot be used. The codes listed in Table II are some of the best codes known. For K = 2, there are also eight codes outperform others, and C2a is one of them. Note that we have not found the MED codes with K = 4. C4a 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-CMM system to the convention uncoded OFDM system with the same spectrum efficiency, we can see the coding gain of the proposed system, which is also given in Table III. For spectrum efficiency η = 1 bit/s/Hz or 2 bit/s/Hz, when the MED code is used, i.e., Df2 ree = 4Eb , there is no coding gain. However, for η = 3 bit/s/Hz, the OFDM-CMM system can have about 3.6dB gain compared to the OFDM system with 8PSK. Similarly, for 4bit/s/Hz system, OFDM-CMM using spreading code C4a has about 1.7dB gain compared to the 16QAM OFDM system. As is 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-CMM system in frequency selective fading channel. They are also provided in Table III. Obviously, the best codes in AWGN channel are not necessarily the best codes in the fading channel. For example, the code C1c have both the maximum free Euclidean distance and the maximum free Hamming distance, but the code C2a only has Lf ree = 1 though has Df2 ree = 4Eb . For K > 1, most of the codes listed in the table have diversity gain, but the MHD codes have not been found. The minimum product distances Df2 p of the codes are also provided in Table III. According to (20) and (21), we can calculate the gain of the OFDM-CMM over the conventional OFDM system with the same spectrum efficiency and the same order of diversity. For example, with diversity order equals to 1, OFDM-CMM system with spreading code C2a can gain about 6dB relative to the OFDM system with QPSK. Similarly, C1c has no coding gain when obtaining diversity order 4; C3a has coding gain about 3.6dB gain relative to 8PSK OFDM when achieving two-order diversity; and C4a

0

0

10

10

−1

10

−1

10

−2

10

−2

10

−3

BER

BER

10 −3

10

−4

10 −4

10

−5

10 OFDM, BPSK, Theoretical OFDM−CMM, C1a

−5

10

−6

10

OFDM−CMM, C

1b

10

3a

OFDM−CMM, C1c

−6

0

2

4

OFDM, BPSK, Theoretical OFDM, 8PSK OFDM, 16QAM OFDM−CMM, C OFDM−CMM, C

4a

−7

6 8 E / N (dB) b

10

12

14

0

10

0

2

4

6 8 E / N (dB) b

Fig. 4. Performance of the OFDM-CMM system with spectrum efficiency η = 1 bit/s/Hz in AWGN channel, N = 4, K = 1

10

12

14

0

Fig. 6. Performance of the OFDM-CMM system with spectrum efficiency η = 3, 4 bit/s/Hz in AWGN channel, N = 4, K = 3, 4

0

10

−1

10

−2

BER

10

−3

10

−4

10

OFDM, QPSK, Theoretical OFDM−CMM, C2a

−5

10

OFDM−CMM, C2c OFDM−CMM, C2d

−6

10

0

2

4

6 Eb / N0 (dB)

8

10

12

Fig. 5. Performance of the OFDM-CMM system with spectrum efficiency η = 2 bit/s/Hz in AWGN channel, N = 4, K = 2

can gain about 4dB over 16QAM OFDM with two-order diversity. Note that, C3a and C4a perform almost the same because they have equal Lf ree and Df2 p , as will be verified by the simulation results in the next section. V. S IMULATION R ESULTS The performance of the proposed OFDM-CMM system is evaluated by numerical simulations, both in AWGN channel and frequency selective Rayleigh fading channel. The spreading codes for the system are selected from Table II. We compare the OFDM-CMM system with the conventional OFDM system with 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. For the situation of the frequency selective fading channel, the flat fading on each subcarriers is assumed to be independent. And ideal channel estimation is used in the simulations, i.e., the fading coefficients are supposed to be known at the receiver. The comparison of the OFDM-CMM system and the conventional OFDM in AWGN channel is shown in Fig. 4-6. As

is predicted by the previous analysis, the MED codes perform asymptotically the same as the theoretical result for BPSK 2 = 4Eb . Some of the codes, modulation, which have Dmin such as C1c and C2a , have their BER-SNR curves overlapped with theoretical BPSK curve in the whole SNR range. As is pointed in the last section, they belong to best codes in AWGN channel. Note that since C1a has Df2 ree = 2Eb , it loses 3dB relative to the MED codes. When the spectrum efficiency of the systems is high, the OFDM-CMM system shows its superiority to the conventional OFDM system, as can be seen in Fig. (6). The OFDM-CMM using C3a gains more than 3dB over the OFDM using 8PSK at high SNR, and using C4a gains about 2dB over OFDM using 16QAM, which can be explained by their parameters Df2 ree given in Table III. Fig. 7-8 show the results for the frequency selective fading channel. The performance of the conventional modulation in independent Rayleigh fading channel with MRC (Maximum Ratio Combining) is also shown in the figures. It can be seen that the diversity gain of the OFDM-CMM 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. It’s shown in Fig. 7 that the code C1c performs the best, and its curve is very close to the theoretical result of the MRC diversity with order four. So the code C1c is the optimum code with spectrum efficiency 1 bit/s/Hz both in AWGN channel and in frequency selective fading channel. On the other hand, other codes simulated have a loss in diversity order. Especially, the code C2a has only one-order diversity, though it performs the best in AWGN channel. For high spectrum efficiency, we found both C3a and C4a can not get four-order diversity, but can only have order two. Nevertheless, they can obtain coding gain relative to convention MRC diversity with order two. Note that C3a and C4a perform almost the same at high SNR, as is predicted in the last section. Fig. 8 also shows the ML-detection performance of the

VI. C ONCLUSION

0

10

C

1a

C1b

−1

10

C1c C

−2

10

C C

−3

10 BER

C

2a 2b 2c 2d

−4

10

−5

10

−6

10

−7

10

0

5

10

15

20 25 E / N (dB) b

30

35

40

0

Fig. 7. Performance of the OFDM-CMM system with spectrum efficiency η = 1, 2 bit/s/Hz in frequency selective fading channel. The dotted lines are the theoretical results of the MRC diversity with order 1, 2, 3, 4 from up to down for BPSK.

In this paper, we propose a new multiplexing scheme used in the OFDM-based system, which is called OFDMCMM system. In this scheme, data symbols are spread onto multiple subcarriers by convolutional spreader. And multiple data streams with different spreading codes are summed up to be transmitted simultaneously. This scheme has high spectrum efficiency while achieving significant diversity gain. Moreover, compared to the conventional OFDM system, the OFDMCMM system can obtain obvious coding gain both in AWGN channel and frequency selective fading channel. The spreading codes of the OFDM-CMM system have to be carefully designed. We have searched several good codes with high coding gain and diversity gain. And their performance is evaluated by numerical simulations. The codes found in this paper are dedicated to the BPSK modulation. For high order modulation, new codes should be designed, and the alphabet of the code symbol may have to be enlarged. ACKNOWLEDGMENT The research is supported by National Natural Science Foundation of China (NSFC) under grant no. 90604035.

0

10

8PSK, P=1 8PSK, P=2 16QAM, P=1 16QAM, P=2 OFDM−CDM, 8PSK OFDM−CMM, C

−1

10

R EFERENCES

3a

−2

10

OFDM−CMM, C

BER

4a

−3

10

−4

10

−5

10

−6

10

0

5

10

15

20 25 Eb / N0 (dB)

30

35

40

Fig. 8. Performance of the OFDM-CMM system with spectrum efficiency η = 3, 4 bit/s/Hz and the OFDM-CDM system with 8PSK in frequency selective fading channel. The curves for 8PSK and 16QAM with independent P -path MRC diversity are shown for comparison.

OFDM-CDM system with 8PSK modulation, in which the Walsh-Hardamard code of length four is used. It can only achieve one-order diversity and is inferior to the OFDM-CMM system with the same spectrum efficiency (using C3a ) at high SNR. As regard to the coding gain in the fading channel, we find the values given in Table III are not very accurate. This is because that the gain defined by (20) is based on the approximation of the performance of the OFDM-CMM

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 III, the code C2c outperforms C2b by 2dB while both of them achieve three-order diversity. This is verified by Fig. 7.

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