Performance Enhanced Coded OFDM with Almost Linear Inter leaver over Rayleigh Fading Channels

This paper presents a comprehensive performance analysis of coded orthogonal frequency division multiplexing (COFDM) over the quasi-static multipath Rayleigh fading channels. We first analyze the pairwise error probability (PEP) of COFDM and then consider the union bound on bit error rate (BER) by introducing the notion of diversity guard (DG) and a novel interleaver class, named almost linear interleaver (ALI). A construction of ALI is also introduced with its parameter selection. Simulation results show that the COFDM with ALI outperforms that with random interleaver (RI) and the block interleaver (BI) adopted in the IEEE industry standards.

contention-free property [13]. Among deterministic interleavers, the block interleaver (BI) [14] and linear interleaver (LI) [15] are two simple interleavers. For the interleaver size L, a BI of depth D exists if gcd(L, D) = D while a LI of depth D exists if gcd(L, D) = 1 1 . We may properly design BIs and LIs which perform better than RIs for a small L but for medium to large frame sizes in general. Therefore, to improve the performance of turbo codes for a large frame size, interleaver design for a large L has been extensively studied by many researchers. The almost regular permutation (ARP) [16] and quadratic permutation polynomial (QPP) interleavers [17], [18] are such examples and the QPP is selected for the Long term evaluation (LTE) standard [19]. However, QPP is designed for the tail-biting Turbo code for AWGN channel. For COFDM, IEEE adopted the depth-QN/16 BI in the industry standards 802.11a [20] and IEEE 802.16e [21], where N and 2 Q are the numbers of subcarriers and constellation size, respectively. We have been shown in [22] that the performance of COFDM can be improved by the use of a novel interleaver class, named almost linear interleaver (ALI).
This paper presents a thorough study of the COFDM with ALI. We analyze the bit error rate (BER) performance and consider the interleaver design problem of COFDM over multipath Rayleigh fading channels. Under assumption of the QPSK with Gray labeling and the maximum likelihood (ML) decision, we obtain a union (upper-)bound on BER from pair-wise error probability (PEP). After than, we introduce the notions of diversity guard (DG) to the interleavers which guarantee the diversity of order d f for the codewords of weight d f +G, for a certain G > 0.
For simplification of the union bound, we confine our attention to ALIs which make the union bound is irrelevant to the starting position of codeword errors. Subsequently, a construction of ALI is introduced with its parameter selection for COFDM applications over multipath fading 1 gcd(L, D) denotes the greatest common divisor between L and D. June 7, 2021 DRAFT channels. After verifying the validity of the approximation utilized in the simplification, we performed computer simulations for COFDM systems with 64 and 1024 subcarriers employing CCs adopted in IEEE 802.11a and IEEE 802.16e over the quasi-static multipath Rayleigh fading channels. The simulation results shown that the COFDM with ALI always outperforms that with RI and the BI adopted in the IEEE 802.11a and IEEE 802.16e standards.
The rest of this paper is organized as follows. In Section II, we briefly review COFDM. Section III introduces an upper bound of PEP for COFDM over quasi-static multipath Rayleigh fading channels under assumption of the maximum likelihood (ML) decision. In Section IV, we derive a union bound on BER and its simplification. A construction ALI is proposed in Section V with its parameter selection and the conjecture utilized for the simplification of the union bound is verified in Section VI. Simulaiton results are shown in Section VII. Finally, we conclude this paper in Section Section VIII.

A. Notations
Throughout the paper, we assume the following notations. For a random variable ξ, E{ξ} expresses the expectation of ξ. For two integers a and b, ⌊a/b⌋ denotes the maximum integer not greater than a/b while the minimum integer not smaller than a/b is denoted by ⌈a/b⌉. [

II. REVIEW OF COFDM
The block diagram of the COFDM system with N subcarriers is shown in Fig. 1. The encoded by a rate-R binary CC C with constraint length K to give a length-L codeword c ∈ C for L = QN , Q > 0. The initial state of the encoder is assumed to be 0 K−1 . Next, c is fed to an interleaver Ψ to give the interleaved codeword (ICW) d = (d ℓ ) L−1 ℓ=0 = Ψ(c), and d is partitioned into N length-Q subsequences d n := (d n,q ) Q−1 q=0 for d n,q = d nQ+q , 0 ≤ n < N . Each d n is then mapped to a point of the signal constellation 2 S, µ(d n ) ∈ S, and the resultant symbol vector Finally, a length-N G cyclic prefix (CP) is appended to x and the resultant time domain is transmitted over the channel.
Let h P := (h p ) P −1 p=0 be the channel impulse response and we assume quasi-static Rayleigh fading, that is, h p are mutually independent complex-valued circularly symmetric (CS) Gaussian random variables [23] with E{h p } = 0 and E{|h p | 2 } = σ 2 p > 0 for 0 ≤ p < P . Thus, . The channel is corrupted by complex-valued additive white At the receiver, after the CP removed, the length-N received signal y is transformed by DFT to a frequency domain vector r = yF N . Thus, if we let η := ξF N andh := hF N for h = (h P 0 N −P ), OFDM provides a frequency-domain channel model We assume that the channel vector h and the noise variance N 0 are known at the receiver.
For decoding the transmitted information, assuming all d n,q are equiprobable, we introduce the log-likelihood ratio (LLR) of d n,q as ϕ n,q = ln Pr{d n,q = 1|r} Pr{d n,q = 0|r} The binary vector dn is labeling of the symbol µ(dn).
We consider QPSK modulation (Q = 2) for S = {±1, ±j} with the Gray labeling Since d n,q are equiprobable, all the symbols in S are selected with equal probability.
In general, the decoding rule (1) may not meet the maximum likelihood (ML) decision criterion.
However, the following theorem is proved in Appendix A.

A. PEP bound in terms of SEV
For a givenh, the probability that the symbol s ′ = s − e has a likelihood larger than s is bounded as Since ℜ e diag(h)η H has a Gaussian distribution with mean zero and variance N 0 h diag (e) 2 /2, the conditional PEP is bounded as where Q(x) is the Gaussian Q-function and the Craig's Q-function expression is used [24], [25].
For an SEV e = (e n ) N −1 n=0 , let N (e) := {n |e n ̸ = 0, 0 ≤ n < N } and call it the symbol support set (SSS) of e. As discussed in [26], the covariance matrix ofh takes different forms depending on whether |N (e)| > P or |N (e)| ≤ P . The case |N (e)| > P is true either if there are many symbol errors or if P is small. The occurrence of many symbol errors is considered to be rare in general. The occurrence of a small P is, on the other hand, considered to be an indication of of small path loss as discussed in [27], [28]. Thus, in this paper, we consider the case |N (e)| ≤ P . Since σ 2 p > 0 for 0 ≤ p < P , the Hermitian matrix D(e) is non-singular, and the PEP bound is averaged with respect to the probability ofh(e) as follows P (e) = Eh P (e|h) where we let B(e, α) := diag (|e n | 2 ) n∈N (e) +4N 0 sin 2 α D −1 (e). If we substitute det (B(e, α)) ≥ n∈N (e) |e n | 2 and calculate the integral with respect to α, then we have a simper upper-bound where n C r is the binomial coefficient. We next consider the relationship between ICWEV ν and SEV e = s − s ′ for s = µ(d) and

B. PEP bound in terms of codeword error vectors
. The Gray mapping from d n = (d n,0 , d n,1 ) to s n is represented as Thus, the squared-magnitude of the symbol error e n is, with some abuse of logical and integer arithmetic, given as where ν n is the nth subsequence of ν and w H (ν n ) stands for the Hamming weight of ν n .
Since e n = 0 if and only if ν n = 0 Q , N (e) = {n|ν n ̸ = 0 Q , 0 ≤ n < N }. Thus, we also write N (e) as N (ν). Then, the matrix D(e) is also written as and the bound (3) can be rewritten as We note that the right-hand side of (5) is completely determined by ν = Ψ(ι) and can be written gives the coding gain of ν, respectively. In a moderate SNR region, the inclination of the PEP v.s. SNR curve is determined by the diversity order while the vertical position of the curve is determined by the coding gain [23]. Thus, in a moderate to high SNR region, we should preferentially optimize the diversity order rather than coding gain.

IV. UNION BOUND ON BER AND ITS APPROXIMATIONS
For the codewords c and c ′ whose corresponding information vectors are b and b ′ , respectively, If we let P (c ′ |c) be the PEP that the adversary codeword c ′ is selected for the transmitted codeword c, the BER union (upper-)bound is given by where we used the fact that all the codewords are transmitted with equal probabilities. DRAFT June 7, 2021 Since A(c, c ′ ) = A(ι, 0) for ι = c ⊕ c ′ , we can simply write it as A(ι) and the union bound in terms of ι as For ML decision, P (c ⊕ ι|c) does not depend on c but only on ι and can be written as P (ι).
Thus, we finally have the following union bound expression in terms of ι as Given an interleaver Ψ, we further introduce a subset of E w conditional on the resultant N (ν), which is the set of the weight-w CWEV that give diversity of order of v. Notice |N (ν)| ≤ w H (ν) = w H (ι) and since each subcarrier is maximally capable Q bits, the relationship (7).
The BER union bound (6) is a summation of PEPs, and the inclination of the bound is dominated by the CWEV(s) with the smallest diversity order. Therefore, the diversity order of the BER bound, or diversity order of BER simply, is given by min ι∈E |N (Ψ(ι)) | and is upper-bounded by The union bound is overwhelmed by those terms with the smallest diversity order at sufficiently large SNRs. Thus, if the employed CC is not a pathological one but a standard one as used in June 7, 2021 DRAFT the ordinary COFDM, it is reasonable to neglect the contributions of the CWEVs with diversity orders strictly higher than d f and to approximate the union bound as where P (ν) is upper-bounded by the PEP bound (5) for ν = Ψ(ι). However, application of (5) needs some preliminary discussions and assumptions.

A. Approximation for the interleavers with a positive diversity guard
We introduce the following interleaver class.
Definition 1. For a given interleaver Ψ, we say that a diversity loss occurred for a CWEV ι ∈ E if the PEP P (ν), ν = Ψ(ι), has a diversity order less than d f , and say that the interleaver has a diversity guard (DG) G if it prevent diversity losses for all CWEVs in ι ∈ ∪ G g=0 E d f +g . We denote the collection of interleavers with DG G as Ψ G .
Conjecture 1. For a certain G > 0, the union bound given in (8) can be simplified 4 as In the followings, we assume the employed interleaver is choosen from Ψ G . Notice that each interleaver in Ψ G , G ≥ 0, makes n w H (ν n ) = 1 for ι ∈ E d f . Thus, by substituting (5) to (9), we have where the factor α represents all the factors not dependent on ι.
we can rewrite (10) as where the weight A (ι s (w)) is apparently independent of s and we have writen it simply as be the bit support set (BSS) of w and assume interleaver Ψ maps the ith bit into the bit position ψ(i). Then, the SSS corresponding to the ICWEV ν(s, w) := Ψ(ι s (w)) is given by (11) can be rewritten in temrs of BCWs as

B. A simplified union bound for almost linear interleavers
For further simplification, we consider (12) the bits at positions (s + i) and (s + i ′ ) are interleaved into the subcarriers with distance where ϵ(s, i, i ′ ) ∈ {0, ±1} and the equalities are considered in modulo N .
Let D(w) = D 0 (w). Then, since ) for a sufficiently large N 5 . Obviously, the same results holds for the following interleaver class.

Definition 3. If an interleaver Ψ satisfies
it is called almost linear interleaver (ALI) and we denote the collection of ALIs as Ψ A .
Finally, for the interleaver class Ψ G,A := Ψ G ∩ Ψ A , the union bound (13) can be further simplified as and we can optimize the performance of COFDM by interleavers in the sense to minimize the union bound asΨ = arg min It is worth to note that the restriction s + i < L in (14) comes from the fact that, for the zero-padding scheme, there is no CWEV spreads over (L−1)th to 0th positions and the (almost) linearlity is not required to hold over the positions. 5 We note that, in the case of LI, ϵ(s, i, i ′ ) = 0 for BPSK and hence that Ds(w) = D(w).

V. A CONSTRUCTION OF ALI AND ITS APPLICATION FOR COFDM
In this section, we introduce a construction of ALI, proposed in [22], and discuss its parameter selection for COFDM.

A. Construction of ALI
To consider the projection property of (16), we uniquely represent each integer 0 ≤ i < L using the modulo-A decomposition as i = i 1 A + i 0 , for 0 ≤ i 0 < A and 0 ≤ i 1 < C. Then, the mapping rule can be shown as and, since A and B are relative prime, the last expression is the modulo-C decomposition which is also unique for each integer in 1 ≤ j < L. Thus, the mapping is a bijection and, since  (14) can be shown as Notice that the interleaving specified by (16) includes LI [15] and block interleaver (BI) [14] as two special cases, that are cases gcd

B. Depth Selection of ALI for COFDM
For a given G, let D G be the set of depths which allow the corresponding (L, D)-ALIs to have the DG G. Then, our interleaver design is reduced to the depth selection of ALI aŝ where we may let 1 < D ≤ L/2 from its symmetry to reduce searching complexity and the condition D ∈ D G can be verified by whether In COFDM, however, the full diversity order may be obtained only at high SNRs if the subcarriers with indexes n ∈ N (e) are highly correlative. In order to maximize the diversity order in a moderate SNR region, 1's in ι should be interleaved into the subcarriers which are low correlative each other. For a fixed ν, we have shown in Sec. III that the diversity order of ν is determined by |N (ν)| and from the fact that the subcarrier correlations rely on the subcarrier spacing (SS) ∆n, we define the following depths set. With obove definition, the depth selection of (L, D)-ALI can be modified aŝ

D(∆n) = arg min
and the selection given in (18) is the special case of ∆n = 1.

C. Influence of SS on the depth selection
To evaluate the influence of SS ∆n on the depth selection, assuming QPSK (M = 2) modulated COFDM with N = 64, we observed the change of DG G with varing ∆n for each depth 1 ≤ D ≤ 64. We considered the exponentially decaying Rayleigh fading channels with σ 2 p /σ 2 p−1 = 0.8, 1 ≤ p < 16 and employed the rate-1/2 feedforward CCs adopted in the IEEE 802.11a [20] and IEEE 802.16e [21]. The constraint lengths, free distances, and generator sequences in octal form of the CCs are listed in Table I. The results are summarized in Tables II, III, and IV, for codes I, II,  For a fixed G, on the other hand, we can see from this table that the population of the Gpermissible depths |D G (∆n)| decreases with the increasement of ∆n. A small ∆n allows some depths in D G (∆n) interleave 1's in a CWEV into some subcarriers which are highly correlated each other. With increasement of ∆n, such depths will be discarded from D G (∆n) and the depth set becomes empty at a large ∆n. As the results, with a small ∆n, we can choose a depth from many candidates but the selected depth potentially involves performance losses at a moderate SNR while some proper depths will be discarded from the candidates for a large ∆n. In order to find the proper value of SS ∆n and to make the depth selection applicable to other cases, in the next subsection, we consider its influence in terms of the subcarrier correlation ratio of coherent bandwidth (SCR).

D. Depth selection in terms of SCR
For the exponentially decaying PDP, σ 2 p , 0 ≤ p ≤ P − 1, the normalized correlation [3, p. 99] between subcarriers with SS ∆n is known as    and, for a given t, corresponding SS can be calculated as To find a proper value of t, in Table V, we let G = 2 and listed the depths selected by (19), and for these depths, we compared the required E b /N 0 s to achieve BER 10 −5 in Fig. 2.
Although we could not identify the optimal t from this figure, the optimal t takes value in the rage 0.6 < t opt < 0.9. Thus, we let t = 0.7 and considered the cases N = 64 and 1024 with P = 16 and 128, respectively. The selected depths for the exponentially decaying channels σ 2 p /σ 2 p−1 = 0.8 and 1, for 1 ≤ p < P with P −1 p=0 σ 2 p = 1 are listed in Table VI.

VI. VALIDITY OF THE CONJECTURE
In the previous section, we have approximated the union bound as in (8) and simplified the bound as (9) under the conjecture that, for the interleavers with a positive DG G, the contributions of CWEVs ι ∈ ∪ To consider the validity of the conjecture, we should distinguish the effect of CWEVs ι ∈ Since the former implies diversity losses on the union BER, the employed interleaver should prevent such events completely. On the other hand, the later affect the accuracy of the simplification, hence (9) keeps its validity if

A. The analysis of diversity order
We assume N subcarriers each of which consisting of Q bit positions and, for a weight-w CWEV, consider the occurrence probability of the event ι ∈ E v w , denoted as E(w, v), under random interleaving argument, that is, given c, ψ maps c 0 into a bit position selected with the probability 1 QN and maps c 1 into one of the remaining bit positions with probability 1 QN −1 , and so on. Under such random interleaving, every bit, wherever it is, is mapped to each position with the same probability 1 QN . Then, we can assume ι = (1 w 0 QN −w ) without loss of generality.
Since E(w, v), v ≤ w, is the event that the employed interleaver maps 1 w into v subcarriers, it occurs either interleaver maps the last 1 into a subcarrier on which at least one bit has been mapped previously under the event E(w − 1, v), or maps it into on an empty subcarrier under event E(w − 1, v − 1). Thus, we can obtain the following equality On the other hand, E(w, v) implies that for QN − w empty bit positions, (N − v)Q of them are on the empty subcarriers while remain vQ−w bit positions are on the non-empty subcarriers.
Thus, we have the conditional probabilities By substituting (22) into (21), we derive the following recursive equations as and it can be evaluated numerically with the following boundary conditions To evaluate the occurrence probability of diversity loss for the CWEVs in E w , we let F (w, v) be the event that E v w = ∅ for all CWEVs in E w , and letF (w, v) be the complementary event of F (w, v). Then, since the occurrence probability of F (w, v) is given by where the last inequality can be proved using mathematical induction and is the same with the expection of the population for E v w .

B. Numerical results
Assuming QPSK (Q = 2) modulated COFDM system with N = 64, for the codes in Table I To evaluate the probabilities Pr{F (d f + g, d f − a)} for large g, for the codes listed in Table I, we shown in Fig. 3 the probabilities Pr{E(d f +g, d f −a)} calculated by (23) for g = 0, 1, 2, 3, 4.
We can observe from this figure that each curve is concave. This fact implies that, with increasement of g, Pr{E(d f + g, d f − a)} tends to 0 faster than exponential order. Thus, if we and since the right-hand side is a decreasing function of g, Pr{F (d f + g, d f − a)} tends to 0. Therefore, if we employ an interleaver with DG G to prevent the occurrence of diversity losses for CWEVs ι ∈ E d f +g , g ≤ G, the occrrence probability of diversity losses for CWEVs ι ∈ E d f +g , g > G, is vanishingly small and we can suppress the diversity losses on union BER The confirmation is also carried out by observing |D g (1)| with varying 0 ≤ g ≤ (Q−1)(d f −1) for ALI. Table VII lists |D g (1)| found in the following manner. We let D = {D} L/2 D=1 be the initial depth set and, starting from the first BCW in each W d f +g for g = 0, the depths that yield diversity loss are removed from the set. The process is repeated over the remaining BCWs and for g = 1 so on. As shown in Table VII, the size of depth sets are |D 2 (1)| = 58, 49, and 41 for codes I, II, and III, respectively. Notice |D g (1)| converses to a constant value fast as g increment and remains the value for g > 2, and the fact D d f (1) = D 2 (1) is confirmed for each code.
Therefore, the diversity losses are completely suppressed by use of the ALI with DG G = 2.
On the other hand, the ratio of the (expected) number of CWEVs is given by .0 for Code I 2.9 for Code II 3.8 for Code III comparing with ι ∈ E d f +g for g > 0, the contributions of ι ∈ E d f are dominated part and the approximation of (13) seems acceptable for a practical COFDM with a good designed CC.
Apparently, by taking into account the contribution of the BCWs which result E d f d f +b on the union bound, we can select the depth with more accurate manner. However, the selection of the optimal depth is too complicate and the benefits come from metric modification seems limited.

VII. SIMULATION RESULTS
In this section, the effectiveness of our interleaver design is confirmed by comparison of the BER performances with the ALI for different depths and that with RI and the BI adopted by IEEE industry standard. We also show the superiority of our design by comparisons of BER curves.

A. BER comparisions
In order to verify our depth selection, for the situation 1 in Table VI, we compared BERs at E b /N 0 = 7dB for ALIs with different depths in Figure 4. We obtained each BER plot by accumulating more than 200 bit errors and distinguished G permissible depths with ∆n = 2 for G < 0, G = 0, and G = 1.
We can see from Figure 4 that, for Code I, like at the depths with G < 0, the BER performances potentially degrade seriously at the depths with G = 0 and 1. For the Codes II and III, even a serious impact does not observed for some depths with G = 0 and 1, it may be caused It is worth to note that the optimality of the depth selection is SNR dependent and our design is aimed for the applications working at moderate to high SNR region. Thus, although the ALI with our depth selection does not shown the best BER performance at E b /N 0 = 7dB, it achieves near best performance comparing with other depths.

B. BER comparision
To confirm the effectiveness of our interleaver design, for the situations listed in Table VI, we compared the BER curves of our interleaver design with that of RI 6 and the BIs adopted by 6 We randomly generated RI but fixed through all simulations. On the other hand, for the case N = 1024, although the BI which is widely adopted as industry standards and performs better than RI in situation 3 as shown in Figure 7, we can see from Fig. 8 that the BI performs worse than RI in situation 4 while our design still shows the best performance at a moderate SNR region in both cases.
In situation 3, comparing with the BI, the performance improvement by our designs are about 0.4dB, 0.9dB, and 0.6dB for for codes I, II, and III, respectively, at BER 10 −5 . In the situation 4, our design improves E b /N 0 requirements of RI about 0.2dB, 0.1dB, and 0.2dB, for codes I, II, and III, respectively, at BER 10 −5 and these are increased to about 0.2dB, 0.8dB, and 1.3dB, respectively, for the BI.
Comparing the exponential decaying PDP, the respective interleavers realize a near full diversity on the uniform one and come close to each other in BER curves. We can see from stuation 3 and 4 on the exponential decaying PDP, it gives more chances to obtain the full diversity order.

VIII. CONCLUSION
In this paper, we analyzed the BER performance of COFDM and proposed the use of COFDM with ALI. The simulation results shown that our design outperforms the COFDM with RI and the BI adopted by IEEE industry standards.

A. Optimality of our decoding rule
Since all-zero initial and final states are assumed, the CC can be regarded as the length-QN block code C. For a given h and r, the ML decisionĉ ML and the decisionĉ VA according to our where we let α := G (µ(y), r n ) + G (jµ(y), r n ) + G (−jµ(y), r n ) + G (−µ(y), r n ) and used the fact G (µ(y), r n ) G (−µ(y), r n ) = 1 for the last derivation. Comparing (27) and (28), we can conclude that VA coincides ML decision.