This section is organized as follows. In Section 3.1, the rationale underlying the LA strategy and LPP method is recalled. In Section 3.2, the concept of the κESM ESNR technique for MIMO BIC-OFDM systems with simple ARQ mechanism is briefly summarized. Finally, in Section 3.3, the novel LPP method, named αESM, is derived for HARQ-based MIMO BIC-OFDM systems with bit-level combining.
Rationale of the adaptive HARQ strategy
The approach to follow is to properly choose the parameters of the system described in Section 2.2, e.g., modulation order and coding rate, in order to obtain the best link performance. Such LA strategy can be formalized as a constrained optimization problem where the objective function, representing the system performance metric, is optimized over the constrained set of the available transmission parameters. Specifically, for a packet-oriented system, information-theoretical performance measure based on capacity, which relies on ideal assumptions of Gaussian inputs and infinite length codebooks, is inadequate to give an actual picture of the link performance [26]. More suitable metrics have been recently identified as the packet error rate (PER) and the GP [20, 21, 26], which in turn depends on the PER itself. Therefore, a simple yet effective link performance prediction method is required, accounting for both the CSI as well as the information coming from different techniques that further improve the transmission quality, i.e., the HARQ mechanisms with bit-level combining. In the sequel, we will focus on LPP techniques based on the well-known ESM concept, which has been shown to be the most effective framework to solve this issue, especially for multicarrier systems [22].
Background on the κESM LPP model
In multicarrier systems, where the frequency-selective channel introduces large SNR variations across the subcarriers and practical modulation and coding schemes are adopted, an exact yet manageable expression of the PER reveals to be demanding to derive. Due to these above reasons, ESM techniques are successfully employed, according to which the PER depends on the SNRs on each subcarrier through a scalar value, called ESNR. The latter represents the SNR of a single-carrier equivalent coded system working over AWGN channel, whose performance can be simply evaluated either off-line according to analytical models [27].
Within the ESM framwork, the κESM method, proposed for MIMO BIC-OFDM systems in [23], shows a remarkable tradeoff between accuracy and complexity when ARQ mechanisms are applied without any combining at the receiver. Such technique is based on the in-depth statistical characterization of the soft metrics at the input of the decoder, i.e., the bit LLR metrics \(\Lambda _{k}^{(\ell)}\), which, for the kth transmitted coded bit \(b_{k}^{(\ell)}\) at ℓth PR, reads as
$$ \Lambda^{(\ell)}_{k} = \log \frac { \lambda_{j}\left({b_{\Phi^{(\ell)}(j,n,\nu)}^{(\ell){\prime}}}, z^{(\ell)}_{n,\nu}\right) } { \lambda_{j}\left(b_{\Phi^{(\ell)}(j,n,\nu)}^{(\ell)}, z^{(\ell)}_{n,\nu}\right) } $$
(5)
where Φ(ℓ)(j,n,ν)=k is the mapping function defined in Section 2.2 after (1),
$$ {{\begin{aligned} \lambda_{j}\left(a,z^{(\ell)}_{n,\nu}\right) = \sum\limits_{\tilde{x} \in {\chi_{a}}^{\left(\ell,j,n,\nu\right)}} { \exp\left(-\left|z^{(\ell)}_{n,\nu} - \sqrt{\gamma^{(\ell)}_{n,\nu}} \tilde{x} \right|^2 \right) }, \;\; a\in\left\{b_{k}^{(\ell)},{b_{k}^{(\ell)}}^{\prime}\right\}, \end{aligned}}} $$
(6)
denotes the bit decoding metric, \(b_{k}^{\prime }\) is the complement of b
k
, \(\chi _{a}^{\left (\ell,j,n,\nu \right)}\) represents the subset of all the symbols belonging to the modulation adopted on the subchannel (n,ν), whose jth label bit is equal to a, whereas \(z^{(\ell)}_{n,\nu }\) is the generic entry of the vector z(ℓ) defined in (3) with \(\gamma ^{(\ell)}_{n,\nu }\) given by (4). If coded bit \(b_{k}^{(\ell)}\) is not transmitted, i.e., it is punctured at PR ℓ, note that \(\Lambda ^{(\ell)}_{k}\triangleq 0\). After a few approximations, it is shown in [23] that the PER performance of the coded MIMO BIC-OFDM system over frequency-selective channel is accurately given by that of a coded BPSK system over AWGN channel having SNR equal to the κESM ESNR
$$ \gamma^{(\ell)} \triangleq - \log \left(\frac{1}{{\sum\limits_{(n,\nu)\in{\mathcal{C}}} m_{n,\nu}^{(\ell)} }} \sum\limits_{(n,\nu)\in {\mathcal{C}}} \Omega_{n,\nu}^{(\ell)}\left(m_{n,\nu}^{(\ell)}\right) \right) $$
(7)
where
$$ \Omega_{n,\nu}^{(\ell)} \left(m_{n,\nu}^{(\ell)}\right) \triangleq {\sum\limits_{\mu = 1}^{\sqrt{2^{m_{n,\nu}^{(\ell)}-2}}} {\frac{{{\psi_{m_{n,\nu}^{(\ell)}}}(\mu)}}{{{2^{{m_{n,\nu}^{(\ell)}} - 1}}}} \cdot{{\mathrm{e}}^{- \frac{{{\gamma^{(\ell)}_{n,\nu}}{{\left({\mu \cdot d_{n,\nu}^{(\min)}} \right)}^{2}}}}{4}}}} } $$
(8)
with \(\psi _{m_{n,\nu }^{(\ell)}}\) and \(d^{(\min)}_{n,\nu }\) being the constant values depending on the modulation order adopted on subchannel (n,ν) at PR ℓ. Expression (7) comes from the CMGF \(\kappa _{\Lambda }^{(\ell)} (\hat {s})\triangleq \log \mathrm {E} \left \{ \text {e}^{\hat {s} \Lambda _{k}^{(\ell)}}\right \}\) of the bit LLR metric \(\Lambda _{k}^{(\ell)}\) given by (5) evaluated at the saddlepoint \(\hat {s} = 1/2\) and, specifically, \({\gamma ^{(\ell)} = - \kappa _{\Lambda }^{(\ell)} (\hat {s})}\) [23].
From (7)–(8), it has to be pointed out that γ(ℓ) depends on the modulation order adopted on each subchannel given Γ(ℓ).
The αESM model
In this section, we introduce the concept of aggregate ESNR mapping, or αESM for short, in order to predict the performance of the system of interest under HARQ mechanism. Specifically, by extending to the HARQ context, the method presented in [28] for the estimation of the pairwise error probability (PEP), the key idea of the αESM we will propose is built upon two concepts: (i) the decoding score, a RV whose positive tail probability yields the PEP [28], and (ii) the equivalent binary input output symmetric (BIOS) model of the BICM scheme [2] applied to the MIMO BIC-OFDM system described in Section 2.2. According to the latter, at each PR \(\ell \in \mathcal {L}_{\text {PR}}\) and for each of the \(N^{(\ell)}_{\text {OFDM}}\) symbols during such round, the MIMO BIC-OFDM channel is modeled as a set of
$$ B^{(\ell)}\triangleq\sum\limits_{(n,\nu)\in {\mathcal{C}}} m_{n,\nu}^{(\ell)} $$
(9)
parallel BIOS channels. We recall from Section 2.2 that \(B^{(\ell)}\cdot N^{(\ell)}_{\text {OFDM}}\ge N^{(\ell)}_{c} \). From now on, for the sake of simplicity but w.l.g., we assume that only one OFDM symbol is sufficient for the transmission of the \(N_{c}^{(\ell)}\)-bit-long codeword, so that the dependence on the OFDM symbol index is avoided. In particular, we have \(B^{(\ell)} = N_{c}^{(\ell)}\).
Considering that the exact estimation of the PER for the system at hand is a demanding problem, we will first evaluate the PEP expression, then resort to the standard union bound. The one-to-one mapping between the codeword and the associated vector of modulation symbols allows us to express the PEP as follows. Let \(\mathbf {c}^{(\ell)}\triangleq \left \{c_{1}^{(\ell)},\cdots,c_{N_{c}^{(\ell)}}^{(\ell)}\right \}\) be the reference codeword (corresponding to the transmitted RLC-PDU at the ℓth PR) at the output of the puncturing device and \({\mathbf {c}^{(\ell)}}{\prime }\triangleq \left \{{c_{1}^{(\ell)}}{\prime },\cdots,{c_{N_{\mathrm {c}}^{(\ell)}}^{(\ell){\prime }}}\right \}\) the competing codeword, being \(c^{(\ell)}_{i}\) the ith coded bit after puncturing. Besides, let us define Π(ℓ)(i)=k, \(i=1,\cdots,N_{c}^{(\ell)}\), \(k\in \{1,\cdots,{\bar N}_{c}\}\), as the puncturing mapping such that \(c^{(\ell)}_{i}=b^{(\ell)}_{\Pi ^{(\ell)}(i)}\), where \(b^{(\ell)}_{\Pi ^{(\ell)}(i)}\) is the kth coded bit prior to puncturing. Then, upon denoting the reference and competing codewords as \(\mathbf {b}^{(\ell)}\triangleq \left \{b_{\Pi ^{(\ell)}(1)}^{(\ell)},\cdots,b_{\Pi ^{(\ell)}\left ({N_{\mathrm {c}}^{(\ell)}}\right)}^{(\ell)}\right \}\) and \({\mathbf {b}^{(\ell)}}{\prime }\triangleq \left \{{b_{\Pi ^{(\ell)}(1)}^{(\ell){\prime }}},\cdots,{b_{\Pi ^{(\ell)}\left ({N_{\mathrm {c}}^{(\ell)}}\right)}^{(\ell){\prime }}}\right \}\), respectively, the PEP results as
$$ \text{PEP}~ \left(\mathbf{b}^{(\ell)},{\mathbf{b}^{(\ell)}}{\prime}\right) \triangleq \text{Pr} \left\{ \lambda\left({\mathbf{b}^{(\ell)}}{\prime},\mathbf{z}^{(\ell)}\right) > \lambda\left({\mathbf{b}^{(\ell)}},\mathbf{z}^{(\ell)}\right) \right\}, $$
(10)
where λ(·) is the soft decoding metric depending on the chosen decoding strategy. In the sequel, we will first recall the case where no bit combining is performed [28], and then, we will extend this approach to the bit-level combining receiver, which represents the novel contribution of the work.
No bit combining at the receiver. With reference to the equivalent BIOS model of the MIMO BIC-OFDM system as depicted in Fig. 2, the following observations hold.
-
The input to the ith BIOS channel, 1≤i≤B(ℓ), is the bit \(b_{\Pi ^{(\ell)}(i)}^{(\ell)}\), which is mapped in the jth position of the label of the QAM symbol \(x^{(\ell)}_{n,\nu }\) sent on subchannel (n,ν), being Φ(ℓ)(j,n,ν)=k.
-
The output is the bit log-likelihood metric \(\Lambda _{k}^{(\ell)}\), also named bit score, evaluated as in (5).
-
The decoder metric for the reference codeword b(ℓ) is the BICM maximum a posteriori metric results as [28]
$$ \lambda ({{b}^{(\ell)}},{{z}^{(\ell)}}) = \prod\limits_{(n,\nu) \in {\mathcal{C}}} {\prod\limits_{j = 1}^{m_{n,\nu}^{(\ell)}} {\lambda_j \left({b^{(\ell)}_{\Phi^{(\ell)}(j,n,\nu)}},{z}^{(\ell)}_{n,\nu}\right)} }, $$
(11)
where λ
j
(·,·) is the decoding metric associated to bit \(b^{(\ell)}_{\Phi ^{(\ell)}(j,n,\nu)}\), evaluated according to (6), whereas that one for the competing codeword b(ℓ)′ is obtained as in (11) by simply replacing b(ℓ) with b(ℓ)′.
Hence, the pairwise decoding score (PDS) relevant to the transmitted codeword b(ℓ) with respect to b(ℓ)′ can be written asFootnote 2
$$ \Lambda_{{\text{PW}}}^{(\ell)} \triangleq \sum\limits_{(n,\nu) \in {\mathcal{C}}} {\sum\limits_{j = 1}^{m_{n,\nu}^{(\ell)}} {{\Lambda}_{\Phi^{(\ell)} (j,n,\nu)}^{(\ell)}} }, $$
(12)
where the LLR bit metric \({{\Lambda }_{\Phi ^{(\ell)} (j,n,\nu)}^{(\ell)}}\) is defined by (5). Therefore, upon plugging (11) evaluated for both b(ℓ) and b(ℓ)′ in the PEP expression (10), after some algebra, we obtain
$$ \text{PEP}\left(\mathbf{b}^{(\ell)},\mathbf{b}^{{(\ell)}'}\right)=\text{Pr}\left(\Lambda_{{\text{PW}}}^{(\ell)}>0 \right). $$
(13)
Bit-level combining at the receiver. The optimal receiver that accounts for the combination of all the received copies should perform a joint decoding of the pairwise decoding scores over all the possible L transmissions. However, it would result in an unfeasible complexity, exponentially increasing with L [29]. On the other side, exploiting the bit-level combining offers an effective trade-off between performance and complexity [30]. Accordingly, this is the approach we will pursue in the sequel. The decoding metric in (11) shall now account for the recombination mechanism up to the PR ℓ. At every PR indeed, the actual bit scores are evaluated as in (5) and, for each bit k, added to the bit scores evaluated during the previous PRs. Thus, the output of the equivalent BIOS channel is now the aggregate bit score
$$ {\mathcal{L}}_{k}^{(\ell)} \triangleq {{q}_{k}^{(\ell)}}^{\mathrm{T}}\boldsymbol{\Lambda}_{k}^{(\ell)}, $$
(14)
where \(\boldsymbol {\Lambda }_{k}^{(\ell)} \triangleq \left [\Lambda _{k}^{(1)},\cdots,\Lambda _{k}^{(\ell)} \right ]^{\mathrm {T}}\) collects the per-round bit scores of the coded bit k up to PR ℓ and \(\mathbf {q}_{k}^{(\ell)}\triangleq \left [q_{k}^{(1)},\cdots,q_{k}^{(\ell)} \right ]^{\mathrm {T}} \in \{0,1\}^{\ell }\) is the puncturing vector, that is, \(q_{k}^{(i)}=1\) if bit k has been transmitted at round i, otherwise 0 if it has been punctured. In turn, the aggregate PDS at round ℓ is given by
$$ {\mathcal{L}}_{{\text{PW}}}^{(\ell)} = \sum\limits_{(n,\nu) \in {\mathcal{C}}} \sum\limits_{i=1}^{\ell} {\sum\limits_{j = 1}^{m_{n,\nu}^{(i)}} {q^{(i)}_{\Phi (j,n,\nu)} \Lambda^{(i)}_{\Phi (j,n,\nu)}} }. $$
(15)
Then, after some algebra, the PEP using bit-level combining at the receiver results as
$$ \text{PEP}(\mathbf{b}^{(\ell)},\mathbf{b}^{{(\ell)}{\prime}})=\text{Pr}\left({\mathcal{L}}_{{\text{PW}}}^{(\ell)}>0 \right). $$
(16)
Let us now define the CMGF of the bit score \({\mathcal {L}}_{k}^{(\ell)}\) as
$$ {\kappa_{{\mathcal{L}}}^{(\ell)}}(s) \triangleq \log \left({{\mathrm{E}}\left\{ {{\mathrm{e}^{s{\mathcal{L}}_{k}^{(\ell)}}}} \right\}} \right) $$
(17)
where the expectation is done w.r.t. all the random variables, and rely on the following assumption:
The above is motivated by the fact that at each PR, a random subset of the coded bit is selected among the ones at the input of the puncturing device. As a consequence of A2, the puncturing pattern can be designated as q(ℓ)=[q(1),⋯,q(ℓ)]T. Then, exploiting the law of total probability, from (14), the CMGF (17) turns out to be
$$ {{\begin{aligned} {\kappa_{{\mathcal{L}}}^{(\ell)}}(s) = \log \left({\sum\limits_{{\bar{\mathbf{q}}^{(\ell)}} \in {{\mathcal{Q}}^{(\ell)}}} {\Pr \left({{{q}^{(\ell)}} = {{\bar{\mathbf{q}}}^{(\ell)}}} \right){{\prod\limits_{i = 1}^\ell {\left[ {{\mathrm{E}}\left\{ {{\mathrm{e}^{s\Lambda_{k}^{(i)}}}} \right\}} \right]^{{{\bar q}^{(i)}}}} }}}} \right), \end{aligned}}} $$
(18)
where \({\bar {\mathbf {q}}^{(\ell)}}\triangleq \left [\bar q^{(1)},\cdots,\bar q^{(\ell)}\right ]^{\mathrm {T}}\), \({\mathcal {Q}}^{(\ell)}\) is the set of all the possible puncturing patterns \(\bar {\mathbf {q}}^{(\ell)}\) over the first ℓ PRs. Further, recalling that \(\kappa _{\Lambda }^{(\ell)} (\hat s)\triangleq \log \mathrm {E} \left \{ \text {e }^{\hat s\Lambda _{k}^{(\ell)}}\right \}\), (18) can be rewritten as
$$ {\kappa_{{\mathcal{L}}}^{(\ell)}}(s)= \log \left({\sum\limits_{{\bar{\mathbf{q}}^{(\ell)}} \in {{\mathcal{Q}}^{(\ell)}}} {\Pr \left({{{q}^{(\ell)}} = {\bar {\mathbf{q}}^{(\ell)}}} \right){{\prod\limits_{i = 1}^\ell {\left[ {{\mathrm{e}^{{\kappa_{\Lambda}^{(i)}}(s)}}} \right]^{{{\bar q}^{(i)}}}} }}}} \right). $$
(19)
Following the line of reasoning about the no bit combining case previously recalled [28], in case of sufficiently long interleaving and linear binary code, the per-round bit scores \(\Lambda _{k}^{(\ell)}\) are, to a practical extent, i.i.d RVs and independent of q(ℓ). Hence, resorting to the so-called Gaussian approximation, the PEP can be approximated by [31]
$$ {\text{PEP}}(d) \simeq Q\left({\sqrt { - 2d{\kappa_{{\mathcal{L}}}^{(\ell)}}(\hat s)}} \right), $$
(20)
where d is the Hamming distance between b(ℓ) and b(ℓ)′ and \(\hat s\) represents the saddle point, with \(\hat s = 1/2\) for BIOS channels [31].
The above expression (20) can be seen as the PEP of an equivalent coded BPSK system operating over AWGN channel with SNR equal to \(-\kappa _{\mathcal {L}}^{(\ell)}(\hat {s})\). Thus, using (19) and exploiting the first equality in (7), we can eventually define the aggregate effective SNR, or αESNR for short, as
$$ \Gamma _\alpha^{(\ell)} \triangleq - \log \left({\sum\limits_{{\bar {\mathbf{q}}^{(\ell)}} \in {{\mathcal{Q}}^{(\ell)}}} {\Pr \left({{{q}^{(\ell)}} = {\bar{\mathbf{q}}^{(\ell)}}} \right)\prod\limits_{i = 1}^\ell {{{\left[ {{\mathrm{e}^{- {\gamma^{(i)}}}}} \right]}^{{{\bar q}^{(i)}}}}}} } \right), $$
(21)
where \(\gamma ^{(i)} \triangleq -\kappa _{\Lambda }^{(i)}(\hat s)\), 1≤i≤ℓ, is the ESNR relevant to the ith HARQ round, derived in [23] and reported in (7).
In conclusion, (21) can be property rearranged, leading to the result stated in the following.
Theorem 1
The αESM \(\Gamma _{\alpha }^{(\ell)}\) can be lower-bounded as
$$ \Gamma_\alpha^{(\ell)} \ge g\left(\Gamma_\alpha^{(\ell-1)},\xi^{(\ell)}\right) + f\left(\gamma^{(\ell)},\xi^{(\ell)}\right), \;\;\; 1 < \ell \le L, $$
(22)
where r(ℓ) is the coding rate employed at PR ℓ, 1≤ℓ≤L, \(\Gamma _{\alpha }^{(1)}=\gamma ^{(1)}\), R(1)=r(1),
$$ \xi^{(\ell)} \triangleq \frac{r^{(\ell)}}{R^{(\ell-1)}},\quad R^{(\ell)} \triangleq \min\{R^{(\ell-1)},r^{(\ell)}\}, \;\;\; 1 < \ell \le L, $$
(23)
and
$$ {{\begin{array}{*{20}{c}} {g(x,a) = \left\{ {\begin{array}{ll} { - \log\left[1 + a({\mathrm{e}^{- x}} - 1)\right],} & {{r^{(\ell)}} \le {R^{(\ell - 1)}}} \\ {x,} & {{r^{(\ell)}} > {R^{(\ell - 1)}}} \\ \end{array},} \right.} \;\;\;1 < \ell \le L, \\ {f(x,a) = \left\{ {\begin{array}{ll} {x,} & {{r^{(\ell)}} \le {R^{(\ell - 1)}}} \\ { - \log \left[1 + \frac{1}{a}({\mathrm{e}^{- x}} - 1)\right],} & {{r^{(\ell)}} > {R^{(\ell - 1)}}} \\ \end{array},} \right.} \;\;\;1 < \ell \le L. \\ \end{array}}} $$
(24)
Proof
See Appendix A. □
Remark
In order to evaluate the tightness of the lower bound of in Theorem 1, the relative error \(\delta _{\alpha } \triangleq \left (\Gamma _{\alpha }^{(\ell)}-\bar {\Gamma }_{\alpha }^{(\ell)}\right)/\Gamma _{\alpha }^{(\ell)}\) is depicted in Fig. 3, with \(\bar {\Gamma }_{\alpha }^{(\ell)}\) being the right-hand side of (22), i.e., the lower bound on the true αESM value, while the exact expression (21) is evaluated numerically, as a function of the PRs ℓ∈[1,8]. Specifically, for a given value of ℓ, \(\Gamma _{\alpha }^{(\ell)}\) is averaged over Navg=104 independent realizations. At each realization: the sequence of coding rates \(\{r^{(i)}\}_{i=1}^{\ell }\), thanks to which the puncturing patterns probability in (21) are evaluated, is randomly drawn from the set of available coding rates \(\mathcal {D}_{r}\); the sequence of ESNRs \(\{\gamma ^{(i)}\}_{i=1}^{\ell }\) is drawn as \(\left.\gamma ^{(i)}\right |_{\text {dB}} \in {\mathcal {U}}\left [-3,3\right ]\). The lower bound (22) is evaluated for the above sets \(\{r^{(i)}\}_{i=1}^{\ell }\) and \(\{\gamma ^{(i)}\}_{i=1}^{\ell }\) and then averaged over the Navg realizations. In Fig. 3, despite the derived lower bound gets looser for higher L, it can be considered tight for more practical values of L, i.e., at least for L≤5. In particular, it can be noted that it is very accurate up to L=3, where we have δ
α
≤0.07.
Upon defining \(\boldsymbol {\varphi }^{(\ell)} \triangleq \{{m^{(\ell)}},r^{(\ell)}\}\) as the MCS at PR ℓ, with \(m_{n,\nu }^{(\ell)} = m^{(\ell)}\), \(\forall (n,\nu) \in \mathcal {C}\), \(m^{(\ell)} \in {\mathcal {D}}_{m}\) and \(r^{(\ell)} \in {\mathcal {D}}_{r}\), so that \(\boldsymbol {\varphi }^{(\ell)} \in {\mathcal {D}}_{\boldsymbol {\varphi }s} \triangleq {\mathcal {D}}_{m} \times {\mathcal {D}}_{r}\) is the set of the allowable MCSs, a few comments are now discussed.
-
1.
Updating \(\Gamma _{\alpha }^{(\ell)}\) through (22) requires only (i) the aggregate quantities \(\Gamma _{\alpha }^{(\ell -1)}\) and R(ℓ−1) related to the previous (ℓ−1)th step, (ii) together with the κESNR γ(ℓ), which is evaluated at the current ℓth PR according to (7), based on the current SNRs Υ(ℓ) and MCS φ(ℓ). Accordingly, \(\boldsymbol {\sigma }^{(\ell)} \triangleq \left \{\Gamma _{\alpha }^{(\ell)},R^{(\ell)}\right \}\) can be defined as the “state” of the HARQ scheme we are processing. Hence, the αESNR \(\Gamma _{\alpha }^{(\ell)}\) at the ℓth PR depends only on the state σ(ℓ−1) (related to the past retransmissions up to the (ℓ−1)th one), the current SNRs Υ(ℓ), both known at PR ℓ at the transmitter, and the MCS φ(ℓ), which stands for the optimization parameter to find in order to improve the link performance. Thus, the αESM can be written as \(\Gamma _{\alpha }^{(\ell)}(\boldsymbol {\varphi }^{(\ell)} | (\boldsymbol {\sigma }^{(\ell -1)},\boldsymbol {\Upsilon }^{(\ell)}))\), whereas the κESM in (7) can be expressed as γ(ℓ)(φ(ℓ)|Υ(ℓ)). The update recursion is depicted in Fig. 4, where the selector output is \((x,y,a)=\left (\gamma ^{(\ell)},\Gamma _{\alpha }^{(\ell -1)},R^{(\ell -1)}/r^{(\ell)}\right)\) if r(ℓ)>R(ℓ−1) or \((x,y,a)=\left (\Gamma _{\alpha }^{(\ell -1)},\gamma ^{(\ell)}, r^{(\ell)}/R^{(\ell -1)}\right)\) if r(ℓ)≤R(ℓ−1).
-
2.
The PER performance of the MIMO BIC-OFDM system over frequency-selective fading channel with HARQ and packet combing mechanism can be approximated up to round ℓ as
$$ \begin{aligned} &\text{PER}\left(\boldsymbol{\varphi}^{(1)},\cdots,\boldsymbol{\varphi}^{(\ell)},\boldsymbol{\Upsilon}^{(1)},\cdots, \boldsymbol{\Upsilon}^{(\ell)}\right) \simeq \Psi_{r^{(\ell)}}\\ &\left(\Gamma_\alpha^{(\ell)}(\boldsymbol{\varphi}^{(\ell)} | (\boldsymbol{\sigma}^{(\ell-1)},\boldsymbol{\Upsilon}^{(\ell)}))\right), \end{aligned} $$
(25)
where \(\Psi _{r^{(\ell)}}\left (\Gamma _{\alpha }^{(\ell)}(\boldsymbol {\varphi }^{(\ell)} | (\boldsymbol {\sigma }^{(\ell -1)},\boldsymbol {\Upsilon }^{(\ell)}))\right)\) is the PER of the equivalent coded binary BPSK system over AWGN channel operating at SNR \(\Gamma _{\alpha }^{(\ell)}(\boldsymbol {\varphi }^{(\ell)} |(\boldsymbol {\sigma }^{(\ell -1)},\boldsymbol {\Upsilon }^{(\ell)}))\). It can be noted that such PER is a monotone decreasing and convex function in the SNR region of interest [27].
-
3.
It can be shown that the lower-bound (22) is exactly met when r(ℓ)≤R(ℓ−1), 1≤ℓ≤L, i.e., if the coding rate decreases along the retransmissions.
-
4.
Under the assumption that the coding rate is not adapted, i.e., r(j)=r(j−1), 2≤j≤ℓ, then \(\Gamma _{\alpha }^{(\ell)} = \sum _{j=0}^{\ell {\gamma ^{(j)}}}\), thus meaning that the aggregate ESNR of the HARQ mechanism is obtained as expected by accumulating the ESNRs evaluated at each PR.