Crosslayer link adaptation for goodput optimization in MIMO BICOFDM systems
 Riccardo Andreotti^{1},
 Vincenzo Lottici^{2} and
 Filippo Giannetti^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s136380171008y
© The Author(s) 2017
Received: 18 August 2017
Accepted: 17 December 2017
Published: 4 January 2018
Abstract
This work proposes a novel crosslayer link performance prediction (LPP) model and link adaptation (LA) strategy for softdecoded multipleinput multipleoutput (MIMO) bitinterleaved coded orthogonal frequency division multiplexing (BICOFDM) systems employing hybrid automatic repeat request (HARQ) protocols. The derived LPP, exploiting the concept of effective signaltonoise ratio mapping (ESM) to model system performance over frequencyselective channels, does not only account for the actual channel state information at the transmitter and the adoption of practical modulation and coding schemes (MCSs), but also for the effect of the HARQ mechanism with bitlevel combining at the receiver. Such method, named aggregated ESM, or αESM for short, exhibits an accurate performance prediction combined with a closedform solution, enabling a flexible LA strategy, that selects at every protocol round the MCS maximizing the expected goodput (EGP), i.e., the number of correctly received bits per unit of time. The analytical expression of the EGP is derived capitalizing on the αESM and resorting to the renewal theory. Simulation results carried out in realistic wireless scenarios corroborate our theoretical claims and show the performance gain obtained by the proposed αESMbased LA strategy when compared with the best LA algorithms proposed so far for the same kind of systems.
Keywords
1 Introduction
To meet the demanding need for ever increasing data rate and reliability, orthogonal frequency division multiplexing (OFDM), bitinterleaved coded modulation (BICM) [2], spatial multiplexing (SM) via multipleinput multipleoutput (MIMO) [3], adaptive modulation and coding (AMC) [4], and hybrid automatic repeat request (HARQ) [5] are wellknown techniques currently adopted, as advanced LTE (LTEA) [6], and envisaged to be exploited in the future wireless systems [7]. To be specific, the HARQ technique combines the automatic repeat request (ARQ) mechanism with both the channel coding error correction and the error detection capability of the cyclic redundancy check (CRC) [5]. If the CRC is successfully detected, the packet is correctly received and an acknowledgement (ACK) is fed back to the transmitter. Conversely, a CRC failure means that the received packet is affected by uncorrected errors and a nonACK (NACK) is sent back. In the latter condition, a retransmission of the corrupted packet is performed according to one of the following HARQ strategies [8]: (i) type I or Chase Combining (CC), the packet is retransmitted using the same redundancy; (ii) type II with partial Incremental Redundancy (IR), only a subset of previously unsent redundancy is transmitted; and (iii) type II with full IR, the systematic bits plus a different set of coded bits than those previously transmitted are sent. The HARQ mechanism potentials are fully exploited when the receiver suitably combines, e.g., using maximum ratio combining (MRC), the currently retransmitted packet with the previously unsuccessfully received ones, thus building a single packet whose reliability is more and more increased [5]. HARQ combining can be performed either on the received symbols or, should a different symbol mapping be employed in each transmission round, at bitlevel, i.e., by accumulating the bit loglikelihood ratio (LLR) metrics [9].
Background and related works. In the literature, a considerable effort has been put in quantifying the performance limits for HARQbased transmissions, mainly focusing on the ergodic capacity and outage probability [10–12]. In [10], an informationtheoretical study about the throughput of HARQ signaling schemes is given for the Gaussian collision channel. Then, starting from [10, 11] presents a mutual information (MI) based analysis of the longterm average transmitted rate achieved by HARQ in a blockfading scenario, which allows to adjust the rate so that a target outage probability is not exceeded. In [12], the optimal tradeoff among throughput, diversity gain, and delay is derived for an ARQ blockfading MIMO channel with discrete signal constellations.
In order to further enhance the system performance, the HARQ approach can be made adaptive by applying link adaptation (LA) strategies that do not only account for the information coming not only from the physical layer, but also from the higher layer schemes based on packet combining, so obtaining a crosslayer optimization of the link resource utilization. Most of the works considering such an issue, however, focus on theoretical performance limits based on capacity and channel outage probability, as in [13–17]. Specifically, [13] investigates the problem of power allocation for rate maximization under quantized channel state information (CSI) feedback, [14] adapts the transmission rates using the outdated CSI, whereas [15] proposes two power allocation schemes: one minimizes the transmitted power under a given packet drop rate constraint and the other minimizes the packet drop rate under the available power constraint. Note that in [13–15], IR HARQ is considered to optimize performance under narrowband fading channels. In [16], user and power are jointly selected in a multiuser contest under slowfading channels and outdated CSI to maximize system goodput (GP), whereas [17] proposes a user, rate, and power allocation policy for GP optimization in multiuser OFDM systems with ACKNACK feedback.
Only few recent works, however, consider practical modulation and coding schemes (MCSs). In [18], the outlined AMC algorithm maximizes the spectral efficiency under truncated HARQ for narrowband fading channels. A power minimization problem under individual user GP constraint is tackled in [19] for an orthogonal frequency division multiple access (OFDMA) network employing type II HARQ and only statistical knowledge of CSI. The work in [20] proposes the selection of the MCS to maximize GP performance in MIMOOFDM systems under CC HARQ, where the packet error rate is evaluated through the exponential effective SNR mapping (ESM) method (EESNR). A similar approach is proposed in [21], although the physical layer performance is modeled using the MI based effective SNR (MIESM).
 1.The proposed LPP model, named aggregated ESM, or αESM, relies on the ESM concept [22], which enables the prediction of the performance of a multicarrier system affected by frequencyselective fading by compressing all the persubchannel (identified by the pair subcarrier and spatial stream) SNRs into a scalar value representing the SNR of a coded equivalent binary system working over additive white Gaussian noise (AWGN) channel.

The LPP we put forward exhibits an accurate performance prediction combined with a closedform solution which makes it eligible for practical implementation of LA algorithms. Indeed, at the generic protocol round (PR) ℓ of a given packet, the αESM is obtained recursively, by combining the aggregated effective SNR (ESNR), that stores the performance up to the previous retransmission (step ℓ−1), with the actual ESNR at PR ℓ, which depends on the current CSI and choice of the MCS.

 2.The proposed αESM is derived from the ESM method originally proposed in [23] as κESM, by taking into account the persubchannel SNRs along with the HARQ mechanism. The key idea of the αESM method is to properly combine together the bit LLR metrics relevant to the retransmissions of the same packet, with the result of increasing decoding reliability. Specifically, the combined bit LLR metrics are characterized following an accurate method based on the cumulant moment generating function (CMGF).

The αESM is shown to overcome the limitations exhibited by [20, 21], where the MCS used in the subsequent retransmission is identical with that originally chosen, in that the LPP works with CC only. Conversely, since the proposed method has the inherent possibility of choosing the MCS optimizing the GP metric within the retransmissions of the same packet, as a result, it enables a much more flexible LA strategy.

 3.
The formulation of the GP at the transmitter, named expected goodput (EGP), is derived resorting to the renewal theory framework [24] and the longterm channel static assumption [20, 21]. The goal is, indeed, to obtain a reliable performance metric that can lead to a manageable LA optimization problem. Towards this end, both theoretical and numerical analyses are employed throughout the paper to corroborate our claims and findings.
 4.
Finally, simulation results carried out over realistic wireless channels testify the advantages obtained employing the proposed LA strategy based on the αESM, when compared with the best algorithms known so far.
Organization. The rest of the paper is organized as follows. Section 2 describes the HARQ retransmission mechanism and the MIMO BICOFDM system. In Section 3, after a brief rationale and review of the κESM LPP, the proposed αESM model is derived. Section 4 derives the EGP formulation and describes the proposed GPoriented (GO) LA strategy. Finally, Section 5 illustrates the numerical results, whereas in Section 6, a few conclusions are drawn.
Notations. Matrices are in uppercase bold, column vectors are in lowercase bold, [·]^{T} is the transpose of a matrix or a vector, a_{i,j} represents the entry (i,j) of the matrix A, × is the Cartesian product, calligraphic symbols, e.g., \({\mathcal {A}}\), represent sets, \({\mathcal {A}}\) is the cardinality of \({\mathcal {A}}\), \({\mathcal {A}}(i)\) is the ith element of \({\mathcal {A}}\), ⌈·⌉ denotes the ceil function, and E_{ x }{·} is the statistical expectation with respect to (w.r.t.) the random variable (RV) x.
2 System model
In this section, we first describe the HARQ retransmission protocol. Then, the MIMO BICOFDM signalling system is outlined.
2.1 HARQ retransmission protocol
After the transmission of each packet^{1}, the receiver sends back a 1bit feedback about the successful (ACK) or unsuccessful (NACK) packet reception. Whenever a NACK is received, the transmitter sends again the packet by encoding it with either the same puncturing pattern, a different subset of redundancy bits, or a tradeoff, according to the type of HARQ. This goes on until the transmitter receives an ACK or the maximum number of retransmissions L is reached. For both cases, the packet is removed from the buffer and the transmitter moves on sending the subsequent ones. At the receiver side, according to the HARQ scheme, for a given packet, the previously unsuccessfully received copies are stored and combined with the new received ones, thus creating more reliable metrics [5]. Since at each PR a different symbol mapping per subchannel may be applied, it is not possible to precombine received symbols. Hence, the packet combining strategy consists of accumulating the bit LLR metrics [9], as explained in detail in the following sections.
2.2 MIMO BICOFDM system

we assume the CSI \(\mathbf {H}^{(\ell)}_{n}\), \(\forall n \in {\mathcal {N}}\), to be known at the transmitter side.
Specifically, with reference to Fig. 1, the interleaved sequence of punctured coded bits is subdivided into subsequences of \(m_{n,\nu }^{(\ell)}\) bits each, which are graymapped onto the unitenergy symbols \({x}^{(\ell)}_{n,\nu } \in 2^{m_{n,\nu }^{(\ell)}}\)QAM constellation, i.e., one symbol per available subchannel \((n,\nu)\in {\mathcal {C}} \triangleq \{(n,\nu)1\le n\le N, 1\le \nu \le M\}\), with \(m_{n,\nu }^{(\ell)} \in {{\mathcal {D}}}_{m} = \left \{2, 4,\cdots,m_{\max } \right \}\).
Further, let us denote Φ^{(ℓ)}(·,·,·) as a function mapping the punctured \(N_{c}^{(\ell)}\) coded bits, out of the \({\bar N}_{c}\) coded bits at the output of the mother code, into the label bits of the QAM symbols transmitted on the available subchannels, summarizing the puncturing, interleaving, and QAM mapping functions. Specifically, Φ^{(ℓ)}(j,n,ν)=k means that the coded bit \(b^{(\ell)}_{k}\), \(k\in \left \{1,\cdots,{\bar {N}}_{c}\right \}\) occupies the jth position, \(j=1,\cdots,m_{n,\nu }^{(\ell)}\), within the label of the \(2^{m_{n,\nu }^{(\ell)}}\)QAM symbol sent on the νth spatial stream, ν=1,⋯,M, of the nth subcarrier, n=1,⋯,N.
According to the SM approach, each sequence of QAM symbols \(\mathbf {x}^{(\ell)}_{n}\triangleq \left [x^{(\ell)}_{n,1},\cdots,x^{(\ell)}_{n,{N_{T}}}\right ]^{\mathrm {T}}\) is preprocessed obtaining \({\tilde {\mathbf {x}}}^{(\ell)}_{n}\triangleq {\mathbf {V}_{n}^{(\ell)}}{\mathbf {x}}^{(\ell)}_{n}\), where \({\tilde {\mathbf {x}}}^{(\ell)}_{n}\triangleq \left [{\tilde x}^{(\ell)}_{n,1},\cdots,{\tilde x}^{(\ell)}_{n,{N_{T}}}\right ]^{\mathrm {T}}\), \(\forall n \in {\mathcal {N}}\). It is worth noting that \(x^{(\ell)}_{n,\nu }\triangleq 0\) for ν=M+1,⋯,N_{ T }, if N_{ T }>N_{ R }, \(\forall n \in \mathcal {N}\), being C subchannels available for transmission.
After that, the sequences \({\tilde {\mathbf {x}}}^{(\ell)}_{n}\), \(\forall n \in {\mathcal {N}}\), are mapped onto the frequency symbols \(\mathbf {y}_{\nu }^{(\ell)}\triangleq \left [{\tilde x}^{(\ell)}_{1,\nu },\cdots,{\tilde x}^{(\ell)}_{N,\nu }\right ]^{\mathrm {T}}\), for ν=1,⋯,N_{ T }, to which conventional inverse discrete Fourier transform (DFT), paralleltoserial conversion, and cyclic prefix (CP) insertion are applied. The resulting signal is then transmitted over a MIMO frequencyselective blockfading channel, using \(N^{(\ell)}_{\text {OFDM}}\triangleq \left \lceil {N_{c}^{(\ell)}}/{\sum _{(n,\nu)\in \mathcal {C}m_{n,\nu }^{(\ell)}}} \right \rceil \) OFDM symbols.
Finally, the receiver evaluates the soft metrics, followed by deinterleaving and decoding.
3 Link performance prediction for HARQbased MIMO BICOFDM systems
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 BICOFDM systems with simple ARQ mechanism is briefly summarized. Finally, in Section 3.3, the novel LPP method, named αESM, is derived for HARQbased MIMO BICOFDM systems with bitlevel combining.
3.1 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 packetoriented system, informationtheoretical 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 bitlevel combining. In the sequel, we will focus on LPP techniques based on the wellknown ESM concept, which has been shown to be the most effective framework to solve this issue, especially for multicarrier systems [22].
3.2 Background on the κESM LPP model
In multicarrier systems, where the frequencyselective 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 singlecarrier equivalent coded system working over AWGN channel, whose performance can be simply evaluated either offline according to analytical models [27].
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)}.
3.3 The αESM model
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 \(N_{c}^{(\ell)}\)bitlong codeword, so that the dependence on the OFDM symbol index is avoided. In particular, we have \(B^{(\ell)} = N_{c}^{(\ell)}\).
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 bitlevel combining receiver, which represents the novel contribution of the work.

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 loglikelihood 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)}^{′}.

the pattern \(\mathbf {q}^{(\ell)}_{k}\) can be modeled as a sequence of ℓ independent and identically distributed (i.i.d.) binary RVs taking values 0 or 1, independently of the bit index k.
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].
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
Proof
See Appendix A. □
Remark
 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 BICOFDM system over frequencyselective fading channel with HARQ and packet combing mechanism can be approximated up to round ℓ as$$ \begin{aligned} &\text{PER}\left(\boldsymbol{\varphi}^{(1)},\cdots,\boldsymbol{\varphi}^{(\ell)},\boldsymbol{\Upsilon}^{(1)},\cdots, \boldsymbol{\Upsilon}^{(\ell)}\right) \simeq \Psi_{r^{(\ell)}}\\ &\left(\Gamma_\alpha^{(\ell)}(\boldsymbol{\varphi}^{(\ell)}  (\boldsymbol{\sigma}^{(\ell1)},\boldsymbol{\Upsilon}^{(\ell)}))\right), \end{aligned} $$(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 lowerbound (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.
4 Link adaptation for EGP optimization
4.1 Expected goodput formulation
Capitalizing on the results gained in the previous section, let us now derive the expression of the EGP metric at the generic PR ℓ. Toward this end, we resort to the renewal theory [24], which was first introduced in [32] to analyze the throughput performance of a HARQ system, under the assumptions of error and delayfree feedback channel and infinitelength buffer.
As an initial step, let us assume that at the ℓth PR, the system has previously experienced ℓ−1 unsuccessful packet transmission attempts and there are still L−ℓ+1 PRs available. Then, let us define a renewal event as the following occurrence: the system stops transmitting the current packet because either an ACK is received or because the PR limit L is reached. Let \(\left \{X_{i}^{(\ell)}\right \}\) be independent identically distributed nonnegative RVs, denoting the time elapsed between the renewal event i and i+1, i.e., the interrenewal time, and \(\left \{Z^{(\ell)}_{i}\right \}\) a sequence of independent positive random rewards earned at every renewal event.
Theorem 2
Proof
From the renewal theory [24]. □
Remark
Theorem 2 states that the accumulated reward over time equals the ratio between the expected reward \(\mathrm {E}\left \{Z^{(\ell)}_{i}\right \}\) and the expected time \(\mathrm {E}\left \{X^{(\ell)}_{i}\right \}\) in which such reward is earnead.
In light of Theorem 2 and the αESM model derived in Section 3.3, the EGP metric can be formulated as follows.
Theorem 3
denoting the time interval required to transmit a packet of \(N^{(\ell)}_{\mathrm {c}}=N_{\mathrm {s}}/r^{(\ell)}\) coded bit employing MCS φ^{(ℓ)}, and T_{ B } being the OFDM symbol duration.
Proof
See Appendix B. □
 1.Thanks to longterm static channel assumption:

at PR ℓ, each packet experiences the current channel condition Υ^{(ℓ)}) over its possible future retransmissions, then φ^{(ℓ+j)}=φ^{(ℓ)}, j∈[0,L−ℓ].
Therefore, at the ℓth PR, the ESNRs \(\Gamma _{\alpha }^{(\ell)}, \Gamma _{\alpha }^{(\ell +1)},\cdots, \Gamma _{\alpha }^{(\ell +L)}\) are only function of φ^{(ℓ)}given the status (σ^{(ℓ−1)},Υ^{(ℓ)}), i.e., we can write \(\Gamma _{\alpha }^{(\ell +j)}\left (\boldsymbol {\varphi }^{(\ell)}  {\boldsymbol {\sigma }^{(\ell  1)}},\boldsymbol {\Upsilon }^{(\ell)} \right)\), j∈[0,L−ℓ].

 2.
Assumption A3 may seem counterintuitive. Indeed, if the channel does not change, there would not be the need to adapt the MCS at each retransmission. However, the channel does change from PR to PR and, everytime, the corresponding metric is fed back to the transmitter (see assumption A1). The latter exploits this information to evaluate the EGP and adapt the MCS for the current retransmission. As a matter of fact, it is only for the sake of evaluating the EGP that the channel is assumed, during the following PRs, to be constant and equal to the current one, so as to obtain a manageable expression for the EGP.
 3.
The UPD expression (28) is obtained assuming independent PER among the PRs, even though they are related by the recursive αESM expression. Such an assumption is confirmed in Section 5, where numerical results obtained over realistic wireless channels show that the proposed LA strategy, optimizing the EGP, outperforms the best LA known so far.
 4.Recalling remark 1) and approximating the αESM \(\Gamma _{\alpha }^{(\ell)}(\boldsymbol {\varphi }^{(\ell)}  (\boldsymbol {\sigma }^{(\ell 1)},\boldsymbol {\Upsilon }^{(\ell)}))\) with the lower bound given by Theorem 1, we have$$ \begin{aligned} &\Gamma_\alpha^{(\ell+j)}\left(\boldsymbol{\varphi}^{(\ell)}  \left(\boldsymbol{\sigma}^{(\ell1)},\boldsymbol{\Upsilon}^{(\ell)}\right)\right)\\ &= g\left(\Gamma_\alpha^{(\ell1)},\xi^{(\ell)}\right) + (j+1)~f\left[\!\gamma^{(\ell)}\left(\boldsymbol{\varphi}^{(\ell)}\boldsymbol{\Upsilon}^{(\ell)}\right),\xi^{(\ell)}\right],\\ & \le j \le L\ell. \end{aligned} $$(31)
Expression (31) can be simply shown by induction upon noting that, due to remark 1), we have φ^{(ℓ+j)}=φ^{(ℓ)} and hence r^{(ℓ+j)}=r^{(ℓ)} and γ^{(ℓ+j)}=γ^{(ℓ)}, for j∈[0,L−ℓ].
Thus, remark 4) paves the way for the following proposition.
Proposition 1
 1.
In view of the normalization by the OFDM signal bandwidth W, the EGP in (32) can be read as a spectral efficiency metric measured in (bit/s/Hz);
 2.
Due to (31), it is apparent that the EGP depends on the MCS only, which has to be optimized according to the AMC optimization problem (OP) outlined in the next section.
4.2 GoodputorientedAMC (GOAMC) OP
The AMC OP whose objective function is given by the EGP (32) is summarized in the following proposition.
Proposition 2
■
The OP (35) can be easily solved through an exhaustive search over all the pairs of modulation order and coding rate \(\boldsymbol {\varphi } \in {\mathcal {D}}_{\boldsymbol {\varphi }s}\). Since all the quantities to be evaluated have a closedform expression, it can be pointed out that the complexity of the GOAMC OP simply reduces to \(\mathcal {O}(\mathcal {D}_{\boldsymbol {\varphi }s}) = \mathcal {O}(\mathcal {D}_{m}\cdot \mathcal {D}_{r})\), i.e., linear with the allowable MCS pairs.
5 Simulation results
Parameters and features of the HARQbased MIMO BICOFDM system
Parameter/feature  Symbol  Value/description 

RLCPDU  
RLCPDU length  N_{p}+N_{CRC}  1056 bits 
OFDM  
No. of active subcarriers  N  1320 
FFT size  N _{FFT}  2048 
CP length  N _{CP}  160 samples 
Modulation and coding  
Bits per subcarrier  \({\mathcal {D}}_{m}\)  {2,4,6} 
Code type  PCCC turbocode  
Mother code rate  1/3  
Punctured code rates  \(\mathcal {D}_{r}\)  {78,120,193,308,449,602,378, 
490,616,466,567,666,772,873,  
948}/1024  
Transmitted power  P  40 dBm 
Bandwidth  W  20 MHz 
Multiantenna configuration  N_{ R }×N_{ T }  
4x4 (MIMO) in Fig. 6  
8x8 (MIMO) in Fig. 7  
ARQ  
ARQ scheme  Multiplechannel Stop & Wait  
No. of PRs  L  10 
Parameters and features of the wireless propagation channel model
Parameter/feature  Value/description 

Pathloss model  NLOS urban scenario [IEEE 802.16] 
Carrier frequency  2 GHz 
Base station height  12.5 m 
Mobile terminal height  1.5 m 
Noise power level  − 100 dBm 
Longterm fading model  Lognormal distribution 
Variance of the shadowing  6 dB 
Shortterm fading model  3 GPP typical urban channel 
List of acronyms
Acronym  Meaning 

ACK  Acknowledgement 
αESM  Aggregated effective SNR mapping 
AMC  Adaptive modulation and coding 
ARQ  Automatic repeat request 
AWGN  Additive white Gaussian noise 
BIC  Bitinterleaved coded 
BICM  Bitinterleaved coded modulation 
BIOS  Binary input output symmetric 
BPSK  Binary phase shift keying 
CC  Chase combining 
CCDF  Complementary cumulative distribution function 
CMGF  Cumulant moment generating function 
CP  Cyclic prefix 
CRC  Cyclic redundancy check 
CSI  Channel state information 
DFT  Discrete Fourier transform 
EESNR  Exponential effective SNR 
EGP  Expected goodput 
ESM  Effective SNR mapping 
ESNR  Effective SNR 
GO  Goodputoriented 
GP  Goodput 
HARQ  Hybrid automatic repeat request 
HEESM  HARQ EESM 
HMIESM  HARQ MIESM 
IR  Incremental redundancy 
LA  Link adaptation 
LLR  Loglikelihood ratio 
LPP  Link performance prediction 
MCS  Modulation and coding scheme 
MI  Mutual information 
MIESM  Mutual information based effective SNR 
MIMO  Multipleinput multipleoutput 
MRC  Maximum ratio combining 
OFDM  Orthogonal frequency division multiplexing 
OFDMA  Orthogonal frequency division multiple access 
OP  Optimization problem 
PDS  Pairwise decoding score 
PDU  Protocol data unit 
PEP  Pairwise error probability 
PER  Packet error rate 
PR  Protocol round 
RLC  Radio link control 
RV  Random variable 
SISO  Singleinput singleoutput 
SM  Spatial multiplexing 
SNR  Signaltonoise ratio 
UE  User equipment 
UPD  Unsuccessful packet decoding 
Finally, we outline the computational complexity of the proposed method. To this end, we take as the reference the proposed αESM method and the HEESM, since they represent the best and worst case, respectively, as apparent from Figs. 5, 6 and 7. The HEESM method has a closed form and is based on the logarithm of a sum of negative exponential functions, as can be seen in Eq. (15) of [20]. Also, the αESM method, for a given PR, has a closed form and is based on the logarithm of a sum of negative exponential functions, as can be seen from Eq. (8), and the recursive Eq. (22). The complexity required by the recursive equation can be considered negligible when compared to the evaluation of the corresponding ESNR, in that the functions g(·,·) and f(·,·) defined in (24) can be properly calculated using a lookup table. Therefore, their computational complexity at each PR can be considered comparable. The only difference is that, while the HEESM is evaluated only at the first PR, the αESM is reevaluated PR by PR. Thus, its complexity increases linearly with the number of PRs. Since this number is limited (usually below 10), the increment of complexity w.r.t. the HEESM is lower (or at most equal to) one order of magnitude, but with a great gain in performance, as previously shown.
6 Conclusion
This paper presented an innovative crosslayer LPP methodology, named αESM, suited for packetoriented MIMO BICOFDM transmissions, which accounts for CSI, practical MCSs, and HARQ. The proposed αESM suitably extends the κESM method so to account also for the HARQ mechanism with bitlevel combining at the receiver. The proposed LPP method gives an accurate closedform solution and the possibility to enable a flexible LA strategy, where at each PR the MCS that maximizes the GP performance at the UE is selected based on the information about the past transmissions and actual CSI. In particular, the formulation of the GP at the transmitter, named EGP, is derived resorting to the renewal theory. Simulation results carried out over realistic wireless channels demonstrate that LA strategy based on the αESM method outperforms the best known algorithms proposed so far, providing gains of about 5 and 7.5 dB in SISO and up to 11 dB in MIMO configurations, respectively. An interesting followup of this work consists in moving the focus from the LPP method itself, applied here to the reference conventional (MIMO)OFDM system, to its extension to more advanced transmission schemes such as (MIMO)OFDM with spatial modulation [35], index mapping [36], or UFMC [37].
7 Appendix
7.1 A. Proof of Theorem 1
where we exploit the relationship R^{(ℓ)}=r^{(1)} due to (23) and the assumption of increasing coding rate.
where R^{(ℓ)}=r^{(ℓ)} due to the assumption of decreasing coding rate.
7.2 B. Proof of Theorem 3
where the first term on the right hand side (RHS) is the time elapsed over the previous ℓ−1 failed transmissions, which is a known quantity at the ℓth PR; T_{u}(φ^{(j)}) is defined in Eq. (30), whereas ℓ≤ℓ_{ i }≤L is a RV depending on the number of packet transmissions after which the renewal event happens.
Besides, since we are interested in correctly receiving the N_{p} information bits out of the \(N_{\mathrm {c}}^{(\ell)}\) transmitted ones, the reward is \(Z^{(\ell)}_{i}=N_{\mathrm {p}}/W\) if the renewal is due to a successful decoding; otherwise \(Z^{(\ell)}_{i}=0\).
Evaluation of (53) would require the knowledge of the channel p.d.f. for all the possible cases of interest, which is unrealistic in practice. Therefore, as usual in these cases [20, 21],let us adopt the longterm static channel assumption given as A3, i.e., the packet experiences the current channel conditions Υ^{(ℓ)} throughout its possible future retransmissions.
It follows that \(\mathrm {E}_{\boldsymbol {\Upsilon }s^{(\ell +k)}}\left \{\Psi _{r^{(\ell +k)}}\left (\Gamma _{\alpha }^{(\ell +k)}\left (\boldsymbol {\varphi }^{(\ell +j)}\right.\right.\right.\left.\left.\left. \left (\boldsymbol {\sigma }^{(\ell +k1)},\boldsymbol {\Upsilon }^{(\ell +k)}\right)\right) \right)\right \}\) is replaced by \(\Psi _{r^{(\ell)}}\left (\Gamma _{\alpha }^{(\ell +k)}\right.\left.\left (\boldsymbol {\varphi }^{(\ell)}\left (\boldsymbol {\sigma }^{(\ell 1)},\boldsymbol {\Upsilon }^{(\ell)}\right)\right) \right)\) in (53), and, accordingly, φ^{(ℓ+j)}=φ^{(ℓ)}, implying T_{u}(φ^{(ℓ+j)})=(j+1)T_{u}(φ^{(ℓ)}), ∀j∈{0,⋯,L−ℓ}. Finally, upon plugging (51)–(53) in (26) after the substitutions listed above, the EGP formulation (27) follows.
Notes
Declarations
Acknowledgements
This work has been partially supported by the PRA 2016 research project 5GIOTTO funded by the University of Pisa and by SVI.I.C.T.PRECIP. project, in the framework of Tuscany’s “Programma Attuativo Regionale,” cofunded by “Fondo per lo Sviluppo e la Coesione” (FSC) and Italy’s Ministry for Education, University and Research (MIUR), Decreto Regionale n.3506, 28/07/2015.
The authors would like to thank Prof. Luc Vandendorpe and Ivan Stupia, PhD, from Université catolique de Louvain, LouvainlaNeuve, Belgium, for the fruitful discussions and their helpful suggestions.
Authors’ contributions
RA worked on the derivation of both the link performance prediction method for HARQbased MIMO BICOFDM systems and the link adaptation for EGP optimization. He also run numerical simulations which provided numerical results. VL contributed to the introduction’s background and to the bibliographical survey on related works. Also, he contributed to the analytical derivation of the link performance prediction method for HARQbased MIMO BICOFDM systems and to the interpretation of the numerical results. FG provided the system model description and the definitions of the performance metrics for the proposed algorithms. He also contributed to the interpretation and to the comments of the numerical results. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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