F Cochachin

E Boutillon

Senior Member, IEEE D Declercq

Senior Member

Sign-Preserving Min-Sum Decoders

Keywords: Error correction, Low-Density Parity-Check (LDPC) codes, Density Evolution, Sign-Preserving

This paper proposes a new finite precision iterative decoder for low-density parity-check (LDPC) codes. The proposed decoder, named Sign-Preserving Min-Sum (SP-MS), significantly improves the decoding performance compared to the classical Offset Min-Sum (OMS) decoder when messages are quantized on q = 2, 3, or 4 bits. The particularity of the SP-MS decoder is that messages cannot take the 0 value, and can fully benefit from the q bits of precision. The optimization of the SP-MS decoder is investigated in the asymptotic limit of the code length using density evolution (DE). Our study shows that 3-bit SP-MS decoders can achieve the same error-correcting performance as 5-bit OMS decoders, and 2-bit SP-MS decoders outperform 3-bit OMS decoders. The finite-length simulations confirm the conclusions of the DE analysis for several LDPC codes. Our SP-MS decoder shows a signal-to-noise ratio (SNR) gain up to 0.43 dB, with a memory/wire reduction of up to 40%, compared to the OMS decoder. Moreover, the SP-MS decoder converges faster and uses fewer iterations than the OMS decoder, with an improvement of up to 83.3% of the average decoding throughput. On an FPGA, the SP-MS decoder reduces resource utilization by up to 56% compared to the OMS decoder.

sign of messages. The SP-MS decoder forbids the 0 value in its message alphabet during the iterative decoding, meaning that a message cannot be erased. Also, unlike traditional quantized decoders, the proposed decoder uses all the possible values of the alphabet generated for a given precision and uses the sign of the incoming messages to increase the reliability of the outgoing messages at variable nodes. We investigate SP-MS decoders defined over small message alphabets constructed from q = 2, 3, or 4 bits of precision. We also quantize the Log-Likelihood Ratios (LLRs) on small alphabets constructed from q ch = 3 or 4 bits of precision.

In [START_REF] Cochachin | Optimization of Sign-Preserving Noise-Aided Min-Sum Decoders with Density Evolution[END_REF], the Sign-Preserving Noise-Aided Min-Sum (SP-NA-MS) decoders were examined for the case of regular LDPC codes considering the same precision (3 and 4 bits) for both the messages and the LLRs. In this paper, we modify and extend the work of [START_REF] Cochachin | Optimization of Sign-Preserving Noise-Aided Min-Sum Decoders with Density Evolution[END_REF] from regular to irregular LDPC codes. Moreover, we study the performance of SP-MS decoders in which the messages are quantized with one bit less than the LLRs. We also propose an offset model that depends on the magnitude of unsaturated variable-to-check messages, and optimize the offset values with Density Evolution (DE). Finally, we propose a method to improve the performance of SP-MS decoders in the error floor region, when the precision of the messages is lower than the precision of the LLRs.

Our study of the SP-MS decoders with Density Evolution shows that (q ch = 4, q = 3)-bit SP-MS decoders have the same convergence threshold as classical (q ch = 5, q = 5)-bit OMS decoders. Also, we show that (q ch = 3, q = 2)-bit SP-MS decoders have better error correction performance than classical (q ch = 3, q = 3)-bit OMS decoders. These conclusions are corroborated by finite length Monte Carlo (MC) simulations, and we obtain signal-to-noise ratio (SNR) gains up to 0.43 dB compared to the MS/OMS decoders.

Comparing the hardware complexity of the SP-MS and the OMS decoders, we observe that the SP-MS decoder allows us to greatly reduce the complexity of the message update rules when the precision of the messages is reduced. When using the SP-MS, a memory/wire reduction of up to 40% is achieved. Also, the SP-MS uses fewer iterations (converges faster) with an improvement of up to 83.3% of the average decoding throughput. Comparing the synthesis results on a Xilinx FPGA of the SP-MS and the OMS decoders, we observe that the SP-MS decoder has a significant improvement in clock frequency by up to 30%, and reduces resource utilization by up to 56%.

These results open new possibilities for massive parallel implementation of LDPC decoders.

The outline of the paper is as follows. Section II introduces the basic notions of quantized decoders and LDPC codes. Section III explains how to preserve the sign of the messages in May 21, 2021 DRAFT SP-MS decoders, and how to optimize the SP-MS decoders with DE. Section IV presents the results of the asymptotic analysis of regular and irregular LDPC codes. In section V, finite length performance of the SP-MS decoders and the MS/OMS decoders are presented. In section VI, we propose a method to mitigate the early appearance of the error floor of SP-MS decoders. In Section VII, we discussed the hardware complexity of the SP-MS decoders and we present the synthesis results on a Xilinx FPGA of the SP-MS decoders and the OMS decoders. In Section VIII, we compare the SP-MS decoder with other decoders proposed in the literature. λ i x i-1 , and ρ(x) = dc,max j=1

ρ j x j-1 .
The parameters λ i (respectively ρ j ) indicate the fraction of edges connected to degree i VNs (respectively degree j CNs) [START_REF] Richardson | The Capacity of Low-Density Parity-Check Codes Under Message-Passing Decoding[END_REF]. For regular codes, the polynomials are reduced to monomials,

λ(x) = x dv-1 and ρ(x) = x dc-1 .
Let x = (x 1 , ..., x N ) ∈ {0, 1} N be a codeword that satisfies Hx T = 0. In this paper, x is mapped by the Binary Phase-Shift Keying (BPSK) modulation and transmitted over the BI-AWGN channel with noise variance σ 2 . The channel output y = (y 1 , ..., y N ) is modeled by

y n = (1 -2x n ) + z n for n = 1, . . . , N
, where z n is a sequence of independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and variance σ 2 . The decoder produces the vector x = (x 1 , ..., xN ) ∈ {0, 1} N that is an estimation of x. To check whether x is a valid codeword, we verify that the syndrome vector is all-zero, i.e. H xT = 0.

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The LLRs that can be computed at the channel output are equal to:

LLR(y n ) = log Pr(y n | x n = 0) Pr(y n | x n = 1) = 2y n σ 2 ∀n = 1, . . . , N. (1)
For quantized decoders, the LLRs have to be quantized and saturated. Let A L denote the decoder input alphabet, defined as A L = {-N ch , ..., -1, 0, +1, ..., +N ch }, composed of 2N ch + 1 states, with N ch = 2 (q ch -1) -1 and where q ch is the number of precision bits used to quantize the LLRs.

Let the quantizer be defined by

Q : R → A L Q (a) = S ( α a + 0.5 , N ch ) , (2)
where

I n = Q (LLR(y n )) ∀n = 1, . . . , N. (3)
A MP decoder exchanges messages between VNs and CNs along the edges of the Tanner Graph.

During each iteration, the VN update (VNU) and CN update (CNU) compute outgoing messages from all incoming messages.

For classical quantized decoders, the finite message alphabet A M is defined as A M = {-N q , ..., -1, 0, +1, ..., +N q }, and consists of 2N q + 1 states, with N q = 2 (q-1) -1 and where q is the number of quantization bits for the messages. Typically, the message alphabet is equal to the decoder input alphabet, i.e. q ch = q and A M = A L . Let ∈ N denote the iteration number.

Let also m The update rule at a CNU is given by

m ( ) cm→vn = Ψ c m ( ) v→cm v∈V(cm)\{vn} =   v∈V(cm)\{vn} sign m ( ) v→cm   min v∈V(cm)\{vn} m ( ) v→cm . (4)
Let m ( +1),U vn→cm denote the unsaturated variable-to-check message in the ( + 1) th iteration, defined as

m ( +1),U vn→cm = I n + c∈V(vn)\{cm} m ( ) c→vn .
The alphabet of the unsaturated variable-to-check message m ( +1),U vn→cm , denoted A U , is defined as

A U = {-N q (d v -1) -N ch , ..., -1, 0, +1, ..., +N q (d v -1) + N ch }.
Then, the update rule at a VNU is expressed as

m ( +1) vn→cm = Ψ v I n , m ( ) c→vn c∈V(vn)\{cm} = Λ m ( +1),U vn→cm , ϕ v , (5)
where the function Λ(.) is defined by Λ(a,

ϕ v ) = sign(a) S (max(|a| -ϕ v , 0), N q ).
The CNU (4) and VNU (5) define the classical OMS decoder with offset value ϕ v ∈ {+1, ..., +(N q -2)}, applied at the VN 1 , where the special case of ϕ v = 0 corresponds to the classical MS decoder. The discrete functions Ψ v and Ψ c satisfy the symmetry conditions presented in [START_REF] Richardson | The Capacity of Low-Density Parity-Check Codes Under Message-Passing Decoding[END_REF].

Let γ ( ) = (γ ( ) 1 , ..., γ ( )
N ) denote the a posteriori probabilities (APP) in the th iteration with alphabet A app = {-N q d v -N ch , ..., -1, 0, +1, ..., +N q d v + N ch }. The APP γ

γ ( ) n = Ψ a I n , m ( ) c→vn c∈V(vn) = I n + c∈V(vn) m ( ) c→vn . (6)
From the APP, xn can be computed as xn = (1sign(γ

( ) n ))/2 if γ ( ) n = 0, otherwise, xn = (1 -sign(I n ))/2, for n = 1, ..., N .
At the initialization stage of MS-based decoders ( = 0), variable-to-check messages are initialized using m (0) vn→cm = S(I n , N q ) where c m ∈ V(v n ), for n = 1, ..., N ; the saturation function S is required only when q < q ch , i.e., N q < N ch . Analyzing (4) and ( 5), we can see that the message alphabet A M of classical quantized decoders uses only 2 q -1 levels out of a total of 2 q levels achievable with q precision bits. For example, using q = 3 bits of precision, only 7 levels among 8 levels are used in A M .

III. SIGN-PRESERVING MIN-SUM DECODERS

In the classical MS-based decoders, the value of the messages can be zero. In that case, the erased message, i.e. m ( +1) vn→cm = 0, does not carry any information and does not participate in the convergence of the decoder. In this paper, we propose a new type of decoder, with a modified VNU using a sign preserving factor, so that the VNU never generates erased messages.

A. Quantization used for SP-MS Decoders

Using the sign-and-magnitude representation, one can obtain discrete alphabets that are symmetric and composed of 2 q ch states. The decoder input alphabet for SP-MS decoders is defined as B L = {-N ch , ..., -1, -0, +0, +1, ..., +N ch }, with N ch = 2 (q ch -1) -1.

Similarly, the message alphabet for SP-MS decoders denoted by B M is defined as B M = {-N q , ..., -1, -0, +0, +1, ..., +N q }, with N q = 2 (q-1) -1. The sign of a message m ∈ B M indicates the estimated bit value while the magnitude |m| represents its reliability. In this paper, it is assumed that 2 ≤ q ≤ q ch . The alphabets B L and B M can be easily implemented in hardware because each value of B L and B M has a natural (sign, magnitude) binary representation. An example of the binary representation of A M and B M for q = 3 is shown in Table I, showing that -0 is represented by 100 2s , +0 is represented by 000 2s , etc., where the index "2s" indicates the (sign, magnitude) format.

In order to obtain the quantized version of the LLRs belonging to B L , the quantization process defined in ( 2) is replaced by

Q * (a) = (sign(a), S ( α |a| , N ch )) , (7)
The quantized LLRs

I n = Q * (LLR(y n )) ∈ B L for n = 1, .
. . , N are used to initialize the decoder. In the initialization stage, i.e. at = 0, the variable-to-check messages m ( ) vn→cm are computed as m (0) vn→cm = S(I n , N q ) where c m ∈ V(v n ), for n = 1, ..., N . Note that the operation of saturation S is required only when q < q ch , i.e., N q < N ch .

m ∈ AM q = 3 bits m ∈ BM q = 3 bits (sign(m), |m|) -3 101 -3 111 (-1,3) -2 110 -2 110 (-1,2) -1 111 -1 101 (-1,1) unused 100 -0 100 (-1,0) 0 000 +0 000 (+1,0) +1 001 +1 001 (+1,1) +2 010 +2 010 (+1,2) +3 011 +3 011 (+1,3)

B. Sign-Preserving Min-Sum Decoders

Let us now define the VNU and CNU update rules for SP-MS decoders. The discrete functions

Ψ v , Ψ c
, and Ψ a of SP-MS decoders are defined in their modified alphabets, i.e.

Ψ v : B L × B (dv-1) M → B M , Ψ c : B (dc-1) M → B M , and Ψ a : B L × B (dv)
M → B app . The update rule at the CNU of SP-MS decoders is given by

m ( ) cm→vn = Ψ c m ( ) v→cm v∈V(cm)\{vn} =   v∈V(cm)\{vn} sign m ( ) v→cm , min v∈V(cm)\{vn} m ( ) v→cm   . ( 8
)
The CNU computes outgoing messages that always belong to B M based on all the incoming messages that also belong to B M . It is to be noted that ( 8) is identical to (4).

In the case of the VNU, the update rule of classical MS decoders [START_REF]IEEE Standard for Ethernet[END_REF] does not guarantee that the outgoing message belongs to B M . We therefore modify the VNU rule to ensure that the outgoing message will always belong to B M . Let us denote by µ ( ) vn→cm the sign-preserving factor for the variable-to-check message from VN v n to CN c m , defined as

µ ( ) vn→cm = ξ sign(I n ) + c∈V(vn)\{cm} sign m ( ) c→vn , (9)
where the value of ξ depends on the value of the column-weight

d v of a VN v n : ξ =          0, if d v = 2, 1, if d v > 2 and d v is odd, 2, if d v > 2 and d v is even. (10) µ ( ) vn→cm takes its values in {-1, +1} for d v = 2, in {-d v , -d v + 2, ..., -1, +1, ..., +d v } for d v odd and d v > 2, and in {-d v -1, -d v + 1, . . . , -1, +1, ..., +d v + 1} for d v even and d v > 2.
Thus, µ

m ( +1),U vn→cm = µ ( ) vn→cm 2 + I n + c∈V(vn)\{cm} m ( ) c→vn . (11)
The alphabet of m

( +1),U vn→cm is given by B U = {-N q (d v -1) -N ch -(d v -1 + ξ)/2, ..., -1.5, -0.5, +0.5, +1.5, ..., +N q (d v -1)+N ch +(d v -1+ξ)/2}. We note that (µ ( ) vn→cm )/2 can be written as (µ ( ) vn→cm )/2 = (µ ( ) vn→cm )/2 + 0.5. The fractional part of (µ ( ) vn→cm )/2, which is 0.5, prevents m ( +1),U
vn→cm from being zero and guarantees that a sign is always assigned to the variable-to-check message. The integer part of (µ ( ) vn→cm )/2 increases the reliability of the message m ( +1),U vn→cm and thus helps the decoder to converge faster. Then, the update rule at a VNU of the SP-MS decoder with offset value ϕ v is given by

m ( +1) vn→cm = Ψ v I n , m ( ) c→vn c∈V(vn)\{cm} = sign m ( +1),U vn→cm , S max m ( +1),U vn→cm -ϕ v , 0 , N q . ( 12
)
The APP update at a VN v n of the SP-MS decoder is defined as follows

γ ( ) n = Ψ a I n , m ( ) c→vn c∈V(vn) = I n + 1 2 ξ sign(I n ) + c∈V(vn) m ( ) c→vn + 1 2 sign m ( ) c→vn . (13)
The alphabet of APPs for SP-MS decoders is given by

B app = {-N q d v -N ch -(d v + ξ)/2), ..., -1, 0, +1, ..., +N q d v + N ch + (d v + ξ)/2)}.
From the APP, xn can be computed as

xn = (1 -sign(I n ))/2 if γ ( ) n = 0, otherwise, xn = (1 -sign(γ ( ) n ))/2 for n = 1, ..., N .

C. Optimization of Sign-Preserving Min-Sum Decoders

In order to optimize Sign-Preserving Min-Sum decoders, we analyze the offset value ϕ v . In [START_REF] Chung | On the design of Low-Density Parity-Check Codes within 0.0045 dB of the Shannon limit[END_REF], we can see that the offset value ϕ v is applied to m ( +1),U vn→cm , and the value of the offset is the same for any value of the message. We can further improve the performance by replacing the constant value ϕ v by an offset whose value depends on m ( +1),U vn→cm . To simplify the notations in this section, we use m u to denote any m

( +1),U vn→cm ∈ B U and m to denote any m ( +1) vn→cm ∈ B M .
To perform the optimization, we consider three offset values that are denoted by ϕ = (ϕ s , ϕ a , ϕ 0 ), where ϕ s is applied to |m u | = N q + 0.5, ϕ a is applied to |m u | ∈ {2.5, ..., N q -0.5}, and ϕ 0 is applied to |m u | = 1.5. It is not necessary to consider an offset value for the values |m u | > N q +0.5 because those values will be saturated to |N q |. Let Υ : ϕ = (ϕ s , ϕ a , ϕ 0 ) be an offset model used at VNs. The offset corrected message is obtained as:

m = (sign(m u ), S (max ( |m u | -b, 0) , N q )) , (14)
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m u +3.5 +2.5 +1.5 +0.5 m +3 +2 +1 +0 ϕ s = 1 ϕ a = 1 ϕ 0 = 0 Figure 1: The offset model Υ for (ϕ s = 1, ϕ a = 1, ϕ 0 = 0).
where the offset b ∈ {ϕ s , ϕ a , ϕ 0 } depends on the magnitude of m u .

As an example, Υ : ϕ = (ϕ s = 1, ϕ a = 1, ϕ 0 = 0) is depicted in Figure 1 for (q = 3, N q = 3).

Let us discuss the case of ϕ s = ϕ a = ϕ 0 . The SP-MS decoder with offset ϕ v = 1 can be obtained as a special case of Υ setting ϕ = (1, 1, 1); similarly, the SP-MS without offset, i.e.

ϕ v = 0, can be obtained with the setting ϕ = (0, 0, 0).

The offset values ϕ s and ϕ 0 are applied to the extreme values of the message alphabet B M . Therefore, ϕ s and ϕ 0 have to be analyzed differently from ϕ a .

In the case of very low message precision q = 2, Υ is defined only by one offset value ϕ = (ϕ s ).

For this special case, the message alphabet, which is only composed of four different values, is

given by B M = {-1, -0, +0, +1}, with N q = 1.

D. Density Evolution for Sign-Preserving Min-Sum Decoders

The goal of DE [START_REF] Richardson | The Capacity of Low-Density Parity-Check Codes Under Message-Passing Decoding[END_REF]- [START_REF] Ngassa | Density Evolution and Functional Threshold for the Noisy Min-Sum Decoder[END_REF] is to recursively compute the probability mass function (PMF) of the messages in the Tanner graph along the iterations. DE enables us to predict whether an ensemble of LDPC codes, parametrized by its degree distribution, decoded with a given MP decoder, converges to zero error probability in the limit of infinite block length.

For the BI-AWGN channel, the DE threshold δ is the maximum value of the standard deviation σ or the minimum SNR for which the decoder converges to a zero error probability. As in [START_REF] Cochachin | Optimization of Sign-Preserving Noise-Aided Min-Sum Decoders with Density Evolution[END_REF],

[37], δ can be expressed as δ db = 10 log 10

1 2 R σ * 2
, where σ * = δ and R is the rate of the code. In this paper, the details of the DE equations are not presented, and we refer to [START_REF] Cochachin | Noise-Against-Noise Decoder[END_REF] for a complete presentation. The DE threshold δ depends on the degree distribution (λ(x), ρ(x)) of the code, but also on the quantized decoder parameters: the number of precision bits (q ch , q), the channel gain factor α, and the offset values (ϕ s , ϕ a , ϕ 0 ). We use DE to jointly optimize the offset values (ϕ s , ϕ a , ϕ 0 ) and the channel gain factor α for a fixed precision (q ch , q) and a fixed degree distribution (λ(x), ρ(x)) as follows

(ϕ * s , ϕ * a , ϕ * 0 , α * ) = arg min (ϕs,ϕa,ϕ 0 ,α) {δ db (λ(x), ρ(x), q ch , q, α, ϕ s , ϕ a , ϕ 0 )} . ( 15
)
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The optimization of the offset values of the model Υ and the channel gain factor α is made using a greedy algorithm that computes a local maximum of the DE threshold. For MS and OMS decoders, the optimization ( 15) is reduced to the optimum channel gain factor α * that is computed by performing a grid-search.

IV. ASYMPTOTIC ANALYSIS OF SIGN-PRESERVING MIN-SUM DECODERS

In this section, we present the asymptotic analysis of SP-MS decoders for regular and irregular LDPC codes with the values of the sign preserving factor ξ defined in [START_REF] Kschischang | Factor Graphs and the Sum-Product Algorithm[END_REF].

A. Regular LDPC codes

For all results presented in this section, we consider four ensembles of (d v , d c )-regular LDPC codes with the following parameters:

(i) (d v = 3, d c = 6
) and code rate R = 0.5, (ii)

(d v = 4, d c = 64) and R = 0.94, (iii) (d v = 5, d c = 20
) and R = 0.75, and (iv)

(d v = 6, d c = 32
) and R = 0.8413. We also consider quantized decoders with precision

(q ch , q) ∈ {(5, 5), (4, 4), (4, 3), (3, 3), (3, 2)}.
DE thresholds of the classical MS and OMS decoders are given in Table II, which shows that the OMS is almost always superior to the MS for the cases involved, except when q ch = q = 3 for regular 1) First, the DE thresholds of SP-MS decoders are almost always better than the DE thresholds of classical decoders when the same precision for the messages and the LLRs is used (q ch = q). An exception appears for the regular (d v = 6, d c = 32) LDPC code, we observe a degradation of around 0.01 dB.

(d v =
2) Second, the DE thresholds of (q ch , q = q ch -1)-bit SP-MS decoders are almost equal or equal to the DE thresholds of (q ch , q = q ch )-bit SP-MS decoders for d v ∈ {4, 5, 6}. In the case of From these observations, we can conclude that the SP-MS decoders can be implemented using lower precision for the messages than for the LLRs, with q = q ch -1, with a negligible impact in the decoding performance.

d v = 3,
3) Third, the DE gains of SP-MS decoders are significant for low precision (q ch = 3, q = 3) and very low precision (q ch = 3, q = 2), while the DE gains are smaller for the largest May 21, 2021 DRAFT precision (q ch = 4, q = 4) and (q ch = 4, q = 3). The largest gain obtained is around 0.279 dB for the regular (d v = 3, d c = 6) LDPC code and precision (q ch = 3, q = 3).

We thus conclude that the SP-MS decoders are more effective when the decoders are implemented in low precision.

4) A final remark comes from the interpretation of the optimum ϕ * obtained through the DE analysis. For the case of precision q ≥ 3, we have ϕ * = (ϕ * s , ϕ * a , ϕ * 0 ) = (1, 1, 0) for regular d v = 3 LDPC code. The value ϕ * 0 = 0 means that the offset is not applied to the message m u = ±1.5, hence m u = ±1.5 is mapped to m = ±1.

For SP-MS decoders, when using the precision q ≥ 3 and regular d v > 3 LDPC codes, we always have ϕ * = (ϕ * s , ϕ * a , ϕ * 0 ) = (1, 1, 1) which corresponds to the SP-MS decoder with offset ϕ v = 1. In the case of very low precision q = 2 we obtain ϕ * = (ϕ * s ) = 1 for regular d v = 5 and d v = 6 LDPC codes, and ϕ * = (ϕ * s ) = 0 for d v = 3 and d v = 4. These results indicate that the optimal offset correction depends on both the amplitudes of the messages and the VN degree.

B. Irregular LDPC codes

In the case of regular LDPC codes, we have observed that the optimum offset values (ϕ * s , ϕ * a , ϕ * 0 ) depend on the VN degree. Therefore, in order to optimize the SP-MS decoders for the irregular LDPC codes, we extend our optimization approach by considering an offset model Υ with different offset values for the different VN degrees. The precision considered in this section is

q ch = q ∈ {3, 4}. Let Υ (2) : ϕ (2) = ϕ (2) s , ϕ (2)
a , ϕ

(2) 0 denote the offset model for the VNs of degree d v = 2, and let Υ (3) :

ϕ (3) = ϕ (3) s , ϕ (3)
a , ϕ

Υ (≥4) : ϕ (≥4) = ϕ (≥4) s , ϕ (≥4) a , ϕ (≥4) 0 .
The optimization of the offset values for an irregular LDPC code with distribution (λ(x), ρ(x)) is performed by the maximization of the DE thresholds: 2) ,ϕ (3) ,ϕ (≥4) ,α)

ϕ (2) * , ϕ (3) * , ϕ (≥4) * , α * = arg min (ϕ (
{δ db } . (16)
For our analysis, we consider the ensemble of irregular LDPC codes that follows the distribution of the R = 1/2 LDPC code described in the WIMAX standard [START_REF]IEEE Standard for Local and metropolitan area networks -Part 16: Air Interface for Fixed and Mobile Broadband Wireless Access Systems -Amendment 2: Physical and Medium Access Control Layers for Combined Fixed and Mobile Operation in Licensed Bands and Corrigendum 1[END_REF]. The degree distribution of The DE thresholds of SP-MS decoders are summarized in Table VI, where we indicate the optimum channel gain factor α * and the optimum offset values ϕ (2) * , ϕ (3) * , ϕ (≥4) * . These results confirm the conclusions of the regular LDPC codes analysis: (i) the DE thresholds of SP-MS decoders are better than the DE thresholds of MS and OMS decoders, (ii) the optimum value of ϕ * 0 is 0 for d v = 3 VNs and for precision q ∈ {3, 4}, and (iii) the optimum value of ϕ (≥4) * is ϕ (≥4) * = (1, 1, 1) for precision q ∈ {3, 4}. Another conclusion can be derived from this table. The DE analysis shows that the offset should not be applied on degree d v = 2 VNs, since we always obtain (ϕ

(2) s , ϕ (2)
a , ϕ

0 ) = (0, 0, 0). This observation, combined with the fact that the optimum values of ϕ (≥4) * are always 1, leads to the conclusion that the offset in SP-MS decoders must be chosen carefully for irregular LDPC codes.

Finally, the gains of SP-MS decoders for irregular codes are larger than for the regular codes with a gain of 0.4307 dB for lower precision q = 3 and a gain of 0.4359 dB for the largest precision q = 4. QC-LDPC code. The results show that for low precision messages q = 3 and for regular

d v = 3
LDPC codes, the MS decoder is better than the OMS decoder. This result is not surprising since the DE thresholds in Table II are better for MS than for OMS with q = 3 bits of precision. The same observation holds for the regular (d v = 4, d c = 64) QC-LDPC code and precision q = 3.

The gains/losses of the SP-MS decoders compared with the MS/OMS measured at FER = 10 -2 are reported in Table VII, which shows that the Monte Carlo simulations are congruent with the values predicted by the DE thresholds.

The FER performance curves plotted in Figure 2 and Figure 3 show that the (4, 3)-bit SP-MS decoders and the (3, 2)-bit SP-MS decoders exhibit poor performance due to an early error floor.

A method to mitigate the early appearance of the error floor is proposed in section VI. ⊥ SNR gains are obtained by comparing the (qch, q = qch -1)-bit SP-MS and the (qch, q = qch)-bit MS/OMS. VII.

From Figure 5, we can see that the (4, 3)-bit OMS decoder has the same performance as the (4, 4)-bit OMS decoder. We also observe that the (3, 2)-bit OMS decoder exhibits a very high error floor. In addition, the (3, 2)-bit SP-MS decoder mitigates the early appearance of the error floor and it greatly outperforms the (3, 2)-bit OMS decoder and the (3, 3)-bit OMS decoder, thus the SP-MS decoder is an excellent candidate for a hardware implementation.

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B. Convergence Performance Analysis

FER convergence results are presented in Figure 6 for the IEEE 802.3 ETHERNET code. This figure shows that the FER curves for the SP-MS decoders decrease faster than for the MS and the OMS decoders. This figure also shows that the BP decoder converges faster than all quantized decoders during the first 6 iterations; after 7 iterations, the FER curve of the BP decoder decreases slowly. It is to be noted that from iteration 7 to iteration 20, the (q ch = 4, q)-bit SP-MS decoders show better performance than the BP decoder.

We note that the FER convergence performance of the (q ch , q = q ch -1)-bit SP-MS and the (q ch , q = q ch )-bit SP-MS are almost equal or equal. In addition, after 10 iterations of decoding, only marginal FER improvement is obtained for the (q ch = 4, q)-bit SP-MS. The maximum number of decoding iterations can thus be set to 10 without significant performance degradation.

Similarly, the maximum number of iterations for the (q ch = 3, q)-bit SP-MS can be set to 15.

Finally, compared to the MS/OMS decoders, the SP-MS decoder uses fewer iterations to reach the same FER. For example, at E b /N 0 = 4.75 dB, the (4, 3)-bit SP-MS and the (4, 4)-bit SP-MS reach FER = 10 -6 with 6 iterations, while the (4, 4)-bit OMS uses 9 iterations. The average number of decoding iterations for the ETHERNET code is shown in Figure 7. For low precision q ch = 3, we note that the SP-MS decoder not only has a better performance than the MS/OMS decoder, it also has the lowest average number of decoding iterations, which leads to lower latency and a higher average throughput. At E b /N 0 = 4.5 dB, the average number of decoding iterations is 5.5 for the OMS whereas it is only 3 for the SP-MS. This reduction of When comparing the FER performance, the FER convergence, and the average number of iterations, the (q ch , q = q ch -1)-bit SP-MS decoder offers the best trade-off between performance and complexity. For applications that require low hardware complexity, the (3, 2)-bit SP-MS decoder is the best option. For applications where hardware complexity is not an issue and decoding performance is privileged, the (4, 3)-bit SP-MS decoder is the best choice.

We also performed MC simulations for the following LDPC codes:

C. Performance of Irregular LDPC codes

Figure 8 shows the simulation results for the WIMAX rate 1/2 LDPC code, for a maximum of 100 iterations. We observe that the SNR gains in the waterfall region are congruent with the gains predicted by the DE analysis, with a 0.40 dB gain for q = 3 and a 0.40 dB gain for q = 4, at FER = 10 -2 .

Additionally, the (3, 3)-bit SP-MS decoder has the same FER performance as the (4, 4)-bit MS decoder. In the waterfall region, the (4, 4)-bit SP-MS has the same FER performance as the (5, 5)-bit OMS. 3 For a fully parallel architecture, the average decoding throughput is given by Tavg = N F/Lavg (in Mbit/s), where F (in MHz) is the clock frequency and Lavg is the average number of decoding iterations. Note that F is constant for all decoders. 14), i.e. the VNU will propagate an incorrect message despite the correct values of the check-to-variable (or incoming in this case) messages. This kind of VNU error propagation has a negative impact on the decoding process and contributes to the observed high error floor. A similar behavior can be observed for other configurations such as m ( +1)

c 3 →vn = +1. In -3 -2 -1 -0 +0 m ( ) c2→vn \m ( ) c1→vn -1 -0 +0 +1 -1 -0 +0 +1 -1 -0 +0 +1 -1 -0 +0 +1 -1 -0 +0 +1 -1 -5 -4 -3 -2 -4 -3 -2 -1 -3 -2 -1 0 -2 -1 0 1 1 1 -1 0 0 0 1 1 1 2 2 2 -0 -4 -3 -2 -1 -3 -2 -1 0 -2 -1 0 1 1 1 -1 0 1 1 1 2 2 2 0 0 0 1 1 1 2 2 2 3 3 3 +0 -3 -2 -1 0 -2 -1 0 1 1 1 -1 0 1 1 1 2 2 2 0 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 4 4 4 +1 -2 -1 0 1 1 1 -1 0 1 1 1 2 2 2 0 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 4 4
c 1 →vn = m ( ) c 2 →vn = +1, and ϕ v = 1 (ϕ s = ϕ v ) we obtain m ( +1) vn→c 3 = Ψ v (-3, +1, +1) = -0 using (
vn→c 3 = Ψ v I n = -2, m ( ) c 1 →vn = +1, m ( ) c 2 →vn = +0 = -0 or m ( +1) vn→c 3 = Ψ v I n = -2, m ( ) c 1 →vn = +0, m ( ) c 2 →vn = +0 = -0.

B. Mitigation of the Early Appearance of the Error Floor

To mitigate the appearance of the early error floor, we propose a modification of the VN update that involves changing the maximum amplitude of the check-to-variable messages from N q to Ω > N q after a given number of iterations L m . The amplitude N q is used for the first L m -1 iterations, and then a larger amplitude Ω > N q is used from iteration L m until the end of decoding. Then the modified VNU rule is given by m

( +1) vn→cm = Ψ v I n , m ( ) c→vn c∈V(vn)\{cm} = sign m ( +1),U vn→cm , S max m ( +1),U vn→cm -b, 0 , N q . ( 17
)
where b ∈ {ϕ s , ϕ a , ϕ 0 } is the offset and the unsaturated variable-to-check message m ( +1),U vn→cm is redefined as:

m ( +1),U vn→cm = µ ( ) vn→cm 2 + I n + c∈V(vn)\{cm} w ( ) c→vn , (18)
May 21, 2021 DRAFT with the value of w ( ) c→vn depending on the iteration:

w ( ) c→vn =          m ( ) c→vn , if < L m , m ( ) c→vn , if ≥ L m and m ( ) c→vn < N q , sign m ( ) c→vn , Ω , if ≥ L m and m ( ) c→vn = N q . (19)
In addition, the APP computation at a VN v n is redefined as follows

γ ( ) n = Ψ a I n , w ( ) c→vn c∈V(vn) = I n + 1 2 ξ sign(I n ) + c∈V(vn) w ( ) c→vn + 1 2 sign w ( ) c→vn . (20)
The variable-to-check messages m ( +1)

vn→cm ∈ B M are computed using [START_REF] Zhang | An Efficient 10GBASE-T Ethernet LDPC Decoder Design With Low Error Floors[END_REF], and the CNU generates check-to-variable messages m ( +1) cm→vn ∈ B M with (8).

Table IX: Optimal values to implement the (q ch , q = q ch -1)-bit SP-MS decoders.

(dv = 3, dc = 6) (dv = 4, dc = 64) (q ch , q) Ω Lm α * (ϕ * s , ϕ * a , ϕ * 0 ) Ω Lm α * (ϕ * s , ϕ * a , ϕ * 0 ) (3, 2) 2 0 0.6 (1, -, -) 2 13 0.50 (1, -, -) (4, 3) 4 0 1.16 (1, 1, 0) 5 13 0.95 (1, 1, 1)
Table X: All values that can take γ The optimal values of α * , ϕ * = (ϕ * s , ϕ * a , ϕ * 0 ), and the new parameters Ω and L m , computed with Monte Carlo simulations, are reported in Table IX, and those values are used to implement the (q ch , q = q ch -1)-bit SP-MS decoders for the d v = 3 and d v = 4 LDPC codes.

( ) n = Ψ a I n , w ( ) c 1 →vn , w ( ) c 2 →vn , w ( ) c 3 →vn , with w ( ) c 3 →vn = +2. In -3 -2 -1 -0 +0 w ( ) c2→vn \w ( ) c1→vn -2 -0 +0 +2 -2 -0 +0 +2 -2 -0 +0 +2 -2 -0 +0 +2 -2 -0 +0 +2 -2 -6 -4 -3 -1 -5 -3 -2 0 -4 -2 -1 1 1 1 -3 -1 0 2 2 2 -2 0 0 0 1 1 1 3 3 3 -0 -4 -2 -1 1 1 1 -3 -1 0 2 2 2 -2 0 1 1 1 3 3 3 -1 1
To illustrate how the proposed method helps the decoder, we use the example of the (3, 2)-bit SP-MS decoder and consider the parameters presented in Table IX. The new values of the APP γ ( ) n computed after iteration L m are shown in Table X. The circled numbers correspond to the new configurations for which we can estimate the correct value of the bit with our modification.

As an example for I n = -3, we observe that from a single configuration ( up to 40% in the size of the memory can be achieved by using q = 3 bits message precision instead of q = 5 bits.

B. Synthesis results on FPGA

In this section, we present the synthesis result of a fully parallel architecture on the Xilinx The FPGA resource utilization of the OMS and SP-MS decoders is listed in Table XI. In addition, the maximum clock frequency that each decoder can reach is listed. From the results obtained, we can see that the maximum clock frequency of (q ch , q = q ch -1)-bit SP-MS is higher compared to the maximum clock frequency of the (q ch , q = q ch )-bit OMS, we observe an increase in the clock frequency of up to 30%. Reduced precision for the messages entails reduced complexity of the VNUs and CNUs and reduced wires in implementation, and thus a reduced critical path, which promotes a higher clock frequency (i.e. the throughput is increased).

When comparing the use of FPGA resources, we can clearly see that the (q ch , q = q ch -1)-bit SP-MS decoders use less resources than the (q ch , q = q ch )-bit OMS decoders. A large savings of FPGA resources can be observed: around 27% of slice registers and 56% of slice LUTs for low precision (q ch = 3, q = 2) compared to precision (q ch = 3, q = 3), and 20% of slice registers and 48% of slice LUTs for precision (q ch = 4, q = 3) compared to precision (q ch = 4, q = 4). These results confirm that the complexity of update rules is reduced when the precision of messages is reduced. From all these results, we can conclude that the SP-MS decoder is better overall than the MS/OMS decoder. Not only does the SP-MS decoder have better performance, it also reduces the complexity of the VN and CN processing, and finally it has a better convergence speed.

May 21, 2021 DRAFT

  Finally, Section IX concludes this paper. II. BASIC NOTIONS OF CLASSICAL QUANTIZED DECODERS AND LDPC CODES An LDPC code is a linear block code defined by a sparse parity-check matrix H = [h mn ] composed of M rows by N columns, with M < N . The usual graphical representation of an LDPC code is a Tanner graph, which is a bipartite graph G composed of two types of nodes: the variable nodes (VNs) v n , n = 1, ..., N and the check nodes (CNs) c m , m = 1, ..., M . A VN in the Tanner graph corresponds to a column of H, and a CN corresponds to a row of H. An edge connecting CN c m to VN v n exists if and only if h mn = 0. Let V(v) denote the set of neighbors of a VN v, and V(c) denote the set of neighbors of a CN c. The degree of a node is the number of its neighbors in G. A code is said to have a regular column-weight d v if all VNs v have the same degree d v . Similarly, if all CNs c have the same degree d c , a code is said to have a regular row-weight d c . In case of irregular LDPC codes, the nodes can have different connection degrees, defining an irregularity distribution, which is usually characterized by the two polynomials λ(x) = dv,max i=1

  . depicts the floor function and S(b, N ch ) is the saturation function clipping the value of b in the interval [-N ch , N ch ], i.e. S(b, N ch ) = min(max(b, -N ch ), +N ch ). The parameter α is called channel gain factor and is used to increase or decrease the amplitude of LLRs at the decoder input. The value of α can be seen as an extra parameter in the quantized decoder that can be analyzed and optimized for better performance. With these notations, we define the quantized version of the LLRs that initialize the MP decoder by the vector I = (I 1 , ..., I N ) ∈ A N L , with

  M denote the variable-to-check message sent from VN v to CN c in the th iteration, and m ( ) c→v ∈ A M denote the check-to-variable message sent from CN c to VN v in the th iteration. Let us briefly recall the VNU and CNU equations for the quantized MS-based decoders, before introducing the SP-MS decoders. For this purpose, let Ψ v : A L × A (dv-1) M → A M denote the discrete function used for the update at a VN v of degree d v , and let Ψ c : A (dc-1) M → A M denote the discrete function used for the update at a CN c of degree d c . May 21, 2021 DRAFT

  is associated to a VN v n , n = 1, 2, . . . , N . Let Ψ a : A L × A (dv) M → A app denote the discrete function used for the APP computation at a VN v of degree d v . The function Ψ a satisfies the same conditions of symmetry as the function Ψ v . With these notations, the APP computation at a VN v n is given by

  ( ) vn→cm is always an odd number. May 21, 2021 DRAFT Let us now redefine the unsaturated variable-to-check message m ( +1),U vn→cm as

3 , d c = 6 )

 36 LDPC codes and regular (d v = 4, d c = 64) LDPC codes.

  the DE threshold of the (3, 2)-bit SP-MS decoder is significantly worse than the DE threshold of the (3, 3)-bit SP-MS decoder.We can observe that the performance of the (4, 3)-bit SP-MS decoder can come close to the performance of the (5, 5)-bit OMS decoder, within 0.05 dB for d v ∈ {4, 5, 6}. We also observe that the (3, 2)-bit SP-MS decoders show a better performance than the (3, 3)-bit MS and (3, 3)-bit OMS decoders; the only exception appears for d v = 3.

  model for the VNs of degree d v = 3.

Finally, we decide

  to use the same model for all other VNs with degrees d v ≥ 4, denoted

  LENGTH PERFORMANCE OF SIGN-PRESERVING MIN-SUM DECODERSIn this section, we present the frame error rate (FER) performance of classical MS, classical OMS, and SP-MS decoders over the BI-AWGN channel.A. Performance of Regular LDPC codesTo corroborate the asymptotic results obtained by DE, we present the Monte Carlo simulations for (i) the (d v = 4, d c = 64)-regular QC-LDPC code with N = 8960, rate R = 0.94 and circulant size L = 140 for Flash Memory [8], and (ii) the (d v = 6, d c = 32)-regular LDPC codes with R = 0.8413 for the IEEE 802.3 ETHERNET code [5]. In addition, the PEG algorithm from [39] is used to design the following regular QC-LDPC codes: (iii) a (d v = 3, d c = 6)regular QC-LDPC code with length N = 1296, rate R = 1/2 and circulant size L = 54, and (iv) a (d v = 5, d c = 20)-regular QC-LDPC code with length N = 10240, rate R = 3/4, and circulant size L = 512. For all codes, the Belief Propagation decoder performance is shown as a benchmark 2 . A maximum of 100 iterations have been set for d v = 3 LDPC codes, while for the case of d v ∈ {4, 5, 6} LDPC codes, a maximum number of 30 iterations has been used. For each simulated SNR, the FER is estimated with at least 100 frame errors.

Figure 2

 2 Figure 2 shows the FER performance comparison between the classical MS, classical OMS, and SP-MS decoders, for three precisions of messages q ∈ {2, 3, 4}, and for the regular (d v = 3, d c = 6) QC-LDPC code. Figure 3 draws the same curves for the regular (d v = 4, d c = 64)

2

  The simulation results for the BP decoder were obtained with the open-source simulator AFF3CT: A Fast Forward Error Correction Toolbox, 2020. [Online]. Available: https://aff3ct.github.io/index.html May 21, 2021 DRAFT E b /N 0 (dB) 6)-regular LDPC, N = 1296 BP, float-point (5, 5)-bit OMS, α = 2.625 (3, 3)-bit MS, α =0.9375 (4, 4)-bit MS, α =2 (3, 3)-bit OMS, α =1.0625 (4, 4)-bit OMS, α =1.875 (3, 3)-bit SP-MS, α = 0.95 (3, 2)-bit SP-MS, α = 0.48 (4, 4)-bit SP-MS, α = 1.79 (4, 3)-bit SP-MS, α = 1.16

Figure 2 :

 2 Figure 2: FER performance for (3, 6)-regular QC-LDPC code.

Figure 3 :

 3 Figure 3: FER performance for (4, 64)-regular QC-LDPC code.

*

  Results obtained when comparing the (4, 3)-bit SP-MS decoder with the (5, 5)-bit OMS decoder. 20)-regular LDPC, N = 10240 BP, float-point (5, 5)-bit OMS, α = 1.61 (3, 3)-bit MS, α =0.56 (4, 4)-bit MS, α =1.4 (3, 3)-bit OMS, α =0.92 (4, 4)-bit OMS, α =1.39 (3, 3)-bit SP-MS, α = 0.89 (3, 2)-bit SP-MS, α = 0.88 (4, 4)-bit SP-MS, α = 1.39 (4, 3)-bit SP-MS, α = 1.42

Figure 4 :Figure 5 :

 45 Figure 4: FER performance for (5, 20)-regular QC-LDPC code.

Figure 6 :

 6 Figure 6: FER convergence comparison on the ETHERNET code at E b /N 0 = 4.75 dB.

Figure 7 :

 7 Figure 7: Average number of iterations of the ETHERNET code with different decoders.

  (d v = 4, d c = 8, N = 1296), (d v = 5, d c = 10, N = 1280), (d v = 3, d c = 12, N = 1296), (d v = 4, d c = 16, N = 1296), and (d v = 5, d c = 20, N = 1280), and we obtained similar conclusions.

I 3

 3 n ∈ {-3, -2, -1, -0, +0}. For I n = -3 we obtain γ ( ) n > 0 (x n = 0) only when m →vn = +1, i.e. one can estimate the correct value of the bit only if all messages have the maximum allowed value (+N q ). For the case of I n = -2 (respectively I n = -1) we can see 3 (respectively 6) configurations to correctly estimate the bit.The effect of low precision messages is even more pronounced on the message update at the VNU. Considering I n = -3, m ( )

vn→c 3

 3 for I n = -3, m ( ) c 1 →vn = m ( ) c 2 →vn = +1, and ϕ v = 1 (ϕ * s = ϕ v ),May 21, 2021 DRAFT

  XC7V2000T-1FLG1925 FPGA chip for the IEEE 802.3 ETHERNET code. The main available resources of the FPGA are: (i) 2,443,200 Slice Registers, (ii) 1,221,600 Slice LUTs, and (iii) 351,321 LUT-FF pairs.

Table I :

 I Binary representation of the quantized values.

	Classical Decoder	Sign-Preserving Decoder
	May 21, 2021	DRAFT

Table II :

 II DE thresholds of classical MS (ϕ v = 0) and OMS (ϕ v = 1) decoders.In TableIII, we indicate the optimal channel gain factor α * , the optimal offset values ϕ * = and the DE thresholds δ db obtained with[START_REF] Chen | Reduced-Complexity Decoding of LDPC Codes[END_REF] for optimized SP-MS decoders. We also show the DE gain/loss compared to the best DE thresholds of MS/OMS decoders indicated in bold in TableII, for the same precision of the LLRs (q ch ). We report in TableIVthe DE losses of the (4, 3)-bit SP-MS decoder with respect to the (5, 5)-bit OMS decoder. Several conclusions can be derived from this analysis.

			(dv = 3, dc = 6)	(dv = 4, dc = 64)	(dv = 5, dc = 20)	(dv = 6, dc = 32)
	(qch, q) ϕv	α *	δdb	α *	δdb	α *	δdb	α *	δdb
	(3, 3)	0 1	0.9375 1.0625	1.789 2.204	0.50 0.69	4.760 4.904	0.56 0.92	3.645 3.231	0.455 0.84	4.081 3.593
	(4, 4)	0 1	2.0 1.875	1.644 1.348	1.06 1.15	4.572 4.421	1.40 1.39	3.392 2.762	1.035 1.28	3.815 3.169
	(5, 5)	0 1	4.0 2.625	1.613 1.215	2.20 1.73	4.534 4.338	2.47 1.61	3.337 2.724	1.985 1.45	3.751 3.140
	(ϕ * s , ϕ * a , ϕ * 0 ), May 21, 2021									DRAFT

Table III :

 III SP-MS decoders for the (d v , d c )-regular LDPC codes. DE gains are obtained by comparing the (qch, q = qch -1)-bit SP-MS and the (qch, q = qch)-bit MS/OMS.

	(dv, dc) (qch, q)	α *	(ϕ * s , ϕ * a , ϕ * 0 )	δdb	DE (dB) gain/loss	(dv, dc) (qch, q)	α *	(ϕ * s , ϕ * a , ϕ * 0 )	δdb	DE (dB) gain/loss
		(3, 3)	0.95	(1, 1, 0)	1.510	0.279		(3, 3)	0.89	(1, 1, 1)	3.014	0.217
	(3, 6)	(3, 2) (4, 4) (4, 3)	0.48 1.79 1.16	(0, -, -) (1, 1, 0)	1.932 1.269 1.391	-0.143 ⊥ 0.079 -0.043 ⊥	(5, 20)	(3, 2) (4, 4) (4, 3)	0.88 1.39 1.42	(1, -, -) (1, 1, 1)	3.022 2.741 2.738	0.209 ⊥ 0.021 0.024 ⊥
		(3, 3)	0.57	(1, 1, 1)	4.612	0.148		(3, 3)	0.74	(1, 1, 1)	3.396	0.197
	(4, 64)	(3, 2) (4, 4) (4, 3)	0.57 1.04 1.05	(0, -, -) (1, 1, 1)	4.624 4.379 4.379	0.136 ⊥ 0.042 0.042 ⊥	(6, 32)	(3, 2) (4, 4) (4, 3)	0.74 1.18 1.22	(1, -, -) (1, 1, 1)	3.398 3.179 3.174	0.195 ⊥ -0.010 -0.005 ⊥
	⊥

Table IV :

 IV DE losses obtained by comparing the (4, 3)-bit SP-MS and the (5, 5)-bit OMS.

	(dv, dc) (q ch , q)	α *	(ϕ * s , ϕ * a , ϕ * 0 )	δ db	DE loss (dB)
	(3, 6)	(4, 3)	1.16 †	(1, 1, 0)	1.391	-0.176
	(4, 64)	(4, 3)	1.05 †	(1, 1, 1)	4.379	-0.041
	(5, 20)	(4, 3)	1.42	(1, 1, 1)	2.738	-0.014
	(6, 32)	(4, 3)	1.22	(1, 1, 1)	3.174	-0.034

†

The channel gain factor α is further optimized in section VI.

Table V :

 V DE thresholds of MS and OMS decoders for the WIMAX degree distribution + 28 76 x 6 . For this distribution, we indicate in Table V the DE thresholds of the MS and OMS decoders.

	R	(qch, q) ϕv	α *	δdb
		(3, 3)	0 1	0.44 1.8310 0.40 5.2283
	1/2	(4, 4)	0 1	1.07 1.3941 0.80 2.8140
		(5, 5)	0 1	2.30 1.3013 1.55 1.1828
	the WIMAX code is λ(x) = 22 76 x + 24 76 x 2 + 30 76 x 5 and ρ(x) = 48 76 x 5
	May 21, 2021				DRAFT

Table VI :

 VI DE thresholds of SP-MS decoders for the WIMAX degree distribution.

	R	(qch, q)	α *	dv	(ϕ * s , ϕ * a , ϕ * 0 )	δdb	DE gain (dB)	SNR gain (dB) @ FER = 10 -2
				2	(0,0,0)
		(3, 3)	0.65	3	(0,0,0)	1.4003	0.4307	0.40
	1/2			≥ 4 2	(1,1,1) (0,0,0)
		(4, 4)	1.24	3	(0,1,0)	0.9582	0.4359	0.40
				≥ 4	(1,1,1)

Table VII :

 VII DE gains and SNR gains of SP-MS decoders for the (d v , d c )-regular LDPC codes.

				DE	SNR				DE	SNR
	(dv, dc) (qch, q)	α *	gain/loss	gain/loss (dB)	(dv, dc) (qch, q)	α *	gain/loss	gain/loss (dB)
				(dB)	@ FER = 10 -2				(dB)	@ FER = 10 -2
		(3, 3)	0.95	0.279	0.26		(3, 3)	0.89	0.217	0.22
		(3, 2)	0.48 † -0.143 ⊥	-0.13 ⊥		(3, 2)	0.88	0.209 ⊥	0.20 ⊥
	(3, 6)	(4, 4)	1.79	0.079	0.06	(5, 20)	(4, 4)	1.39	0.021	0.02
		(4, 3)	1.16 † -0.043 ⊥ -0.176 *	0.01 ⊥ -0.14 *		(4, 3)	1.42	0.024 ⊥ -0.014 *	0.02 ⊥ -0.015 *
		(3, 3)	0.57	0.148	0.16		(3, 3)	0.74	0.197	0.20
		(3, 2)	0.57 †	0.136 ⊥	0.13 ⊥		(3, 2)	0.74	0.195 ⊥	0.20 ⊥
	(4, 64)	(4, 4)	1.04	0.042	0.05	(6, 32)	(4, 4)	1.18	-0.010	0.0
		(4, 3)	1.05 †	0.042 ⊥ -0.041	0.03 ⊥		(4, 3)	1.22	-0.005 ⊥	0.0 ⊥

* -0.07 * -0.034 * -0.028 * † The channel gain factor α is further optimized with Monte Carlo simulations (see section VI).

Table VIII :

 VIII All values that can take γ

	( ) n = Ψ a I n , m ( ) c 1 →vn , m ( ) c 2 →vn , m ( ) c 3 →vn , with m

( )

  Table VIII), we now have 5 configurations (Table X) to correctly estimate the bit. Calculating the variableto-check message m

Table XI :

 XI Synthesis results on FPGA for the IEEE 802.3 ETHERNET code.

	Decoder (qch, q)	Maximum Clock	Number of slice	Number of slice	Number of fully	Wires
			Frequency (MHz)		registers		LUTs	used LUT-FF pairs	reduction
	OMS	(3, 3)	111.100	(0.0%)	45061	(0.0%)	463739	(0.0%)	38917	(0.0%)	0.0%
	SP-MS	(3, 2)	133.582	(+20.24%) 32773	(-27.27%) 204091	(-55.99%) 32773	(-15.78%)	-33.33%
	OMS	(4, 4)	87.790	(0.0%)	59397	(0.0%)	666300	(0.0%)	51205	(0.0%)	0.0%
	SP-MS	(4, 3)	113.814	(+29.64%) 47109	(-20.68%) 345913	(-48.08%) 47109	(-7.99%)	-25.0%

Note that in the literature, ϕv is often applied at the CN. Applying the offset at the VN or at the CN is equivalent only when the saturation function is not used since mini=1,...,n(|xi| -ϕv) = mini=1,...,n(|xi|) -ϕv.May

21, 2021 DRAFT

The size of memory required to store the check-to-variable messages of a CN of degree dc is: dc + log 2 (dc) + 1 + 2(q -1). May 21, 2021 DRAFT

May 21, 2021 DRAFT

VI. MITIGATION OF THE EARLY APPEARANCE OF THE ERROR FLOOR

In this section we propose a method to mitigate the early appearance of the error floor of the SP-MS decoder when the precision of the messages is lower than the precision of the LLRs, i.e. when q = q ch -1.

A. Early Appearance of the Error Floor

Our MC simulations of regular LDPC codes shows that the (q ch , q = q ch -1)-bit SP-MS decoders have almost the same performance as the (q ch , q = q ch )-bit SP-MS decoder for d v ≥ 5 LDPC codes. In the case of d v = 3 and d v = 4 LDPC codes, the (q ch , q = q ch -1)-bit SP-MS decoder exhibits a very high error floor as shown in Figure 2 and Figure 3.

The early appearance of this error floor is not due to trapping sets, but to the effect of low message precision on the VNU computation: the maximum absolute value of check-to-variable messages is too small compared to the maximum absolute value of the LLRs. This effect is due to configurations where the sum of check-to-variable messages does not exceed an erroneous high LLR value and therefore cannot correct the bit error. Let us take the example of the (3, 2)-bit SP-MS decoder, with N ch = 3, N q = 1, and the case of a VN v n connected to three check-nodes c 1 , c 2 , and c 3 . For this example, we assume that

x n = 0 is the value of the bit associated with variable-node v n , hence, the estimated bit xn is correct (

Figure 9 shows the simulation results of the (q ch , q = q ch -1)-bit SP-MS decoders for d v ∈ {3, 4} LDPC codes. We can see that the proposed modification helps greatly to lower the error floor. The FER performance curves obtained using the optimal values presented in Table IX, plotted in Figure 10, show that the (4, 3)-bit SP-MS decoder is slightly better than the (4, 4)-bit OMS decoder. In addition, the (3, 2)-bit SP-MS decoder is better than the (3, 3)-bit MS/OMS decoder for the d v = 4 LDPC code. In the case of the d v = 3 LDPC code, we observe a loss of performance of the (3, 2)-bit SP-MS compared with the (3, 3)-bit MS.

The proposed method to mitigate the appearance of the early error floor can be easily adapted to the case of MS/OMS decoders. First the unsaturated variable-to-check message is redefined as m

c→vn . The check-to-variable message is computed using (4). For example, we obtain with MC simulation that (Ω = 2, L m = 12, α * = 0.68, ϕ v = 1) optimizes the FER performance of the (3, 2)-bit OMS decoder for the IEEE 802.3 ETHERNET code; we have observed that the Figure 10: Comparison of the performance of (q ch , q = q ch -1)-bit SP-MS decoders with the MS and OMS decoders after lowering the error floor.

VII. HARDWARE COMPLEXITY OF SIGN-PRESERVING MIN-SUM DECODERS

A. Reduction of Wires and Memory

The decoding performance results indicate that the (q ch , q = q ch -1)-bit SP-MS decoder is a good candidate for hardware implementation. The memory and the number of wires used in an implementation is the same for the MS, OMS, and SP-MS decoders when the same precision is used (q ch = q). Considering a fully parallel architecture, when the precision q goes from 3 bits to 2 bits, the reduction of wires is 33.33%, whereas when the precision q goes from 4 bits to 3 bits, a reduction of 25% of wires is obtained. For a layered architecture, a memory reduction of up to 33.33% can be obtained when check-to-variable messages are stored. In the case of storing the variable-to-check messages in the compressed format 4 of [START_REF] Wang | A Memory Efficient Partially Parallel Decoder Architecture for Quasi-Cyclic LDPC Codes[END_REF] that requires the storage of the signs, first and second minima, and the index of the first minima, a memory reduction of up to 15.38% can be achieved for the regular

The FER results show that the (4, 3)-bit SP-MS can achieve the performance of the (5, 5)-bit OMS, thus promoting a significant reduction of the wires (by 40%), with negligible performance degradation (0.028 dB for d v = 6, see Table VII for other values). In addition, a reduction of

VIII. COMPARISON WITH OTHER STUDIES

Many decoders have been proposed in the literature for the IEEE 802.3 ETHERNET code, and some of them are listed in Table XII. In Figure 11, we compare the error correction performance of the SP-MS decoders and the decoders listed in Table XII. The BER/FER curves of the decoders listed in Table XII were taken directly from the cited papers. All SP-MS decoders exhibit better BER/FER performance than the (4, 3)-bit finite alphabet decoder [START_REF] Ghanaatian | A 588-Gb/s LDPC Decoder Based on Finite-Alphabet Message Passing[END_REF], the (2, 2)-bit APP based MS decoder [START_REF] Cui | Improved Low-Complexity Low-Density Parity-Check Decoding[END_REF], and the (5, 1)-bit delayed stochastic decoder [START_REF] Naderi | Delayed Stochastic Decoding of LDPC Codes[END_REF].

Additionally, the (3, 2)-bit SP-MS decoder has almost the same performance as the (5, 5)-bit split-row-16 MS decoder [START_REF] Mohsenin | A Low-Complexity Message-Passing Algorithm for Reduced Routing Congestion in LDPC Decoders[END_REF] and the (4, 3)-bit normalized probabilistic MS decoder [START_REF] Cheng | A Fully Parallel LDPC Decoder Architecture Using Probabilistic Min-Sum Algorithm for High-Throughput Applications[END_REF] with a much lower hardware complexity. Comparing the (3, 2)-bit SP-MS decoder with the (6, 1)-bit improved differential binary decoder [START_REF] Cushon | High-Throughput Energy-Efficient LDPC Decoders Using Differential Binary Message Passing[END_REF] and the (6, 1)-bit MTFM-based stochastic decoder [START_REF] Sharifi Tehrani | Majority-Based Tracking Forecast Memories for Stochastic LDPC Decoding[END_REF],

we observe that all three decoders have almost the same performance. It is worth mentioning that the (3, 2)-bit SP-MS decoder uses only 20 iterations whereas the 1-bit LDPC decoders [START_REF] Sharifi Tehrani | Majority-Based Tracking Forecast Memories for Stochastic LDPC Decoding[END_REF], [START_REF] Naderi | Delayed Stochastic Decoding of LDPC Codes[END_REF], [START_REF] Cushon | High-Throughput Energy-Efficient LDPC Decoders Using Differential Binary Message Passing[END_REF] use more than 300 iterations.

As a last remark, we observe that the SP-MS decoders exhibit an error floor at a FER level of 10 -8 due to trapping sets.

IX. CONCLUSION

In this paper, we have proposed a new message-passing iterative LDPC decoder which uses a sign-preserving factor that helps the decoder keeping the sign information of extrinsic messages during the VNU processing. The sign-preserving factor also helps increasing the reliability of variable-to-check messages and thus helps the finite precision iterative decoder improving the error-correcting performance. We have also proposed an offset model that depends on the magnitude of unsaturated variable-to-check messages. Density Evolution was used to optimize


Antoine Ayache
email: antoine.ayache@univ-lille.fr

Florent Bouly
email: florent.bouly@univ-lille.fr

On local path behavior of Surgailis multifractional processes

Keywords: Gaussian processes, variable Hurst parameter, local and pointwise Hölder regularity, local self-similarity AMS Subject Classification (2010). 60G17, 60G22, 60H05

Multifractional processes are stochastic processes with non-stationary increments whose local regularity and self-similarity properties change from point to point. The paradigmatic example of them is the classical Multifractional Brownian Motions (MBM) {M(t)} t∈R of Benassi, Jaffard, Lévy Véhel, Peltier and Roux, which was constructed in the mid 90's just by replacing the constant Hurst parameter H of the well-known Fractional Brownian Motion by a deterministic function H(t) having some smoothness. More then 10 years later, using a different construction method, which basically relies on nonhomogeneous fractional integration and differentiation operators, Surgailis introduced two non-classical Gaussian multifactional processes denoted by {X(t)} t∈R and {Y (t)} t∈R .

In our article, under a rather weak condition on the functional parameter H(•), we show that {M(t)} t∈R and {X(t)} t∈R as well as {M(t)} t∈R and {Y (t)} t∈R only differ by a part which is locally more regular than {M(t)} t∈R itself. Thus it turns out that the two non-classical multifractional processes {X(t)} t∈R and {Y (t)} t∈R have exactly the same local path behavior as that of the classical MBM {M(t)} t∈R .

= {B H (t)} t∈R N . Though FBF is a useful model, a serious limitation of it comes from the fact that local behavior of its paths does not change from point to point. More precisely, roughness of paths of a continuous nowhere differentiable real-valued stochastic field {Z(t)} t∈R N around some fixed point τ ∈ R N is usually measured through the pointwise Hölder exponent at τ ρ Z (τ ) := sup r ∈ [0, 1] ; lim sup

t→τ |Z(t) -Z(τ )| |t -τ | r < +∞ , (1.1)
or through the local Hölder exponent at τ ρ Z (τ ) := sup r ∈ [0, 1] ; lim sup (t ,t )→(τ,τ )

|Z(t ) -Z(t )| |t -t | r < +∞ . (1.2)
Observe that, one always has that ρ Z (τ ) ≤ ρ Z (τ ) , for all τ ∈ R N .

(1.3)

Local roughness of paths of the FBF {B H (t)} t∈R N does not change from point to point since it is known (see for instance [START_REF] Xiao | Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields[END_REF][START_REF] Berman | Gaussian sample functions: uniform dimension and Hölder conditions nowhere[END_REF][START_REF]Local nondeterminism and local times of Gaussian processes[END_REF][START_REF] Pitt | Local times for Gaussian vector fields[END_REF][START_REF] Ayache | Multifractional stochastic fields : wavelet strategies in multifractional frameworks[END_REF]) that there exists an universal event Ω * of probability 1 such that one has

ρ B H (τ, ω) = ρ B H (τ, ω) = H , for all (τ, ω) ∈ R N × Ω * .
In order to overcome this limitation of FBF, it has been proposed in [START_REF] Benassi | Elliptic Gaussian random processes[END_REF][START_REF] Peltier | Multifractional Brownian motion: definition and preliminary results[END_REF] to replace its constant Hurst parameter H by H(t), where H(•) denotes a continuous function on R N with values in some compact interval included in (0, 1). This idea has led to Multifractional Brownian Field {M(t)} t∈R N which is more commonly called multivariate Multifractional Brownian Motion (MBM). Throughout our article we always assume that the continuous function H(•) satisfies the usual condition: ρ H (τ ) > H(τ ) , for all τ ∈ R N , (1.4) where ρ H (τ ) denotes the local Hölder exponent of the function H(•) at τ . Under the condition (1.4), it has shown (see [START_REF] Ayache | Wavelet construction of Generalized Multifractional Processes[END_REF][START_REF] Ayache | Multifractional stochastic fields : wavelet strategies in multifractional frameworks[END_REF]) that there exists an universal event Ω * * of probability 1 such that one has ρ M (τ, ω) = ρ M (τ, ω) = H(τ ) , for all (τ, ω) ∈ R N × Ω * * .

(1.5) Also, under the same condition, it has shown (see [START_REF] Benassi | Elliptic Gaussian random processes[END_REF][START_REF] Ayache | Multifractional stochastic fields : wavelet strategies in multifractional frameworks[END_REF]) that, at every fixed point τ ∈ R N , the stochastic field {M(t)} t∈R N is strongly locally asymptotically self-similar of exponent H(τ ). This means that, for some positive constant c(τ ) and for any fixed positive real number T , when λ → 0 + , the stochastic field

λ -H(τ ) (M(τ +λu)-M(τ )) u∈R N converges in distribution to {c(τ )B H(τ ) (u)} u∈R N in C [-T, T ] N the
Banach space of the real-valued continuous functions on the cube [-T, T ] N equipped with the uniform norm.

From now on, we assume that N = 1 and that H(•) is a continuous function on the real line with values in some compact interval included in the open interval (1/2, 1). Since we are mainly concerned with the non-classical Gaussian multifractional processes introduced by Surgailis in his article [START_REF] Surgailis | Nonhomogeneous fractional integration and multifractional processes[END_REF], it is convenient to use from now on the same notations as in this article. Therefore, we denote by α(•) the continuous function from R into [α inf , α sup ] ⊂ (0, 1/2) defined as:

α(x) = H(x) -1/2 , for all x ∈ R. (1.6)
Similarly to the article [START_REF] Surgailis | Nonhomogeneous fractional integration and multifractional processes[END_REF], we always suppose that α(•) satisfies the uniform Dini condition:

sup t∈R 1 -1 |α(t) -α(t + u)| |u| du < +∞ , (1.7)
and also the condition:

ρ α (τ ) > α(τ ) + 1/2 , for all τ ∈ R. (1.8)
Observe that (1.8) is nothing else than the condition (1.4) with N = 1 expressed in terms of the function α(•).

We are now going to give, in terms of the function α(•), the precise definition of the classical Gaussian MBM {M(t)} t∈R initially introduced in [START_REF] Peltier | Multifractional Brownian motion: definition and preliminary results[END_REF], as well as those of the two non-classical Surgailis Gaussian multifractional processes {X(t)} t∈R and {Y (t)} t∈R constructed in [START_REF] Surgailis | Nonhomogeneous fractional integration and multifractional processes[END_REF]. To this end, we make use of the usual convention: for all (y, θ) ∈ R 2 , one has (y) θ + := y θ if y > 0 and (y) θ + = 0 else.

(1.9)

The classical MBM {M(t)} t∈R with continuous paths is defined, for each t ∈ R, through the Wiener integral:

M(t) := R 1 Γ(1 + α(t)) (t -s) α(t) + -(-s) α(t) + dB(s) , (1.10)
where Γ(•) is the well-known "Gamma" function defined as: Γ(x) := +∞ 0 y x-1 e -y dy , for each x ∈ (0, +∞).

Notice that it easily follows from (1.9), (1.10), and the equality Γ(x+1) = xΓ(x), for all x ∈ (0, +∞), that, for every t ∈ R, one has

M(t) = R t 0 1 Γ(α(t)) (τ -s) α(t)-1 + dτ dB(s) . (1.11)
The two non-classical Surgailis multifractional processes {X(t)} t∈R and {Y (t)} t∈R with continuous paths are, for every t ∈ R, defined through the Wiener integrals:

X(t) := R t 0 1 Γ(α(τ )) (τ -s) α(τ )-1 + e H-(s,τ ) dτ dB(s) (1.12)
and

Y (t) := R 1 Γ(1 + α(s)) (t -s) α(s) + e -H+(s,t) -(-s) α(s) + e -H+(s,0) dB(s) , (1.13)
where, for all real numbers s and t satisfying s < t, one has set

H -(s, t) := t s α(u) -α(t) t -u du and H + (s, t) := t s α(s) -α(v) v -s dv. (1.
ρ R (τ, ω) > α(τ ) + 1/2 = H(τ ) = ρ M (τ, ω) , for all (τ, ω) ∈ R × Ω . (1.17)
Theorem 1.2 Assume that the conditions (1.7) and (1.8) are satisfied. Then, there exists an universal event Ω of probability 1 such that one has

ρ D (τ, ω) > α(τ ) + 1/2 = H(τ ) = ρ M (τ, ω) , for all (τ, ω) ∈ R × Ω . (1.
ρ Y (τ, ω) = ρ Y (τ, ω) = α(τ ) + 1/2 = H(τ ) , for all (τ, ω) ∈ R × Ω .
Corollary 1.5 Assume that the conditions (1.7) and (1.8) are satisfied. Then, at every fixed point τ ∈ R, the stochastic process {X(t)} t∈R is strongly locally asymptotically self-similar of exponent H(τ ) = α(τ ) + 1/2. More precisely, for some positive constant c(τ ) and for any fixed positive real number T , when λ → 0 + , the stochastic process λ -H(τ ) (X(τ + λu) -X(τ )) u∈R converges in distribution to {c(τ )B H(τ ) (u)} u∈R in C [-T, T ] the Banach space of the realvalued continuous functions over the compact [-T, T ] equipped with the uniform norm.

Proof It follows from (1.15) that, for each fixed positive real number λ, the stochastic process λ -H(τ ) (X(τ +λu)-X(τ )) u∈R N can be expressed as the sum of the two processes λ -H(τ ) (M(τ +λu)-M(τ )) u∈R N and λ -H(τ ) (R(τ +λu)-R(τ )) u∈R N . One already knows from [START_REF] Benassi | Elliptic Gaussian random processes[END_REF][START_REF] Ayache | Multifractional stochastic fields : wavelet strategies in multifractional frameworks[END_REF] that the process λ -H(τ

) (M(τ + λu)-M(τ )) u∈R N converges in distribution to {c(τ )B H(τ ) (u)} u∈R in C [-T, T ]
when λ → 0 + . Thus, for proving the corollary it is enough to show that the process λ -H(τ ) (R(τ + λu) -R(τ )) u∈R N , viewed as a random variable with values in the space C [-T, T ] , converges almost surely to 0 in this space when λ → 0 + . The latter fact results from Theorem 1.1, (1.3) and (1.1) which entail that, for each ω ∈ Ω (the same event of probability 1 as in Theorem 1.1), there are 3 positive finite constants C 0 (ω), ε 0 (ω) and η 0 (ω) such that, for all real number v satisfying |v| ≤ η 0 (ω), one has R(τ

+ v, ω) -R(τ, ω) ≤ C 0 (ω)|v| H(τ )+ε0(ω) .
Corollary 1.6 Assume that the conditions (1.7) and (1.8) are satisfied. Then, at every fixed point τ ∈ R N , the stochastic process {Y (t)} t∈R is strongly locally asymptotically self-similar of exponent H(τ ) = α(τ ) + 1/2.

The proof of Corollary 1.6 is skipped since it is very similar to that of Corollary 1.5 except that Theorem 1.2 and (1.16) have to be used instead of Theorem 1.1 and (1.15).

Proof of Theorem 1.1

Let us first point out that the proof of Theorems 1.1 mainly relies on the following proposition which is a classical result derived from the equivalence of Gaussian moments and the well-known Kolmogorov-Chentsov Hölder continuity theorem (see e.g. [START_REF] Karatzas | Brownian motion and stochastic calculus[END_REF][START_REF] Khoshnevisan | Multiparameter processes: an introduction to random fields[END_REF]). Proposition 2.1 Let {Z(t)} t∈R be a real-valued Gaussian process with continuous1 paths. Suppose that, for some compact interval I ⊂ R and for some constants c > 0 and ζ ∈ (0, 1], the inequality

E|Z(t ) -Z(t )| 2 ≤ c|t -t | 2ζ
holds for all t , t ∈ I. Then, with probability 1, the paths of {Z(t)} t∈R satisfy on I a uniform Hölder condition of any order β ∈ (0, ζ). More precisely, there exists Ω I an event of probability 1, which a priori depends on I, such that one has

sup t ,t ∈I |Z(t , ω) -Z(t , ω)| |t -t | β < +∞ , for all (ω, β) ∈ Ω I × (0, ζ). (2.1)
Remark 2.2 For proving Theorem 1.1 it is enough to show that, for all fixed t 0 ∈ R, there are 3 constants ε t0 > 0, η t0 > 0 and c t0 ≥ 0, which may depend on t 0 , such that

E|R(t ) -R(t )| 2 ≤ c t0 |t -t | 2(α(t0)+1/2+3εt 0 ) , for all t , t ∈ I(t 0 , η t0 /2), (2.2)
where

I(t 0 , η t0 /2) := [t 0 -η t0 /2, t 0 + η t0 /2]. Indeed, in view of Proposition 2.1, it results from (2.
2) that there exists Ω t0 an event of probability 1 such that sup t ,t ∈I(t0,ηt 0 /2)

|R(t , ω) -R(t , ω)| |t -t | α(t0)+1/2+2εt 0 < +∞ , for all ω ∈ Ω t0 . (2.3)
Observe that, using the continuity at t 0 of the function α(•), one can choose η t0 small enough so that

|α(t) -α(t 0 )| ≤ ε t0 , for all t ∈ I(t 0 , η t0 /2). (2.4)
Next, let I(t 0 , η t0 /2) be the open interval, containing t 0 and included in I(t 0 , η t0 /2), defined as I(t 0 , η t0 /2) := (t 0 -η t0 /2, t 0 + η t0 /2). One clearly has that R = t0∈R I(t 0 , η t0 /2). Therefore, the local compactness of R implies that

R = m∈N I(t 0,m , η t0,m /2) , (2.5)
for some sequence (t 0,m ) m∈N of real numbers. Next, one denotes by Ω the event of probabiliy 1 defined as

Ω := m∈N Ω t0,m ∩ Ω * * ,
where Ω * * is the same event of probability 1 as in (1.5) with N = 1. Let (τ, ω) ∈ R × Ω be arbitrary and fixed. One knows from (2.5) that there are m(τ ) ∈ N and η(τ ) ∈ (0, 1) such that

I τ, η(τ ) ⊂ I t 0, m(τ ) , η t 0, m(τ ) /2 . (2.6)
Thus, it follows from (2.6), (2.4) and (2.3) that

sup t ,t ∈I(τ, η(τ )) |R(t , ω) -R(t , ω)| |t -t | α(τ )+1/2+εt 0, m(τ ) ≤ sup t ,t ∈I(t 0, m(τ ) ,ηt 0, m(τ ) /2) |R(t , ω) -R(t , ω)| |t -t | α(t 0, m(τ ) )+1/2+2εt 0, m(τ ) < +∞ . (2.7)
Finally, (2.7), (1.2), (1.5), and (1.6) imply that (1.17) is satisfied.

From now on, the goal is to prove that, for any fixed t 0 ∈ R, the inequality (2.2) holds. To this end, one will make an extensive use of the following lemma borrowed from [START_REF] Surgailis | Nonhomogeneous fractional integration and multifractional processes[END_REF].

Lemma 2.3 One assumes that the continuous function α(•) satisfies the condition (1.7), and one denotes by α sup the constant in the interval [α inf , α sup ] ⊂ (0, 1/2) defined as:

α sup := lim sup t-s→+∞ 1 t -s t s α(u)du .
Then, for all fixed (strictly) positive real numbers ε and ν, there is a constant C (which depends on ε and ν) such that the inequalities

(t -s) α(t) e H-(s,t) ≤ C (t -s) αsup+ε (2.8)
and

(t -s) α(s) e -H+(s,t) ≤ C (t -s) αsup+ε (2.9)
hold for all real numbers s and t satisfying t -s ≥ ν.

Notice that, one knows from (1.8) that the open interval (α(t 0 )+1/2, ρ α (t 0 )) is non-empty. Let γ ∈ (α(t 0 ) + 1/2, ρ α (t 0 )) be arbitrary and fixed. Then, one can derive from the definition of local Hölder exponent (see (1.2)) that there are two constants k α ≥ 0 and δ ∈ (0, 1/2], such that one has

|α(x) -α(y)| ≤ k α |x -y| γ , for all x, y ∈ I(t 0 , 2δ) := [t 0 -2δ, t 0 + 2δ].
(2.10)

Remark 2.4 In all the sequel, one assumes that the two arbitrary and fixed positive real numbers ε and η are small enough so that they satisfy the following 4 conditions:

ε < 8 -1 × min α inf , 1/2 -α sup , γ -(α(t 0 ) + 1/2) , (2.11)
η < δ/2 ≤ 1/4 , (2.12
)

η 2ε ≤ δ/2 -η
σ 2 R (t, h) := E|R(t + h) -R(t)| 2 . (2.16)
It can easily be seen that for proving (2.2), it is enough to show that there exists a positive finite constant C 0 not depending on t and h such that one has

σ 2 R (t, h) ≤ C 0 h 2(α(t0)+1/2+ε) .
(2.17) Using (2.16), (1.15), (1.11), (1.12) and the isometry property of Wiener integral, one gets that

σ 2 R (t, h) = R t+h 0 1 Γ(α(t + h)) (τ -s) α(t+h)-1 + dτ - t 0 1 Γ(α(t)) (τ -s) α(t)-1 + dτ - t+h t 1 Γ(α(τ )) (τ -s) α(τ )-1 + e H-(s,τ ) dτ 2 ds = R t+h 0 1 Γ(α(t + h)) (τ -s) α(t+h)-1 + - 1 Γ(α(t)) (τ -s) α(t)-1 + dτ + t+h t 1 Γ(α(t)) (τ -s) α(t)-1 + - 1 Γ(α(τ )) (τ -s) α(τ )-1 + e H-(s,τ ) dτ 2 ds.
Then, it follows from the inequality

(u + v) 2 ≤ 2u 2 + 2v 2 , for all (u, v) ∈ R 2 , (2.18) that σ 2 R (t, h) ≤ 2λ 1 (t, h) + 2λ 2 (t, h) , (2.19)
where

λ 1 (t, h) := R t+h 0 1 Γ(α(t + h)) (τ -s) α(t+h)-1 + dτ - 1 Γ(α(t)) (τ -s) α(t)-1 + dτ 2 ds (2.20)
and

λ 2 (t, h) := R t+h t 1 Γ(α(t)) (τ -s) α(t)-1 + - 1 Γ(α(τ )) (τ -s) α(τ )-1 + e H-(s,τ ) dτ 2 ds.
(2.21)

The following lemma provides an appropriate upper bound for λ 1 (t, h).

Lemma 2.6 There is a constant C 1 , not depending on t and h, such that

λ 1 (t, h) ≤ C 1 h 2γ . (2.22) Proof Let us set λ 1 1 (t, h) := t+h -∞ t+h 0 1 Γ(α(t + h)) - 1 Γ(α(t)) (τ -s) α(t+h)-1 + dτ 2 ds (2.23) and λ 2 1 (t, h) := t+h -∞ t+h 0 1 Γ(α(t)) (τ -s) α(t+h)-1 + -(τ -s) α(t)-1 + dτ 2 ds .
(2.24)

Then one can derive from (2.20), (2.23), (2.24) and (2.18) that

λ 1 (t, h) ≤ 2 λ 1 1 (t, h) + λ 2 1 (t, h) . (2.25)
Let us first show that one has for some constant c 0 , not depending on t and h, λ

1 1 (t, h) ≤ c 0 h 2γ . (2.26) Applying on the interval [α(t) ∧ α(t + h), α(t) ∨ α(t + h)] ⊆ [α inf , α sup ] ⊂ (0, 1/2)
the mean value theorem to the infinitely differentiable positive function x → 1/Γ(x), and using the inequality (2.15), one obtains that

1 Γ(α(t + h)) - 1 Γ(α(t)) ≤ c 1 h γ , (2.27)
where the positive constant c 1 does not depend on t and h. Then combining (2.23) and (2.27) with inequality α(t + h) ≥ α inf one gets that

λ 1 1 (t, h) ≤ c 2 1 h 2γ t+h -∞ t+h 0 (τ -s) α(t+h)-1 + dτ 2 ds ≤ c 2 1 h 2γ α(t + h) 2 t+h -∞ (t + h -s) α(t+h) -(-s) α(t+h) + 2 ds ≤ c 2 1 h 2γ α 2 inf +∞ 0 s α(t+h) -(s -t -h) α(t+h) + 2
ds.

(2.28)

Next, one studies two cases t + h ≥ 0 and t + h < 0. In the case where t + h ≥ 0 one has

+∞ 0 s α(t+h) -(s -t -h) α(t+h) + 2 ds = t+h 0 s 2α(t+h) ds + +∞ 0 (s + t + h) α(t+h) -s α(t+h) 2 ds = (t + h) 2α(t+h)+1 2α(t + h) + 1 + (t + h) 2α(t+h)+1 +∞ 0 (s + 1) α(t+h) -s α(t+h) 2 ds.
(2.29)

Next, observe that, for each fixed real number s ≥ 1, by applying on the interval [s, s + 1] the mean value theorem to the infinitely differentiable function x → x α(t+h) , one obtains that (s + 1) α(t+h) -s α(t+h) ≤ s αsup-1 , for all s ≥ 1.

Therefore, one has that

+∞ 0 (s + 1) α(t+h) -s α(t+h) 2 ds ≤ 1 0 ds + +∞ 1 s 2αsup-2 ds = c 2 . (2.30)
Next, one denotes by c 0 the constant defined as:

c 0 := c 2 1 α 2 inf (|t 0 | + η) 2α inf +1 + (|t 0 | + η) 2αsup+1 1 2α inf + 1 + c 2 .
Then, using (2.28), (2.29), (2.30) and the fact t

+ h ∈ I(t 0 , η/2) := [t 0 -η/2, t 0 + η/2],
s α(t+h) -(s -t -h) α(t+h) + 2 ds = (-(t + h)) 2α(t+h)+1 +∞ 0 (s + 1) α(t+h) -s α(t+h) 2 ds ≤ c 2 (|t 0 | + η) 2α inf +1 + (|t 0 | + η) 2αsup+1 (
λ 2 1 (t, h) ≤ c 2 4 |α(t + h) -α(t)| 2 Γ 2 inf t+h -∞ t+h 0 (τ -s) α inf -1-ε + + (τ -s) αsup-1+ε + dτ 2 ds ≤ c 2 4 k 2 α (α inf -ε) 2 Γ 2 inf +∞ 0 s α inf -ε -(s -t -h) α inf -ε + + s αsup+ε -(s -t -h) αsup+ε + 2 ds h 2γ .
(2.35)

In the case where t + h ≥ 0, one has

+∞ 0 s α inf -ε -(s -t -h) α inf -ε + + s αsup+ε -(s -t -h) αsup+ε + 2 ds = t+h 0 s α inf -ε + s αsup+ε 2 ds + +∞ 0 (s + t + h) α inf -ε -s α inf -ε + (s + t + h) αsup+ε -s αsup+ε 2 ds ≤ 2 |t0|+η 0 s 2(α inf -ε) + s 2(αsup+ε) ds + 2 +∞ 0 (s + t + h) α inf -ε -s α inf -ε 2 ds + 2 +∞ 0 (s + t + h) αsup+ε -s αsup+ε 2 ds ≤ 4 1 + |t 0 | + η 2 + 2(t + h) 2(α inf -ε)+1 +∞ 0 (s + 1) α inf -ε -s α inf -ε 2 ds + 2(t + h) 2(αsup+ε)+1 +∞ 0 (s + 1) αsup+ε -s αsup+ε 2 ds ≤ c 5 , (2.36)
where c 5 is the finite constant, not depending on t and h, defined as:

c 5 := 4 1 + |t 0 | + η 2 1 + +∞ 0 (s + 1) α inf -ε -s α inf -ε 2 ds + +∞ 0 (s + 1) αsup+ε -s αsup+ε 2 ds .
In the case where t + h < 0, one has

+∞ 0 s α inf -ε -(s -t -h) α inf -ε + + s αsup+ε -(s -t -h) αsup-ε + 2 ds ≤ 2(-(t + h)) 2(α inf -ε)+1 +∞ 0 (s + 1) α inf -ε -s α inf -ε 2 ds + 2(-(t + h)) 2(αsup+ε)+1 +∞ 0 (s + 1) αsup+ε -s αsup+ε 2 ds ≤ c 5 .
(2.37) Let us now focus on λ 2 (t, h) defined in (2.21). Using the inequality

(u + v + w) 2 ≤ 3(u 2 + v 2 + w 2 ), for all (u, v, w) ∈ R 3 , (2.38) one has that λ 2 (t, h) ≤ 3λ 1 2 (t, h) + 3 Γ -2 inf λ 2 2 (t, h) + 3 Γ -2 inf λ 3 2 (t, h), (2.39)
where

λ 1 2 (t, h) := R t+h t 1 Γ(α(t)) - 1 Γ(α(τ )) (τ -s) α(t)-1 + dτ 2 ds , (2.40) λ 2 2 (t, h) := R t+h t (τ -s) α(t)-1 + -(τ -s) α(τ )-1 + e H-(s,τ ) dτ 2 ds (2.41) and λ 3 2 (t, h) := R t+h t (τ -s) α(t)-1 + e H-(s,τ ) -1 dτ 2 ds . (2.42)
In view of (2.39), our next goal is to obtain three lemmas which will allow us to conveniently bound from above λ 1 2 (t, h), λ 2 2 (t, h) and λ 3 2 (t, h). Lemma 2.7 There is a constant C 1 2 , not depending on t and h, such that

λ 1 2 (t, h) ≤ C 1 2 h 2(γ+α inf )+1 .
Proof Similarly to (2.27), it can be shown that there is a constant c 1 , not depending on t, h and τ , such that, for all τ ∈ [t, t + h], one has

1 Γ(α(t)) - 1 Γ(α(τ )) ≤ c 1 |t -τ | γ ≤ c 1 h γ . ( 2
α(t)-1 + dτ 2 ds = 1 α(t) 2 +∞ 0 (s + h) α(t) -s α(t) 2 ds + h 0 s 2α(t) ds ≤ h 2α inf +1 α 2 inf +∞ 0 (s + 1) α(t) -s α(t) 2 ds + 1 2α inf + 1 ≤ h 2α inf +1 α 2 inf c 2 + 1 2α inf + 1 , ( 2
(τ -s) α(t)-1 -(τ -s) α(τ )-1 ≤ c 1 h γ (τ -s) α(t0)-1-2ε + (τ -s) α(t0)-1+2ε , (2.46)
where the constant c 1 does not depend on t and h. Next, one sets

K -(t 0 ) := sup |H -(a, b)| , (a, b) ∈ I(t 0 , δ) and a < b .
(2.47)

Observe that K -(t 0 ) is a finite constant. Indeed, one can derive from (1.14) and (2.10) that, for all (a, b) ∈ I(t 0 , δ) satisfying a < b, one has

|H -(a, b)| ≤ b a |α(u) -α(b)| b -u du ≤ k α b a du (b -u) 1-γ = k α 2δ 0 dv v 1-γ < +∞.
It results from (2.41), (2.46), (2.47), the inequalities (τ -s) + ≤ 4η ≤ 1, for all (τ, s) ∈ I(t 0 , 2η) 2 ⊂ I(t 0 , δ) 2 (see (2.12)), and the inequality

τ -s ≥ η/2 , for all (τ, s) ∈ R 2 s.t. τ ∈ [t, t + h] ⊆ I(t 0 , η/2) and s ≤ t + h -3η/2 , (2.48) that λ 2 2 (t, h) ≤ c 2 1 h 2γ t+h -∞ t+h t (τ -s) α(t0)-1-2ε + + (τ -s) α(t0)-1+2ε + e H-(s,τ ) dτ 2 ds ≤ c 2 1 4e 2K-(t0) t+h t+h-3η/2 t+h t (τ -s) α(t0)-1-2ε + dτ 2 ds + 1 + (2/η) 4ε 2 t+h-3η/2 -∞ t+h t (τ -s) α(t0)-1+2ε e H-(s,τ ) dτ 2 ds h 2γ .
(2.49)

Let us now prove that each one of the two integrals in the right-hand side of the last inequality can be bounded from above by a finite constant not depending on t and h. In view of (2.11) one has

t+h t+h-3η/2 t+h t (τ -s) α(t0)-1-2ε + dτ 2 ds ≤ 1 (α inf -2ε) 2 t+h t+h-3η/2 (t + h -s) α(t0)-2ε -(t -s) α(t0)-2ε + 2 ds ≤ 1 (α inf -2ε) 2 t+h t+h-3η/2 (t + h -s) 2α(t0)-4ε ds = 1 (α inf -2ε) 2 3η/2 0 z 2α(t0)-4ε dz := c 2 .
(2.50)

On the other hand, one can derive from (2.48), Lemma 2.3 (with ν = η/2) and (2.14) that

t+h-3η/2 -∞ t+h t (τ -s) α(t0)-1+2ε e H-(s,τ ) dτ 2 ds = t+h-3η/2 -∞ t+h t (τ -s) α(t0)-α(τ ) (τ -s) α(τ )-1+2ε e H-(s,τ ) dτ 2 ds ≤ C 2 t+h-3η/2 -∞ t+h t (τ -s) αsup-1+4ε + (τ -s) αsup-1+2ε dτ 2 ds ≤ c 3 , (2.51)
where C is the same finite constant as in (2.8), and c 3 is the finite constant not depending on t and h defined as:

c 3 := C α sup 2 +∞ 0 (s + 1) αsup+4ε -s αsup+4ε + (s + 1) αsup+2ε -s αsup+2ε 2 ds.
Finally, putting together (2.49), (2.50) and (2.51) one obtains the lemma.

Lemma 2.9 There is a constant C 3 2 , not depending on t and h, such that

λ 3 2 (t, h) ≤ C 3 2 h 2(α(t0)+1/2+ε) .
Proof One can derive from (2.42) that

λ 3 2 (t, h) = µ 1 (t, h) + µ 2 (t, h) + µ 3 (t, h), (2.52)
where

µ 1 (t, h) := t-δ/2 -∞ t+h t (τ -s) α(t)-1 e H-(s,τ ) -1 dτ 2 ds , (2.53) µ 2 (t, h) := t-h 2ε t-δ/2 t+h t (τ -s) α(t)-1 e H-(s,τ ) -
µ 1 (t, h) ≤ 2 t-δ/2 -∞ t+h t (τ -s) α(t)-1 e H-(s,τ ) dτ 2 ds + 2 t-δ/2 -∞ t+h t (τ -s) α(t)-1 dτ 2 ds ≤ 2C 2 (2/δ) 4ε t-δ/2 -∞ t+h t (τ -s) αsup+2ε-1 dτ 2 ds + 2 t-δ/2 -∞ t+h t (τ -s) α(t)-1 dτ 2 ds ≤ 2C 2 (2/δ) 4ε (α sup + 2ε) 2 +∞ δ/2 (s + h) αsup+2ε -s αsup+2ε 2 ds + 2 α(t) 2 +∞ δ/2 (s + h) α(t) -s α(t) 2 ds ≤ 2C 2 (2/δ) 4ε +∞ δ/2 s 2αsup+4ε-2 ds + 2 +∞ δ/2 s 2αsup-2 ds h 2 .
(2.56)

Next, observe that it follows from (2.12) and the inclusion [t, t + h] ⊆ I(t 0 , η/2) that [t -δ/2, t -h 2ε ] ⊂ I(t 0 , δ). Thus, using (2.54), (2.47), the mean value theorem and (2.11), one gets that

µ 2 (t, h) ≤ (e K-(t0) + 1) 2 t-h 2ε t-δ/2 t+h t (τ -s) α(t)-1 dτ 2 ds ≤ (e K-(t0) + 1) 2 α(t) 2 δ/2 h 2ε (s + h) α(t) -s α(t) 2 ds ≤ e K-(t0) + 1 2 h 2 δ/2 h 2ε s 2α(t)-2 ds ≤ e K-(t0) + 1 2 1 -2α sup h 2-2ε ≤ e K-(t0) + 1 2 1 -2α sup h 2(α(t0)+1/2+ε) .
(2.57)

In order to bound from above µ 3 (t, h), one denotes by E -(t 0 ) the finite constant defined as:

E -(t 0 ) := sup e x -1 x , x ∈ R and 0 < |x| ≤ K -(t 0 ) . ( 2
∈ [t -h 2ε , t + h] × [t, t + h] satisfying s < τ , one has e H-(s,τ ) -1 ≤ E -(t 0 ) |H -(s, τ )| ≤ E -(t 0 ) k α τ s (τ -u) γ-1 du = E -(t 0 ) k α γ (τ -s) γ .
(2.59) Thus, one can derive from (2.55) and (2.59) that

µ 3 (t, h) ≤ E -(t 0 ) k α γ(γ + α inf ) 2 t+h t-h 2ε (t + h -s) γ+α(t) -(t -s) γ+α(t) + 2 ds .
(2.60) Moreover, standard computations, the mean value theorem, the inequality γ > 1/2 and the inequalities 0 < α inf ≤ α sup allow to show that

t+h t-h 2ε (t + h -s) γ+α(t) -(t -s) γ+α(t) + 2 ds ≤ h 2ε 0 (h + s) γ+α(t) -s γ+α(t) 2 ds + h 0 s 2(γ+α(t)) ds ≤ h 2 (γ + α(t)) 2 h 2ε 0 (s + 1) 2(γ+α(t)-1) + s -2(1-γ-α(t)) ds + h 2(γ+α(t))+1 ≤ h 2 (γ + α sup + 1) 2 1 0 (s + 1) 2(γ+αsup-1) + s -2(1-γ-α inf ) ds + 1 . (2.61)
Then (2.60) and (2.61) entail that, for some constant c 1 not depending on t and h, one has

µ 3 (t, h) ≤ c 1 h 2 . (2.62) Finally, putting together (2.52), (2.56), (2.57), (2.62) and (2.11)
one obtains the lemma.

We are now in a position to prove the inequality (2.17).

Remark 2.10 Combining (2.39) and (2.11) with Lemmas 2.7, 2.8 and 2.9 one obtains that

λ 2 (t, h) ≤ C 2 h 2(α(t0)+1/2+ε) , (2.63)
where the constant

C 2 := 3(C 1 2 +Γ -2 inf C 2 2 +Γ -2 inf C 3 2
). Thus, it results from (2.19), Lemma 2.6, (2.63) and (2.11) that

σ 2 R (t, h) ≤ 2(C 1 + C 2 )h 2(α(t0)+1/2+ε
) , which shows that (2.17) is satisfied.

3 Proof of Theorem 1.2 Remark 3.1 By arguing as in Remark 2.2 it turns out that for proving Theorem 1.2 it is enough to show that, for all fixed t 0 ∈ R, there are 3 constants ε t0 > 0, η t0 > 0 and c t0 ≥ 0, which may depend on t 0 , such that

E|D(t ) -D(t )| 2 ≤ c t0 |t -t | 2(α(t0)+1/2+3εt 0 )
, for all t , t ∈ I(t 0 , η t0 /2).

(3.1) Remark 3.2 In all the sequel, one assumes that t 0 ∈ R is arbitrary and fixed and that δ ∈ (0, 1/2] is the same as in (2.10). Also, one assumes that the two arbitrary and fixed positive real numbers ε and η are small enough so that they satisfy the 3 conditions (2.11), (2.12) and (2.14).

Remark 3.3

In all the sequel one denotes by t and t + h, where h ∈ (0, 1], two arbitrary real numbers belonging to the interval I(t 0 , η/2), and one sets

σ 2 D (t, h) := E|D(t + h) -D(t)| 2 . (3.2)
It can easily be seen that for proving (3.1), it is enough to show that there exists a positive finite constant C 0 not depending on t and h such that one has

σ 2 D (t, h) ≤ C 0 h 2(α(t0)+1/2+ε) . (3.3)
Using (3.2), (1.16), (1.13), (1.10), the isometry property of Wiener integral, and (2.18) one gets that

σ 2 D (t, h) ≤ 2Λ 0 (t, h) + 2Λ 1 (t, h) + Λ 2 (t, h) + Λ 3 (t, h) , (3.4)
where

Λ 0 (t, h) := t-η -∞ 1 Γ(1 + α(t + h)) (t + h -s) α(t+h) -(-s) α(t+h) + (3.5)
- 1 Γ(1 + α(t)) (t -s) α(t) -(-s) α(t) + 2 ds , Λ 1 (t, h) := t-η -∞ 1 Γ(1 + α(s)) (t + h -s) α(s) e -H+(s,t+h) - 1 Γ(1 + α(s)) (t -s) α(s) e -H+(s,t) 2 ds , (3.6)
Λ 2 (t, h) := t t-η 1 Γ(1 + α(s)) (t + h -s) α(s) e -H+(s,t+h) - 1 Γ(1 + α(s)) (t -s) α(s) e -H+(s,t) (3.7)
- 1 Γ(1 + α(t + h)) (t + h -s) α(t+h) + 1 Γ(1 + α(t)) (t -s) α(t) 2 ds and Λ 3 (t, h) := t+h t 1 Γ(1 + α(s)) (t + h -s) α(s) e -H+(s,t+h) - 1 Γ(1 + α(t + h)) (t + h -s) α(t+h) 2 ds. (3.8)
The following lemma provides an appropriate upper bound for Λ 0 (t, h).

Lemma 3.4 There is a constant C 0 , not depending on t and h, such that

Λ 0 (t, h) ≤ C 0 h 2γ . (3.9)
Proof It follows from (3.5), (2.20), (2.18), (2.34) and the equality Γ(x + 1) = xΓ(x), for all x ∈ (0, +∞), that

Λ 0 (t, h) ≤ 2λ 1 (t, h) + 2Γ -2 inf Λ 0 (t, h) , (3.10)
where

Λ 0 (t, h) := t-η -∞ (t + h -s) α(t) -(t -s) α(t) 2 ds .
Moreover, using the mean value theorem, one has that

Λ 0 (t, h) ≤ c h 2 , (3.11)
where the finite constant c := +∞ η s 2(αsup-1) + s 2(α inf -1) ds. Finally, putting together (3.10), (2.22) and (3.11), one gets (3.9).

The following lemma provides an appropriate upper bound for Λ 1 (t, h). Lemma 3.5 There is a constant C 1 , not depending on t and h, such that

Λ 1 (t, h) ≤ C 1 h 2 .
(3.12)

Proof One can derive from (3.6), (2.34) and (2.18) that

Λ 1 (t, h) ≤ 2 Γ -2 inf Λ 1 1 (t, h) + Λ 2 1 (t, h) , (3.13)
where Λ Let us first focus on Λ 1 1 (t, h). It can easily be seen that e -H+(-s,t+h) -e -H+(-s,t) = e -H+(-s,t+h) 1 -e H+(-s,t+h)-H+(- (t + h + s) 2α(-s) e -H+(-s,t+h) -e -H+(-s,t) 2 ds

≤ c 1 h 2 +∞ -t+η
(t + h + s) 2α(-s) e -2H+(-s,t+h) (s + t) 2 ds.

Thus, using (2.9), in which t and s are replaced by t + h and -s, and the fact that h ∈ (0, 1], one gets that Λ (t + h -s) α(s) e -H+(s,t+h) -1 Γ(1 + α(s)) (t -s) α(s) e -H+(s,t)

(3.22) - 1 Γ(1 + α(t + h)) (t + h -s) α(t+h) + 1 Γ(1 + α(t)) (t -s) α(t) 2 ds and Λ 2 2 (t, h) := t t-ηh 1/2 1 Γ(1 + α(s)) (t + h -s) α(s) e -H+(s,t+h) - 1 Γ(1 + α(s)) (t -s) α(s) e -H+(s,t) (3.23) - 1 Γ(1 + α(t + h)) (t + h -s) α(t+h) + 1 Γ(1 + α(t)) (t -s) α(t) 2 ds.
The following lemma provides an appropriate upper bound for Λ 1 2 (t, h) defined in (3.22).

Λ 1 2 (t, h) ≤ C 1 2 h 2γ + h 2(α(t0)+1/2)+7ε . (3.24)
Proof It follows from (3.22), (2.38), (2.34) and the equality Γ(x + 1) = xΓ(x), for all x ∈ (0, +∞), that

Λ 1 2 (t, h) ≤ 3 Γ -2 inf Λ 1,1 2 (t, h) + Γ -2 inf Λ 1,2 2 (t, h) + Λ 1,3 2 (t, h) , (3.25)
where Λ

:= t-ηh 1/2 t-η 1 Γ(1 + α(t + h)) (t + h -s) α(t+h) - 1 Γ(1 + α(t)) (t -s) α(t) 2 ds. (3.28)
Let us first focus on Λ 1,1 2 (t, h). One clearly has that e -H+(s,t+h) -e -H+(s,t) 2 = e -2H+(s,t+h) 1 -e H+(s,t+h)-H+(

(s, t + h) -H + (s, t)| ≤ t+h t |α(s) -α(v)| v -s dv ≤ k α γ -1 (t + h -s) γ -(t -s) γ ≤ k α η γ-1 h (1+γ)/2 (3.30) and |H + (s, t + h)| ≤ t+h s |α(s) -α(v)| v -s dv ≤ k α t+h s (v -s) γ-1 dv = k α γ -1 (t + h -s) γ ≤ k α γ -1 (η + 1) γ . ( 3
Λ 1,3 2 (t, h) ≤ 3 t-ηh 1/2 t-η 1 Γ(1 + α(t + h)) - 1 Γ(1 + α(t)) 2 (t + h -s) 2α(t+h) ds + 3 Γ -2 inf t-ηh 1/2 t-η (t + h -s) α(t+h) -(t -s) α(t+h) 2 ds + 3 Γ -2 inf t-ηh 1/2 t-η (t -s) α(t+h) -(t -s) α(t) 2 ds . (3.34)
Rather similarly to (2.26), it can be shown that, for some constant c 3 not depending on t and h, one has

t-ηh 1/2 t-η 1 Γ(1 + α(t + h)) - 1 Γ(1 + α(t)) 2 (t + h -s) 2α(t+h) ds ≤ c 3 h 2γ .
(3.35) Rather similarly to (3.33), it can be shown that for some constant c 4 not depending on t and h, one has

t-ηh 1/2 t-η (t + h -s) α(t+h) -(t -s) α(t+h) 2 ds ≤ c 4 h 2(α(t0
2 2 (t, h) ≤ 2 Λ 2,0 2 (t, h) + Λ 2,1 2 (t, h) , (3.39)
where, for j = 0 or j = 1,

Λ 2,j 2 (t, h) := t t-ηh 1/2 1 Γ(1 + α(t + jh)) (t + jh -s) α(t+jh) (3.40) - 1 Γ(1 + α(s)) (t + jh -s) α(s) e -H+(s,t+jh) 2 ds .
In view of (3.39), in order to show that (3.38) is satisfied, it is enough to prove that, for each j ∈ {0, 1}, the following inequality, in which c denotes a (finite) constant not depending on t and h, holds We are now in a position to prove the inequality (3.3).

Λ 2,j 2 (t, h) ≤ c h 2(α(t0)+1/2)+6ε . ( 3
2,j 2 (t, h) ≤ 3 ϕ 1,j (t, h) + Γ -2 inf ϕ 2,j (t, h) + Γ -2 inf ϕ 3,j (t, h) , (3.42)
Remark 3.9 Putting together (3.4), (3.9), (3.12), (3.21), (3.24), (3.38), (3.50), the fact that h ∈ (0, 1], the fact that γ ∈ (1/2, 1) and (2.11), it follows that (3.3) is satisfied.

  d

1

  Introduction and statement of the main resultsLet H ∈ (0, 1), the Fractional Brownian Field (FBF) of Hurst parameter H, which is also called multivariate Fractional Brownian Motion, is a real-valued 1 centred continuous Gaussian field on R N denoted by {B H (t)} t∈R N having, for all t , t ∈ R N , the covariance:Cov(B H (t ), B H (t )) = E(B H (t )B H (t )) = c(H) |t | 2H + |t | 2H -|t -t | 2H ,where c(H) is a positive constant only depending on H, and | • | is the Euclidian norm. One refers to e.g. Chapter 1 of the book[START_REF] Ayache | Multifractional stochastic fields : wavelet strategies in multifractional frameworks[END_REF] for a detailed presentation FBF. Notice that, up to a multiplicative constant, {B H (t)} t∈R N is in distribution the unique Gaussian field which satisfies the following three fundamental properties: self-similaritity that is for all fixed positive real number a one has {B H (at)} t∈R N d = {a H B H (t)} t∈R N ,where the symbol d = means equality of finite-dimensional distributions; stationarity of increments, that is for each fixed t ∈ R N one has {B H (t + t) -B H (t)} t∈R N d = {B H (t)} t∈R N ; and isotropy, that is for every fixed orthogonal matrix Q of size N one has {B H (Qt)} t∈R N

  it follows that (2.26) is satisfied. Let us now turn to the case where t + h < 0. In this case one has

	+∞
	0

  .45) where the constant c 2 , which does not depend on t and h, is the same constant as in (2.30). Finally combining (2.44) and (2.45) one obtains the lemma.

	Lemma 2.8 There is a constant C 2 2 , not depending on t and h, such that
	λ 2 2 (t, h) ≤ C 2 2 h 2γ .
	Proof First observe that using the mean value theorem, (2.14), (2.33) and
	(2.15) , one has for all τ ∈ [t, t + h] and s < τ

  1 dτ

	and
	µ 3 (t, h) :=	t+h t-h 2ε	t	t+h	(τ -s)	α(t)-1 +	e H-(s,τ ) -1 dτ	2	ds .	(2.55)
	Observe that one knows from (2.13) and the inequalities 0 < h ≤ η, that
	t -h 2ε > t -δ/2. Using (2.53), (2.18), Lemma 2.3 with ν = δ/2, (2.14), the
	mean value theorem and (2.11), one obtains that
								2
									ds ,	(2.54)

  (t, h) defined in (3.27), one can derive from (3.31), the mean value theorem, the inclusion [t -η, t -ηh 1/2 ] ⊂ I(t 0 , 2η), (2.14) and (2.11) that

	As regards Λ 1,2 2 Λ 1,2 2 (t, h) ≤ c 2 h 2	t-ηh 1/2	(t -s) 2(α(t0)-ε-1) ds ≤ c 2 h 2(α(t0)+1/2)+7ε , (3.33)
		t-η
	where c 2 and c 2 are two (finite) constants not depending on t and h. As regards Λ 1,3 2 (t, h) defined in (3.28), one can derive from (2.38), and (2.34)
	that
				.31)
	Thus, putting together (3.26), (3.29), (3.30), (3.17), (3.31) and α(s) ∈ [α inf , α sup ],
	one gets, for some (finite) constant c 1 not depending on t and h, that
			Λ 1,1 2 (t, h) ≤ c 1 h 1+γ .	(3.32)

  ≤ c 1 h 1/2(1+2γ+2α(t0)-2ε) ≤ c 1 h 2(α(t0)+1/2)+7ε , where c 1 and c 1 are two (finite) constants not depending on t and h. Next, observe that, similarly to(3.31), it can be shown that there exists a (finite) constant c 2 , not depending on t, h and s, such that, for all s ∈ [t -ηh 1/2 , t], one has|H + (s, t + jh)| ≤ c 2 h γ/2 .(t + jh -s) 2α(t+jh) ds ≤ c 3 h 2(α(t0)+1/2)+7ε ,(3.48) where c 3 and c 3 are two (finite) constants not depending on t and h. Next, using (3.45), (3.47), the mean value theorem, (2.15), (2.33), (2.14) and (2.11), one gets thatϕ 3,j (t, h) ≤ c 4 h γ ≤ c 4 h 1/2(1+2γ+2α(t0)-4ε) ≤ c 4 h 2(α(t0)+1/2)+6ε ,where c 4 and c 4 are two (finite) constants not depending on t and h. Finally, putting together (3.42), (3.46), (3.48) and (3.49), one obtains (3.41).The following lemma provides an appropriate upper bound for Λ 3 (t, h) defined in (3.8). There is a constant C 3 , not depending on t and h, such thatΛ 3 (t, h) ≤ C 3 h 2+14ε .(3.50) -s) 2α(s+t) e -2H+(s+t,t+h) ds ,Observe that, similarly to (3.31), it can be shown that there exists a (finite) constant c 1 , not depending on t, h and s, such that, for all s ∈ [0, h], one has|H + (s + t, t + h)| ≤ c 1 h γ . (3.55) One can derive from (3.52), (3.55), the mean value theorem and (2.15) that -s) 2α(s+t) ds ≤ c 2 h 2(γ+α inf )+1 , (3.56) where c 2 is a (finite) constant not depending on t and h. Moreover, it follows from (3.53), (3.55) and (3.17) that -s) 2α(t+s) ds ≤ c 3 h 2(γ+α inf )+1 , (3.57) where c 3 is a (finite) constant not depending on t and h. Furthermore, using (3.54), the mean value theorem, (2.15), (2.33) and (2.11) one has thatΛ 3 3 (t, h) ≤ c 4 h 2γ h 0 s 2(α inf -ε) ds ≤ c 4 h 2(γ+α inf -ε)+1 , (3.58) where c 4 is a (finite) constant not depending on t and h. Finally, putting together (3.51), (3.56), (3.57), (3.58), the inequality 2γ > 1 and the inequality α inf > 8ε (see (2.11)), one obtains (3.50).

						h
	(3.47) Thus, one can derive from (3.44), (3.47), (3.17), (2.12), (2.14) and (2.11) that ϕ 2,j (t, h) ≤ c 3 h γ t t-ηh 1/2 Λ 1 3 (t, h) ≤ c 2 h 2γ 0 h (h Λ 2 3 (t, h) ≤ c 3 h 2γ (h
						0
						t
							(t + jh -s) 2α(t0)-4ε ds	(3.49)
						t-ηh 1/2
	t t-ηh 1/2 Lemma 3.8 Proof It follows from (3.8), (2.38) and (2.34) that 1 Γ(1 + α(t + jh)) -1 Γ(1 + α(s))	2	(t + jh -s) 2α(t+jh) ds , (3.43)
	ϕ 2,j (t, h) :=	t Λ 3 (t, h) ≤ 3 Λ 1 (t + jh -s) 2α(t+jh) 1 -e -H+(s,t+jh) 2 ds 3 (t, h) + Γ -2 inf Λ 2 3 (t, h) + Γ -2 inf Λ 3 3 (t, h) ,	(3.44) (3.51)
		t-ηh 1/2
	where and ϕ 3,j (t, h) := Λ 1 3 (t, h) := 0	h	t	1 Γ(1 + α(s + t))	-	1 Γ(1 + α(t + h))
			t-ηh 1/2			(3.52)
							(3.45)
	It results from (3.43), the mean value theorem, (2.15), (2.12), (2.14) and (2.11) Λ 2 3 (t, h) := 2 ds (3.53)
	that and
	ϕ 1,j (t, h) ≤ c 1 Λ 3 3 (t, h) :=	0	t t-ηh 1/2 h (h -s) α(t+s) -(h -s) α(t+h) 2 (t + jh -s) 2γ+2α(t+jh) ds	ds .	(3.46) (3.54)

where

ϕ 1,j (t, h) := (t + jh -s) α(t+h) -(t + jh -s) α(s) 2

e -2H+(s,t+jh) ds .

2

(h h 0 (h -s) 2α(t+s) e -H+(s+t,t+h) -1

For the sake of simplicity, we can make this continuity assumption since we already know that the paths of the Gaussian processes {R(t)} t∈R and {D(t)} t∈R , defined through (1.15) and (1.16), are continuous functions and we are interested in their local Hölder regularity. Notice that when the assumption of continuity of the paths of {Z(t)} t∈R is dropped then (2.1) holds for a well-chosen modification { Z(t)} t∈I of {Z(t)} t∈I .