Keywords

1 Introduction

It is well known that saving energy is the main concern in wireless sensor networks since the size of the sensors is too small for them to have been designed with enough energy for long time operation. Recently, many approaches including multiple-input multiple-output (MIMO) communication and multi-hop schemes have been developed to improve the energy efficiency. In fact, it is difficult to implement multiple antennas at a small sensor node in a realistic environment for the MIMO approach. Therefore, cooperative MIMO schemes have been designed [1, 2], which allow single antenna nodes to achieve MIMO capability. Energy efficiency has been done to explain that the cooperative MIMO outperforms the SISO (single-input single-output) after a certain distance [3]. But the use of cooperative MIMO in a multi-hop network making the transmission more efficient is still a hot reach topic that should be sought.

In this paper, in order to take full advantage of these approaches, a new scheme is proposed which involves the joint utilization of cooperative MIMO and multi-hop techniques. Moreover, the modulation constellation size is also considered in this scheme for a more practical case. Also, it is further demonstrated that the energy efficiency performance of the proposed scheme can be improved.

The remainder of the paper is organized as follows. In Sect. 2, the fundamentals and system model are introduced. In Sect. 3, the multi-hop cooperative MIMO wireless sensor network is presented; and the quantitative analysis on the performance of the proposed scheme is presented in Sect. 4. Finally, in Sect. 5, the paper is concluded with a brief summary.

2 Fundamentals and System Model

A wireless sensor network composed of n clusters and a destination is shown in Fig. 1b where the cooperative MIMO communication technique Fig. 1a is applied to a multi-hop scheme for saving energy. In the cluster, the longest distance amongst the nodes is defined as d long . The long-haul distance between the nearest nodes of different cluster is defined as d i (i = 1, 2, …n) which is assumed much larger than d long .

Fig. 1
figure 1

An energy efficient wireless sensor network is constructed with a cooperative MIMO communications and b multi-hop techniques

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In order to evaluate the performance of the proposed scheme, the energy consumption is first discussed. From [1] it is known that the total average power consumption can be categorized into two main components: the power consumption of all the power amplifiers P PA and the power consumption of all the circuit blocks P c . Considering the optimized transmission time T on, the total energy consumption per bit can defined as follows:

$$ E_{{{\rm{bt}}}} = (1 + \alpha )\mathop {E_{b} }\limits_{{}}^{ - } \times \frac{{\left( {4\pi d} \right)^{2} }}{{G_{t} G_{r} \lambda ^{2} }}M_{l} N_{f} + P_{c} T_{{on}} /L $$
(1)

where \( \alpha = {\raise0.7ex\hbox{$\xi $} \!\mathord{\left/ {\vphantom {\xi {\eta - 1}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\eta - 1}$}} \) with \( \xi \) is the peak to average ratio (PAR); \( \eta \) is the drain efficiency of the RF power amplifiers; \( \mathop {E_{b} }\limits^{ - } \) is the average energy per bit required for a given bit error rate (BER); d is the transmission distance; G t and G r are the transmitter and receiver antenna gains, respectively; \( \lambda \) is the carrier wavelength; M l is the link margin compensating the hardware process variations and other additive interference or background noise; and N f is the receiver noise. It should be noted that N f is given by N f  = N r /N 0, where N r is the power spectral density (PSD) of the total effective noise at the receiver input and N 0 is the single-sided thermal noise PSD at room temperature with N 0 = −171 dBm/Hz. Assume that the transmitter buffer length is L bits. In Eq. 1, \( \mathop {E_{b} }\limits^{ - } \) is defined by the BER and constellation size b. The average BER can be obtained as follows [4]:

$$ \mathop {P_{b} }\limits^{ - } \approx \varepsilon_{H} \left( {\frac{4}{b}\left( {1 - \frac{1}{{2^{\frac{b}{2}} }}} \right)Q\left( {\sqrt {\frac{3b}{M - 1}\gamma_{b} } } \right)} \right) $$
(2)

for b \( \ge \)2; and for b = 1, Eq. 1 can be simplified as follows:

$$ \mathop {P_{b} }\limits_{{}}^{ - } \approx \varepsilon_{H} \left[ {Q(\sqrt {2\gamma_{b} } )} \right] $$
(3)

where \( \varepsilon _{H} \left[ \; \right] \) denotes the expectation given the channel H; \( \gamma_{b} \) is the instantaneous received signal-to-noise ratio (SNR); Q( ) is the Q-function. The Chernoff bound can be applied to obtain and the upper bound for \( \mathop {E_{b} }\limits^{ - } \) as follows:

$$ \mathop {E_{\text{b}} }\limits^{ - } \le \frac{2}{3} \times \left( {\frac{{\mathop {P_{b} }\limits^{ - } }}{{\frac{4}{b}\left( {1 - \frac{1}{{2^{\frac{b}{2}} }}} \right)}}} \right)^{{ - \frac{1}{{M_{t} M_{r} }}}} \frac{{2^{b} - 1}}{b}M_{t} N_{0} $$
(4)

By substituting Eq. 4 into Eq. 1 the total energy consumption per bit can rewritten as follows:

$$ E_{\text{bt}} = (1 + \alpha ) \times \frac{2}{3} \times \left( {\frac{{\mathop {P_{b} }\limits^{ - } }}{{\frac{4}{b}\left( {1 - \frac{1}{{2^{\frac{b}{2}} }}} \right)}}} \right)^{{ - \frac{1}{{M_{t} M_{r} }}}} \times \frac{{2^{b} - 1}}{b}M_{t} N_{0} \times \frac{{\left( {4\pi d} \right)^{2} }}{{G_{t} G_{r} \lambda^{2} }}M_{l} N_{f} + \frac{{P_{c} }}{bB} $$
(5)

where B is the modulation bandwidth; M t and M r are the number of transmitter and receiver antennas, respectively.

3 Multi-Hop Network Analysis and Calculation

In this section the cooperative MIMO and multi-hop schemes is considered jointly as shown in Fig. 1. Let d i represent the optimal transmission distance and b i represent the optimal constellation size. Then, for a transmission distance d i , the energy consumption per bit of long-haul can be defined as follows:

$$ \begin{gathered} E_{\text{bt}} (d_{i} ) = (1 + \alpha ) \times \frac{2}{3}\left( {\frac{{\mathop {p_{b} }\limits^{ - } }}{{\frac{4}{{b_{i} }}\left( {1 - \frac{1}{{2^{{\frac{{b_{i} }}{2}}} }}} \right)}}} \right)^{{ - \frac{1}{{M_{t} M_{r} }}}} \times \frac{{2^{{b_{i} }} - 1}}{{b_{i} }}M_{t} N_{0} \times \frac{{\left( {4\pi d_{i} } \right)^{2} }}{{G_{t} G_{r} \lambda^{2} }}M_{l} N_{f} + \frac{{P_{c} }}{{b_{i} B}} \hfill \\ \, \hfill \\ \end{gathered} $$
(6)

There can be various scenarios, however, in this paper, it is assumed that two sensor nodes group together making a cluster since this assumption on the number of nodes in a cluster can reduce the complexity of the calculation. Nevertheless, it is important to note that the calculation can in principle be extended to any cluster size. For a scenario where all nodes are transmitting, the total energy consumption can be defined as follows:

$$ E_{total} = E_{local} + 2\sum\limits_{i = 1}^{n} {(n + 1 - i)E_{\text{bt}} (d_{i} + 2d_{long} )} N_{i} $$
(7)

where each sensor node assumes to transmit N i bits. E local is the local energy consumption; and \( 2\sum\limits_{i = 1}^{n} {(n + 1 - i)E_{\text{bt}} (d_{i} + 2d_{long} )} N_{i} \) is defined as the long-haul energy consumption. Using the cooperative communication scheme proposed in [1, 2] the local energy consumption of the proposed scheme E local can be defined as follows:

$$ E_{local} = \sum\limits_{i = 1}^{n} {\left[ {\sum\limits_{i = 1}^{{M_{t} }} {N_{i} } E_{i}^{t} + \sum\limits_{j = 1}^{{M_{r} - 1}} {N_{s} } n_{r} E_{j}^{r} } \right]} $$
(8)

where E t i , E r j denotes the energy cost per bit for local transmission for the transmitter and receiver, respectively; N s is the total number of symbols; n r is the number of bits after quantizing a symbol at the receiver.

On the other hand, in order to minimize the total energy consumption, in this paper, the proposed method tries to optimize the hop distance between the clusters. Let d i be the optimal transmission distance. First, through observation of the transmission distance in the proposed model, the constraint \( \sum\limits_{i = 1}^{n} {d_{i} = d - n} d_{long} \) is set. Along with the constraint, the cost function can be defined as follows:

$$ \Upphi = E_{local} + 2\sum\limits_{i = 1}^{n} {(n + 1 - i)E_{\text{bt}} (d_{i} + 2d_{long} )} N_{i} + w(d - nd_{long} - \sum\limits_{i = 1}^{n} {d_{i} )} . $$
(9)

To Minimize E total under the constraint \( \sum\limits_{i = 1}^{n} {d_{i} = d - n} d_{long} \), the partial derivatives with respect to d i are taken and set them equal to 0 as follows:

$$ \frac{\partial \Upphi }{{\partial d_{i} }} = 4E \times (n + 1 - i)(d_{i} + 2d_{long} ) - w = 0 $$
(10)

where E is \( (1 + \alpha ) \times \frac{2}{3} \times \left( {\frac{{\mathop {p_{b} }\limits^{ - } }}{{\frac{4}{{b_{i} }}\left( {1 - \frac{1}{{2^{{\frac{{b_{i} }}{2}}} }}} \right)}}} \right)^{{ - \frac{1}{{M_{t} M_{r} }}}} \times \frac{{2^{{b_{i} }} - 1}}{{b_{i} }}M_{t} N_{0} \times \frac{{\left( {4\pi } \right)^{2} }}{{G_{t} G_{r} \lambda^{2} }}M_{l} N_{f} N_{i}; \) and w is a Langrage’s multiplier.

Solving for d i in Eq. 10, the hop distance between the clusters can be obtained as follows:

$$ d_{i} = \left(\frac{w}{4E(n + 1 - i)}\right) - 2d_{long} $$
(11)

where w can be obtained by using the constraint \( \sum\limits_{i = 1}^{n} {d_{i} = d - n} d_{long} \). Then Eq. 11 can be rewritten as follows:

$$ d_{i} = \frac{{d + nd_{long} }}{{\sum\limits_{f = 1}^{n} {(\frac{1}{n + 1 - f})(n + 1 - i)} }} - 2d_{long} . $$
(12)

4 Numerical Results

In this section, a quantitative analysis on the performance of the proposed method is presented. There are various scenarios to be considered, however, for the preliminary study, some to the parameters are set in such a way to simplify the complexity of the equations and also the computation.

Now, suppose that n = 10, d = 1000 m, and d long  = 2 m. In Fig. 2, the optimal transmission distances are obtained using Eq. 12 and the results are plotted. The figure shows that for the clusters located farther from the destination the hop distances or lengths for them also increase. The reduction of energy consumption can be obtained using this scheme instead of the equidistance scheme. In order to compare the performance of the proposed method with the equidistance scheme, one set of typical parameters are set and used as follows:

Fig. 2
figure 2

Optimal distances versus hop-length numbers

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B = 10 kHz, f c  = 2.5 GHz, P mix  = 30.3 mW, P filt  = 2.5 mW, P filr  = 2.5 mW, P LNA  = 20 mW, P synth  = 50 mW, M l  = 40 dB, N f  = 10 dB, G t G r  = 5 dBi and η = 0.35, N i  = 20 kb, n r  = 10.

For simplicity, Alamouti scheme [5] × the brute-force simulation method proposed in [6] the optimal constellation size b is obtained for different transmission distances as listed in Table 1.

Table 1 Optimal constellation size b versus different transmission distances
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For the evaluation of the performance, an equidistance scheme is calculated in order to compare with the proposed scheme. As already referred, the proposed scheme is the use of cooperative MIMO in optimal hop-length wireless sensor network with consideration of modulation constellation size, extra training overhead requirement, and data aggregation energy. In Fig. 3 the total energy consumption per bit is plotted as a calculation aim for the proposed scheme and equidistance scheme. It can be seen that the majority of clusters in the proposed scheme have less total energy consumption per bit when compared with the equidistance scheme, i.e. the proposed scheme has a better performance. After calculating energy consumption of each cluster and adding all the values together, the results show that the proposed scheme offers a total energy saving of about 29.5 %.

Fig. 3
figure 3

Total energy consumption per bit in every cluster for different schemes (n = 10, d = 1000 m, d long  = 2 m).

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5 Conclusion

A multi-hop scheme based on a cooperative MIMO has been proposed. The feasibility of using this scheme for optimization of the performance of the wireless sensor networks is validated numerically by measurement of the total energy consumption. The results demonstrate that the proposed scheme offers a total energy saving of around 29.5 % after taking into account with modulation constellation size and transmission distance compared with the traditional scheme. Therefore it is concluded that the proposed scheme can be applied in wireless sensor networks for the reduction of energy consumption when the prime concern is to extend the network life time.