On the Diversity of Hybrid

On the Diversity of Hybrid

Narrowband-PLC/Wireless Communications for Smart Grids

Mostafa Sayed,Student Member, IEEE,Theodoros A.

Tsiftsis,Senior Member, IEEE, and Naofal Al-Dhahir, Fellow, IEEE

Abstract—Narrowband powerline communications (NB-PLC) and unlicensed wireless communications are two leading commu- nications technologies for the emerging Smart Grid applications.

The channel and noise statistics experienced by powerline and wireless transmissions are independent and of a non-identical nature. In this paper, we exploit the diversity provided by the simultaneous transmission of the same information signal over powerline and wireless links to enhance the overall system reliability. In particular, we propose efficient techniques to combine the received signals of the NB-PLC and wireless links for both coherent and differential modulation schemes while considering the impulsive nature of the noise on both links. In addition, we derive closed-form expressions for the average bit- error-rate of the proposed combining techniques. Furthermore, we present simulation results that quantify the performance gains achieved by our proposed receive diversity combining techniques compared to conventional combining techniques.

Index Terms—Impulsive Noise, Powerline Communications, Receive Diversity, Smart Grids, Wireless Communications.

I. INTRODUCTION

Smart Grids are supported by heterogeneous networks that employ both wireless and powerline communication (PLC) technologies since no single solution fits all scenarios [1]. In particular, the two leading contenders for smart-meter two- way wireless communications in the unlicensed 902−928 MHz frequency band in the US are the IEEE 802.15.4g and the emerging IEEE 802.11ah standards. In addition, several PLC standards have been developed for the Smart Grid based on narrowband powerline communication (NB-PLC) in the 3 −500 kHz band (e.g. PRIME, G3, IEEE 1901.2, ITU- T G.hnem). NB-PLC is used for last-mile communications between smart meters at the customer sites and data concentra- tors, which are deployed by local utilities on medium-voltage (MV) or low-voltage (LV) powerlines [2]–[7].

A major design challenge in Smart Grid communications is the presence of strong noise and interference. For instance,

Manuscript received Jul. 27, 2016; revised Feb. 19, 2017; accepted Apr.

7, 2017. Date of publication XXXX XX, 20XX; date of current version XXXX XX, 20XX. The associate editor coordinating the review of this manuscript and approving it for publication was A. Bletsas. This paper was presented in part at the IEEE Global Communications Conference (Globecom), Austin, TX, Dec. 2014 and the IEEE International Conference on Smart Grid Communications (SmartGridComm), Miami, FL, Nov. 2015. This work is supported by a grant from the Semiconductor Research Corporation under SRC GRC Task ID 1836.133 with liaisons Texas Instruments and NXP Semiconductors.

M. Sayed and N. Al-Dhahir are with the Department of Electrical En- gineering, University of Texas at Dallas, Richardson, TX 75080 (e-mail:

{mostafa.ibrahim, aldhahir}@utdallas.edu).

T. A. Tsiftsis is with the School of Engineering, Nazarbayev University, Astana 010000, Kazakhstan (email: theodoros.tsiftsis@nu.edu.kz).

in the unlicensed902−928 MHz band, the wireless interfer- ence is primarily generated from uncoordinated transmissions.

Non-interoperable neighboring devices, operating in the same frequency band, interfere with each other due to coexis- tence issues among different standards. Such uncoordinated interference is impulsive in nature and can be characterized by statistical models such as the Gaussian mixture (GM), Middleton Class-A (MCA) and symmetric alpha stable (SαS) models [8]. In NB-PLC, over the unlicensed3−500kHz band, the dominant interference is a combination of narrowband interference and periodic impulsive noise that is synchronous to half of an AC cycle. Typical sources of the noise and interference include non-linear power electronic devices such as inverters, DC-DC converters, and long-wave broadcast stations whose energy is coupled to the powerlines in the 3−500 kHz band. Throughout this paper, we use the term

“noise” to refer to the combined effect of both noise and interference.

The use of the unlicensed wireless frequency band for Smart Grid applications is one of the use cases of both the IEEE 802.15.4g and the emerging IEEE 802.11ah standards.

For instance, the IEEE 802.11ah standard has included the application of sensors and meters as one of the major use cases [9]–[11]. The coverage of the access point up to1 km is required whereas at least 100 kbps data rate is assumed for the above use cases. For such application, a 1 MHz bandwidth with a factor of2repetition and1/4overall coding rate (including the repetition) is included in the standard so that the transmission range can be increased. The data rate for this scenario is calculated to be 150 kbps [10], [11]

including the guard band (25%) and the cyclic prefix (CP) overhead (20%). Hence, for such a case, the bandwidth and the data rate of the wireless link are comparable to those of the NB-PLC link in the3−500 kHz band. Furthermore, for a 1 MHz channel bandwidth, there are 26 channels in the 902−928 MHz band. In addition, the total bandwidth of the sub-1 GHz band is very narrow compared to the 100 MHz bandwidth in the 2.4 GHz band. Moreover, multiple applications including proprietary ones are operating in the sub-1 GHz band and given the limited number of channels available, these applications might cause high interference to each other. In particular, a critical challenge faced by the IEEE 802.11ah standard is the scarcity of the available spectra in the sub-1 GHz ISM bands [9]. Spatial antenna diversity is a well-known solution to tackle the problem of enhancing the system reliability without sacrificing the spectral efficiency.

Similar to antenna diversity, our proposed NB-PLC/wireless

Combiner

Smart Meter Data Concentrator

OFDM

Modulator DAC UP-

Converter

Input

Bits PLC

Channel

M

OFDM DeModulator Down- ADC

Converter

Output Bits

ADC OFDM

DeModulator

Decoder

D

902-928 MHz

3-500 kHz

Soft Bits Encoder

Figure 1. System Block Diagram for NB-PLC/Wireless Diversity.

Table I

KEY VARIABLES USED THROUGHOUT THE PAPER

Variable Definition Variable Definition

l The OFDM block index Y_p,k^l , Y_w,k^l The frequency domain received symbol on the NB-PLC and the wireless links, respectively k The OFDM subchannel index H_p,k^l , H_w,k^l The frequency domain channel on the NB-PLC and the

wireless links, respectively

N The total number of OFDM subchannels Z_p,k^l , Z_w,k^l The frequency domain noise on the NB-PLC and the wireless links, respectively

N^′ The number of active OFDM subchannels σ²_p, σ²_w The average noise power on PLC and wireless links, respectively

L The number of OFDM blocks within half the AC cycle

period ˜σ²_p,lk,σ˜_w,lk² The average noise power per subchannel on PLC and wireless links, respectively

N_R The number of temporal regions for the PLC link noise

within half the AC cycle period ˇσ²_p,lk,σˇ_w,lk² The frequency-domain instantaneous noise power on PLC and wireless links, respectively

M The number of Gaussian mixture (GM) time-domain

noise states on the wireless link σ_w,m² The noise variance of them-th time-domain GM noise state (out ofM states)

X_k^l The frequency domain transmitted symbol σ¯²_w,i The noise variance of thei-th frequency-domain GM noise state (out ofN+ 1states)

media diversity is also motivated by the need to enhance the system reliability at the same spectral efficiency.

Different from conventional spatial diversity scenarios (e.g.

antenna diversity in wireless systems), simultaneous PLC and wireless transmissions experience noise signals with indepen- dent and non-identical statistics, which we refer to as an asymmetric diversity scenario. This motivates the need for new receive diversity combining techniques that take into account the asymmetric nature of the noise over the diversity branches.

Previous studies on PLC/wireless diversity combining include [12], [13]. However, their investigations considered in-home broadband (BB) PLC transmissions in the 2−30 MHz band and wireless transmissions in the 2.4 GHz band, assuming MCA noise for the BB-PLC link and additive white Gaussian noise (AWGN) for the wireless link, which have different noise characteristics from those encountered by NB-PLC and wireless communications in the unlicensed 902−928 MHz band. Furthermore, in [13], the authors presented a comparison between modulation diversity and coding diversity techniques for parallel BB-PLC and wireless transmission in terms of the total throughput. In modulation diversity, the same coded information bits are transmitted over both links while in coding diversity, the coded information bits are demultiplexed over the two links by transmitting half of the information bits over each link. Coding diversity is shown in [13] to achieve a higher throughput than modulation diversity. On the other hand, at the same data rate, modulation diversity achieves a higher diversity gain than coding diversity. In this paper, assuming orthogonal frequency division multiplexing (OFDM) transmis- sion, we propose efficient modulation diversity receive com- bining techniques for hybrid NB-PLC and unlicensed wireless

transmission that are suitable for the asymmetric impulsive characteristics of the noise on both links. It is noteworthy that modulation diversity is a more suitable technique than coding diversity for the proposed hybrid NB-PLC/wireless diversity since reliability (and not high throughput) is the main goal for outdoor smart meter communications. The main contributions of the present paper are summarized as

• For coherent modulation schemes, we propose two re- ceive diversity combining techniques that take into ac- count the asymmetric impulsive nature of the noise on both the PLC and the unlicensed wireless links. The pro- posed techniques are based on the instantaneous signal- to-noise ratio (SNR) and the noise power spectral density (PSD). The performance of the proposed techniques is compared to the performance of the conventional average- SNR-based combining. The proposed techniques exhibit different performance/complexity tradeoffs.

• For differential modulation schemes, we study the per- formance of two receive diversity combining techniques, which are the average SNR combining and the equal- gain combining. We show that, since the average SNRs of the two links are not necessarily equal, each one of the two combining techniques is suitable for a certain SNR regime.

• For binary phase-shift keying (BPSK), we derive an uncoded bit-error-rate (BER) expression for the proposed combining techniques for coherent modulation under realistic noise and channel models for both links.

• For differential BPSK (DBPSK), we derive an uncoded BER expression for the PLC link with cyclostationary Gaussian noise and a measurement-based channel model.

• For DBPSK, we derive an uncoded BER expression for the unlicensed wireless link with GM impulsive noise and a Rayleigh fading channel model.

• For DBPSK, we derive an uncoded BER expression for the investigated receive diversity combining techniques for differential modulation.

The remainder of this paper is organized as follows. In the next section, we present the system model including the noise and the channel model assumptions. In Section III, we describe our proposed NB-PLC/wireless receive diversity combining techniques for coherent modulation. In Section IV, we analyze the performance of the proposed coherent mod- ulation combining techniques. In Section V, we describe the proposed differential modulation receive diversity combining techniques. In Section VI, the performance of the proposed differential modulation combining techniques is analyzed.

Numerical results are presented in Section VII to compare the performance of the proposed combining techniques. Finally, the paper is concluded in Section VIII. The key variables used in the paper are summarized in Table I.

II. SYSTEMMODEL

As shown in Fig. 1, we assume OFDM transmission for both the NB-PLC and wireless links. At the transmit side, the same information signal is sent over both links simultaneously.

At the receive side, the received signals from the two links are combined by first calculating the log-likelihood ratios (LLRs or soft bits) for each branch and then adding them using appropriate weights. The combined soft bits are then fed to a detector that applies hard decisions on them to obtain estimates for the transmitted information bits. It is worth noting that the combining is performed at the bit (LLR)-level in order to allow the two links to use different signal constellation sizes, fast Fourier transform (FFT) sizes, cyclic prefix lengths or different sampling rates, as long as both links have the same average bit rate at the combiner input.

The received symbols at the combiner input for the k-th subchannel at the l-th OFDM block for the NB-PLC link, denoted as Y_p,k^l , and the wireless link, denoted as Y_w,k^l , are given by Y_p,k^l =H_p,k^l X_k^l +Z_p,k^l , Y_w,k^l =H_w,k^l X_k^l +Z_w,k^l , whereX_k^l is the transmitted symbol with varianceEs=NbEb, where Nb is the number of bits per symbol and Eb is the average energy per bit. Z_p,k^l and Z_w,k^l are complex random variables (RVs) with zero mean and variances σ²_p and σ²_w, respectively, that represent the frequency-domain noise on the NB-PLC and wireless links, respectively. H_p,k^l andH_w,k^l represent the frequency-domain complex channel coefficients of the PLC and wireless links, respectively. Next, we state and justify our assumptions regarding the noise and the channel models for the NB-PLC and wireless links.

A. NB-PLC Link Noise Model

The generation of the impulsive noise process that best fits actual measurements is presented in [14]. The noise process in NB-PLC is a cyclostationary noise process with a period of half the AC cycle that is divided into NR temporal regions where the noise over each region can be assumed to be a stationary process. Each region is characterized by

a discrete-time linear time-invariant (LTI) filter h_j(n). The average noise power in each region is given by E(

|z(n)|²)

=

||h_j(n)||², n ∈ Ij, where E(.) denotes the expectation operation andIjdenotes the set of indexes of the noise sample that belong to regionj. The noise model is then parameterized by: the number of stationary regions NR, the region intervals {Ij : 1 ≤ j ≤ NR}, and the LTI filters {hj(n) : 1 ≤ j≤NR}, which are represented by their corresponding noise PSDs, denoted by {σ˜_p,jk² : 1 ≤ j ≤ NR}, obtained from field measurements¹. Hence, given the noise region index, the probability density function (PDF) of Z_p,k^l is Gaussian with zero mean and variance σ˜²_p,lk = ˜σ_p,jk² , ∀l ∈ Lj, where Lj

denotes the set of OFDM block indexes that belong to thej-th noise region, i.e.

Z_p,k^l |j∼ CN(

0,σ˜_p,jk² )

. (1)

σ²_p can be written in terms of σ˜²_p,jk as σ²_p =

1 N^′

∑N^′−1

k=0 R1σ˜²_p,1k+R2σ˜²_p,2k+R3σ˜²_p,3k, whereRj denotes the time-percentage of the j-th noise region relative to the noise cyclostationarity period.

B. Wireless Link Noise Model

To the best of the authors’ knowledge, there are no papers in the literature that discuss the noise modeling in the902−928 MHz (sub-1 GHz) unlicensed wireless band. However, we assumed the analogy between the sub-1 GHz and the 2.4 GHz frequency bands since both are unlicensed bands that have similar operating communication technologies, which are mainly WiFi- or ZigBee-based standards. In particular, ZigBee, Bluetooth, 6LoWPAN, Wi-Fi and RF4CE are widely used2.4GHz solutions. Sub-1 GHz standards-based solutions include ZigBee (currently the only protocol offering both 2.4 GHz and sub-1 GHz versions in the 868 MHz and 900 MHz bands), IEEE 802.11ah WiFi standard, EnOcean for automation systems, and ONE-NET for control applications [15], [16]. Hence, we followed the work in [17, Sec 3.3]

proposed for the2.4 GHz band which modeled the noise as a two-component Gaussian mixture random process.

It is well-known that most of the unlicensed wireless standards adopt a random multiple access (MAC) protocol, which avoids collisions by sensing the medium and waiting for random back-off time when there is an ongoing transmission.

However, what still causes the interference in this case is the presence of uncoordinated transmissions by non-interoperable devices. In such a case, the devices within the same coverage area might follow different standards, for different applica- tions, that lack coexistence mechanisms between them. An example for such a scenario in the sub-1 GHz band is the presence of devices that follow the IEEE 802.11ah WiFi standard and other non-interoperable devices that follow the IEEE 802.15.4g ZigBee standard, although both standards adopt a carrier sense multiple access (CSMA) with collision avoidance MAC protocol. Furthermore, it is well known that the2.4GHz band is currently better in terms of interoperability than the sub-1 GHz since more global standards are being currently introduced for the sub-1 GHz, but most companies deploy proprietary protocols [15], [16].

1For more details about the NB-PLC noise generation please refer to [4].

Various statistical models have been proposed to capture the statistics of the noise that affects the uncoordinated wireless transmissions in the unlicensed frequency bands including the GM, the MCA, and the SαS models. Given that the MCA PDF is a special case of the GM PDF and that the SαS RV can also be approximated by a GM RV, we assume that the noise in the wireless link is modeled as a GM random process [8]. Next, we present the time-domain and frequency-domain statistics of the noise on the wireless link.

The PDF of the GM distribution is a weighted sum of a set of Gaussian PDFs. The PDF of a GM RVzis given byp(z) =

∑M−1

m=0αw,m/(πσ²_w,m) exp(

−|z|²/σ_w,m² )

,whereαw,mis the probability of them-th Gaussian state, andM is the number of states. Each state has a noise varianceσ_w,m² where the average noise variance over all states is σ_w². We assume that the state with indexm= 0represents the thermal noise component. In practice, only two terms of the GM PDF are enough to fit the impulsive noise to the GM model [8].

The noise in the frequency domain is given by Zk = 1/√

N∑N−1

n=0 ζkn, k = 0, . . . , N − 1, where N is the FFT size and ζkn = zne^−j2πNkn. Given the state of the n-th noise sample, the PDF of ζkn is Gaussian with zero mean and variance σ_w,m² , i.e. ζ_kn|m ∼ CN(

0, σ²_w,m) . Hence, for the special case of M = 2, the PDF of Z_k is given byZ_k∼∑N

i=0

(_N

i

)αⁱ_w,0α^N_w,1⁻ⁱCN( 0,¯σ_w,i² )

[18], where

¯

σ_w,i² = 1 N

[iσ_w,0² + (N−i)σ²_w,1]

. In this case, the PDF of the frequency-domain noise also follows a GM distribution with N+ 1 states. In summary, given the frequency-domain noise state index, the PDF of Z_w,k^l is Gaussian with zero mean and varianceσ˜_w,lk² = ¯σ_w,i² . The last statement hold true provided that theN noise samples added to the OFDM block (without CP) of indexl haveisamples that belong to the first state of the GM distribution (the thermal noise state with prior probabilityαo and varianceσ²_o), i.e.

Z_w,k^l |i ∼ CN( 0,σ¯_w,i² )

. (2)

σ_w² can be written in terms of σ¯²_w,i as σ²_w =

∑N i=0

(_N

i

)αⁱ_w,0α^N_w,1⁻ⁱσ¯²_w,i.

C. Channel Models

For the PLC link, we adopt a channel model based on our laboratory measurements for low-voltage (LV) powerline.

We measured the channel impulse response (CIR) by sending a known periodic training sequence from one end of the powerline and then estimating the CIR from the received signal at the other end. The measured CIR was found to be periodic with a period of around half of the AC cycle. Similar results for the periodicity of the NB-PLC CIR are reported in [2], [19], [20]. For more details about the measurement-based channel model we adopt in this paper, please refer to [4]. For the wireless link, we assume a Rayleigh fading model since it is widely accepted to capture small-scale fading effects on non- line-of-sight signal propagation in wireless environments.

III. NB-PLC/WIRELESSRECEIVECOMBININGFOR

COHERENTMODULATION

In this section, we highlight the differences between the conventional receive diversity combining scenarios and the

proposed NB-PLC/wireless receive diversity, and present our proposed combining techniques for the coherent modulation schemes.

A. Average SNR Combining (ASC)

Given that the noise statistics on both links are asymmetric and that the NB-PLC noise model is non-stationary and is based on field measurements for the noise PSD, it is challenging to derive the optimal maximal ratio combining (MRC) scheme. Furthermore, in the wireless link, the optimal sufficient statistic for signal detection in the presence of GM noise is computationally intensive [21]. A sub-optimal version of the MRC scheme can be implemented by assuming that the noise on both the NB-PLC and wireless links follows a white Gaussian distribution. In this case, the combined log-likelihood (LL) function for thek-th subchannel of thel-th OFDM block can be expressed as

LL(X_k^l) = log[ p(

Y_p,k^l |H_p,k^l X_k^l) p(

Y_w,k^l |H_w,k^l X_k^l)]

= −|Y_p,k^l −H_p,k^l X_k^l|²

σ_p² −|Y_w,k^l −H_w,k^l X_k^l|² σ²_w ,

(3) where σ²_p and σ²_w are the average noise powers (variances) of the NB-PLC and the wireless links, respectively. Without loss of generality, assuming BPSK modulation, the combined LLR for the k-th subchannel of thel-th OFDM block can be expressed as LLRlk = LL(X_k^l = 1)−LL(X_k^l = −1) = LLRw,lk +LLRp,lk. Since the average SNR is inversely proportional to the average noise power, we refer to the combining technique described by (3) as the average SNR combining(ASC) technique.

B. Instantaneous SNR Combining (ISC)

From (3), we note that the contribution of each link to the combined LLR is inherently weighted by the inverse of the average noise power on that link. However, due to the impulsive nature of the noise on both the NB-PLC and the wireless links, the noise power level exhibits rapid variations over both time and frequency. As a result, the average noise power can be a highly sub-optimal combining metric for such noise characteristics.

The rapid variations of the instantaneous noise power over the frequency subchannels across several OFDM blocks are evident in Fig. 2. Moreover, the noise power is shown to have a high peak-to-average ratio (PAR), which is higher on the PLC link (around 21 dB) than on the wireless link (around 14 dB). It is worth mentioning that the PAR for the AWGN scenario is around10dB. Hence, to capture the instantaneous noise power variations, we propose using the instantaneous noise powers, or equivalently the instantaneous SNRs, as the combining weights. Thus, the LL function can be written as LLISC(X_k^l) = −|Y_p,k^l −H_p,k^l X_k^l|²

ˇ

σ²_p,lk −|Y_w,k^l −H_w,k^l X_k^l|² ˇ

σ_w,lk² , (4) whereˇσ_lk² is the instantaneous noise power for thel-th OFDM block at the k-th subchannel. σ_p² and σ_w² can be written in terms of σˇ_p,lk² and σˇ²_w,lk as σ_p² = El,k

[ ˇ σ_p,lk²

]

and σ²_w =

Symbol Index _×10⁴

0 0.5 1 1.5 2 2.5 3 3.5

Noise Power

0 10 20 30 40 50 60

Symbol Index ×10⁴

0 0.5 1 1.5 2 2.5 3 3.5

Noise Power

0 2 4 6 8 10 12 14 16 18 20

Figure 2. Noise Power in PLC link (top) and wireless link (bottom).

El,k

[ ˇ σ_w,lk²

]

, respectively, where El,k[.] denotes the expecta- tion over the OFDM blocks and the frequency subchannels. To estimate the instantaneous noise power, comb-type pilots can be inserted periodically within the active OFDM subchannels to estimate the noise power at the pilot locations. Then, linear interpolation can be used to estimate the noise power at the non-pilot subchannel locations. More accurate techniques for estimating the instantaneous power of the impulsive noise can be found in the literature. Among others, key papers for cyclostationary impulsive noise power estimation include [3], [22]. However, we believe that comparing the performance of different impulsive noise power estimation techniques is out of the scope of this paper. Nonetheless, we included the performance of the instantaneous SNR combining to show that it has a superior performance to other combining techniques in presence of impulsive noise, as will be shown in the numerical results presented in Section VII.B, and also to motivate future research in impulsive noise power estimation since the available techniques in the literature are either too complex for a practical implementation and/or require very high pilot overhead.

C. PSD Combining (PSDC)

We propose using the average noise power per OFDM subchannel, or equivalently the noise PSD, as the combining metric, since the noise PSD is easier to estimate than the instantaneous noise power. In this case, the LL function can be expressed as

LLPSDC(X_k^l) = −|Y_p,k^l −H_p,k^l X_k^l|²

˜ σ_p,lk²

− |Y_w,k^l −H_w,k^l X_k^l|²

˜

σ²_w,lk , (5) whereσ˜²_lkis the average noise power for thel-th OFDM block at the k-th subchannel. It is worth mentioning that the noise PSD might vary from one OFDM block to another in the PLC link since the noise has multiple stationary regions with different PSDs, while for the wireless link the PSD is the same for all OFDM blocks. This is the reason why we added the subscript l to the average noise power per subchannel ˜σ_lk².

Next, we present a simple and effective technique to estimate the noise PSD from the received signal power.

D. Noise PSD Estimation

The PLC noise is cyclostationary with a period of half the AC cycle. Within each period there are three temporal regions where the PLC noise is stationary over each of them, i.e. the noise PSD is fixed over each of the three regions.

Hence, assuming knowledge of the noise region boundaries, the PLC noise PSD for each temporal region can be estimated separately from other regions. In other words, the noise PSD for each noise region is estimated only from the OFDM blocks that belong to this noise region. On the other hand, the wireless link noise, which is assumed to be of GM distribution, is stationary and hence all OFDM blocks can be utilized for noise PSD estimation. Furthermore, the PSD of the noise in the wireless link is shown in the numerical results presented in Section VII.B to be flat over frequency and equal to the GM average noise varianceσ²_w, and hence the noise PSD estimation is not needed for the wireless link. In the following, we explain our proposed noise PSD estimation technique.

The power of the received k-th subchannel of the l-th OFDM block can be written as

|Y_k^l|² = |H_k^lX_k^l|²+|Z_k^l|²+ 2Re[

H_k^lX_k^lZ_k^l∗] . (6) where Re(.) denotes the real part operator. Averaging over

|Y_k^l|², we get E(

|Y_k^l|²)

= E(

|H_k^l|²) E(

|X_k^l|²) +E(

|Z_k^l|²) + 2Re[

E( H_k^lX_k^l)

E( Z_k^l∗)]

. (7)

Since E( Z_k^l∗)

= 0, then E(

|Y_k^l|²)

reduces to E(

|Y_k^l|²)

= E(

|H_k^l|²) E(

|X_k^l|²) +E(

|Z_k^l|²)

. Setting E(

|X_k^l|²)

= 1, we get

˜

σ_lk² =E(

|Z_k^l|²)

=E(

|Y_k^l|²)

−E(

|H_k^l|²)

. (8) Hence, from (8), for a certain subchannel k, the noise power can be estimated by subtracting the average channel power from the average received symbol power. Broadly speak- ing, the averaging length (required for a certain noise PSD estimation accuracy), which is the only parameter for the noise power estimation technique, is dependent on the channel PDF, and the channel autocorrelation function (over time) and its associated channel coherence time in the sense that the averaging length should be long enough such that the channel gain time average approaches E(

|H_k^l|²)

. Furthermore, the averaging time duration has to be long enough to suppress the term E(

Z_k^l∗)

in (7) to obtain an accurate estimate for the noise PSD. However, the receiver can start decoding the received data, using some initial combining weights, while the averaging is running and does not have to wait for the averaging to converge.

There is an important aspect to note about the channel characteristics in the NB-PLC link. The NB-PLC channel is a deterministic channel, that is either fixed over all OFDM blocks or periodic over one (or half) AC cycle [2]. Hence, for such a case, the averaging of the channel over one AC cycle would be sufficient to obtain the average power for the channel gain per subchannel. However, the averaging length over the received signal power should be longer than one AC cycle

to ensure that E( Z_k^l∗)

= 0 and, consequently, that the time average of the received signal power approaches E(

|Y_k^l|²) . For PLC noise, the PSD estimation technique can be im- plemented using three moving-average filters, one for each noise region, where each filter keeps averaging the received frequency domain signal power, following (8), over the filter’s corresponding temporal region until this region ends. Once we hit the next region boundary, the next filter is turned on while the currently running filter is paused. Then, we keep multiplexing over the three filters until the convergence of the noise PSD estimation is reached. One method is to terminate the averaging after a certain number of OFDM blocks is reached. Another method is to keep monitoring the change in the estimated noise PSD and terminate the averaging if the change is less than a certain threshold. It is worth mentioning that the current NB-PLC environment is an outdoor environment where the PLC network topology is almost static. Hence, the noise PSD, which depends on the network topology and the connected components within the network, is expected to be fixed. However, in general, we can periodically re-enable the noise PSD estimation to account for any changes in the PLC network. In this case, the re-enable period should be much longer than the noise PSD estimation convergence time. In addition, the PSD combining technique can be set up to use the latest noise PSD estimate until the convergence of the new PSD estimate.

A question that arises here is how to estimate the noise region boundaries. In this regard, in [23], we developed a technique for the noise region boundary estimation and studied its performance in terms of the missed detection rate (MDR) of the noise region boundaries.

IV. PERFORMANCEANALYSISFORCOHERENT

MODULATION

In this section, first we present analytical expressions for the uncoded average BER for both the PLC and wireless links.

Then, we derive an expression for the uncoded average BER of our proposed PSDC technique assuming perfect PSD knowl- edge. We note that the performance of the ASC technique can be viewed as a special case of the PSDC technique when the noise power is constant over all subchannels. On the other hand, the performance of the ISC technique is dependent on the statistics of the instantaneous noise power, which makes it complex to analyze.

A. PLC Link

The average BER of the PLC link can be written as Pb(E) = 1/(LN^′)∑L−1

l=0

∑N^′−1

k=0 P_b,k^l (E), where P_b,k^l (E) is the average BER corresponding to the l-th OFDM block and the k-th subchannel. For BPSK modulation, P_b,k^l (E) is given by P_b,k^l (E) = Q

(√

2γ^l_p,k )

[24], where γ_p,k^l = E_b|H_p,k^l |²/˜σ_p,lk² is the SNR for thek-th subchannel and thel- th OFDM block andQ(.)is the Gaussian-Qfunction defined by [24, Eq. (2.3-10)].

B. Wireless Link

The BER of the wireless link can be written as Pb(E) =

∑N i=0

(_N

i

)αⁱ_w,0α^N_w,1⁻ⁱP_b,i(E),whereP_b,i(E), for BPSK mod-

ulation and Rayleigh fading, is given by P_b,i(E) = ¹₂ (

1−

√λ_w,i/(1 +λ_w,i) )

, where λ_w,i = ˜E_b,h/σ¯_w,i² ,E˜_b,h=E_bE_h andEh=E|H|².

C. NB-PLC/Wireless PSD Combining

Assuming BPSK modulation, the frequency-domain re- ceived symbol over thek-th OFDM subchannel and thel-th OFDM block after PSD combining, denoted asY^l

PSDC,k, can be represented as²

YPSDC^l ,k =

[|H_p,k^l |²

˜

σ²_p,lk +|H_w,k^l |²

˜ σ_w,lk²

] X_k^l

+



 (

H_p,k^l )_∗

˜

σ²_p,lk Z_p,k^l + (

H_w,k^l )_∗

˜

σ_w,lk² Z_w,k^l



. (9)

The noise term in (9) is complex Gaussian with PDF CN(

0, (

γ_p,k^l +γ_w,k^l )

/Eb

)

, whereγ_p,k^l =Eb|H_p,k^l |²/˜σ²_p,lk, and γ_w,k^l = E_b|H_w,k^l |²/˜σ²_w,lk = ˜E_b,h|H_w,k^l |²/(E_h¯σ_w,i² ) =

|H_w,k^l |²λ_w,i/E_h. Hence, the conditional BER givenγ^l_p,k and γ_w,k^l can be written as P_b,k,i^l (E) = Pb

(

E|γ_p,k^l , γ_w,k^l )

= Q

(√

2 (

γ_p,k^l +γ_w,k^l ))

. Thus, the PSDC average BER can be expressed as

Pb(E) = 1 LN^′

L∑−1 l=0

N∑^′−1 k=0

∑N i=0

(N i

)

αⁱ_w,0α^N_w,1⁻ⁱ

× ˆ _∞

0

P_b,k,i^l (E)pγ_w,i(t)dt, (10) wherepγ_w,i(t) = 1/λw,iexp [−t/λw,i] assuming a Rayleigh fading channel on the wireless link. Hence, Pb(E) can be simplified as

Pb(E) = 1 LN

L∑−1 l=0

N∑−1 k=0

∑N i=0

(N i

)

αⁱ_w,0α^N_w,1⁻ⁱJ_k,i^l , (11a)

J_k,i^l = 1 λw,i

ˆ _∞

0

Q (√

2 (

γ_p,k^l +t ))

exp (

− t λw,i

) dt.

(11b) Using integration by parts,J_k,i^l can be obtained as

J_k,i^l = Q (√

2γ_p,k^l )

+ 1/(2

√

π(1 + 1/λw,i)) exp( η^l_k,i)

× Γ(

1/2, ηl,k,i+γ_p,k^l )

, (12)

where η_k,i^l = γ_p,k^l /λw,i and Γ (., .) is the upper incomplete gamma function defined in [25, Eq. (8.350.2)]. For large values ofη^l_k,i, the exponential function in the expression ofJ_k,i^l tends to infinity while the incomplete gamma function tends to zero, which makes J_k,i^l indeterminate. To avoid this situation, for high values of η_k,i^l , we evaluate (J_k,i^l ) using the asymptotic expansion for the upper incomplete gamma function given in [25, Eq. (8.357)]. Hence, (11a) reduces to (see Appendix A for details)

2The expression in (9) is the symbol-level combining counterpart to the LLR-level combining expression in (5), where both expressions lead to the same uncoded BER.

-5 0 5 10 15 20 25 30 35 40 Eb/No(dB)

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Average BER

PLC-Theory PLC-Simulation Wireless-Theory Wireless-Simulation PSD Combining-Theory PSD Combining-Simulation

Figure 3. Average BER vsEb/Noin dB for BPSK modulation for the PLC link, the wireless link, and the PSDC technique.

Pb(E) ≈













 1 LN

∑L−1 l=0

∑N−1 k=0

∑N i=0

(_N

i

)αⁱ₀

×α^N₁⁻ⁱJ_k,i^l , η^l_k,i< η_Γ^T 1

LN

∑L−1 l=0

∑N−1 k=0

∑N i=0

(_N

i

)αⁱ₀

×α^N₁⁻ⁱJ˜_k,i^l , η^l_k,i≥η_Γ^T

,(13a)

J˜_k,i^l = Q (√

2γ_p,k^l )

+ exp

(−γ_p,k^l ) 2µw,i

√ πγ^l_p,k

( 1− 1

2η^l_k,i 3 4η_k,i^{l 2}

− 15

8η_k,i^{l 3} + 105 16η_k,i^{l 4}

)

, (13b)

where η_Γ^T is set to 50 (see Appendix A for justification).

Fig. 3 shows that the derived average BER expressions of the PLC link, the wireless link and the PSD combining match the simulation results. The system parameters used to generate Fig. 3 are listed in Section VII.

V. NB-PLC/WIRELESSCOMBININGFORDIFFERENTIAL

MODULATION

For OFDM transmission, there are two types of differential modulation schemes: frequency-domain differential modula- tion (FD-DM) and time-domain differential modulation (TD- DM). In FD-DM, the information is carried in the phase difference between two consecutive OFDM subchannels of the same OFDM block. On the other hand, in TD-DM, the information is carried in the phase difference between two OFDM subchannels of two consecutive OFDM blocks at the same frequency subchannel. The performance of FD-DM and TD-DM depends on the channel’s frequency and time coherence characteristics, respectively.

Let Yν and Yν′ denote the two received symbols used for the demodulation of the information symbolXν. Thus,Yνand Yν^′ can be written asYν =HνUν+Zν, Yν^′ =Hν^′Uν^′+Zν^′, where Xν =UνU_ν^∗′, ν =lN^′+k, andν^′ =lN^′+k−1 or ν^′= (l−1)N^′+kfor FD-DM and TD-DM, respectively. For differential PSK modulation, a sufficient statistic for symbol detection is given by [24] ∠Dν = ∠(YνY_ν^∗′), where ∠(.) denotes the angle operation. Assuming the phase difference between Hν and Hν^′ to be very small, Dν can be approxi- mated asD_ν≊|H_ν||H_ν′|X_ν+H_ν^∗_′U_ν^∗_′Z_ν+H_νU_νZ_ν^∗_′+Z_ν^∗_′Z_ν.

In this case, ∠D_ν = ∠X_ν +ϕ_ν, where ϕ_ν represents the noise term. It is noteworthy that, for differential modulation, channel knowledge is not required for information symbol decoding, and thus pilot transmission is not needed. Let Dˆν =e^j∠Dν =Xν+ ˆZν, whereZˆν represents the noise term inDˆν, which is assumed to be Gaussian. Hence, we can write the LL function for detectingXνasLL(Xν) =−|Dˆν−Xν|². Due to the absence of channel knowledge for differential detection, we cannot use receive combining techniques that rely on the received SNR (either the instantaneous SNR or the average SNR per subchannel) since it requires channel knowl- edge, where the instantaneous SNR and the average SNR per subchannel are defined as E_s|H_ν|²/|Z_ν|² and E_s|H_ν|²/σ˜_ν², respectively, whereE_s is the variance ofU_ν andU_ν′. Hence, we consider two alternative techniques for combining the LLRs of the two links; namely, the equal gain combining (EGC) and the ASC techniques, which have the combined LL functions

LLEGC(X_ν) = LL_p(X_ν) +LL_w(X_ν)

= −|Dˆp,ν−Xν|²− |Dˆw,ν−Xν|²,(14) LLASC(Xν) = 1

σ²_pLLp(Xν) + 1

σ²_wLLw(Xν)

= − 1

σ_p²|Dˆp,ν−Xν|²− 1

σ_w²|Dˆw,ν −Xν|². (15) VI. PERFORMANCEANALYSISFORDIFFERENTIAL

MODULATIONS

In this section, we derive uncoded average BER expressions for differential modulation over the PLC link, the wireless link and the investigated PLC/wireless differential combining techniques, namely, the EGC and ASC techniques.

It follows from Section II that, different from the AWGN model, the noise models for both the NB-PLC and the unli- censed wireless links might cause the two symbols involved in the differential detection to experience noise processes with different variances. Let R = [R₁, R₂]^t represent the two symbols used for the differential detection. Moreover, let R¯ = [R¯1, R¯2

]t

denote the mean of R and Λ denote the covariance matrix ofR, which is defined as

Λ=

[ σ11 σ12

σ^∗₁₂ σ22

]

. (16)

Assuming R1 and R2 to be jointly Gaussian, their joint PDF can be written as P(R) ∼ N(0,Λ) = π⁻²|Λ|⁻¹exp

[−(

R−R¯)H

Λ⁻¹(

R−R¯)]

. Considering DBPSK modulation and assuming “0” is sent, which means no change in phase, with a zero initial (reference) phase, we can write the optimal detection metric as [24]

DB=Re(R1R^∗₂). (17) The detection metric in (17) is a special case of the general quadratic detector given by [26] D = A|R1|²+B|R2|²+ CR1R^∗₂ +C^∗R^∗₁R2, where A = B = 0 and C = 1/2 for the differential detector. The average BER of the general quadratic detector is studied in [26] by analyzing the expres- sionP_b(E; ¯R,Λ) =P r{

D <0; ¯R,Λ}

. Following a similar