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New Approach to Improvement and Measurement of the Performance of PHY Layer Links of WSN Anatoliy Platonov, Senior Member, IEEE, and Ievgen Zaitsev

Abstract— This paper presents a new approach to the improvement of the sensor node - base station (SNBS) links of physical layer of wireless sensor networks performance. The key idea is the consideration of SNBS as the remote measurement and estimation systems and application of the corresponding analytical tools. It is shown that the main difficulty in the improvement of digital SNBS performance is impossibility to formulate and solve the optimization task using well developed methods of the optimal estimation theory. In turn, the SNBS transmitting the signals formed by the sensors using analog pulse-amplitude transmitter adjusted over the feedback channel permit the formulation of analytical form of the mean square error (MSE) and solution of optimization task. The obtained optimal transmission–reception algorithm enables designing of a low energy-size-cost analog SNBS, which transmit the signals with minimal MSE and a bit rate equal to the capacity of the link. Moreover, the MSE of transmission determines the information characteristics of the links, as well as enables a development of the unified methods of the SNBS real performance evaluation and measurement. Index Terms— Analog data transmission, information characteristics, mean square error (MSE), optimization, performance measurement, physical (PHY) layer, wireless sensor nets.

I. I NTRODUCTION

T

HE main requirements for wireless sensor network (WSN) are reliable data delivery, greater coverage, unrestricted placement of the sensors, minimal energy consumption and cost, as well as optimal utilization of the channel resources [1]–[3]. The realization of these tasks directly depends on the performance of communication links of physical (PHY) layer of WSN and the cost paid for its achievement. No analytical tools currently exist that would allow systematic design of optimal links, which could provide the fastest and most accurate data transmission from the sensor nodes (SN) to the base stations (BS) at the possibly greater distance under the given level of reliability. This is caused by the multicriterion evaluation of the performance of digital communication systems (DCS), including digital SNBS links [2]–[4] that creates multiple tradeoffs [3]–[5]. Moreover, depending on the optional application and requirements to DCS, designers employ different criteria or groups of criterions, which change the goals and formulations of optimization tasks [4], [5].

Manuscript received December 14, 2013; revised May 24, 2014; accepted May 25, 2014. This work was supported in part by the Polish National Centre for Research and Development through the GEKON Project under Grant 214108 and in part by the National Fund for Environmental Protection and Water Management. The Associate Editor coordinating the review process was Dr. Dario Petri. The authors are with the Institute of Electronic Systems, Faculty of Electronic and Information Technologies, Warsaw University of Technology, Warsaw 00-665, Poland (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TIM.2014.2330491

Fig. 1.

General block diagram of the SNBS link.

These and other problems discussed in Section II make the development of regular approach for optimization of SNBS impossible. Recently, the optimization of digital SNBS (and DCS) has been carried out as a half-heuristic search for optimal balance between the chosen criterions [3]–[5]. The result is not optimal utilizing the spectral and power resources of SNBS, which causes a reduction of the range of reliable data transmission, greater energy consumption, redundant complexity, and cost of the SN. The lack of universal criterion, variety of different and nonoptimal variants of digital SNBS realization, and different scenarios of application do not allow the elaboration of a unified approach to the evaluation and measurement of their performance. As it is noted in [6], measuring the PHY layer performance of WSN is an extremely expensive, complex, and time-consuming task, which can be solved in different ways, depending on the type of SNBS and scenario. The results of measurements give only partial information about the real performance of SNBS and degree of utilization of their power and spectral resources. More and more publications appear, which emphasize the lack of the theory permitting further improvement of the PHY layer performance, as well as highlighting the necessity to uncover new solutions enabling its development [5], [7]. The goal of this paper is to discuss new possibilities in development of the unified criterions and methods of PHY layer links (SNBS) optimization using the approach developed in [8]–[12] on the basis of works [13]–[15]. Metrological and information characteristics of the considered SNBS and methods of their measurement are also examined. To facilitate the understanding of the new approach, a brief description of the key ideas, the results, and the history of research which have been conducted is given below. A. General Description of New Approach and Its Advantages From a general point of view, SNBS (Fig. 1) are a particular case of remote measurement systems. They also can be considered as remote estimation systems, which deliver

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estimates xˆ of the input signal x, formed by the sensor, to the addressee. Finally, SNBS are the communication channels, which deliver information from the sensors to BS. This means that SNBS and WSN optimization theory may employ criteria, methods of the analysis, mathematical tools, and results of the measurement, estimation and communication theories, as well as of the information theory. The latter would permit a substantial extension of the analytical and experimental possibilities of the research. Research results [8]–[12] show that the majority of aforementioned problems can be solved, if the analog signals formed by the sensors are transmitted from SN to BS using only adaptive pulse-amplitude modulator (PAM) adjusted by the signals formed in BS and delivered to SN over the feedback channel. No digitizing and coding units should be involved. At the first sight, the turning back to analog transmission looks rather strange—it is well known that amplitude modulation is the least efficient and least noise resistant method of signal transmission. Nevertheless, as it is shown in [8]–[12], just the transition from the digital to analog transmission enables the development of accurate mathematical basis for the designing of a top-efficient SNBS transmitting the signals from SN to BS with a bit rate equal the capacity of the link and maximal power-bandwidth (P-B) efficiency. This result is unachievable for SNBS with digital SN transmitters. Moreover, analog transmission permits the development of unified methods of the adequate analytical evaluation and measurement of SNBS performance. One should add that the absence of the digitizing and coding units in the analog transmitters of SN radically reduces their complexity, energy consumption, size, and cost. The potential advantages of the analog transmission were noticed far ago and it was the subject of many years of intensive research, now almost forgotten. B. Genesis of Researches in Optimal Analog Transmission Elias [16] showed that analog communication systems with feedback channels (AFCS) may transmit signals with a bit rate equal to the capacity of the forward channels without coding. This paper initiated a great cycle of research in optimal AFCS theory, which continued through to the middle of 1970s ([17]–[21] and others). The basic criterion of transmission quality was the MSE of transmission. The results of researches conducted had undeniably proved the possibility of designing ideal AFCS, which could transmit the signals without coding with the theoretically achievable accuracy, as well as a bit rate and P-B efficiency attaining Shannon’s limits. However, by the beginning of the 1970s, this research practically ceased. Analysis of the reasons of the loss of interest in AFCS theory showed that the main cause was the lack of practical results after many years of work. In turn, impressive successes in DCS theory, technology and applications, great number of new tasks with granted implementation of solutions, as well as active financing of works redirected the attention of researchers from AFCS to DCS. Nowadays, the mass production of WSN (which all have feedback channels) has restored interest in AFCS theory. However, until now, no research has come up with practical results.

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Deeper analysis showed that the direct reason for the lack of implementation of research conducted in the 1950s and 1960s (and which continued to block further research) was the common usage of linear models of the forward transmitters. Linear models do not allow consideration of the always limited output range of the real transmitters, which is a source of their possible saturation (see also Section III). In [10], [11], and [14], it was shown that saturation aggravates the quality of transmission in a way similar to the appearance of large bit-error rate (BER) in digital systems (Section III-B). This effect was not taken into account in earlier and later theoretical research in AFCS optimization, which made the results inapplicable. II. C RITERIONS OF SNBS L INKS P ERFORMANCE A. Role of MSE in SNBS Analysis and Optimization The role of MSE in communications was stressed yet in [24] where the main task of communication systems as being the delivery of the signals from the source to destination with the maximal accuracy (fidelity) given source rate was determined. According to [23], the quality of transmission of the signals from analog sources is to be evaluated by the mean distance between the input signals x and their estimates xˆ formed by the receiver that is by the MSE of transmission P = E[(x − x) ˆ 2 ]. For the discrete sources, Shannon proposed evaluation of the transmission quality by the bit rate and the measures based on the frequency of errors in received sequences of bits. In the theory and practice of measurements, MSE (determined by the mean value and standard deviation of the measurand) is the commonly accepted basic criterion of measurement quality [24]. Of great importance is that every analytical evaluation of the MSE can be verified experimentally. This allows the objective evaluation and comparison of the performance of different measurement systems, as well as of the efficiency of the used method of measurement. The MSE can easily be computed in simulation experiments, which enables a fast and low-cost verification of analytical results, as well as a definition of the parameters and characteristics of the modeled system and input signals, which most influences the quality of measurements. It is worth accenting that the metrology disposes well developed methods of MSE measuring for different systems used in different scenarios. As it was shown in [8]–[12], analog SNBS (further SNBS, without adjective) allow formulation of analytical form of the MSE. Being built on the mathematical models of the main components of SNBS, signals, and noises, MSE is a function of their parameters and statistical characteristics. This permits the formulation and solving of the optimization task by the methods of applied Bayesian estimation theory [13]–[15], [22]. The obtained results permit to design optimal SNBS minimizing MSE of transmission and to determine the minimal MSE (MMSE) value as a function of the main influencing MMSE factors. The SNBS links designed in this way would transmit the signals with the maximal accuracy, bit rate, and P-B efficiency ([8]–[12], Sections III and IV). Using MSE, one may also develop unified and adequate methods of the measurement of SNBS performance (Section V). One

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. PLATONOV AND ZAITSEV: NEW APPROACH TO IMPROVEMENT AND MEASUREMENT OF THE PERFORMANCE OF PHY LAYER LINKS

should add that the values of MSE and MMSE depend on the scenario of SNBS application, and the results of the MSE measurement can be used for positioning the SN and BS in a way, which additionally improves performance of the links, as well as for their adjustment and calibration. Moreover, results of [11] and [12] show that MSE of transmission is the basic measure of SNBS performance and determines the bit rate and P-B efficiency of transmission of the input signals. In development of these papers, in Sections IV and V, we show that the bit rate and P-B efficiency are not sufficient characteristics of SNBS performance. Namely, fact of transmission of the input signals at maximal bit rate and P-B efficiency does not reflect the changes of MSE during the sample transmission. For this cause, evaluation of SNBS performance using information measures gives only partial information about real quality of transmission of the signals formed by the sensors. Nevertheless, MSE is not used as the standard analytical and practical criterion of PHY layer links (and DCS) performance. What is the reason for this?

B. Difficulties in Digital SNBS Performance Evaluation The answer to the formulated question is in the impossibility of describing the digitizing, coding, and modulating (keying) units of the transmitter by way of the continuous input-output models. This does not permit formulation of MSE in a form, which would allow any constructive solution of the optimization task: the search for a minimum MSE in the infinite set of possible methods of coding, digitizing, and modulation (keying and signaling) is principally an unsolvable task. One should add that the codes of the samples of an input signal recovered by BS contain erroneous bits, and it is not known–the less or the more signing bits of the code are corrupted. The statistics of the weights of the corrupted bits depends on the statistics of the values of the input signal and cannot be assessed as well. This makes the analytical estimation of the values of the MSE of a transmission impossible. For these reasons, MSE is practically useless for the analytical evaluation of SNBS performance and is not used in DCS theory. Nowadays, BER, bit rate [bit/s], and P-B efficiency of transmission [4], [5], [25] are used as the main criterions of DCS and digital SNBS performance. The upper boundaries of these measures for the channels with additive Gaussian white noise (AWGN) are known and used as the reference measures in the evaluation of SNBS performance. The capacity of the channel is determined by Shannon’s formula W sign (1) = F0 log2 (1 + Q 2 ) C = F0 log2 1 + Nξ F0 where W sign is the power of the signal at the channel output, Nξ /2 is the double side spectral power density of AWGN, 2F0 is the channel bandwidth, and Q2 =

W sign W sign = = SNR Ch 2 Nξ F0 σξ

(2)

Fig. 2.

3

Block diagram of analog SNBS with adjusted PAM modulator.

is the signal-to-noise ratio (SNR) at the channel output. One should notice that according to Shannon’s result [26], the feedback channel does not change the capacity of the link. The power (P-) efficiency of the transmission (the energy per bit) E bit = W sign T bit = W sign /R [J/bit] is determined as the energy required for the transmission of one bit of information over the channel (T bit = 1/2R is the duration of a single bit transmission, and R is the bit rate [bit/s]). The spectral or bandwidth (B-) efficiency R/F0 describes the number of bits transmitted per second per one hertz of the channel bandwidth. Upper values of the power and bandwidth efficiencies are connected by the relationship [5], [25], [27] F0 FC Q2 E bit 2 0 −1 = . = Nξ C log2 (1 + Q 2 )

(3)

Equation (3) directly follows from (1) and is more convenient for applications to digital systems. The performance of SNBS is assessed by the closeness of the measured values of the bit rate and P-B efficiency of the transmission to the boundaries (1), (3). The BER values are computed independently [25], [27]. Real performance of digital SNBS and DCS never achieves the boundaries (1), (3). Optimization of the links, as it was noted in Section I, is carried out in a half-heuristic way as a search for the optimal balance between the different tradeoffs: 1) P-B tradeoff; 2) tradeoffs between the code length, bit rate, BER, channel bandwidth and other tradeoffs conditioned by the practical requirements [4], [5]. Moreover, the results presented in [11] (see also Section VI-B) show that BER, bit rate (throughput), and P-B efficiency are not complete measures of transmission quality. This permits the claim that all known digital SNBS links are not optimal and do not completely utilize the resources of their transmitters, channels, and receivers. The P-B efficiency of transmission is therefore lower than that which is potentially achievable. The listed problems disappear in the theory and design of optimal analog SNBS. III. O PTIMIZATION OF A NALOG SNBS The block scheme of analog SNBS is presented in Fig. 2. The basic principles of SNBS optimization are described in [8]–[11] and follow from the more general approach to optimization of adaptive estimation and measurement systems with feedback [14], [28], [29]. It is worth noting that the formulation of MSE-based optimization task considered in these papers is close to those given in [17]–[21]. The single



but principal difference is that the search for the minimum MSE is carried out taking into account possible saturation of the SN transmitter. To provide the basis for the further discussion, mathematical backgrounds and main results of the new approach to SNBS optimization are briefly presented in the points A–C below. A. Mathematical Model of SNBS The forward and feedback channels of SNBS are assumed to be linear, memoryless, and channel noises ξk , ηk are AWGN. We also assume that the signals x t formed by the sensor of SN are low-frequency stationary Gaussian processes with the known mean value x 0 , variance σ02 , and baseband [0, F]. Transmitting unit [(TU) in Fig. 2] includes the sample-andhold block S&H, adaptive PAM modulator + M1, and high-frequency transmitter (not shown). Each sample x (m) , (m = 1, 2, . . .) is transmitted in the same way, independently and in n = T /t0 = F0 /F cycles, (T = 1/2F is the sampling period, t0 = 1/2F0 is the duration of one cycle of transmission, 2F0 is the bandwidth of channel Ch1, and bandwidth of Ch2 is not narrower than 2F0 ). In this case, analysis of SNBS can be reduced to consideration of a single sample transmission, which permits the omission of index m in notations x (m) in the models and relationships discussed below. For each kth cycle of transmission, (k = 1, . . . , n), subtracting unit forms a difference signal ek = x − Bˆ k routed to the input of amplifier of modulator M1. Value Bˆ k describes the control signal computed in BS and transmitted to TU over linear feedback channel T2–Ch2–R2 with AWGN ηk and bandwidth 2F0 , (T2 and R2 denote the feedback transmitter and receiver). Independently from the method of transmission and characteristics of the feedback channel, the signal received by TU in the kth cycle can be written in the form: Bˆ k = Bk + vk

(4)

where vk describes the transmission errors. The linearity of the feedback channel and AWGN-type of the noise ηt permit consideration of the transmission errors vk as AWGN with definite variance σv2 , which depends on the quality of the channel. Taking into account (4), difference signal at the modulator input can be written in the form: ek = x − Bk + vk . Like in [8]–[11], it is assumed that TU employs double-side band suppressed carrier PAM. Modulation depth Mˆ k is being set, in each cycle, to the values determined by the solution of optimization task. Saturation appears if the amplified signal Mˆ k ek exceeds the saturation level of the carrier generator. In this case, the amplitude of the emitted signals can be described by the relationship Mˆ k ek if Mˆ k |ek | ≤ 1 sk = A0 (5) sign (ek ) if Mˆ k |ek | > 1 where A0 is the saturation level of the emitter (maximal amplitude of emitted signals). Dependence between the conditional mean E(sk |x) of the emitted signal sk and value of the input signal x is shown in Fig. 3.

Fig. 3.

Static characteristic of adjusted modulator transmitter.

The signal s˜k = Ask + ξk received at the output of the channel Ch1 is processed in the receiver-demodulator DM1, and demodulated signal if Mˆ k |ek | ≤ 1 Mˆ k ek + ξk y˜k = A (6) sign (ek ) if Mˆ k |ek | > 1 is routed to the input of the digital signal processing unit (DSP) of BS. Parameter A = γ0 A0 /r in (6) describes the amplitude of the received signal, γ0 is the gain of the channel Ch1, and r is the distance between SN and BS. In each cycle of the sample transmission, the DSP unit computes the current estimate xˆk = xˆk ( y˜1k ) of the sample according to the Kalman equation

xˆk = xˆk−1 + L k y˜k − E y˜k y˜1k−1 (7) where y˜1k−1 = ( y˜1 , . . . , y˜k−1 ) denotes the sequence of signals delivered to the DSP in previous cycles, E( y˜k | y˜1k−1 ) is the conditional average, and the difference y˜k − E( y˜k | y˜1k−1 ) is the prediction error of the demodulated signal. Parameter L k in (7) determines the rate of convergence of estimates xˆk to the real value x of the transmitted sample, and values L k are also determined by the solution of the optimization task. Apart from the estimate xˆk ( y˜1k ), DSPU computes the controlling signal Bk+1 = Bk+1 ( y˜1k ) transmitted to TU, and resets the gain L k to the value L k+1 . Delivering the signal Bk+1 to the transmitter and setting the modulator parameters to the values Bˆ k+1 = Bk+1 + vk+1 and Mˆ k+1 finalize the cycle and the next (k + 1)th cycle begins. After n cycles, final estimate xˆn of the sample is routed to the addressee and SNBS begins transmission of the next sample. The initial values Bˆ 1 , Mˆ 1 are determined by the mean value x 0 , variance σ02 of the input signal, and saturation factor α, which establishes permissible level μ of the probability of the transmitter’s saturation (see below). B. Influence of Saturation Errors on Transmission Quality For each cycle k = 1, . . . , n, quality of estimates xˆk = xˆk ( y˜1k ) is assessed by MSE Pk = E[(x − xˆ k )2 ]. Models (5)–(7) permit the formulation of the explicit form of MSE for each k = 1, . . . , n. Its values will depend on the parameters Mˆ k , L k and controls Bk = Bk ( y˜1k−1 ). As it was shown in [8]–[11], minimizing the MSE over Mˆ k , L k , Bˆ k omitting possible saturation of SN transmitter makes the obtained results not applicable in practice independently from the constraints on the mean energy or peak values of the signals emitted by


the SN transmitters. The reason for this is as follows: the sufficiently rare cases of saturation (with a probability μ 1) may increase the MSE of the estimates xˆn formed by BS only by the values of O(nμ) order, that is to say the MSE is insensitive to the rare cases of saturation. The same concerns the evaluations of the values of the mean emitted energy. This seemed to be sufficient for the application of linear models of the transmitters in [17]–[21] and other papers. However, saturation with a probability μ 1 per cycle of transmission will cause the appearance about nμ percent of distorted estimates xˆn at the output of SNBS. Each estimate is presented by the binary code word, and the distortion of nμ estimates means the appearance of nμ percent of erroneous code words in the sequences at the SNBS output. Thus, saturation of the transmitter causes the appearance of the word error rate (WER) of transmission similar to the BER in digital FCS. The linear model of the modulator-transmitter does not permit to evaluate and regulate the values of WER, which can be unacceptably large. Taking into account that in digital SNBS acceptable BER is always not greater than 10−4 , WER in the analog SNBS should not have greater values either. The omission of this factor was the main reason for the difficulties in the implementation of the results of [17]–[21] and others. The said above permits the conclusion as follows: The MSE of transmission is the main but not full characteristic of SNBS performance and should be completed by the probability μ of the saturation of their forward transmitters. Remark 1: The same problem appears in the optimization of feedback measurement systems with the adaptive sensing elements, and optimization task also should be solved taking into account the possible saturation of the sensors [28], [29]. C. Formulation and Solution of SNBS Optimization Task Model (5) permits the computation of the analytical form of the probability of saturation for each k = 1, . . . , n and formulation of conditions maintaining it at the given level μ, [8]–[11]

= Pr Mˆ k |x − Bˆ k | > 1 y˜1k−1 < μ. (8) Pr over k Formula (8) represents the statistical fitting condition [13], which determines the set of permissible values of the parameters Mˆ k , Bˆ k = Bˆ k ( y˜1k−1 ) eliminating the possible saturation of SN transmitter with a probability not smaller than 1 − μ. This condition permits the formulation of a SNBS optimization task in a form directly taking into account saturation of the link: One should find estimates xˆk = xˆk ( y˜1k ), controls Bk = Bk ( y˜1k−1 ) and parameters of modulator Mˆ k , Bˆ k , which minimize the MSE of estimates Pk = E[(x − xˆk )2 ] for each k = 1, . . . , n under given probability of the transmitter saturation μ. The solution of this task is carried out sequentially, i.e., for each k = 1, . . . , n, minimal MSE is searched for assuming that previous MSE have minimal values. The result is a group of relationships [8]–[11] determining, for each k = 1, . . . , n,

optimal parameters of modulator (6) and algorithm (7) Bˆ k = Bk + vk ; Bk = E x| y˜1k−1 = xˆk−1 ; 1 Mˆ k = 2 α σv + Pk−1

5

(9)

where Bˆ 1 = x 0 ; Mˆ 1 = 1/ασ0 and α is the saturation factor determined by the equation: (α) = (1 − μ)/2, where (α) is Gaussian error function. Equation (7) takes the form xˆk = xˆk−1 + L k y˜k where gains L k have the values 1 −1 1 − Pk Pk−1 Lk = A Mˆ k

(10)

(11)

and Pk is MMSE of transmission determined by the recursion

σv2 2 −1 2 Pk−1 1+ Q Pk = (1 + Q ) σv2 + Pk−1 (1 + Q 2 )σv2 + Pk−1 Pk−1 (1 + Q 2 ) σv2 + Pk−1 Pk−1 = . Pk−1 1 + Q2 (1 + Q 2 )σv2 + Pk−1 =

(12)

Initial values for (9)–(12) are: xˆ 0 = x 0 , P0 = σ02 ; parameter Q 2 describes SNR at the forward channel output Wksign A2 Mˆ k2 σv2 + Pk−1 A 0 γ0 2 1 2 = Q = noise = W rα Nξ F0 σξ2 = SNRCh1 .

(13)

Value W sign = (A/α)2 = (A0 γ0 /αr )2 in (13) is the mean power of information component A Mˆ k ek of demodulated signal y˜k . Formulas (5)–(7) and (9)–(13) determine the optimal transmission–reception algorithm permitting the claim as follows: Optimal SNBS designed on the basis of this algorithm would transmit the signals from the sensors to BS with minimal MSE in each cycle, and saturation errors eliminated at the given confidence level 1 − μ. Remark 2: One should note that according to (12), the low boundary of the MSE of transmission depends only on SNR Q 2 at the forward channel output and the variance σv2 of feedback transmission errors. In turn, SNR (13) depends not only on the power of the emitted signals and AWGN of the channel Ch1, but also on the distance r between the SN and BS and the channel gain γ0 that is on the scenario of SNBS application. The latter permits the evaluation of optimal SNBS performance (values of MMSE) measuring SNR Q 2 or computing its values using independently measured values r , γ0 . σξ2 . IV. U PPER B OUNDARIES OF O PTIMAL SNBS P ERFORMANCE The results presented below are obtained under usual in communications assumption that the bandwidth 2F0 of the forward channel is given (allocated).



A. MMSE of Transmission In practice, supplied SN emits low power signals, whereas BS transmitter T2 may emit sufficiently powerful signals and directed antennas can be used. In this case, the feedback transmission errors will be relatively small and the initial value of SNR at the modulator input SNRinp = σ02 /σv2 in the real conditions will be much greater than SNR Ch1 = Q 2 at the demodulator output. According to (12), Pk monotonically diminishes for greater k, and there should exist the initial interval 1 ≤ n ≤ n ∗ , where the following inequality is valid: inp

SNRn = Pn−1 /σv2 1 + Q 2 = 1 + SNRCh1 .

the characteristics of SNBS are considered as the functions of duration of the sample transmission, i.e., of the number of cycles). The B-efficiency of transmission is determined by the values RnSNBS /F0 [bit/s/Hz]. A general definition (see [25]) and formulas (13) and (17) permit the finding of the following expression for the P-efficiency of transmission for an SNBS link: E nbit SNBS W sign W sign Tn = = = Nξ Nξ RnSNBS Nξ I (X, Xˆ n )

(14)

Taking into account (14), formula (12) for the MMSE can be approximated by the relationship [8]–[11]: for 1 ≤ n ≤ n ∗ σ 2 (1 + Q 2 )−n Pn = 02 (15) σv (n − n ∗ + 1)−1 for n > n ∗ . The threshold number of cycles n ∗ corresponds to the moment when the MMSE of estimates attains the value equal to the variance of feedback transmission errors: Pn∗ = σv2 . This equation and (15) give the following expression for n ∗ : inp σ02 log2 SNR1 1 F0 ∗ = n = log = ∗. 2 log2 (1 + Q 2 ) σv2 F log2 (1 + SNRCh1 )

Let us emphasize that formulas (17) and (18) are valid for arbitrary, not only optimal SNBS transmitting the signals with MSE Pn and Pn unambiguously determines the corresponding bit rate and P-B efficiencies of the transmission. For optimal SNBS, the substitution of MMSE (15) into (17), (18) gives the relationships: RnSNBS = CnSNBS ⎧ 2 Ch1 for 1 ≤ n ≤ n ∗ ⎪ ⎨ F0 log2 (1 + Q ) = C σ02 = F0 (19) ∗ ⎪ for n > n ∗ ⎩ n log2 σ 2 (n − n + 1) v

(16) B. Bit Rate and Efficiency of Transmission Over SNBS Link The analysis of algorithm (5)–(7), (9)–(13) has shown [8]–[10] that, in optimal analog SNBS, bit rate RnCh1 and P-B efficiencies (E nbit /Nξ , RnCh1 /F0 ) of transmission over the forward channel are constant and attain Shannon boundaries (1), (3) independently from the number of cycles. One should emphasize that regardless of the bit rate and P-B efficiency of SNBS attaining limits (1), (3), the MMSE Pn of the final estimates x n of the samples monotonically diminishes for the greater n beginning from the first cycle. The latter permits the claim as follows. Limits (1), (3) do not reflect the real performance of the analog SNBS (MSE of the transmission), and proximity of the information characteristics of transmission over the forward channel to limits (1), (3) is necessary but not complete measure of the SNBS performance. In turn, MMSE (15) determines the limit information characteristics of SNBS link as a whole (as a generalized communication channel), and these characteristics coincide with (1), (3) only for 1 ≤ n ≤ n ∗ . Really, using definition of the bit rate and known in the information theory results, one may obtain the relationship RnSNBS =

σ2 σ2 I (X; Xˆ n ) F0 log2 0 [bit/s] = Fn log2 0 = Tn Pn n Pn (17)

where I (X; Xˆ n ) = H (X) − H (X| Xˆ n ) = 1/2 log2 (σ02 /Pn ) is the amount of information in the final estimates xˆn about the values x of input samples [19, Sec. 9.7], and Tn = nt0 = n/2F0 (index n in Tn , Fn is added to accent that

n Q2 [J/bit]. σ2 log2 0 Pn (18)

E nbit SNBS Nξ

⎧ bit Ch1 E Q2 ⎪ ⎪ = ⎪ ⎪ ⎪ Nξ log2 (1 + Q 2 ) ⎪ ⎨ n Q2 =

⎪ ⎪ ⎪ σ02 ⎪ ∗ ⎪ ⎪ (n − n + 1) ⎩ log2 σν2

for 1 ≤ n ≤ n ∗ for n ≥ n ∗ .

(20)

Using (19), (20) one can show the following: For 1≤ n ≤ n ∗ , optimal SNBS transmits the samples ideally, with the bit rate equal to the capacity of the forward channel (1) and maximal P-efficiency (minimal energy per bit, E nbit S N B S ) determined by (3). For n > n ∗ , capacity of optimal SNBS monotonically diminishes, and limit P-efficiency of SNBS decreases as well. This effect is caused by the exponentially fast suppression, at the interval 1 ≤ n ≤ n ∗ , of the influence of the forward channel noise ξk , (1 < k < n) on the MSE Pn of estimates. For n = n ∗ , Pn attains the value Pn = σv2 and, for n > n ∗ , does not depend on σξ2 . This means that during n ∗ cycles of the sample transmission optimal adjusting of the transmitter suppresses the influence of AWGN ξk on the estimates xˆn . For n > n ∗ , diminution of Pn is the result of recovering the information component x − xˆk−1 of the signal ek = x − xˆk−1 + vk transmitted over the ideal forward channel. Similar effect should appear also in digital SNBS and in DCS without feedback, if the MSE of transmission will be close to the power of quantization noise caused by the A/C conversion of the input signals. Remark 3: Taking into account that n = F0 /Fn determines the expansion of the input signal spectrum, (19) and (20) lead to an unexpected conclusion: the spectrum expansion greater than n ∗ times lowers the limit information characteristics of the analog SNBS although the MMSE continues to diminish. In other words, the too long transmission of the samples of


7

low frequency signals improves the quality of transmission but decreases the efficiency of SNBS. The transmission shorter than in n ∗ cycles, even with the bit rate and efficiency (1), (3), will result in not full utilization of the SNBS resources. C. Data Transmission Security Under fulfilled condition (8), and Bˆ k = xˆk−1 + vk , (6) can be written in the valid, for each k = 1, . . . , n, form: y˜k = A Mˆ k ek + ξk = A Mˆ k (x − xˆk−1 + vk ) + ξk .

(21)

Using (9), (21), and expression ek = x − xˆk−1 + vk for the signal at the modulator M1 input, one can get the formulas: E ek | y˜1k−1 = xˆk−1 − Bk = 0, E ek ei | y˜1k−1 = σv2 + Pk−1 δki , (22) E y˜k | y˜1k−1 = A Mˆ k (xˆk−1 − Bk ) = 0, E y˜k y˜i | y˜1k−1 = σξ2 (1 + Q 2 )δki . (23) According to (22), signals ek = x − xˆk−1 + vk at the modulator input are the sequences of random noncorrelated values. As it follows from (23), modulator M1 transforms them into realizations of the stationary AWGN. Thus, signals formed by the sensor are transmitted to BS in the form of realization of AWGN. To recover the transmitted data using the intercepted signal from the SN, the nonauthorized user should know the method of its forming (21), algorithm (10) and their parameters, which depend on SNR Q 2 at the receiver of BS. In turn, values Q 2 depend on the scenario of SNBS application and are individual for every SNBS link. The lack of this information makes direct interception of the origin signals impossible. The only possibility to intercept the data is the interception of the signals xˆk−1 transmitted over the feedback channel. Therefore, security of data transmission depends only on the level of the feedback transmission protection, which can be provided in the way known in WSN design. V. M EASUREMENT OF SNBS C HARACTERISTICS As it was shown above, MSE is not only the basic criterion of analog SNBS performance, but also determines the bit rate and P-B efficiencies of transmission, and MMSE (12) determines their theoretically achievable limit values. This means that measurement of these characteristics can be reduced to the measurement of the MSE of transmission with the following substitution of the result to (17) and (18). In turn, the measurement of the MSE of arbitrary (analog or digital) SNBS can be carried out using random Gaussian testing signal with the baseband [−F, F], mean value x 0 , and variance σ02 close to the characteristics of signals formed by the sensor of SN in the given scenario. Registering (in decimal (m) form) the samples x (m) , and their estimates xˆn computed by the receiver (m = 1, . . . , M), one may compute the empirical MSE according to the known formula [24]: Pˆn =

M 2 1 (m) x n − xˆn(m) . M −1 m=1

(24)

Fig. 4. Dependences of the limit bandwidth efficiency of optimal SNBS on the number of cycles under different amplitudes A0 of the emitted signals.

2 Fig. 5. Changes of the energy per bit as the function of SNRCh1 out = Q (in [dB]) after 1, 10, and 25 cycles of the sample transmission.

For M ∼ 103 and greater, values Pk and Pˆk will differ by the values of O(M −1 ) order that is value Pˆn will be an adequate estimate of the real value of MSE provided by the given SNBS link. Substitution of Pˆn to (17) and (18) will give a full set of the basic characteristics of SNBS performance. Both the proposed approach to the evaluation of SNBS performance and theoretical results of this paper were verified in simulation experiments carried out using full model of the optimal SNBS designed on the basis of algorithm (5)–(7), (9)–(13). The plots in Fig. 4 (obtained under parameters σ02 = 1; σξ2 = 10−5 ; σv2 = 10−8 ; γ = 1; α = 5; r = 5; M = 5000 and digitally generated signals and noises) present the dependences of the B-efficiency of transmission on the number of cycles for four different amplitudes A0 of the emitted signals. The empirical values (discrete points) practically coincided with the corresponding theoretical values (continuous and dash lines). This confirms the validity of both theoretical results and the proposed method of the measurement of SNBS performance. The plots also clearly illustrate the threshold effect – diminution of SNBS capacity if the number of cycles exceeds the threshold number n ∗ . In the prethreshold intervals, the capacities of SNBS and forward channel are the same. The plots in Fig. 5 illustrate the dependences of the energy per bit E nbit SNBS on the SNR Q 2 (in decibel) after 1, 10, and 25 cycles of the sample transmission. For n < n ∗ ≈ 4, both theoretical and empirical values of E nSNBS coincide with the corresponding points of ideal dependence (3), (continuous



line in Fig. 5 identical to that presented in [27, Sec. 8.10]). Transmission longer than in the n ∗ cycle increases the measured values E nbit SNBS in concordance with (18) and (20). Appearing under Q 2 < −5 [dB] fluctuations of the measured values of the energy per bit are caused by their high sensitivity to even small fluctuations of the values Pˆn . VI. C ONCLUSION The results of research show that difficulties in the development of the systematic analytical approach to improvement of the PHY layer links performance are caused by application of digital techniques of the data transmission. It is shown that the analog low frequency signals formed by the sensors can be transmitted by the analog SNBS faster, more reliably and efficiently than by the digital SNBS designed for the same aims. Transition to the analog transmission enables the formulation of the MSE of transmission and solution of the MSEbased optimization task. The derived optimal transmission– reception algorithms contain all necessary information for designing top-performance optimal SNBS which transmit the signals from SN to BS with maximal accuracy, bit rate, and PB efficiency. Moreover, the absence of digitizing and coding units in the transmitters of SN substantially decreases the energy consumption of the sensor nodes, as well as their cost and size. The AFCS (SNBS in this paper) were the subject of intensive researches until the beginning of the 1970s. The reasons for their termination are explained in Section I.B. This was an important moment in the development of communication theory which, together with the rapid development of the computer and digital transmission technologies, determined its digital future. The absence of an alternative compelled the researchers and industry to use digital technologies in digital SNBS design and manufacturing. The results presented in this paper can be considered as a continuation of the research conducted in the 1950–1960s. It is worth noting that the results presented in Sections IV and V coincide, for 1 ≤ n ≤ n ∗ , with the results of works [17]–[21] and some of the later works: for example, (15) coincides with [17, eq. (21)], [20, eq. (15)], [30, eq. (3.14)]; in turn, (16) coincides with [31, eq. (38)], the list can be continued. Summarizing, one may say the following: 1) The results presented in this paper provide an analytical basis for the designing of new classes of top efficient PHY layer links with low energy-size-cost SN transmitters. These results also enable the development of unified and adequate methods of the analytical evaluation and measurement of SNBS performance (Sections IV and V), as well as of the methods of their optimal installation and adjusting (future research). One should emphasize that the universality of the MSE-based methods of performance measurements may substantially facilitate the evaluation and comparison of the real performance of digital SNBS now evaluated mainly by the throughput (output bit rate) and BER [2], [25].

2) WSN with optimal analog SNBS links would provide better quality, rate and reliability of data transmission, as well as will be less energy consuming, cheaper, and optimally utilizing the allocated frequency resources. Moreover, information delivered by the analog SNBS will be distorted only by the channel noises (no quantization noise). This means that formulas (12) and (15)–(20) also determine the limit information characteristics of digital SNBS with feedback channels. 3) The possibility of analytical accessing the upper boundaries of performance and designing real SNBS links of the same or close to the upper boundary performance enables the development of analytical tools, which would permit the optimization and design of WSN most efficiently employing their spectral and power resources (initial results in this direction are given in [32]). 4) The practical implementation of the presented results, including the development of practical methods of the SNBS performance measurement, will not be too complex – the majority of the appearing questions and tasks are solved in digital communications theory. R EFERENCES [1] L. Doherty, J. Simon, and T. Watteyne, “Wireless sensor network challenges and solutions,” Microw. J., vol. 55, pp. 22–34, Aug. 2012. [2] N. Hunn, Essentials of Short-Range Wireless. Cambridge, U.K.: Cambridge Univ. Press, 2010. [3] C. Buratti, A. Conti, D. Dardari, and R. Verdone, “An overview on wireless sensor networks technology and evolution,” Sensors, vol. 9, no. 9, pp. 6869–6896, Aug. 2009. [4] J. Akhtman and L. Hanzo, “Power versus bandwidth efficiency in wireless networks: From economic sustainability to green radio,” China Commun., vol. 7, no. 2, pp. 6–15, 2010. [5] Y. Chen, S. Zhang, S. Xu, and G. Y. Li, “Fundamental trade-offs on green wireless networks,” IEEE Commun. Mag., vol. 49, no. 6, pp. 30–37, Jun. 2011. [6] S. Caban, J. A. Garcia Naya, and M. Rupp, “Measuring the physical layer performance of wireless communication systems: Part 33 in a series of tutorials on instrumentation and measurement,” IEEE Instrum. Meas. Mag., vol. 14, no. 10, pp. 8–17, Oct. 2011. [7] M. Dohler, R. W. Heath, A. Lozano, C. B. Papadias, and R. A. Valecuela, “Is the PHY layer dead?” IEEE Commun. Mag., vol. 49, no. 4, pp. 159–165, Apr. 2011. [8] A. A. Platonov, “Optimization of adaptive communication systems with feedback channels,” in Proc. IEEE Wireless Commun. Netw. Conf. (WCNC), Apr. 2009. [9] A. Platonov, “Unemployed possibilities of analog communications,” Telecommun. News, no. 6, pp. 286–289, Jul. 2011. [10] A. Platonov, “Analysis of power-bandwidth efficiency and MSE of transmission as the measures of communication systems performance,” Telecommun. News, no. 4, pp. 116–119, Apr. 2012. [11] A. Platonov, “Capacity and power-bandwidth efficiency of wireless adaptive feedback communication systems,” IEEE Commun. Lett., vol. 16, no. 5, pp. 573–576, May 2012. [12] A. Platonov and I. Zaitsev, “Performance of PHY layer links of WSN: Criterions and ways of improvement,” in Proc. IEEE Int. Workshop Meas. Netw., Naples, Italy, 2013, pp. 52–57. [13] A. A. Platonov, “Optimal identification of regression-type processes under adaptively controlled observations,” IEEE Trans. Sign. Process., vol. 42, no. 9, pp. 2280–2291, Sep. 1994. [14] A. Platonov, Analytical Methods in Analog-Digital Adaptive Estimation Systems Design (in Polish), Warsaw, Poland: Warsaw Univ. Technol., 2006. [15] A. Platonov, “Concurrent software–hardware optimisation of adaptive estimating, identifying and filtering systems,” Kybernetes, vol. 37, no. 5, pp. 590–697, 2008. [16] P. Elias, “Channel capacity without coding,” Res. Lab. Electron., MIT, Cambridge, MA, USA, Tech. Rep. 43, pp. 90–93, 1956.


[17] T. J. Goblick, Jr., “Theoretical limitations on the transmission of data from analog sources,” IEEE Trans. Inf. Theory, vol. 1, no. 4, pp. 558–567, Oct. 1965. [18] T. Kailath, “An application of Shannon’s rate-distortion theory to analog communication over feedback channels,” Proc. IEEE, vol. 55, no. 6, pp. 1102–1103, Jun. 1967. [19] R. G. Gallager, Information Theory and Reliable Communication. New York, NY, USA: Wiley, 1968. [20] J. Schalkwijk and L. Bluestein, “Transmission of analog waveforms through channels with feedback (Corresp.),” IEEE Trans. Inf. Theory, vol. 13, no. 4, pp. 617–619, Oct. 1966. [21] S. Butman, “A general formulation of linear feedback communication systems with solutions,” IEEE Trans. Inf. Theory, vol. 15, no. 3, pp. 392–400, May 1969. [22] H. L. Van Trees, Detection, Estimation and Modulation Theory. New York, NY, USA: Wiley, 1972. [23] C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, no. 3, pp. 379–423, Jul. 1948. [24] Evaluation of Measurement Data—Guide to the Expression of Uncertainty in Measurement, document JCGM 100 and BIPM, 2008. [25] B. Sklar, Digital Communications: Fundamentals and Applications. Englewood Cliffs, NJ, USA: Prentice-Hall, 2001. [26] C. E. Shannon, “The zero-error capacity of a noisy channel,” IRE Trans. Inf. Theory, vol. 2, no. 3, pp. 8–19, Sep. 1956. [27] J. S. Lee, CDMA Systems Handbook. Boston, MA, USA: Artech House, 1998. [28] A. A. Platonov and J. Szabatin, “Analog-digital systems for adaptive measurements and parameter estimation of noisy processes,” IEEE Trans. Instrum. Meas., vol. 45, no. 1, pp. 60–69, Feb. 1996. [29] A. A. Platonow, J. Szabatin, and K. Je˛drzejewski, “Optimal synthesis of smart measurement systems with adaptive correction of drifts and setting errors of the sensor’s working point,” IEEE Trans. Instrum. Meas., vol. 47, no. 1, pp. 659–665, Jun. 1998. [30] A. D. Wyner, “Fundamental limits in information theory,” Proc. IEEE, vol. 69, no. 2, pp. 234–251, Feb. 1981. [31] E. Baccarelli and R. Cuzani, “Linear feedback communication systems with Markov sources: Optimal system design and performance evaluation,” IEEE Trans. Inf. Theory, vol. 41, no. 6, pp. 1868–1876, Nov. 1995. [32] P. Elias, “Networks of Gaussian channels with applications to feedback systems,” IEEE Trans. Inf. Theory, vol. 13, no. 3, pp. 493–501, Jul. 1967.

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Anatoliy Platonov (M’91–SM’08) received the M.S. (Hons.) and Ph.D degrees in physics from Lomonosov Moscow State University, Moscow, Russia, in 1969 and 1973, respectively, and the D.Sc. degree in signal processing and electronics from the Warsaw University of Technology, Warsaw, Poland, in 2007. He was with the Faculty of Applied Mathematics, Moscow Institute of Electronics and Mathematics, Moscow, Chair of Cybernetics, from 1972 to 1988, where he got the sc. title of Associate Professor in Cybernetics in 1980. In 1988, he joined the Institute of Electronic Systems at the Warsaw University of Technology, Warsaw, Poland, as the invited Associate Professor, where he is currently a Professor. He was the Head and Chief Researcher of many ministry and university grants. He currently conducts the research group in adaptive processing of information and signals. He has more than 170 publications in the materials of national and international conferences and journals, has authored a monograph Analytical Methods in Analog-Digital Adaptive Estimation Systems Design (in Polish), and holds patents. His current research interests include the information theory, optimization and design of adaptive ADC, top-efficient estimation, measurement and comminication systems with feedback, including the analysis of the upper boundaries of systems performance, and conditions of their achievement.

Ievgen Zaitsev was born in Kiev, Ukraine, in 1987. He received the B.S. (Hons.) and M.S. (Hons.) degrees in telecommunications from the National Technical University of Ukraine (Kyiv Polytechnic Institute), Kiev, in 2008 and 2010, respectively. He is currently pursuing the Ph.D. degree with the Warsaw University of Technology, Warsaw, Poland. He is the author and co-author of eight publications on the power lines EMC modeling and measurement, and adaptive communication systems, published in the materials of national and international conferences and journals. His current research interests include electromagnetic compatibility, adaptive wireless communication systems, electronics, and related areas.