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In this paper, we derive feedback power control strategies for blockfaded multiple access schemes with correlated sources and joint channel decoding (JCD). In particular, upon the derivation of the feasible signaltonoise ratio (SNR) region for the considered multiple access schemes, i.e., the multidimensional SNR region where errorfree communications are, in principle, possible, two feedback power control strategies are proposed: (i) a classical feedback power control strategy, which aims at equalizing all link SNRs at the access point (AP), and (ii) an innovative optimized feedback power control strategy, which tries to make the network operational point fall in the feasible SNR region at the lowest overall transmit energy consumption. These strategies will be referred to as “balanced SNR” and “unbalanced SNR,” respectively. While they require, in principle, an unlimited power control range at the sources, we also propose practical versions with a limited power control range. We preliminary consider a scenario with orthogonal links and ideal feedback. Then, we analyze the robustness of the proposed power control strategies to possible nonidealities, in terms of residual multiple access interference and noisy feedback channels. Finally, we successfully apply the proposed feedback power control strategies to a limiting case of the class of considered multiple access schemes, namely a central estimating officer (CEO) scenario, where the sensors observe noisy versions of a common binary information sequence and the AP's goal is to estimate this sequence by properly fusing the softoutput information output by the JCD algorithm.
Wireless multiple access schemes, where correlated signals, observed at different nodes, need to be transferred to one or more collectors, model several communication scenarios. For example, these schemes apply to wireless sensor networks, where a set of nodes collect and transmit correlated data to a common sink [
An alternative solution to distributed source coding is based on joint source channel coding (JSCC) schemes, where the correlated sources are not source encoded but only channel encoded. If one compares a JSCC system with a system based on source/channel coding separation with the same information rate, the channel codes used in a JSCC scheme must be less powerful (i.e., they have higher rates). In fact, this apparent weakness is compensated by exploiting the source correlation at the decoder, which jointly recovers the information signals of all sources. For this reason, this approach is also referred to as joint channel decoding (JCD). In this case, it can be shown that the final system performance can approach the theoretical limits. This approach has attracted the attention of several researchers in the recent past, also because of its implementation simplicity [
In the introduced scenario, we study the performance of wireless multiple access schemes, with binary correlated sources communicating to an AP and with block faded communication links. It is well known that the presence of blockfaded channels may dramatically degrade the performance of wireless multiple access systems, unless some countermeasures are taken at the transmitters to protect highly faded links. For instance, the performance of multiple access schemes can be improved by the use of “feedback.” In general terms, the AP can provide the sources with supplementary information (e.g., on the links' states) to allow them to counteract the effects of fading. From an informationtheoretic viewpoint, while feedback does not increase the capacity of a memoryless channel with one sender and one receiver [
In this paper, we refer to block faded multiple access schemes with correlated sources and JCD. In particular, we consider serially concatenated convolutional coding (SCCCing) or lowdensity paritycheck (LDPC) coding at the sources. We first investigate, in the absence of non idealities (besides fading), feedback power control strategies which can guarantee theoretically errorfree communications, i.e., that the system operational point lies in the feasible signaltonoise ratio (SNR) region of the multiple access channel. In this context, we first propose a classical feedback power control strategy, which tends to balance (i.e., equalize) the link SNRs and is optimal in traditional transmitting scenarios where the sources are independent. Then, we derive an innovative optimized power control strategy, which makes the system operate in the feasible SNR region at the lowest transmit energy consumption. It will be shown that the latter strategy leads to “unbalanced” target SNRs at the correlated sensors and, to the best of our knowledge, this is a novel result. The impact of possible nonidealities, in terms of residual multiple access interference and noisy feedback channels, on the performance of multiple access schemes using the proposed feedback power control strategies is also investigated. Even in this case, it will be shown that the unbalanced SNR strategy is still to be preferred. Finally, we apply the proposed feedback power control strategies to a socalled central estimating officer (CEO) problem, which can be interpreted as a limiting case of the general class of considered multiple access scenarios [
This paper is structured as follows. In Section 2., we describe the considered multiple access scheme. In Section 3., we first derive the power control strategies with unlimited transmit power, using both balanced SNR and unbalanced SNR strategies. Section 4. is devoted to the description of the JCD iterative decoding scheme at the AP and to the simulationbased performance analysis of the proposed multiple access schemes with feedback power control. In Section 5., we investigate the robustness of the proposed feedback power control strategies with respect to errors in the power control commands and the possible presence of residual multiple access interference. In Section 6., the proposed framework for multiple access schemes is extended to encompass, at the AP, the presence of information fusion after feedback power control, i.e., to a CEO scenario. Finally, in Section 7. we provide concluding remarks.
Consider
According to the chosen correlation model, the apriori joint probability mass function (PMF) of the information signals at the inputs of the
In
We preliminary consider a system with orthogonal links. This is meaningful for wireless sensor networking scenarios with reservationbased medium access control (MAC) protocols, such as time/frequency division multiple access (TDMA/FDMA). The use of these protocols allows to represent the multiple access channel as a set of parallel orthogonal channels [
Under the above assumptions, after matched filtering and carrierphase estimation the real observable at the AP, relative to a transmitted sample, can be expressed as
Upon reception of the signals transmitted from all sources, the goal of the AP is to reconstruct each information signal by exploiting the source correlation. In order to do this, JCD schemes for twosource scenarios with systematic channel coding at each source and iterative decoding at the AP have been proposed [
In
It is well known that distributed source coding allows to reduce the amount of data to be transmitted to the AP without needing extra intersensor communications. In particular, the performance achievable by distributed source coding is identical to that which could be achieved if the sources were encoded jointly. The SW theorem allows to determine the achievable rate region for the case of separate lossless encoding of correlated sources. Denoting by
By exploiting the well known relation between joint and conditional entropies [
In [
As discussed in Subsection 2.1., according to the considered blockfaded sensorAP channel model, the instantaneous perlink SNR at the AP is subject to perpacket fading fluctuations due to the timevarying nature of the channel. A feedback power control strategy for a multiple access scheme consists of a rule, depending on the (ideally perfectly estimated) links' statuses, according to which a power control command is sent, by the AP, to each sensor. Equivalently, the power control strategy is based on the determination of proper target SNRs at the AP, denoted as
The classical approach for power control in multiple access systems tries to “balance” the SNRs at the AP over all possible links. More precisely, the AP fixes a
If multiple access schemes with uncorrelated sensors are considered, i.e.,
We now derive an optimized a transmit power allocation strategy which allows to achieve a feasible operational point at the lowest overall energy cost. In this case, the power control strategy can be cast into the following optimization problem with respect to the unknown vector of link capacities
We now make a comment on the power control strategies described so far. The AP carries out its optimization strategy determining, after solving of (
Before evaluating extensively, through simulations, the performance of the proposed feedback power control strategies in the following section, in
In subfigure (a), we show the results obtained by applying the optimized (unbalanced) power control strategy. The target SNRs at the AP (shown as green bars) are the following:
In subfigure (b), we show the results obtained by applying the balanced SNR power control strategy. The minimum common target SNR at the AP, given by (
The proposed illustrative comparison shows the benefits which can be obtained by properly unbalancing the target SNRs in the various links according to the actual channel conditions. In fact, in both power control strategies, setting the target SNRs at the transmitters as indicated makes the network operational point fall in the feasible SNR region, thus allowing theoretically errorfree communications. The unbalanced SNR power control strategy, however, guarantees a given performance level at a lower energy cost than that required by the balanced SNR power control strategy.
The proposed feedback power control strategies require that a source might need to increase, in principle, its transmit energy without limit–for instance, this might be the case over a link characterized by an extremely small fading coefficient. Moreover, the proposed power control schemes assume the presence of an ideal communication (transmission and reception) scheme which achieves the system capacity bounds. Therefore, the power allocation strategies proposed so far will lead to reference performance results. In the remainder of this subsection, practical versions of the balanced SNR and unbalanced SNR feedback power control strategies are proposed, such that (i) the sources can adapt their transmit energies within a
For both practical versions of the proposed power control schemes, we assume that each node sends a pilot signal to the AP at a fixed initial SNR defined as follows:
In particular, we assume that the maximum perbit energy variation which can be carried out by a source is Δ
It is worth noting that the proposed power control scheme compares favorably with classical power control schemes (see, for example, [
As for the reliability of the feedback power control command, in the presence of quasistatic channels it is reasonable to assume errorfree feedback channels. More precisely, this can be obtained by protecting the power control commands through the use of lowrate channel codes, e.g., repetition codes. On the basis of these considerations, in Section 4. we will consider errorfree (ideal) feedback channels. However, the impact of noisy feedback channels will be investigated in Subsection 5.2..
As described in Section 2., the information sequences are separately encoded using the same channel code (either an LDPC code or a SCCC) and transmitted over the communication links. In all cases, we assume that the common coding rate at the sources is
Under the assumption of perfect channel state information at the receiver, the channel LLR, relative to the
The total LLR relative to the
It is immediate to observe that the proposed iterative JCD scheme has a complexity (measured in terms of basic operations, such as additions and multiplications) which, for a single external iteration, is a linear function of the number
We remark that the above complexity level is realistic in the presence of digitization and decoding. In the case of real sources, one should perform more complicated (and computationally heavier) marginalization process to convert bit probabilities into symbol probabilities, and vice versa (see, for example, [
In LDPCcoded scenarios, each of the source sequences is encoded using (i) a regular (3,6) LDPC code or (ii) an irregular LDPC code with double diagonal (DD) square submatrix in the parity check matrix [
In turbolike coded scenarios, we consider a SCCC given by the concatenation, through a bit random interleaver, of an outer convolutional code with an inner convolutional code [
In all cases (LDPCcoded and SCCCed), we directly compare scenarios without feedback power control (W/o PC) and with feedback power control (W PC). Moreover, in the W PC case the performance is evaluated considering the two proposed power control strategies (balanced SNR and unbalanced SNR) and different values of the number
In
In
In
In
In this section, we investigate the robustness of the proposed feedback power control strategies against possible nonidealities. In particular, we analyze the performance in the presence of (i) mutual interference between the transmitting nodes (i.e., when the multiple access links are not perfectly orthogonal) and (ii) noisy feedback channels. Without loss of generality, a simple illustrative scenario with
In order to investigate the impact of nonorthogonality between the communication links, we consider the presence of mutual interference between the transmitted signals. Since an accurate characterization of the multiple access interference is beyond the scope of this paper, the residual interference is simply modeled as AWGN. Under this assumption, referring to
In
In
Referring to the feedback power control commands (given by binary sequences) described in Section 3., we assume that each bit of the “up/down” binary sequence can be “flipped” with probability
In
In
So far we have been considering general scenarios where the sources are correlated. A particular case of these scenarios can be observed when the
Under the assumption of a single common source, we consider the scheme shown in
In the remainder of this section, we first derive the fusion rule to be used in the corresponding block, and then we investigate the performance of the feedback power control strategies in the presence of information fusion.
Denote as
Finally, the fusion rule (
It is of interest to evaluate the probability of decision error when the channel SNR becomes very high. In this case, in fact, the iterative JCD scheme allows to recover perfectly the effectively transmitted sequence, denoted as
In
In
In this paper, we have derived feedback power control strategies and evaluated their impacts on the performance of blockfaded multiple access schemes with JCD. In all cases, the use of feedback power control is expedient to set the network operational point in the feasible SNR region. First, we have derived a classical power control strategy which tries to equalize the link SNRs at the AP. Then, we have derived an innovative optimized feedback power control strategy, which allows the system operational point to lie in the feasible SNR region at the lowest overall transmit energy cost. In this case, the SNRs are typically unbalanced. Our results show that both feedback power control strategies significantly improve the performance, with respect to schemes without power control. In particular, the unbalanced SNR feedback power control strategy guarantees a performance better than that with the balanced SNR power control strategy, and this is more pronounced when a proper channel code is used (namely, a properly designed SCCC). We have then analyzed the robustness of the proposed power control strategies in the presence of nonidealities, in terms of residual multiple access interference and noisy feedback channels. Our results show that even in nonideal scenarios the best feedback power control strategy is to unbalance the target SNRs. Finally, we have applied the proposed feedback power control strategies to a limiting case of the considered multiple access scheme, obtaining a CEO scenario where the
Multiple access scheme with feedback.
Feasible region in a scenario with two sources and
Illustrative comparison, in terms of target SNRs at the AP and at the transmitters, between (a) ideal unbalanced SNR and (b) ideal balanced SNR feedback power control strategies in a scenario with
Iterative JCD scheme at the AP in the presence of
Average BER, as a function of the average SNR at the sources, considering: (a) regular LDPCcoded schemes, (b) DD LDPCcoded schemes, and (c) SCCCed schemes. The performance in the presence of the balanced SNR power control strategy is compared with that associated to the absence of power control.
Average BER, as a function of the average SNR at the sources, considering: (a) regular LDPCcoded schemes, (b) DD LDPCcoded schemes, and (c) SCCCed schemes. Both balanced SNR and unbalanced SNR power control strategies are considered.
Outage probability, as a function of the average SNR at the sources, considering: (a) regular LDPCcoded schemes, (b) DD LDPCcoded schemes, and (c) SCCCed schemes. The performance in the presence of the balanced SNR power control strategy is compared with that associated with the absence of power control.
Outage probability, as a function of the average SNR at the sources, considering: (a) regular LDPCcoded schemes, (b) DD LDPCcoded schemes, and (c) SCCCed schemes. Both balanced SNR and unbalanced SNR power control strategies are considered.
Average BER, as a function of the average SNR at the sources, in a scenario with nonorthogonal links, considering: (a) DD LDPCcoded schemes and (b) SCCCed schemes. Two possible values for
Outage probability, as a function of the average SNR at the sources, in a scenario with nonorthogonal links, considering: (a) DD LDPCcoded schemes and (b) SCCCed schemes. Two possible values for
Average BER, as a function of the average SNR at the sources, in a scenario with noisy feedback channels, considering: (a) DD LDPCcoded schemes and (b) SCCCed schemes. Two possible values for
Outage probability, as a function of the average SNR at the sources, in a scenario with noisy feedback channels, considering: (a) DD LDPCcoded schemes and (b) SCCCed schemes. Two possible values for
CEO scenario: multiple access scheme followed by fusion.
Limiting (for large SNR) probability of decision error as a function of the correlation coefficient, for the CEO scenario. Different values for the number of sensors
Error probability, as a function of the average SNR at the sources, for the CEO problem in a scenario with
Practical feedback power control commands and energy corrections. The maximum energy correction is denoted as Δ
Δ 
Binary Feedback Command  

 

−Δ 

…  …  … 

−2Δ 
11 

−Δ 
1 

Δ 
+1 

+2Δ 
+1+1 
…  …  … 

+Δ 

For the sake of notational simplicity, the derivation is carried out considering a single packet transmission act, i.e., we do not use any index to indicate the specific packet.
Note that, for a fixed symbol duration (i.e., transmitting rate), a power variation is in a onetoone correspondence with an energy variation.
Note that the internal iterations in each component subdecoder refer to (i) the iterations between the variable nodes and the check nodes in the presence of LDPC coding and the SP algorithm or (ii) the turbo iterations between convolutional decoders in the presence of turbo coding and the BCJR algorithm (at each convolutional decoder).
Since the a priori probabilities need to be evaluated for the systematic bits, in this case
Note that the number of internal iterations is fixed with SCCCing, whereas it can vary in the LDPC coded case. Note also that the number
Note that the exact statistics of the residual multiple access interference should be better investigated. This goes beyond the scope of this paper. However, the Gaussian approximation allows to have useful insights on the impact of the multiple access interference on the proposed feedback power control strategies.