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Distributed estimation in wireless sensor networks with semiorthogonal MAC
 Jian Su^{1} and
 Ha H. Nguyen^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s136380160716z
© The Author(s) 2016
 Received: 23 February 2016
 Accepted: 27 August 2016
 Published: 8 September 2016
Abstract
This paper is concerned with distributed estimation of a scalar parameter using a wireless sensor network (WSN) that employs a large number of sensors operating under limited bandwidth resource. A semiorthogonal multipleaccess (MA) scheme is proposed to transmit observations from K sensors to a fusion center (FC) via N orthogonal channels, where K≥N. The K sensors are divided into N groups, where the sensors in each group simultaneously transmit on one orthogonal channel (and hence the transmitted signals are directly superimposed at the FC as opposed to be coherently combined). Under such a semiorthogonal multiple access channel (MAC), performance of the linear minimum mean squared error (LMMSE) estimation is analyzed in terms of two indicators: the channel noise suppression capability and the observation noise suppression capability. The analysis is performed for two versions of the proposed semiorthogonal MA scheme: fixed sensor grouping and adaptive sensor grouping. In particular, the semiorthogonal MAC with fixed sensor grouping is shown to have the same channel noise suppression capability and two times the observation noise suppression capability when compared to the orthogonal MAC under the same bandwidth resource. For the semiorthogonal MAC with adaptive sensor grouping, it is determined that N=4 is the most favorable number of orthogonal channels when taking into account both performance and feedback requirement. In particular, the semiorthogonal MAC with adaptive sensor grouping is shown to perform very close to that of the hybrid MAC, while requiring only log2N=2 bits of information feedback instead of the exact channel phase for each sensor.
Keywords
 Wireless sensor networks
 Distributed estimation
 Multiple access channel
 Multiple access scheme
1 Introduction
Wireless sensor networks (WSNs) have found applications in diverse areas such as environmental data gathering [1], industrial monitoring [2] and monitoring of smart electricity grids [3], and mobile robots and autonomous vehicles [4]. Such widespread applications of WSNs are made possible by advances in wireless communications and highspeed lowpower electronics, which makes WSNs inexpensive, compact and versatile [5]. All these applications of WSNs are based on the same fundamental task of sampling (i.e., observing) some signal parameter using sensors geographically distributed over a field and estimating the parameter of interest using a central processing unit (fusion center). Such a signal processing task is generally known as distributed estimation.
To perform distributed estimation using a WSN, each sensor makes an observation of the quantity of interest, generates a local signal, and then sends it to a fusion center (FC) via a wireless fading channel. Based on the data collected from the sensors, the FC produces a final estimate of the desired quantity according to some fusion rule. An important design consideration for a WSN is the transmission method from the sensors to the fusion center, which can be analog or digital. With analog transmission, each sensor amplifies and forwards its observation to the FC. On the other hand, for digital transmission, each sensor performs source and channel coding before transmitting the encoded information over the fading channel (see [6] and references therein). According to the studies in [7–17], analog transmission generally outperforms digital transmission. This is because the fidelity of the source’s parameter is always compromised in the source coding (quantization) process required for digital transmission, while it is preserved with analog transmission. As such, this paper also focuses on distributed estimation in WSNs based on analog transmission.
There are many factors that affect the performance of distributed estimation. These include the accuracy of sensors’ observations (which is usually modeled as observation noise), the available bandwidth and power resources, the fading characteristics of the wireless channels between sensors and the FC, the fusion rule used by the FC, and the type of multiple access channel (MAC)^{1} used to communicate the sensors’ observations to the FC. To date, there are three types of MAC commonly considered for distributed estimation: coherent, orthogonal, and hybrid. For these MACs, it is required that the responses of all wireless channels connecting the sensors to FC be estimated at the FC (typically via the use of training signals) and used in the distributed estimation algorithm. Such channel estimation at the FC is assumed to be perfect for all the MACs considered in this paper. On the other hand, whether the sensors require any channel state information (CSI) depends on the type of MAC.
For the coherent MAC studied in [8], the sensors’ observations are coherently combined and transmitted to the FC on one channel. Although the coherent MAC appears to be very bandwidth efficient, it requires that each sensor needs to know the wireless channel response from it to the FC so that synchronization among sensors can be established. This requirement presents a serious challenge in a practical implementation of the coherent MAC since the channel responses need to be measured at the FC and fed back to the sensors. Such a feedback overhead can be very significant for a large WSN. The impact of imperfect synchronization corresponding to phase errors is investigated in [18], where a masterslave architecture is also proposed to reduce the synchronization overhead. It should also be pointed out that phase modulation is investigated in [19] for coherent transmission of sensor observations to the FC, but without the important consideration of fading.
In contrast to the coherent MAC, for the orthogonal MAC examined in [7], all K sensors in the network transmit their observations to the FC via K orthogonal channels, which can be realized with orthogonal frequencydivision or timedivision multiplexing. The orthogonal MAC does not require synchronization among sensors, and hence is more favorable for implementation. The major disadvantage of the orthogonal MAC is that it requires larger transmission bandwidth or latency to accommodate K orthogonal channels. More recently, a hybrid MAC is investigated in [17], where all sensors are divided into groups and the coherent MAC is used for sensors within each group, whereas the orthogonal MAC is used across different groups. A flexible tradeoff between the coherent and orthogonal MACs can therefore be obtained by changing the number of groups and the number of sensors in each group. However, in such a hybrid MAC, synchronization among sensors within the same group is still required and the amount of channel information feedback from the FC to the sensors is the same as that of the coherent MAC.
This paper proposes and investigates the use of another type of MAC, referred to as a semiorthogonal MAC for distributed estimation. The proposed semiorthogonal multiple access (MA) scheme aims to improve the performance of distributed estimation under a limited bandwidth constraint. Specifically, considered is a scenario where N orthogonal channels are shared by K sensors to transmit the their observations to the FC, where the N is much smaller than K due to bandwidth constraint. While the semiorthogonal MA scheme is designed based on the similar idea of sensor grouping in [17], the key difference is that, in the proposed MA scheme, the sensors in one group transmit simultaneously without the expensive phase synchronization operation. This means that the signals from sensors within one group are directly superimposed instead of coherently combined as in the hybrid MAC.
The proposed semiorthogonal MA scheme can be implemented with either fixed or adaptive sensor grouping. In fixed sensor grouping, each sensor transmits on fixed orthogonal channels. In general, more than one orthogonal channel can be allocated to one sensor. However, it shall be shown that such channel allocation causes correlation among the equivalent channel responses ^{2} and degrades the estimation performance. As such, fixed sensor grouping should be done in such a way that the groups are disjoint. For adaptive sensor grouping, sensors are grouped according to the ranges (i.e., subregions) that their channel phases fall into. The extra cost for implementing adaptive sensor grouping is only log2N bits of feedback information from the FC to each sensor to indicate channel allocation. This amount of feedback overhead is significantly smaller than the phase values (real numbers) of the channel responses required in the coherent and hybrid MA schemes. It will be shown that, compared to fixed sensor grouping, the estimation performance achieved with adaptive grouping is improved by a large margin. In fact, the performance of the semiorthogonal MA scheme with adaptive grouping is very close to the performance of the hybrid MA scheme under the same bandwidth and power constraints and the same number of sensors.
There has been extensive study of distributed estimation using a WSN. To put the novelty and contributions of the current work in context, key papers on distributed estimation that are related to the study in the present paper are discussed next. The seminal work in [20] analyzes the fundamental tradeoffs between the number of sensors, their total transmit power, the number of degree of freedom of the source, the spatiotemporal communication bandwidth, and the endtoend distortion under a coherent MAC. An important result established in [20] is that, for typical situations, the distortion goes down at best like 1/K, where K is the number of sensors. For the case of a simple “Gaussian” sensor network, where a single memoryless Gaussian source is observed by many sensors subject to independent Gaussian observation noises and the sensors are linked to a fusion center via a coherent Gaussian MAC, it is shown in [10] that uncoded transmission is strictly optimal, rather than only in the scaling law sense of 1/K. The system model of distributed estimation considered in the present paper is similar to the simple Gaussian sensor network in [10], albeit the novel semiorthogonal MAC is used instead of the coherent MAC. In fact, it is shown in ([21] Chapter 5) that the semiorthogonal MAC with adaptive sensor grouping achieves the optimal scaling law.
It should be pointed out that designing optimal transmit power/energy allocation strategies for WSNs has been an active area of research in recent years. For example, reference [7] considers a WSN with the orthogonal MAC and derives the optimal power allocation policies in a way that the total distortion is minimized subject to a sum power constraint at the sensors. The work in [22] considers the same orthogonal MAC as [7] but instead of minimizing the total distortion, total transmission power is minimized under distortion constraints. Wu and Wang [23] studies power allocation taking into account sensing noise uncertainty, whereas the optimal power allocation for linear estimation over coherent MAC has been considered in [8]. All the studies on power allocation in [7, 8, 22, 23] focus on sensors equipped with conventional batteries with fixed energy storages. More recently, reference [5] addresses the problem of optimal power allocation to efficiently estimate a random source using distributed wireless sensors equipped with energy harvesting technology. Since this paper focuses on the impact of different MACs, the simple equal power allocation shall be considered throughout.^{3}
Before closing this section, it is important to stress that in a truly distributed estimation framework, the objective is to coordinate all the sensors so that without communicating with one another, they collectively maximize the quality of estimation at the FC [24]. In contrast to such truly distributed estimation, collaborative estimation is considered in [25], where the network is divided into a set of sensor clusters, with collaboration allowed among sensors within the same cluster, but not across clusters. It is shown in [25] that when the channels to the FC are orthogonal and costfree collaboration is possible within each cluster, the optimum collaboration strategy is to perform the inference in each cluster and use the best available channel to transmit the estimated parameter (local message) to the FC. The optimum power allocation among the clusters is also found in [25] and shown to operate in a waterfilling manner. Kar and Varshney [24] considers a similar sensor collaboration paradigm as in [25] but using a coherent MAC for communications from sensors to FC. The authors obtained the optimum cumulative powerdistortion tradeoff when a fixed but otherwise costfree collaboration topology is used and addressed the design of collaborative topologies where finite costs are involved in collaboration. Instead of having all the sensors collaboratively observe an (single) underlying scalar parameter, reference [26] derives an optimum power allocation scheme among collaborative sensors for the case that the sensors observe individual signals that are spatially correlated. While it is intuitively expected that sensor collaboration helps to reduce the distortion of the estimated parameter(s) at the FC, it comes at significant costs of providing reliable communication links among the collaborative sensors as well as extra signal processing (linear combination of shared observations) for each sensor cluster. As such, sensor collaboration is not considered in the present paper.
The remaining of this paper is organized as follows. Section 2 describes the system model of distributed estimation under the coherent, orthogonal, hybrid, and semiorthogonal MACs. Section 3 proposes two sensor grouping approaches for the semiorthogonal MAC. Section 4 analyzes and compare the estimation performance under the various MACs. Section 5 presents numerical results and discussion. Section 6 concludes the paper.
2 System model
where the source signal s and observation noise v _{ i } are treated as random variables with zero mean and variances \({\sigma ^{2}_{s}}\) and \( {\sigma ^{2}_{v}}\), respectively. The observation signaltonoise ratio (SNR) is defined as \(\gamma _{\mathrm {o}} = \frac {\sigma ^{2}_{s}}{\sigma ^{2}_{v}}\).
Using analog modulation, the ith sensor simply amplifies x _{ i } with a gain a _{ i } and transmits the result to the FC. The total transmit power in this WSN is \(P_{\text {tot}}=\sum ^{K}_{i=1} {a^{2}_{i}} \left (\sigma ^{2}_{s} + \sigma ^{2}_{v}\right)\). The communication channels from sensors to the FC are considered to be wireless fading channels. Let \(h_{i}= r_{i} {\mathrm e}^{j \varphi _{i}}\), i=1,…,K, represent the channel response from sensor i to the FC. These channel responses are modeled as independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance, denoted as \(\mathcal {CN}(0,1)\). This also means that the magnitude r _{ i } and phase φ _{ i } are independent random variables with Rayleigh and uniform distributions, respectively. On each wireless fading channel, the transmitted signal is disturbed by additive white Gaussian noise (AWGN). The AWGN sample, denoted as ω, is modeled as a complex Gaussian random variable with zero mean and variance \(\sigma ^{2}_{\omega }\). The channel SNR is defined as \(\gamma _{\mathrm {c}} = \frac {P_{\text {tot}}}{\sigma ^{2}_{\omega }}\).
The received signal at the FC is generally denoted by y=[y _{1},y _{2},…,y _{ N }]. The dimension N and expression of y in terms of a _{ i }, x _{ i }, h _{ i } and ω depend on the type of multiple access channel (MAC) realized from the sensors to the FC. This is elaborated in the below subsections. Regardless of the type of MAC, the task of the FC is to estimate the underlying source signal based on the received signal y. Using the linear minimum mean square error (LMMSE) estimator [27] and assuming that all the channel responses are available at the FC, the estimation of s is \(\check {s} = \mathbf {C}_{s \mathbf {y}} \mathbf {C}^{1}_{\mathbf {y} \mathbf {y}} \, \mathbf {y}\), where C _{ s y } is the covariance between s and y and C _{ y y } is the covariance of y. The corresponding MSE is \(\epsilon ={\sigma _{s}^{2}}\mathbf {C}_{s \mathbf {y}} \mathbf {C}^{1}_{\mathbf {y} \mathbf {y}} \mathbf {C}_{\mathbf {y} s}\), which depends on the specific realizations of channel responses h _{ i }’s. The longterm performance of a WSN is evaluated using the average MSE (AMSE), defined as \(\text {AMSE} = \mathcal {E} \left \{ \epsilon \right \}\), where the expectation is taken over channel response realizations.
2.1 Coherent MAC
2.2 Orthogonal MAC
where \(\bar {r}_{i} = a_{i} r_{i}\), \(\bar {v}_{i} = a_{i} v_{i} r_{i}\) and \(\bar {\omega }_{i} = \mathcal {R} \left \{ \omega {\mathrm e}^{j \varphi _{i}} \right \}\).
2.3 Hybrid MAC
With the hybrid MAC considered in [17], all sensors are divided into groups and the coherent MAC is used for sensors within each group, whereas the orthogonal MAC is used across different groups. This MAC provides a solution for scenarios where there are N (a small number due to bandwidth constraint) orthogonal channels that are shared by K sensors, where K≥N. In this hybrid MAC, to obtain coherent combination in each group, channel phase information feedback from the FC to the sensors is still required and the amount of feedback is the same as that of the coherent MAC. In addition, the required transmission bandwidth in this MAC depends on the number of sensor groups.
2.4 Semiorthogonal MAC
where ω _{ n }’s are the i.i.d. (over n) complex AWGN components with zero mean and variance \(\sigma ^{2}_{\omega }\).
The above phase compensation discards halves of observation noise and channel noise. It is pointed out that the phase compensation of the equivalent channel response is performed at the FC. Therefore, no phase information is needed at the sensors and feedback of channel phase information is not required.
The parameter α indicates the impact of channel noise on the MSE performance. The larger α is, the lesser the impact is.
In this case, the parameter β indicates the impact of observation noise on the MSE performance. The larger β is, the lesser the impact is.
3 Sensor grouping methods for semiorthogonal MAC
3.1 Fixed sensor grouping
Fixed sensor grouping means that the assignment of orthogonal channels, once decided, does not change during the communication phase. When assigning N orthogonal channels to K sensors, where K≥N, an obvious question arises: Should more than one orthogonal channel be assigned to a single sensor and will this improve the MSE performance of distributed estimation?
To answer the above question, let us first examine a simple scenario where there are two orthogonal channels (N=2) with K _{1} sensors transmitting on each of them. Under equal power allocation, the gain factor is \(a_{i} = \bar {a} = \sqrt {\frac {P_{\text {tot}}}{2 K_{1} \left ({\sigma ^{2}_{s}} + {\sigma ^{2}_{v}} \right)}}\). Note that there are M= max(2K _{1}−K,0) sensors that transmit on both orthogonal channels. Treating the equivalent channel responses \(\hat {h}_{1} = \bar {a} \sum \limits ^{K}_{i=1} g^{(i)}_{1} h_{i}\) and \(\hat {h}_{2} = \bar {a} \sum \limits ^{K}_{i=1} g^{(i)}_{2} h_{i}\) as random variables, the correlation coefficient between \(\hat {h}_{1}\) and \(\hat {h}_{2}\) is easily found to be \(\rho =\frac {M}{K_{1}}\). If there is no sensor transmitting on both orthogonal channels, M=0 and the above correlation coefficient will be zero. However, such scenario requires that K=2K _{1} and all K sensors are equally divided into two disjoint groups with sensors in each group transmitting on one orthogonal channel.
Next, define \(\tilde {h}_{1}=\frac {1}{\sqrt {2 K_{1}}} \sum ^{K}_{i=1} g^{(i)}_{1} h_{i}\) and \(\tilde {h}_{2}=\frac {1}{\sqrt {2 K_{1}}} \sum ^{K}_{i=1} g^{(i)}_{2} h_{i}\), which are basically the scaled versions of \(\bar {h}_{1}\) and \(\bar {h}_{2}\) defined in (16). Then, the parameter α is \(\alpha =  \tilde {h}_{1} ^{2} +  \tilde {h}_{2} ^{2}\). Since each of \(\tilde {h}_{1}\) and \(\tilde {h}_{2}\) is a \(1/\sqrt {2K_{1}}\) times the sum of K _{1} i.i.d. complex Gaussian random variables, each with zero mean and unit variance, it is a complex Gaussian random variable with zero mean and variance 1/2. It follows immediately that the expected value of α is \(\mathcal {E}\{\alpha \}=1\). To find the expression for β, express \(\tilde {h}_{1}\) and \(\tilde {h}_{2}\) as \(\tilde {h}_{1}=\frac {m_{1}}{\sqrt {2}} \mathrm {e}^{j \phi _{1}}\) and \(\tilde {h}_{2}=\frac {m_{2}}{\sqrt {2}} \mathrm {e}^{j \phi _{2}}\). Then, Appendix A shows that, when K approaches infinity, \(\beta =\frac {2 \left [ {m^{2}_{1}}  2 \rho \cos \left (\phi _{1}  \phi _{2} \right) m_{1} m_{2} + {m^{2}_{2}} \right ]}{1  \rho ^{2} \cos ^{2} \left (\phi _{1}  \phi _{2} \right)}\). The expectation of β is more tedious to obtain, and it is given in (61) of Appendix A.
Values of \(\mathcal {E}\{\beta \}\) with N=2
ρ  0.05  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  0.95 

Theory  3.984  3.969  3.909  3.812  3.680  3.518  3.333  3.133  2.930  2.741  2.655 
Simulation  3.998  3.987  3.923  3.816  3.689  3.528  3.356  3.150  2.948  2.789  2.703 
For N>2, it is not easy to determine channel allocation among sensors and perform the corresponding correlation analysis among the equivalent channel responses of the orthogonal channels. Instead, the following example of ad hoc channel allocation scheme shall be investigated. Assume that K is an integer multiple of N. If \(\frac {\left (n1\right)K}{N}+K_{1} \leq K\), then the nth orthogonal channel is shared by sensors with indices in the set of \(\left \{ \frac {\left (n1\right)K}{N}+1, \ldots,\frac {\left (n1\right)K}{N}+K_{1} \right \}\). If \(\frac {\left (n1\right)K}{N}+K_{1} > K\), then the set of sensor indices is \(\left \{ 1, \ldots, \frac {\left (n1\right)K}{N}+K_{1}K \right \} \cup \left \{ \frac {\left (n1\right)K}{N}+1, \ldots, K \right \}\). With such a channel assignment, as long as \(K_{1}>\frac {K}{N}\), some sensors will transmit on more than one orthogonal channel and the correlation among the equivalent channel responses of orthogonal channels is not zero. As K _{1} increases from \(\frac {K}{N}\) to K, more and more sensors transmit on more than one orthogonal channel and the correlation among the equivalent channel responses increases from 0 to 1. Therefore, K _{1} can be adjusted for different levels of correlation among the equivalent channel responses.
From the above theoretical derivations and simulation results, it can be concluded that only one orthogonal channel should be assigned to each sensor. In other words, all sensors are divided into disjoint groups and those sensors in the same group transmit on one orthogonal channel. Moreover, the fixed sensor grouping proposed here is such that all sensors are equally divided into groups, i.e., the nth group is \(\Omega _{n} = \left \{ \frac {\left (n1\right)K}{N}+1, \ldots, \frac {nK}{N} \right ]\).
3.2 Adaptive sensor grouping
where β _{ n } is also affected only by the channel responses of sensors transmitting on the nth orthogonal channel, and it can be interpreted as the indicator of the observation noise suppression capability of the nth orthogonal channel.
The above simple observation suggests that if all sensors can be properly grouped according to their channel responses, larger α _{ n } and β _{ n } can be obtained for each orthogonal channel and thus the overall channel noise suppression and observation noise suppression capabilities of the semiorthogonal MAC will be improved.
where ϕ is the phase of the equivalent channel response and \(\tan \phi = \frac {r \sin \vartheta }{r \cos \vartheta + 1}\).
If α _{ n }>1, the added sensor is said to be constructive for channel noise suppression and if β _{ n }>1, the added sensor is constructive for observation noise suppression. Note that if the added sensor transmits on an orthogonal channel alone, then α _{ n }=r ^{2} and β _{ n }=1. This means that if the added sensor is constructive, sensor grouping improves performance of individual sensors (i.e., without being grouped).

If 𝜗 is in region A, i.e., \(0\leq \vartheta <\arccos \left (\frac {r}{2} \right)\), the added sensor is constructive for both channel noise suppression and observation noise suppression. Note that region A includes \(\left [ 0, \frac {\pi }{2} \right ]\), regardless of the value of r.

If 𝜗 is in region B, i.e., \(\arccos \left (\frac {r}{2} \right)\leq \vartheta <\arccos \left (r \right)\), the added sensor is destructive for channel noise suppression, but constructive for observation noise suppression.

If 𝜗 is in region C, i.e., arccos(−r)≤𝜗<π, the added sensor is destructive for both channel noise suppression and observation noise suppression.
At this point, the question raised at the beginning of this section has been answered for grouping two sensors. In summary, if the phase difference between the channel responses of the two sensors is in the region of \(\left [ 0, \frac {\pi }{2} \right ]\), sensor grouping is beneficial. However, if the phase difference is larger than \(\frac {\pi }{2}\), grouping sensors on the same orthogonal channel may be destructive for either channel noise suppression or observation noise suppression, or for both of them, and sensor grouping may give worse performance. Therefore, if the semiorthogonal MAC is employed with N=2 or N=3 and if the whole phase region of 2π is partitioned into N=2 or N=3 equal subregions (each of length π or 2π/3), then grouping the sensors with channel phases in the same subregion to transmit on the same orthogonal channel might not always be beneficial. However, if the semiorthogonal MAC is used with N≥4 and the whole phase region is partitioned into N equal subregions (each of length \(\frac {2 \pi }{N}\)), then grouping the sensors with channel phases in the same subregion always picks constructive sensors in one group and therefore performance improvement is guaranteed^{8}.
In summary, for N≥4, adaptive sensor grouping is always beneficial and is done such that all sensors whose channel phases fall into the same nth region \(\left [\frac {2 \pi \left (n1 \right)}{N}, \frac {2 \pi n}{N}\right ]\) transmit on the nth orthogonal channel. This adaptive sensor grouping is analyzed in more detail in the next section.
4 Performance analysis
In general, with the same number of sensors, different MACs yield different values of \(\mathcal {E} \left (\alpha \right)\) and \(\mathcal {E} \left (\beta \right)\), implying different capabilities of channel noise suppression and observation noise suppression. This section analyzes in detail the estimation performance of the semiorthogonal MAC under fixed and adaptive sensor grouping and also compare with the coherent MAC, orthogonal MAC and hybrid MAC. The analysis and comparison are carried out for equal power allocation, i.e., \(a_{i}=\bar {a}=\sqrt {P_{\text {tot}} / K \left ({\sigma ^{2}_{s}}+{\sigma ^{2}_{v}} \right)}\).
4.1 Orthogonal, coherent, and hybrid MACs
For the orthogonal MAC, it is easily seen that \(\mathcal {E}_{\text {orth}} \left (\alpha \right) = \mathcal {E} \left (\sum ^{K}_{i=1} \frac {\left  h_{i} \right ^{2}}{K} \right) = 1\) and \(\mathcal {E}_{\text {orth}} \left (\beta \right) = K\).
4.2 Semiorthogonal MAC with fixed sensor grouping
For the parameter α, one has \(\mathcal {E}_{\mathrm {semiF}} \left \{ \alpha \right \} = \mathcal {E} \left \{ \sum ^{N}_{n=1} \left  \frac {\sum _{i \in \Omega _{n}} h_{i}}{\sqrt {K}} \right ^{2} \right \} = 1\), which is the same as for the case of the orthogonal MAC.
Values of \(\mathcal {E}_{\mathrm {semiF}} \left \{ \beta \right \}\) with N=4
K  4  8  16  32  64  128 
\(\mathcal {E}_{\mathrm {semiF}} \left \{ \beta \right \}\)  4  5.3  6.4  7.1  7.5  7.8 
When K→∞, it is shown in Appendix B that that β follows a Gamma distribution with parameters a=N and b=2. As a consequence, \(\mathcal {E}_{\mathrm {semiF}} \left \{ \beta \right \}=2N\). This result means that, for a WSN with a large number of sensors, the semiorthogonal MAC is two times better than the orthogonal MAC in the observation noise suppression capability.
4.3 Semiorthogonal MAC with adaptive sensor grouping
5 Numerical results and discussions
Asymptotic performance in terms of \(\mathcal {E} \left \{ \alpha \right \}\) and \(\mathcal {E} \left \{ \beta \right \}\)
Type of MAC  \(\mathcal {E} \left \{ \alpha \right \}\)  \(\mathcal {E} \left \{ \beta \right \}\)  Number of orthogonal channels, N  Required feedback 

Coherent  0.78 K  0.78 K  1  Exact channel phase 
Orthogonal  1  K  K  None 
Hybrid (N=4)  0.20 K  0.78 K  4  Exact channel phase 
Semiorthogonal, fixed (N=4)  1  8  4  None 
Semiorthogonal, adaptive (N=4)  0.16 K  0.78 K  4  log2N bits 
For the semiorthogonal MAC with adaptive sensor grouping and N=4, as K→∞, \(\mathcal {E} \left \{ \alpha \right \}\) and \(\mathcal {E} \left \{ \beta \right \}\) increases in the order of K, and thus the average MSE distortion goes to zero. This phenomenon is the same as those of both the coherent and hybrid MACs. However, for the orthogonal MAC and the semiorthogonal MAC with fixed sensor grouping, the average MSE distortion converges to a fixed value as K increases. This is because \(\mathcal {E} \left \{ \alpha \right \}=1\), regardless of K, for these two MACs.
The semiorthogonal MAC with adaptive sensor grouping can achieve the same performance at low γ _{c} and even better performance at high γ _{c} as compared to the coherent MAC. However, the semiorthogonal MAC requires N=4 times the number of orthogonal channels and about five times the number of sensors. Nevertheless, it does not require channel phase information feedback. Furthermore, the semiorthogonal MAC with adaptive sensor grouping can performs very close to the hybrid MAC. According to the simulation results in Fig. 4, for \(\mathcal {E} \left \{ \alpha \right \}\), the semiorthogonal MAC is better for small K but worse for large K. With about K=16, the two MACs have the same \(\mathcal {E} \left \{ \alpha \right \}\). For \(\mathcal {E} \left \{ \beta \right \}\), the semiorthogonal MAC performs nearly the same as the hybrid MAC for all values of K. Again, it is important to point out that channel phase information feedback is needed in the hybrid MAC.
It is pointed out that Fig. 6 also provides the results for N=2. Compared to N=4, although there are fewer orthogonal channels and thus smaller (overall) noise power for N=2, using N=2 has almost the same channel noise suppression capability as using N=4. In addition, the observation noise suppression capability for N=2 is much weaker than that for N=4. These phenomenons are consistent with the analysis in Section 3.2. In general, N=4 is the best choice for the semiorthogonal MAC with adaptive sensor grouping, which achieves the largest performance improvement while requiring the least transmission bandwidth.
The qualities \(\frac {\mathcal {E}\left \{\alpha \right \}}{K}\) and \(\frac {\mathcal {E} \left \{\beta \right \} }{K}\) are also plotted in Fig. 6 for the hybrid MAC. The advantage of the hybrid MAC over the semiorthogonal MAC with adaptive sensor grouping is most obvious for N=2. Again, this is because with N=2, destructive superposition of signals from two sensors happens in the semiorthogonal MAC, while it is never the case in the hybrid MAC. As N increases, the subregions of channel phases become narrow, and the direct superposition behaves more and more like the coherent combination. For N=8, the two MACs have nearly the same performance.
Regarding the bandwidth requirement (in terms of the number of orthogonal channels), the hybrid and semiorthogonal MACs are much more efficient than the orthogonal MAC. The coherent MAC is the most bandwidth efficient since only one channel is used. For the orthogonal MAC and the semiorthogonal MAC with fixed sensor grouping, no feedback of channel phases from the FC to the sensors is required. For the coherent MAC, due to the requirement of coherent combination among sensors, channel phases need to be transmitted from the FC to the sensors. The exact number of bits used for such information feedback depends on the capability of the feedback channel and the required accuracy, but this is certainly a large amount of overhead. For the hybrid MAC, because coherent combination among sensors in each group is required, the amount of channel information feedback from the FC to the sensors is still the same as that of the coherent MAC. For the semiorthogonal MAC with adaptive sensor grouping, the FC needs to send only log2N bits to inform each sensor the orthogonal channel to transmit on.
With K increasing from K=4 to K=16, the performance improvements are significant, except for the semiorthogonal MAC with fixed sensor grouping. In particular, the performance of the semiorthogonal MAC with adaptive sensor grouping is the same as that of the hybrid MAC, which is consistent with the theoretical derivation, and it is between those of the orthogonal MAC and the coherent MAC.
When K is further increased to K=80, the performance of the semiorthogonal MAC with fixed sensor grouping stays nearly the same as the performance with K=16. On the other hand, the performance of the semiorthogonal MAC with adaptive sensor grouping improves significantly. The semiorthogonal MAC with adaptive sensor grouping and K=80 achieves the same (at low γ _{c}) or even better (at high γ _{c}) performance compared to the coherent MAC with K=16. In addition, with K=80, the hybrid MAC only slightly outperforms the semiorthogonal MAC at low γ _{c}. All the simulation results match with the theoretical analysis presented before.
6 Conclusions
For WSNs consisting of a sufficient large number of sensors but operating under limited bandwidth resource, a novel semiorthogonal multiple access scheme was proposed for transmission from K sensors to the FC over N orthogonal channels, where K≥N. The paper thoroughly analyzed the performance of distributed estimation over such a semiorthogonal MAC with either fixed or adaptive sensor grouping and compared with the performance achieved with other related MACs. Compared to the orthogonal MAC operating under the same bandwidth, the semiorthogonal MAC with fixed sensor grouping has the same channel noise suppression capability, but twice the observation noise suppression capability as K approaches infinity. This is achieved with no requirement of information feedback from the FC to sensors. For the semiorthogonal MAC with adaptive sensor grouping, it is determined that N=4 is the most favorable number of orthogonal channels when taking into account both performance and feedback requirement. In particular, the semiorthogonal MAC with adaptive sensor grouping is shown to perform very close to that of the hybrid MAC under the same bandwidth and number of sensors, while requiring only two bits of information feedback instead of the exact channel phase for each sensor.
The present paper considers estimating a single source signal. In general, when the number of sources increases, the amount of information to transmit from the sensors to the FC increases, which translates to a larger transmission bandwidth, or equivalently a larger number of orthogonal channels. If the sources are uncorrelated, the proposed semiorthogonal transmission framework can be applied individually for each source signal. However, if the source signals are correlated, the correlation information should be used in the development of joint semiorthogonal multiple access schemes and this is left for further research.
7 Endnotes
^{1} To be consistent with existing literature, the term MAC is also used in this paper, although it is more appropriate to use the term “multiple access scheme” when discussing different communication methods between the sensors and FC.
^{2} The definition and meaning of equivalent channel responses will be made clearer in Section 3.
^{3} The interested reader is referred to [28] for a novel power allocation solution under the semiorthogonal MAC, which is shown to improve the estimation performance when compared to equal power allocation, especially at low channel signaltonoise ratios.
^{4} For complex scalars, vectors and matrices, \(\mathcal {R} \left \{ \cdot \right \}\) denotes the real part and \(\mathcal {I} \left \{ \cdot \right \}\) denotes the imaginary part.
^{5} This allocation is similar to the transmission in an overloaded codedivision multiple access (CDMA) systems [29, 30] if one views vector g ^{(i)} as the signature vector of sensor i.
^{6} For random variables, \(\mathcal {E} \left \{ \cdot \right \}\) and \(\mathcal {D} \left \{ \cdot \right \}\) denote expectation and variance, respectively.
^{7} If the channel response of the added sensor is of magnitude larger than 1, then it can be taken as the first sensor and the other sensor is taken as the added sensor.
^{8} Since the phases of wireless channel responses are modeled as uniform, one does not expect any benefit to use nonuniform phase partitions.
8 Appendix A
9 Appendix B
where \(\Gamma \left (a \right) = \int ^{\infty }_{0} x^{a1} \mathrm {e}^{x} {\mathrm d} x\) is the Gamma function. If a is an integer, then Γ(a)=(a−1)!.
Declarations
Acknowledgements
This work was supported by a Discovery Grant from the Natural Sciences and Engineering Council of Canada (NSERC). The authors would like to thank the anonymous reviewers for many constructive comments, which greatly helped in improving the clarity of this paper.
Authors’ contributions
The work was carried out by the first author when she was a graduate student under the academic supervision of the second author. Both authors read and approved the final manuscript.
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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