Cross-layer link adaptation for goodput optimization in MIMO BIC-OFDM systems

2.1 HARQ retransmission protocol

In order to enable reliable and spectral efficient packet transmissions, a HARQ retransmission protocol, with a maximum of L rounds, is jointly designed along with an AMC mechanism. The information to be transmitted is conveyed by packets (typically IP packets) received from the upper layers of the stack, i.e., layer 3 and above, and stored in an infinite-length buffer at the data link layer. At the radio link control (RLC) sublayer, each packet is mapped into a RLC protocol data unit (PDU) made of the three sections: (i) header, with size N_h; (ii) payload, with size N_p; and (iii) CRC for error detection, with size N_CRC. As shown in Fig. 1, at the generic PR \(\ell \in \mathcal {L}_{\text {PR}} \triangleq \{1,\cdots,L\}\), the RLC-PDU is encoded with a code rate \(r^{(\ell)}\in \mathcal {D}_{r} \triangleq \{r_{0}, r_{1}, \cdots, r_{\text {max}}\}\), thus producing \(N^{(\ell)}_{\mathrm {c}}\triangleq N_{\mathrm {s}}/r^{(\ell)}\le {\bar N}_{c}\) coded binary symbols, or coded bits for short, where \(N_{\mathrm {s}} \triangleq N_{\mathrm {h}}+N_{\mathrm {p}}+N_{\text {CRC}}\) and \({\bar N}_{c} \triangleq N_{\mathrm {s}}/r_{0} \), which is the number of coded bits at the output of the mother code, i.e., prior to puncturing. The \(N^{(\ell)}_{\mathrm {c}}\) coded bits are transmitted using the MIMO BIC-OFDM system described in Section 2.2 over the available band W.

After the transmission of each packet¹, the receiver sends back a 1-bit 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 pre-combine 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 BIC-OFDM system

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 gray-mapped onto the unit-energy 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 pre-processed 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), parallel-to-serial conversion, and cyclic prefix (CP) insertion are applied. The resulting signal is then transmitted over a MIMO frequency-selective block-fading 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 de-interleaving and decoding.

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 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].

3.2 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].

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 \(N_{c}^{(\ell)}\)-bit-long 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 bit-level combining receiver, which represents the novel contribution of the work.

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.

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 N_avg=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 N_avg 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.

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 delay-free feedback channel and infinite-length 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 non-negative RVs, denoting the time elapsed between the renewal event i and i+1, i.e., the inter-renewal time, and \(\left \{Z^{(\ell)}_{i}\right \}\) a sequence of independent positive random rewards earned at every renewal event.

Thus, remark 4) paves the way for the following proposition.

4.2 Goodput-oriented-AMC (GO-AMC) OP

The AMC OP whose objective function is given by the EGP (32) is summarized in the following proposition.

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 closed-form expression, it can be pointed out that the complexity of the GO-AMC 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.

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 long-term 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 \(\mathrm {E}_{\boldsymbol {\Upsilon }s^{(\ell +k)}}\left \{\Psi _{r^{(\ell +k)}}\left (\Gamma _{\alpha }^{(\ell +k)}\left (\boldsymbol {\varphi }^{(\ell +j)}|\right.\right.\right.\left.\left.\left. \left (\boldsymbol {\sigma }^{(\ell +k-1)},\boldsymbol {\Upsilon }^{(\ell +k)}\right)\right) \right)\right \}\) 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.