Adaptive coding and modulation using imperfect CSI in cognitive BIC-OFDM systems
- Jeroen Van Hecke^{1}Email authorView ORCID ID profile,
- Paolo Del Fiorentino^{2},
- Riccardo Andreotti^{2},
- Vincenzo Lottici^{2},
- Filippo Giannetti^{2},
- Luc Vandendorpe^{3} and
- Marc Moeneclaey^{1}
https://doi.org/10.1186/s13638-016-0739-5
© The Author(s) 2016
Received: 23 March 2016
Accepted: 20 September 2016
Published: 26 October 2016
Abstract
This work investigates adaptive coding and modulation (ACM) algorithms under the realistic assumption that the available channel state information (CSI) at the transmitter is imperfect due to estimation errors and/or feedback delays. First, we introduce an optimal performance metric for the secondary user (SU) bit-interleaved coded orthogonal frequency division multiplexing (BIC-OFDM) system, called the expected goodput (EGP). By using an accurate modeling approximation, we succeed in deriving a tractable and very accurate approximation for the EGP. This approximate EGP (AEGP) is then used for the derivation of several ACM algorithms which optimize the code rate and bit and energy allocation under a constraint on the interference caused to the PU network. In the numerical results, we show that the AEGP is far more accurate than previous attempts to model the GP in the presence of imperfect CSI. Further, we verify that, in spite of the imperfect nature of the available CSI, the derived ACM algorithms significantly increase the goodput of the SU network, compared to a non-adaptive selection of the transmission parameters.
Keywords
1 Introduction
To meet the demand of high data rates and the increasing amount of traffic, the current and next generation of wireless networks need spectrally efficient solutions such as multicarrier orthogonal frequency division multiplexing (OFDM) transmission, efficient channel coding techniques in the form of bit interleaved coded modulation (BICM) [1], and adaptive coding and modulation (ACM) [2]. To further increase the spectral efficiency, the idea of cognitive radio (CR) [3, 4] has been proposed. This technique allows unlicensed or secondary users (SUs) to transmit over sections of spectrum owned by licensed or primary users (PUs), on the condition that the former do not harm the quality of service (QoS) of the latter.
If channel state information (CSI) is available at the transmitter, ACM can significantly improve the performance of the network by adapting the transmission parameters, such as energy and bit allocation per subcarrier, constellation size, and code rate, to the actual state of the channel. However, in a wireless environment, the CSI at the transmitter, obtained from channel estimates fed back by the receiver, will be imperfect, due to channel estimation errors at the receiver and, in the case of a time-varying channel, the feedback delay on the return channel from the receiver to the transmitter. In [5], the authors show for a single user OFDM system that, even with CSI imperfections at the transmitter, the throughput of the system can be significantly increased by using adaptive modulation. The adaptation algorithms take the CSI imperfections into account, and their performance was shown to improve by having multiple estimates available at the transmitter. This means that, when multiple estimates are available, the network can tolerate larger channel estimation errors or longer delays, while still achieving an acceptable performance level. In [6], this scenario was extended to a multi-user OFDMA-system where the subcarriers are allocated to the user with the best signal-to-noise ratio (SNR) conditions and the number of bits per subcarrier are optimized by maximizing the average throughput. However, the results in [5, 6] were obtained for an uncoded OFDM system; this considerably simplifies the optimization problem (OP) because the probability of a bit error on a subcarrier only depends on the SNR of the considered subcarrier, but the results are of limited use in a practical scenario where channel coding is used.
In recent years, there have been several works [7–12] that studied resource allocation in cognitive underlay networks with imperfect CSI. However, these works used a more theoretic performance metric like the capacity metric or SNR and did not consider the difficult problem of implementing ACM in a practical coded multi-carrier transmission system. Because the bits are coded and the channel is frequency-selective, the throughput of the network depends upon a complicated function of the SNRs of all the subcarriers which are used for the transmission. A technique which allows to simplify the analytical expression for the performance metric is effective SNR mapping (ESM) [13]. This technique transforms the vector of subcarrier SNRs, which affect the codeword, into a scalar SNR. This effective SNR is the operating point at which an equivalent coded system, which uses the same modulation and coding scheme, operating over an additive white Gaussian noise (AWGN) channel, has the same performance as the system under consideration. A very promising mapping function, called the cumulant-generating function-based ESM (κESM), was introduced in [14]. This mapping function combines the simplicity of exponential ESM (EESM) with the accuracy of mutual information ESM (MIESM) [15]. Another advantage is that this mapping function can be used to optimize the coding rate together with the energy and bit allocation per subcarrier.
In [16], EESM has been applied to ACM in a multi-carrier system with feedback delays. The bit allocation per subcarrier and the code rate are selected such that the throughput gets maximized under a certain block error rate constraint. However, because the transmitter is unaware of the fact that the available CSI is delayed, the transmitter sometimes over- or underestimates the actual channel conditions which results in a loss of spectral efficiency. In [17], the throughput of a BIC-OFDM system is optimized under a target packet error rate (PER) constraint, where a packet can consist of multiple OFDM symbols. Also here, the considered adaptation algorithm at the transmitter does not account for CSI imperfections, which leads to a violation of the PER constraint when only delayed CSI is available.
- 1.
Instead of resorting to the often used information-theoretical capacity metric, a more practically relevant metric, i.e., the GP, is optimized, which gives the advantage of allowing the optimization of realistic modulation and coding formats.
- 2.
Unlike the ad hoc approaches used in our previous work [19, 20], we now start from the optimal expression for the EGP. By using the statistical approximation for the effective SNR, which we introduced in [21], we now derive an analytical, tractable approximation for the EGP, which we call the approximate EGP (AEGP). In the numerical results, we show that the AEGP is a far more accurate approximation of the EGP, compared to the metrics used in [19, 20]. To the authors’ knowledge, these works are the first ones which propose to use a practical metric, which takes care of the imperfect CSI, for the optimization of the transmission parameters.
- 3.
In this work, we successfully combine the practical assumption of imperfect CSI with the accurate model of the effective SNR, which results in the AEGP metric. This AEGP metric, which takes care of the imperfect CSI, is proposed as the objective function of an OP to search for the optimal combination of the ACM parameters under the above mentioned transmit energy and interference constraints. By using the AEGP, packet errors or a loss in spectral efficiency by over- or underestimating the actual channel conditions are largely avoided. This differs from the approach taken in [16, 17], where the transmitter is unaware that its CSI is imperfect and only the impact of the imperfect CSI on the performance is investigated.
- 4.
We derive several ACM solutions which optimize the code rate together with uniform or non-uniform bit allocation and uniform or non-uniform energy allocation. The performance of these algorithms is investigated for different types of CSI at the SU transmitter.
- 5.
Although affected by imperfect CSI, extensive simulation runs show that the proposed ACM algorithms allow significant gains compared to non-adaptive ACM schemes. Further, depending on the quality level of the CSI, the resulting GP performance can be very close to that obtainable in scenarios where perfect CSI is employed.
Organization. In Section 2, we describe the cognitive BIC-OFDM system. In Section 3, we introduce the EGP metric and discuss the statistical approximation of the κESM. The ACM algorithms which select the code rate and the energy and bit allocations per subcarrier are derived in Section 4. The accuracy of the EGP metric and the performance of the ACM algorithms are validated in Section 5. The conclusions are presented in Section 6.
Notations. Expectation operator is E[ ·], [ ·]^{T} is the transpose operator, [ ·]^{H} is the Hermitian transpose operator, x∼(0,Σ) refers to a circular symmetric zero-mean Gaussian complex random vector with covariance matrix Σ, and the matrix I denotes the identity matrix. The ith column of the identity matrix is denoted by e _{ i }. The notation (X)_{ i,j } refers to the element on the ith row and jth column of the matrix X, while (x)_{ i } denotes the ith component of the vector x.
2 Cognitive BIC-OFDM system model
where E _{max} is the maximal transmit energy per OFDM symbol. Next, the SU receiver first performs soft demapping, and finally de-interleaves and decodes the packet; the CRC allows to verify whether the packet has been correctly decoded.
Let us arrange the received SNRs into a vector \(\boldsymbol {\Gamma }{\overset {\Delta }{=}} [\gamma _{1},\ldots,\gamma _{N}]\) for further use. We define the transmission mode (TM) \(\mathbf {\phi }{\overset {\Delta }{=}}\{\mathbf {m},r\}\in {\mathcal {D}}_{\mathrm {m}}^{N}\times {\mathcal {D}}_{\mathrm {r}}\), with \(\mathbf {m}{\overset {\Delta }{=}}[m_{1},\ldots,m_{N}]^{T}\). As not all N available subcarriers will necessarily be used for the transmission, we make a distinction between the set {1,…,N} of available subcarriers, and the set \(\mathcal {N\subseteq }\{1,\ldots,N\}\) of active subcarriers. When the kth subcarrier is not active (i.e., \(k\notin \mathcal {N}\)), we have E _{ k }=0 and m _{ k }=0.
Because of noise and/or feedback delays, the CSI available at the transmitter will often be imperfect. To make the description of our proposed approach quite general, we will denote the CSI, which is available at the transmitter about the actual channel realization \(\mathbf {H}{\overset {\Delta }{=}}[H_{1},\ldots,H_{N}]^{T}\), by the vector C S I. We make the assumption that H and C S I are jointly zero-mean circular symmetric Gaussian. It then follows that H conditioned on C S I is Gaussian, with expectation μ _{ H|C S I }=E_{ H }[H|C S I] and covariance matrix C _{ H|C S I }=E_{ H }[H H ^{ H }|C S I]−μ _{ H|C S I } μ H|C S I H; note that μ _{ H|C S I } is the minimum mean-squared error (MMSE) estimate of H based on C S I. Some examples of C S I and the associated statistics are given in the Appendix section.
The signals transmitted in the SU network cause interference at the PU receivers, which should be constrained in order not to affect the PU QoS. Denoting by \(G_{k}^{(q)}\) the channel gain from the SU transmitter to the qth PU receiver, experienced by the kth subcarrier, the interference constraints can be expressed as \(\sum _{k\in {\mathcal {N}}}E_{k}|G_{k}^{(q)}|^{2}\leq {\mathcal {I}}_{q}\) for \(q\in {\mathcal {Q}}{\overset {\Delta }{=}}\{1,\ldots,N_{\text {PU}}\}\).
where we have assumed that the distribution of G ^{(q)} conditioned on \(\mathbf {CSI}_{\text {PU}}^{(q)}\) is Gaussian with mean μ G|C S I _{PU}(q) and covariance matrix C G|C S I _{PU}(q).
In the case that the dynamically allocated energy vector \(\mathbf {E}{\overset {\Delta }{=}}[E_{1},\ldots,E_{N}]^{T}\) leads to an intolerable IP, one can substitute \({\mathcal {I}}_{q}\) in the corresponding interference constraint by \(\kappa _{q}{\mathcal {I}}_{q}\). The scaling factor κ _{ q } is chosen such that IP_{ q } reaches an acceptable value, after finding a new dynamic allocation of the vector E which satisfies the new constraint.
Finally, it is clear that the constraints (5), (8), and (9) all have the same mathematical form. This means that our proposed algorithms are compatible with all these constraints. For the remainder of the paper however, we will consider the average interference constraint (5).
3 Goodput performance metric
where PER(ϕ,Γ) is the packet error rate (PER) corresponding to the selected (ϕ,Γ). Note that the goodput (11) is a function of the actual channel realization H because of (3). As a performance measure of the SU network, we consider the long-term average of the goodput (11) over many channel realizations.
If perfect CSI were available at the transmitter (i.e., the transmitter knows the realizations of its channels to the SU receiver and PU receivers), the optimal way of selecting the transmission mode ϕ and the energy allocation vector E as a function of these realizations is to maximize (11) under the constraints on the SU transmit energy and the interference at the PU receivers, for the given realizations H and \(\{\mathbf {G}^{(q)},q\in \mathcal {Q}\)}. This selection obviously maximizes the long-term average goodput of the system, given by \(\text {GP}_{\text {avg}}=\mathrm {E}_{\mathbf {H},\left \{\mathbf {G}^{(q)},q\in \mathcal {Q}\right \}}\left [\text {GP}\right ]\).
which is the conditional expectation of GP for given C S I and represents the optimal performance metric in terms of GP_{avg} when only imperfect CSI is available at the transmitter.
where d _{ k,min} denotes the minimum Euclidean distance of the constellation used on the kth subcarrier, and α _{ k,n } is a known constant which depends on the chosen constellation.
The expectation w.r.t. Z in (17) can be approximated by means of numerical integration.
4 Goodput optimization
In this section, we consider different algorithms the transmitter can employ to optimize the code rate r, the energy allocation E _{ k } and the bit allocation m _{ k } (\(\forall k\in {\mathcal {N}}\)) such that the AEGP from (17) is maximized, while satisfying the transmit energy constraint (2) and the interference constraints (4) at the PU receivers. These algorithms assume that only imperfect CSI is available at the transmitter.
4.1 Uniform energy and bit allocation
Uniform energy and bit allocation
Optimization of E, m and r |
---|
Set AEGP_{opt}=0 |
Set \(E=\min \left (\min _{q\in \mathcal {Q}}\frac {\mathcal {I}_{q}}{\sum _{l\in {\mathcal {N}}}(|(\boldsymbol {\mu }_{\mathbf {G}|\mathbf {CSI}}^{(q)})_{l}|^{2}+(\mathbf {C}_{\mathbf {G}|\mathbf {CSI}}^{(q)})_{l,l})},\frac {E_{\text {max}}}{|{\mathcal {N}}|}\right)\) |
For \(m\in \mathcal {D}_{\mathrm {m}}\) |
Set \(m_{k}=m,\quad \forall k \in \mathcal {N}\) |
For \(r\in {\mathcal {D}}_{r}\) |
Set AEGP according to (17) |
If AEGP≥AEGP_{opt} Then |
Set AEGP_{opt}=AEGP |
Set r _{opt}=r |
Set m _{opt}=m |
End If |
End For |
End For |
4.2 Optimized energy and uniform bit allocation
According to [28], an OP is convex when both the constraints and the objective function are convex. From (24), it is clear that the constraints are convex, as they are linear in the components of E. Further, the convexity of the objective function follows from the fact that the second derivative of g _{ k,n }(E _{ k }) with respect to E _{ k } can be shown to be non-negative; hence, each term of the objective function is convex, so that the entire objective function is convex as well. Therefore, the OP of (24) can be efficiently solved by using optimization tools such as CVX [29].
For the optimization of the EGP, we slightly adapt the algorithm outlined in Table 1. We start by considering all available subcarriers as active, i.e., \(\mathcal {N}=\{1,\ldots,N\}\). For every possible TM ϕ={m,r} the algorithm computes the approximation (17) of the EGP, using as energy allocation the solution of OP (24). Because the energy allocation now depends on the parameter m, it must now become part of the outer loop of the algorithm. For a given value of m, it might happen that for some k the optimized value of E _{ k } equals 0. In this case, the corresponding subcarriers are removed from the active set \({\mathcal {N}}\) by putting m _{ k }=0, which also removes the large terms with E _{ k }=0 (i.e., γ _{ k }=0) from (15) for the considered bit allocation. Finally, the algorithm selects the TM and the corresponding energy allocation yielding the largest value of the AEGP (17).
4.3 Uniform energy and greedy bit allocation
In this subsection we consider a uniform energy allocation according to (19) and an optimized bit allocation per subcarrier.
where g _{ k,n }(m _{ k },E _{ k }) is given by (23), and the dependence on m _{ k } is shown explicitly. However, this represents a mixed integer programming problem, which is computationally very hard. In order to obtain a computationally efficient solution, we base our algorithm on the iterative suboptimal greedy algorithm described in [30].
In the current iteration, we modify the bit allocation from the previous iteration by adding 2 bits (because we restrict our attention to square QAM constellations, representing an even number of bits) to the subcarrier which leads to the smallest increase of \(\sum _{k\in {\mathcal {N}}}\mathrm {E}_{\mathbf {H}}\left [\Omega _{k}(m_{k},E_{k})|\mathbf {CSI}\right ]\). For the resulting bit and energy allocation, we determine the code rate r which leads to the highest AEGP (17). The iterative algorithm is initialized with m _{ k }=0 for all available subcarriers (yielding M(m)=0) and continues until all N available subcarriers have m _{max} bits (yielding M(m)=m _{max} N), where m _{max} is the largest allowed number of bits in the constellation. At that point, we select the code rate r and the energy and bit allocation which correspond to the value of M(m) for which the AEGP (17) is maximal.
As in this case, the set of active subcarriers equals \(\mathcal {N}\) for both the previous and the current iteration, the uniform energy allocation from (19) satisfies E(m+2e _{ k })=E(m). In the current iteration, the increments \(\delta _{k}^{m_{k}+2}(\mathbf {m})\) are computed for all k∈{1,…,N}; then, the subcarrier k which yields the lowest \(\delta _{k}^{m_{k}+2}(\mathbf {m})\) (k∈{1,…,N}) is selected, and the bit allocation for this subcarrier and M(m) are both increased by 2, compared to the previous iteration.
4.4 Suboptimal joint energy and bit allocation
The greedy bit allocation algorithm introduced in the previous subsection requires the reevaluation of the values of \(\delta _{k}^{m_{k}+2}(\mathbf {m})\) (∀k∈{1,…,N}) each time the set \(\mathcal {N}\) of active subcarriers is modified. The complexity would increase even further if we combined each step of the greedy bit allocation algorithm with the optimized energy allocation introduced in Section 4.2, which requires solving a convex optimization algorithm instead of a simple evaluation of Eq. (19).
Suboptimal joint energy and bit allocation
Optimization of E, m and r |
---|
Set AEGP_{opt}=0 |
Set r and m according to section 4.1 |
Set E according to (24) |
For k∈{1,…,N} |
For \(m_{k}\in \mathcal {D}_{\mathrm {m}}\) |
Set \(\delta _{k}^{m_{k}}\) according to (28) |
End For |
Set \(\delta _{k}^{m_{\text {max}}+2}=\infty \) |
End For |
Set m _{ k }=0 (∀k∈{1,…,N}) |
For M∈{2,4,…,m _{max} N} |
Set \(k=\arg \min \{\delta _{1}^{m_{1}+2},\ldots,\delta _{N}^{m_{N}+2}\}\) |
Set m _{ k }=m _{ k }+2 |
Update \(\mathcal {N}\) |
For \(r\in {\mathcal {D}}_{r}\) |
Set AEGP according to (17) |
If AEGP≥AEGP_{opt} Then |
Set AEGP_{opt}=AEGP |
Set r _{opt}=r |
Set m _{opt}=m |
End If |
End For |
End For |
Set E according to (24) |
5 Numerical results
System parameters
Data subcarriers (N) | 48 |
Sampling rate (1/T) | 5.6 MHz |
FFT size (N _{car}) | 512 |
Length of cyclic prefix (ν) | 64 |
Convolutional code | (133,171)_{8} |
Code rates (\({\mathcal {D}}_{\mathrm {r}}\)) | 1/2, 2/3, 3/4, 5/6 |
Constellation sizes (\({\mathcal {D}}_{\mathrm {m}}\)) | 2, 4, 6 bits |
Information bits (N _{p}) | 1024 |
CRC (N _{CRC}) | 32 |
As a performance indicator for the different resource allocation schemes, we will display (12), which denotes the average of the actual GP w.r.t. the joint probability density function of H, C S I, and C S I _{PU}. This averaging involves the generation of realizations of C S I and C S I _{PU}, from which the corresponding (m,E,r) are computed. For each such realization of (m,E,r), we generate realizations of H according to the conditional distribution p(H|C S I). For each such realization of H, we transmit and decode one packet using the transmission parameters (m,E,r) and verify whether a decoding error has occurred; averaging the indicator of a decoding error over the realizations of H yields E_{ H }[PER(ϕ,Γ)|C S I] corresponding to the considered realization of (m,E,r).
5.1 Accuracy of AEGP
In this subsection, we investigate how accurately the AEGP metric (17) approximates the EGP from (13). As a reference, we compare the accuracy with the predicted GP (PGP) introduced in [20] and the IC- κESM introduced in [19]. The PGP is obtained by neglecting the uncertainty on H given the actual CSI, and is calculated by substituting H by μ _{ H|C S I } in the expression (15) and using this deterministic value of Y to replace the random variable Z in (17). The IC- κESM is an approximation that only applies to delayed CSI. For this reason, we will compare the accuracy of these three metrics for the scenario where the transmitter only has delayed CSI available (see the “Delayed CSI” section in the Appendix). The following simulation parameters are used: SNR=10 dB, \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0\ \text {dB}\), and the value of f _{d} τ _{d} is equal to 0.05.
Accuracy of the AEGP, PGP and IC- κESM metric (SNR =10 dB, \({\mathcal {I}}_{q}/\sigma _{\mathrm {w}}^{2}=0\ \text {dB}\) and f _{d} τ _{d}=0.05)
AEGP | PGP | IC- κESM | |
---|---|---|---|
E[ ε] | 1.87×10^{−2} | 5.48×10^{−1} | 5.57×10^{−1} |
\(\sqrt {\text {Var}[\epsilon ]}\) | 2.07×10^{−2} | 2.45×10^{−1} | 1.95×10^{−1} |
\(\sqrt {\mathrm {E}[\epsilon ^{2}]}\) | 2.79×10^{−2} | 6.00×10^{−1} | 5.90×10^{−1} |
5.2 Uniform energy and bit allocation
The performance of the uniform energy and bit allocation algorithm described in Section 4.1 is investigated. As a reference, we will also show the performance in the case of perfect CSI and also for non-adaptive transmission.
Using this uniform energy allocation, the GP metric (11) is computed for each possible TM { m,r} but with PER(ϕ,Γ) replaced by PER_{ESM}(r,γ). The TM which corresponds to the largest GP is then considered optimal.
For the above energy allocation, the transmitter selects, for the current value of SNR (29), the TM { m,r} which leads to the highest value of E_{ H }[GP], with GP given by (11).
5.3 Optimized energy and uniform bit allocation
5.4 Greedy bit allocation
5.5 Computational complexity
6 Conclusions
In this paper, we have considered adaptive coding and modulation in a cognitive BIC-OFDM system, under the realistic assumption that only imperfect CSI is available. In order to tackle this problem, we introduced an optimum performance metric called the expected goodput (EGP), which is the expectation of the goodput, conditioned on the imperfect CSI.
A major advantage of this metric is that it allows the transmitter to account for the imperfections of the CSI by selecting its transmission parameters such that the best average goodput is achieved. To make the optimization of the code rate, bit and energy allocation tractable, we proposed a very accurate approximation of this performance metric, referred to as approximate EGP (AEGP). The numerical results clearly show that the ACM algorithms based on the AEGP have at least the same performance as the non-adaptive algorithms and, in most cases, clearly outperform them. Finally, we also show that, depending upon the quality of the available CSI, the proposed algorithms can come very close to the performance of algorithms with perfect CSI.
7 Endnotes
^{1} This EGP metric is different from the expected effective goodput metric proposed in [18]. The metric introduced in [18] takes into account the expected transmission time, which can vary because of the possibilities of retransmissions. It has however nothing to do with imperfect CSI which is the focus of the present paper.
^{2} Note that if we have a number of paths L<ν+1, only L diagonal elements of R _{ h } are strictly greater than 0.
8 Appendix
8.1 Examples of different types of CSI at the transmitter
In the following, the impulse response of a generic channel between the SU transmitter and any receiver of the PU or SU network will be denoted by h(m,t), where the delay variable is represented by the discrete time index m associated with a sampling rate 1/T, and the time variability of the channel is indicated by a continuous time index t. Without any loss of generality, we can assume that h(m,t)=0 for m<0 and for m>ν, where ν is defined as the length of the cyclic prefix. For given t, the samples h(m,t) (0≤m≤ν) of the channel impulse vector \(\mathbf {h}(t){\overset {\Delta }{=}}[h(0,t),\ldots,h(\nu,t)]^{T}\) are assumed to be independent circular symmetric zero-mean Gaussian complex random variables; assuming stationarity w.r.t. the variable t, the covariance matrix of h(t) is given by^{2} \(\mathbf {R}_{h}\overset {\Delta }{=}\text {diag}({\sigma _{0}^{2}},\ldots,\sigma _{\nu }^{2})\). The time variations of the channel are described by Jakes’ model [33], which gives \(\mathrm {E}\left [h(m,t+\tau _{\mathrm {d}})h^{*}(m,t)\right ]=\mathrm {J}_{0}(2\pi f_{\mathrm {d}}\tau _{\mathrm {d}}){\sigma _{m}^{2}}\), where J_{0}(x) represents the zeroth-order Bessel function of the first kind, and f _{d} denotes the Doppler spread.
the time-varying frequency response of the channel can then be written as H(t)=F h(t) which has the covariance matrix R _{ H }=F R _{ h } F ^{ H }. The kth component of H(t) denotes the channel gain which affects the kth subcarrier at time instant t.
where \(\mathbf {n}(t)\sim {\mathcal {C}N}(0,\mathbf {C}_{\mathbf {H}|\mathbf {CSI}})\). The probability density function p(H(t)|C S I) is then given by \({\mathcal {C}N}(\boldsymbol {\mu }_{\mathbf {H}|\mathbf {CSI}}(t), \mathbf {C}_{\mathbf {H}|\mathbf {CSI}})\). If only N of the N _{car} subcarriers are available at the transmitter, as is the case in the numerical section, we can define a smaller μ _{ H|C S I } and C _{ H|C S I } which only contain the elements corresponding to the available subcarriers.
8.1.1 Estimated CSI
Note that in the case of perfect estimation (i.e., σ ^{2}=0) we obtain perfect CSI, as (34), (35) and (36) reduce to \(\tilde {\mathbf {H}}(t)=\mathbf {H}(t)\), μ _{ H|C S I }=H(t) and C _{ H|C S I }=0.
8.1.2 Delayed CSI
When τ _{ d }=0, we obtain perfect CSI, as (37) and (38) reduce to μ _{ H|C S I }=H(t) and C _{ H|C S I }=0.
8.1.3 Estimated and delayed CSI
Declarations
Acknowledgements
J. Van Hecke is supported by a Ph. D. fellowship of the Research Foundation Flanders (FWO).
This work was supported by the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications NEWCOM# (Grant agreement no. 318306), and the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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