The soft-demapper block in Figure 4.3 is responsible for calculating from the symbols z[n] the soft information corresponding to each coded bit q of the joint symbol ς[n], meaning, the set{LLRq}lq=0. The LLRs are defined as the logarithm of the ratio of the probabilities
of the qth bit assuming its two legitimate values conditioned to z[n]. According to [133],
the LLRs can be defined as follows
LLRq= log4
P (q = 0|z[n])
P (q = 1|z[n]) (4.1)
where P (q|z[n]) is the conditional probability density function (pdf). Particularizing for the joint constellation M each LLR can be written as follows
LLRq = log P ς∈M(0)q p(z[n]|ς) P ς∈M(1)q p(z[n]|ς) , (4.2)
where M(b)q is the subset of M with the qth bit having the value of b = 0 or b = 1.
By using Bayes’ rule we can obtain the conditional pdf p(z[n]|ς) from P (q|z[n]), which obeys the next Gaussian distribution
p(z[n]|ς) = 1 2πIexp −|z[n] − ς|2 2I , (4.3)
where I is the interference plus noise variance. As well, we use the max-log approximation formulated as
ln(eA+ eB) = max(A, B) + ln(1 + e(−|A−B|))≈ max(A, B), (4.4) explained in [132] for BCJR algorithm in logarithm domain, for the sake of converting multiplications to additions. The losses stemming from max-log approximation are ana- lyzed in [134]. With all this we can derive the LLR for bit q as follows
LLRq = log max ς∈M(0)q p(z[n]|ς) max ς∈M(1)q p(z[n]|ς) =− 1 I ςmin∈M(0) q |z[n] − ς|2− min ς∈M(1)q |z[n] − ς|2 ! . (4.5)
We can observe that this metric is the same as for a coherent system with the advantage that channel information is not necessary because of the scaling laws of m-MIMO. where the channel effects have disappeared due to (2.18) (2.20) and (3.2).
Once we have the set of l LLRs values in LM for each joint symbol ς, then the LLRs
of each user L(u0
l/J consecutive values for each user. This way of separating the individual symbols from the joint symbols is due to the labelling used in the mapping of the joint symbol in connection with each individual user’s labelling at the transmitter. Remember from Section 2.4.2 that the joint labelling is formed by concatenating the individual labelling (an example for building the joint constellation was shown in that section).
From the point of view of the soft information, if we have J=3 users and M =4, where user 1 transmits the pair of consecutive bits{u 1(1), u
(2)
1 }, the user 2 transmits {u
(1)
2 , u
(2)
2 }
and the user 3 transmits {u 3(1), u 3(2)}, then the labelling of the joint symbol is the serial concatenation formulated as {u 1(1), u (2) 1 , u (1) 2 , u (2) 2 , u (1) 3 , u (2) 3 }.
Then, when we have the soft-values LM = “ LLR1, LLR2
user1 , LLR3, LLR4 user2 , LLR5, LLR6 user3 ”, the separation for each individual user gives L(u 1) = ”LLR1, LLR2”, L(u
2) = “LLR3, LLR4”
and L(u 3) = ”LLR5, LLR6”.
In order to employ the EXIT chart the pdf of the a priori LLR values has to be Gaussian, which depends on the interleaver length (Section 4.4.1). We simulate the system shown in Figure 4.2 and Figure 4.3. At each iteration, we plot the histogram of the LLRs for checking their distribution. As an example, we plot the distribution for an iteration in the decoding procedure in Figure 4.4, where we can see that the distribution for the a priori LLRs is Gaussian with zero mean.
4.4. New approach of EXIT chart tool in massive MIMO 83
4.4
New approach of EXIT chart tool in massive
MIMO
The EXtrinsic Information Transfer (EXIT) chart concept was first proposed by Stephan ten Brink in [135], which constitutes a powerful tool for designing and analyzing iteratively decoded systems. A comprehensive tutorial is provided in [133] on the design of systems based on EXIT-curves. However, in this work we present a new approach of EXIT tool focused for m-MIMO.
The purpose of using the EXIT charts is to predict the convergence behavior of the iterative decoder, composed of both parallel or serially concatenated encoders. This is achieved by examining the evolution of the input/output mutual information (MI) ex- change between the constituent decoders in consecutive iterations. The average MI rep- resents the average amount of source information X acquired per received symbol Y , defined by information theory in [136] as
I(X, Y ) =X i,j P (Xi, Yj)· log2 P (Xi/Yj) P (Xi) bits/symbol, (4.6)
where P (Xi, Yj) is the joint probability that Xi was transmitted and Yj was received.
P (Xi) is the probability of transmitting Xi and P (Xi/Yj) is the conditional probability
that the Yj was received given that Xi was transmitted. In order to plot the EXIT chart,
we have to derive the MI between the bit stream and its corresponding L soft-values. It was shown in [137] that the MI between the general equiprobable bits X and their respective LLRs L always obeys
I(X, L) = 1− Z +∞ −∞ p(L|X = +1) · log2[1 + e−L]dL = 1− E{X = +1}log2[1 + e−L], (4.7)
where E{X} is the expected value of X. Henceforth, the bit stream X is associated with the coded bits c or u in Figure 4.2, depending on whether we are talking about outer or inner encoder, respectively. The L-values have to obey a symmetric distribution of p(L|X = ±1) and to satisfy the consistency condition defined in [137]. The closer to 1 MI is, the more accurate the iterative decoding, then the lower BER is. There are two main methods to compute the MI in (4.7): the histogram based approximation [138, 139] and the method based on time averaging [137]. We employ in this work the time averaging method which has the advantage of not requiring any knowledge of the input bit sequence b.
Thus far, the contributions which use EXIT tool to design iteratively decoded systems, built the EXIT-curves based on SNR. Conversely, in this thesis we propose a new approach
to build the EXIT-curve, this is based on the number of antennas R to measure and represent the nature of a m-MIMO system.