4

Broadband In-Home Characterisation and Correlation-Based Modelling

Kaywan Afkhamie, Paola Bisaglia, Arun Nayagam, Fabio Osnato, Deniz Rende, Raffaele Riva and Larry Yonge

CONTENTS

4.1    Introduction

4.2    Description of the MIMO PL Field Test System

4.3    Channel Statistical Analysis

4.3.1    Average Attenuation versus Frequency

4.3.2    Distribution of the Channel Coefficients

4.3.3    RMS-DS versus the Attenuation

4.3.4    MIMO Channel Correlation

4.3.4.1    Spatial Correlation Analysis

4.4    Proposed PL Channel Model

4.4.1    SISO Channel Model

4.4.2    MIMO Channel Model

4.4.2.1    Spatial Correlation Model

4.4.2.2    Simpler Correlation Model

4.5    Characterisation of Noise in MIMO PL Channels

4.5.1    Eigenspread Analysis of MIMO PL Channels

4.5.2    Impact of Noise Correlation on the MIMO PL System

4.6    Conclusions

References

4.1    Introduction^*

The in-home network of the future will be a hybrid network where multimedia contents must be provisioned over a stable broadband backbone (as described in Chapter 15). In agreement with this vision, in the past years, the HomePlug Alliance had released the HomePlug AV 1.1 (HPAV 1.1) specification [4]. This solution allows achieving a maximum throughput of 200 Mbit/s over the power line (PL; see also Chapter 13). Today, the market demand requires a migration to a much higher performance in order to support applications like high-definition multimedia contents and gaming. To accommodate the request for this increase in capacity, reliability and coverage, the HomePlug Alliance has defined the specification for the next generation of PL technology, namely, HomePlug AV 2.0 (HPAV 2.0) (see Chapter 14). One of the major enhancements related to this technology is the introduction of multiple-input multiple-output (MIMO) technique. This technique has already attracted attention in wireless communications (see, e.g.the standards IEEE 802.11n [5] and 3GPP LTE [6]).MIMO applied to PL, as already analysed in the literature, addresses either the channel capacity [7,8] or the performance of different MIMO schemes [9,10] (see Chapter 9). Moreover, MIMO PL channels are also addressed in [11,12,13,14,15,16,17,18]. An introduction to PL channel and noise characterisation is provided in Chapter 1, where the most representative in-home. PL models that can be found in recent literature are summarised. More details on the characteristics of the MIMO channel are presented in this chapter and in Chapter 5. In the literature, the characteristics of the PL noise have not been widely discussed; in many contributions, it is simply assumed to be additive white Gaussian noise (AWGN). The main results suggested that the in-home PL channel capacity can be increased by a factor of around 2 when MIMO techniques are applied.

This chapter addresses the statistical characterisation of MIMO PL channels and noise based on experimental measurements collected in the United States. It presents a MIMO channel modelling approach based on correlation matrices. Statistics of the in-home channel and noise characterisation in Europe is presented in Chapter 5, along with a stochastic channel model based on experimental databases. Analysing a wide set of measurements, the statistical distribution of the most important channel and noise parameters is reported. The considered set of in-home PL channels contains 92 transfer functions measured in the 1.8–88 MHz range, in five North American houses.

This chapter is organised as follows. Section 4.2 details the measurement setup and the procedure to estimate the coefficients of the MIMO PL channels. Section 4.3 presents a statistical analysis of the collected data and a comparison with previous available literature analysis. Based on these results, Section 4.4 proposes a MIMO PL channel model with particular attention to the spatial correlation properties. Section 4.5 concentrates on the characterisation of MIMO PL noise and its effect on channel capacity. Finally, in Section 4.6, conclusions are reported.

The following notation will be used. Vectors and matrices are in bold face. The symbols (⋅)^T, (⋅)^–1, (⋅)^H, (⋅)^*, E[⋅] denote the transposition, the inverse, the Hermitian (conjugate transposition), the conjugate and the expectation, respectively.

4.2    Description of the MIMO PL Field Test System

The apparatus and signalling used to characterise the MIMO PL channel is described in this section. In details, the characterisation of the transfer function of the channel is needed, as well as the noise that is present on the different receiver ports. A typical approach that is used to observe the transfer function of a channel involves the use of a vector network analyser (VNA). The VNA sounds the channel using narrowband tones and computes the various S-parameters. This results in an accurate measurement of the channel transfer function. However, time-domain phenomena cannot be observed using a VNA.

In order to observe time-domain behaviour and to have the ability to collect noise samples, a time-domain characterisation approach has been chosen. A block diagram of the system is presented in Figure 4.1. Although three ports are available for signalling, only two out of the three ports are useful for transmitting simultaneously. This is because on the transmitter side, the voltage on the third port is a linear combination of the voltages on the other two ports (Kirchhoff’s law). In this effort, line–neutral (L-N) and line–protective earth (L-PE) were used as the transmitting wire pairs. On the transmitter side, two independent signals are generated by a Tektronix AWG2021 arbitrary wave-form generator (AWG) and passed through an analogue front end (AFE). The two-port AFE amplifies the signals and provides the coupling circuitry to inject the signal onto the electrical wires. The two time-domain waveforms are packets composed of two portions: the preamble and the payload. Preamble is dedicated to automatic gain control tuning, frame detection and frame synchronisation. The payload portion is orthogonal frequency-division multiplexing (OFDM) modulated, with a carrier spacing of 24.414 kHz (as in the HPAV 1.1 specification [4]) and 4096 carriers. In order to facilitate the estimation of the MIMO channel coefficients, two orthogonal codes are applied to the transmitted signals; hence, during the payload, each OFDM symbol is repeated twice, with the sign fixed by the orthogonal codes, that is, [+1, +1] code applied to the signal transmitted from one port and [+1, -1] code applied to the signal transmitted from the other port. Considering a sampling frequency of 200 MHz, the two signals occupy the frequency range from 0 to 100 MHz. The actual starting frequency is 1.8 MHz, as in HPAV 1.1, and to avoid the noise interference from the FM band, the frequencies above 88 MHz have been masked. Each packet consists of five orthogonal symbol pairs.

FIGURE 4.1
The MIMO power line field test system. (From Rende, D., Nayagam, A., Afkhamie, K., Yonge, L., Riva, R., Veronesi, D., Osnato, F., and Bisaglia, P., Noise correlation and its effects on capacity of inhome MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

To improve the received signal quality, all three of the ports can be used on the receive side, that is, L-N, L-PE and N-PE. In fact, a fourth port is also available if the common-mode signal can be captured [9]. The receiver in the field test system consists of a three-port receiver AFE and coupler module followed by a 200 MSamples/s, 4-channel 16-bit digitizer (CS16200 manufactured by Gage-Applied Inc.). Note that coupling on both the transmitter and receiver sides is capacitive and designed to minimise crosstalk between the individual ports of the AFE. The digitizer is connected to a personal computer. The General Purpose Interface Bus (GPIB) back-channel link is used to load the AWG with the two transmit waveforms that are generated by the computer. The receiver AFE filters and amplifies signals that are coupled from the PL on the L-N, L-PE and N-PE wire pairs, respectively.

The signalling structure is shown in Figure 4.2. The AWG transmits a repeating sequence of three packets. As illustrated in Figure 4.2, the transmitter first performs a MIMO transmission using two transmitter ports, then a single-input single-output (SISO) transmission using L-N wire pairs only and then a SISO transmission using L-PE wire pairs only. This sequence is repeated continuously, and received signals are captured to collect about 16 repetitions of MIMO, SISO L-N and SISO L-PE transmissions. During the L-N-and L-PE-only transmissions, the transmit power is increased by 3 dB to keep the overall transmit power spectral densities (PSDs) even between the MIMO and SISO transmissions. At the output of the transmit coupler, the PSD for each transmit stream in the 1.8–88 MHz band is -67 dBm/Hz during the MIMO packet transmission (first packet in the sequence) and -64 dBm/Hz during the SISO transmissions (second and third packets in the sequence).

FIGURE 4.2
Signalling used in the PL MIMO field test system. (From Rende, D., Nayagam, A., Afkhamie, K., Yonge, L., Riva, R., Veronesi, D., Osnato, F., and Bisaglia, P., Noise correlation and its effects on capacity of inhome MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

As seen in Figure 4.2, there are gaps between the different packets. The received waveform samples in the ‘transmission-free’ gaps between the packets are samples of noise on the PL, and these samples are used to characterise the PL noise.

This test setup was used to collect data from 92 paths distributed equally among five homes in North America. All homes are detached single family homes varying in age from 5 to 25 years and in size from ∼180 to ∼320 m². The extraction of relevant channel and noise parameters from the waveforms collected in the field is described next.

During MIMO transmission, the received samples in the frequency domain on carrier k are given by

$x_k=H_k{a}_k+n_k,$

(4.1)

where

$a_k={\left[a_k^{\left(1\right)},a_k^{\left(2\right)}\right]}^T$ are the symbols transmitted from the two transmitter ports

$x_k={\left[x_k^{\left(1\right)},x_k^{\left(2\right)},x_k^{\left(3\right)},\right]}^T$ are the samples received at the three receiver ports

$n_k={\left[n_k^{\left(1\right)},n_k^{\left(2\right)},n_k^{\left(3\right)}\right]}^T$ are the noise samples at the three receiver ports

H_k is the 3 × 2 channel matrix on carrier k

$H_k=\left[\begin{array}{cc}{H}_k^{\left(1,1\right)}& H_k^{\left(1,2\right)}\\ H_k^{\left(2,1\right)}& H_k^{\left(2,2\right)}\\ H_k^{\left(3,1\right)}& H_k^{\left(3,2\right)}\end{array}\right].$

(4.2)

$H_k^{\left(r,t\right)}$, in the channel matrix, represents the channel from the transmitter port t to receiver port r on carrier k. Note that on the PL the noise is not generally AWGN, that is, n_k is a Gaussian random vector with zero mean and covariance matrix R_n,k ≠ I, where I is the identity matrix $\left(n_k\in N_3\left(0,R_{n,k}\right),\text{with}0={\left[0,0,0\right]}^T\right)$. The noise covariance matrix is defined as $R_{n,k}=E\left[n_k{n}_k^H\right]$.

The linear model in Equation 4.1 can be whitened by pre-multiplying by ${\left(R_{n,k}\right)}^{-1/2}$. This pre-whitening approach is often used in signal processing to extend results on AWGN to coloured noise. Pre-whitening Equation 4.1 yields

${\left(R_{n,k}\right)}^{-1/2}{x}_k={\left(R_{n,k}\right)}^{-1/2}{H}_k{a}_k+w_k,$

(4.3)

where w_k ∈ N₃(0, I). Henceforth, the pre-whitened channel $H_{w,k}={\left(R_{n,k}\right)}^{-1/2}{H}_k$ will be referred to as the composite channel (the effect of noise correlation and channel attenuation is combined into one matrix).

Let $a_k^{\left(t\right)}\left[m\right]$ be the symbol loaded on carrier k with k = 1, 2, …, 4096 on the transmitter port t with t = 1, 2 during the mth OFDM symbol of the payload, with m = 1, 2, …, N. Let $x_k^{\left(r\right)}\left[m\right]$ be the received sample on the carrier k, on the receiver port r with r = 1, 2, 3, during the mth OFDM symbol. Given these definitions and recalling the fact that two orthogonal codes are applied to the transmitted signals, the estimates of the channel coefficients are defined by the following equation:

$H_k^{\left(r,t\right)}=\{\begin{array}{cc}\frac{1}{N/2}{\displaystyle \sum _{m=1}^{N/2}\frac{x_k^{\left(r\right)}\left[2m\right]+x_k^{\left(r\right)}\left[2m-1\right]}{2\cdot a_k^{\left(1\right)}\left[2m\right]}}& \text{when\hspace{0.17em}}t=\text{1},\\ \frac{1}{N/2}{\displaystyle \sum _{m=1}^{N/2}\frac{x_k^{\left(r\right)}\left[2m\right]-x_k^{\left(r\right)}\left[2m-1\right]}{2\cdot a_k^{\left(2\right)}\left[2m\right]}}& \text{when\hspace{0.17em}}t=\text{2}\text{.}\end{array}$

(4.4)

By using Equation 4.4, the assumption made is that the channel does not change from the mth OFDM symbol to the (m+1)th. In our study, the number of transmitted OFDM symbols N is 10. For given values of r and t, by computing the inverse fast Fourier transform (IFFT) of $H_k^{\left(r,t\right)}$, the channel impulse responses $h_u^{\left(r,t\right)}$ are obtained with u = 1, 2, …, L_H. Choosing L_H = 2000, at least 99% of the channel energy is preserved. Considering a sampling frequency of 200 MHz, the sampling time is equal to 5 ns. Thus, the channel impulse response L_H = 2000 is equivalent to 10 μs. In the next section, the channels that were estimated on the 92 paths are analysed.

For the noise samples on a certain carrier, $n_k={\left[n_k^{\left(1\right)},n_k^{\left(2\right)},n_k^{\left(3\right)}\right]}^T$, the noise covariance matrix R_n,k can be empirically obtained as an ensemble average from the noise samples that are captured using the field test system.

4.3    Channel Statistical Analysis

In this section, the estimated channels are analysed in order to provide a statistical description of the most relevant parameters useful to characterise MIMO PL channel responses. As an example, Figure 4.3 shows the time impulse response and the frequency response of a generic SISO channel extracted from one of the MIMO channel captures. In other words, the time and frequency characterisation for one element of the H matrix is shown in Figure 4.3. The dashed line of Figure 4.3b refers to a channel realisation obtained as detailed in Section 4.4.

FIGURE 4.3
Example of a channel realisation: (a) time impulse response and (b) frequency response. (From Veronesi, D., Riva, R., Bisaglia, P., Osnato, F., Afkhamie, K., Nayagam, A., Rende, D., and Yonge, L., Characterization of inhome MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

4.3.1    Average Attenuation versus Frequency

In this section, each MIMO channel is considered as a collection of six SISO channels and the energy of each channel is normalised as follows:

${\overline{H}}_k^{\left(r,t\right)}=\frac{H_k^{\left(r,t\right)}}{\sqrt{{\displaystyle {\sum }_{k=1}^{4096}{\left|H_k^{\left(r,t\right)}\right|}^2}}},$

(4.5)

where ${\overline{H}}_k^{\left(r,t\right)}$ denotes the coefficients of the normalised channel matrix ${\overline{H}}_k$. In Figure 4.4a, the channel frequency response is reported, for all the normalised channels. The super-imposed bold black curve represents the average value. It can be observed that the PL channel gain linearly depends on the frequency with a negative slope. The linear approximation of the average channel gain can be defined as

${\left|H\left(f\right)\right|}_{\text{dB}}=A.f+B$

(4.6)

with A = −1.9819×10^–7 [1/Hz] and B = 1.2578. This linear approximation is shown in Figure 4.4a with the bold straight grey line. It is also interesting to observe how the captured noise can affect the channel measurements. This phenomenon could be clearly seen from one measurement in the figure showing a spike around 65 MHz. In Figure 4.4b, the probability density function (PDF) of the channel gain on a generic carrier is shown. The mean and the standard deviation of this PDF are extracted and then they are used to superimpose the Gaussian distribution. On this carrier, it appears that the truncated Gaussian distribution(shown with the dashed line) fits the PDF of the channel gain. This consideration holds true for all the carriers. Other possible distributions for the observed experimental data would be the Rayleigh or Weibull distributions.

FIGURE 4.4
a) Channel frequency response for the normalised channels and (b) PDF of the amplitude on the 2048th carrier. (From Veronesi, D., Riva, R., Bisaglia, P., Osnato, F., Afkhamie, K., Nayagam, A., Rende, D., and Yonge, L., Characterization of in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright ¡ 2011 IEEE. With permission.)

4.3.2    Distribution of the channel coefficients

To facilitate the evaluation of the statistical properties of the channel impulse response, in our analysis, all the time-domain profiles are aligned on their maximum value. In other words, the channel impulse response is time shifted such that its maximum absolute value appears always at a given time instant. Let us indentify this time instant with the label MaxPos.Figure 4.5 reports the superposition of all the estimated channel impulse responses, where the maximum absolute value of the channel impulse response is placed at the time instant MaxPos = 300.

Analysing the distribution of the real-valued channel coefficient at a given time instant, three statistical distributions could characterise all the channel impulse responses. For the time instants belonging to the range χL = [1, MaxPos − 2], the channel coefficients have a random sign and amplitude values described by a Weibull distribution. For the time instants belonging to the range χ_C = [MaxPos −1,MaxPos +1], the channel coefficients have a random sign and amplitude values described by a Gaussian distribution with non-zero mean. For the time instants belonging to the range χ_R = [MaxPos +2, L_H], the channel coefficients have a Gaussian distribution with zero mean. As an example, Figure 4.6 reports the measured PDFs with a black line and the proposed distributions with a marked grey line for the time instants: (a) MaxPos − 5, (b) MaxPos and (c) MaxPos + 5.

FIGURE 4.5
Superposition of all the estimated channel impulse responses. (From Veronesi, D., Riva, R., Bisaglia, P., Osnato, F., Afkhamie, K., Nayagam, A., Rende, D., and Yonge, L., Characterization of in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

The PDF of a Weibull random variable x is completely characterised by two parameters – the shape parameter k and the scale parameter λ − as

$f\left(x,\text{λ,}k\right)=\{\begin{array}{cc}\frac{k}{\text{λ}}{\left(\frac{x}{\text{λ}}\right)}^{k-1}{e}^{-{\left(x/\text{λ}\right)}^k},& x\ge 0,\\ 0,& x<0.\end{array}$

(4.7)

The shape parameter k is assumed to be a constant value equal to 3/4 and the scale parameter λ is expressed as a function of the time instant:

$\text{λ}\text{=}{e}^{C_0+C_1\cdot p+C_2\cdot p^2+C_3\cdot p^3+C_4\cdot p^4+C_5\cdot p^5}$

(4.8)

with C₀ = − 6.83, C₁ = 9.5 × 10⁻³, C₂ = − 2.25 × 10⁻⁴, C₃ = 2.07 × 10⁻⁶, C₄ = −6.98 × 10⁻⁹ and C₅ = 9.15 × 10^–12.Figure 4.7a shows the value of the scale parameter extracted from the captures with a dashed line and Equation 4.8 with markers, as a function of the time instant.

FIGURE 4.6
PDF of the channel impulse response at three different time instants: (a) MaxPos − 5, (b) MaxPos and (c) MaxPos + 5. (From Veronesi, D., Riva, R., Bisaglia, P., Osnato, F., Afkhamie, K., Nayagam, A., Rende, D., and Yonge, L., Characterization of in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

The Gaussian distribution with non-zero mean is characterised by the mean and standard deviation values reported in Table 4.1. These parameters are extracted from the captures for the time instants belonging to the range [MaxPos − 1, MaxPos + 1].

The Gaussian PDF having zero mean is completely characterised by the standard deviation which can be expressed as a function of the time instant:

${\text{σ}}_p=e^{C_0+C_1\cdot p+C_2\cdot p^2+C_3\cdot p^3+C_4\cdot p^4+C_5\cdot p^5}$

(4.9)

with C₀ = 1.1, C₁ = −1.59 × 10⁻², C₂ = 1.84 × 10⁻⁵, C₃ = −1.37 × 10⁻⁸, C₄ = 5.62 × 10⁻¹² and C₅ = −9.22 × 10⁻¹⁶.Figure 4.7b reports the value of the standard deviation extracted from the captures with a dashed line and Equation 4.9 with markers, as a function of the time instant.

4.3.3    RMS-DS versus the Attenuation

In [19], the relation between the root mean square of the delay spread (RMS-DS) and the average gain of the PL channel is proposed.

Figure 4.8 shows a scatter plot of the PL measures together with the trend lines, calculated using the least squares algorithm. Also in this figure, the MIMO channels are considered as the composition of six independent SISO channels, and they are depicted with circles. In the figure, two distinct ‘clouds’ of attenuation values could be seen. In general, higher attenuation values correspond to longer physical paths. One aspect to be noticed is that the experimental measurements belong to a finite set. It could be expected that increasing the number of collected measurements, the average attenuation could span all the range of values roughly from –70 to –10 dB. In Figure 4.8, the same relation as in [19] (labelled ‘literature’), that is, σ_τ,μs = – 0.01 ⋅ ${\overline{G}}_{\text{dB}}$ (with σ_τ,μs being the RMS-DS and ${\overline{G}}_{\text{dB}}$ being the average channel attenuation), and the equivalent results obtained from the MIMO PL measures (labelled ‘MIMO trend line’) are reported. Note that the measurements analysed in [19] also belong to a set of channel measurements collected in the past years by the HomePlug Alliance.

FIGURE 4.7
(a) Scale parameter versus the time instant, expressed in samples, and (b) standard deviation versus the time instant, expressed in samples. (From Veronesi, D., Riva, R., Bisaglia, P., Osnato, F., Afkhamie, K., Nayagam, A., Rende, D., and Yonge, L., Characterization of in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

TABLE 4.1

Mean and Standard Deviation Values for the Gaussian Distribution with Non-Zero Mean

	Time Instant Samples
	MaxPos − 1	MaxPos	MaxPos + 1
Mean	1.45 × 10−1	2.56 × 10−1	1.47 × 10−1
Standard deviation	6.96 × 10−2	8.16 × 10−2	7.49 × 10−2

Source:  Veronesi, D., Riva, R., Bisaglia, P., Osnato, F., Afkhamie, K., Nayagam, A., Rende, D., and Yonge, L., Characterization of in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.

FIGURE 4.8
RMS-DS versus the average channel gain. (From Veronesi, D., Riva, R., Bisaglia, P., Osnato, F., Afkhamie,K., Nayagam, A., Rende, D., and Yonge, L., Characterization of in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

Based on these results, the relationship between the RMS-DS and the average gain proposed in [19] holds true also for the other five SISO channels that constitute the composite 3 × 2 MIMO channel.

4.3.4    MIMO Channel Correlation

Let us decompose the channel matrix on a given carrier k with the singular value decomposition as $H_k=U_k{S}_k{V}_k^H$. The matrix S_k has only two non-zero entries on the major diagonal. These entries ${\text{λ}}_{H_k}^{\left(i\right)}$ with i = 1, 2 are the singular values of H_k. Similarly to [8], a correlation factor among the channel coefficients on carrier k is introduced:

${\kappa }_k={\left(\frac{\min \left\{{\text{λ}}_{H_k}^{\left(1\right)},{\text{λ}}_{H_k}^{\left(2\right)}\right\}}{\max \left\{{\text{λ}}_{H_k}^{\left(1\right)},{\text{λ}}_{H_k}^{\left(2\right)}\right\}}\right)}^2.$

(4.10)

By definition, when the channel is completely correlated, κ_k = 0; while, when the channel is completely uncorrelated, κ_k = 1. To analyse this factor, Figure 4.9a reports the cumulative distribution function (CDF) of κ_k considering all the measurements and all the frequencies (labelled ‘measured’), while Figure 4.9b reports the average value of κ_k, of all the measurements, versus the frequency (labelled ‘measured’). The dashed line of Figure 4.9 refers to channel realisations obtained as detailed in Section 4.4.

FIGURE 4.9
(a) CDF of the channel correlation and (b) mean value of the channel correlation versus the frequency. (From Veronesi, D., Riva, R., Bisaglia, P., Osnato, F., Afkhamie, K., Nayagam, A., Rende, D., and Yonge, L., Characterization of in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

From Figure 4.9a, it is clear that the estimated channels have a correlation factor κ_k that is not uniformly distributed in the range [0,1], and in 90% of the cases, there is a correlation factor lower than 0.2. For each carrier, the correlation factor has the same PDF (it is obtained considering all the estimated channels). To prove this observation, Figure 4.9b reports the average value of the correlation factor versus the frequency. It is quite evident that the average correlation is almost constant with respect to the frequency. As a consequence, the proposed channel model of Section 4.4 introduces a unique correlation factor for all the carriers.

4.3.4.1    Spatial Correlation Analysis

In this section, the ‘spatial’ properties of MIMO PL channels are investigated, where the term ‘spatial’ refers to the MIMO ports. The generic carrier k is considered, and for simplicity, the index k is omitted where not necessary.

FIGURE 4.10
An example of channel spectrum from one field measurement. The two lines represent the squared singular values of H in logarithmic scale for each carrier. (From Tomasoni, A., Riva, R., and Bellini, S., Spatial correlation analysis and model for in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, March 2012. Copyright © 2012 IEEE. With permission.)

The first step for the analysis of the spatial correlation is to evaluate the singular values of the channel coefficients in the frequency domain. As an example, Figure 4.10 plots the squared singular values, as defined in Section 4.3.4, of the channel H in logarithmic scale, from one field measurement. It is clear from the figure that the channel strongly fades at high frequencies (the Nyquist frequency is 100 MHz, and the channel is measured in the frequency range 1.8–88 MHz with a carrier spacing equal to 24.414 kHz).

The channel taps $h_u^{\left(r,t\right)}$ have null mean (as shown in the previous sections):

$E\left[h_u^{\left(r,t\right)}\right]=0.$

(4.11)

The second-order statistic is more complicated. Indeed, channel taps cannot be considered independent. This can be easily understood thinking on the topology of the network: wires have almost the same length and are likely to cover similar paths. Besides, Kirchhoff’s law must hold at the receiver side. All these effects can be captured by a channel covariance matrix:

$R_h=E\left[vect\left(H\right)vect{\left(H\right)}^H\right]$

(4.12)

with dimensions N_t N_r × N_t N_r providing information about all couples of channel taps, where N_t and N_r stand for the number of transmitter and receiver ports, respectively. The operator vect(⋅) aligns the columns of a matrix with size N_r × N_t to form a column vector of length N_r N_t.

The channel covariance matrix can be written as

$R_h=G^2\cdot R_t\otimes R_r,$

(4.13)

where

G is a constant introducing an overall channel gain

R_t and R_r are the transmitter and receiver correlation matrices

⊗ is the Kronecker product and the normalisation chosen is as follows

$\text{tr}\left(R_t\right)=N_t,$

(4.14)

$\text{tr}\left(R_t\right)=N_r,$

(4.15)

where the operator tr(⋅) denotes the trace.

The computation of the covariance matrix of the transmitter and receiver ports from the field measurements is an interesting aspect in the channel analysis. The average of the covariance matrices along the used carriers is not suitable due to the high dynamic range shown in Figure 4.10. The lower frequencies tend to experience lower attenuation than the higher frequencies, therefore, covariance matrices are better calculated as a weighted average.

Although the study of the channel attenuation is mainly useful to model the spectral properties of MIMO PL channels, it also helps to characterise the spatial properties of the same channels. First, the average power of the samples of H is calculated as

$P_h\left(k\right)=\frac{1}{N_t{N}_r}\text{tr}\left(H^{H}H\right)=\frac{1}{N_t{N}_r}{\displaystyle \sum _{r=1}^{N_r}{\displaystyle \sum _{t=1}^{N_t}{\left|H_k^{\left(r,t\right)}\right|}^2}}.$

(4.16)

Once P_h(k) has been obtained for each carrier, it is used to calculate from the measurements the correlation matrices ${\widehat{R}}_t$ and ${\widehat{R}}_r$ at the transmitter and at the receiver, respectively, as

${\widehat{R}}_t=G_t\cdot {\displaystyle \sum _k\frac{1}{P_h\left(k\right)}{H}_k^T{H}_k^{*},}$

(4.17)

${\widehat{R}}_r=G_r\cdot {\displaystyle \sum _k\frac{1}{P_h\left(k\right)}{H}_k^{*}{H}_k^T,}$

(4.18)

where G_t and G_r are normalising factors, necessary to fulfil Equations 4.14 and 4.15.

Figure 4.11 depicts an example extracted from one field measurement. Spatial correlation matrices highlight pronounced values for the diagonal elements and lower values for the off-diagonal elements. This behaviour should be taken into account during the modelling phase. The behaviours of the eigenvalues and of the diagonal elements of the measured ${\widehat{R}}_t$ and ${\widehat{R}}_r$ are reported in Figures 4.12 and 4.13 with dashed lines (solid lines represent synthetic values, described in the following analysis). Each point on the x-axis represents one of the N = 92 measurements. Horizontal lines are the average values of the corresponding curves.

FIGURE 4.11
An example of channel spatial covariance matrices from one field measurement, obtained with the weighted average over the frequency of Equations 4.17 and 4.18 R_h (a), _t (b) and R_r (c). (From Tomasoni, A., Riva, R., and Bellini, S., Spatial correlation analysis and model for in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, March 2012. Copyright © 2012 IEEE. With permission.)

FIGURE 4.12
Eigenvalues of the channel covariance matrices. R_t (a) and R_r (b) Dashed lines are the actual measurements. Solid lines are the results obtained through model simulations Horizontal lines represent the average value over the N = 92 field measurements. (From Tomasoni, A., Riva, R., and Bellini, S., Spatial correlation analysis and model for in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, March 2012. Copyright © 2012 IEEE. With permission.)

FIGURE 4.13
Diagonal elements of the channel covariance matrices. (a) $R_t^{\left(i,i\right)}$ and (b) $R_r^{\left(i,i\right)}$. Dashed lines are the actual measurements. Solid lines are the results obtained through model simulations. Horizontal lines represent the average value over the N = 92 field measurements. (From Tomasoni, A., Riva, R., and Bellini, S., Spatial correlation analysis and model for in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, March 2012. Copyright © 2012 IEEE. With permission.)

4.4    Proposed PL Channel Model

4.4.1    SISO Channel Model

The PL SISO channel model is obtained by generating each channel impulse response coefficient accordingly to the distributions observed in Section 4.3. Note that until now the results do not take into account the presence of correlation among contiguous samples. In order to introduce a correlation among the generated impulse response coefficients, it should be considered that not all the channel coefficients in the range [1, …, L_H] should be randomly generated. Some of them should be obtained as a combination of the generated ones. Considering the channel coefficients in the range χ_R, it can be observed that samples with large standard deviation values correspond to samples with low time correlation. On the other hand, samples with small standard deviation values correspond to samples with high time correlation. Based on these facts, a suitable set of time instants named ${\text{χ}}_L^{*}$ has been defined and a random variable for each element in this set has been generated; the elements of ${\text{χ}}_L^{*}$ are shown in Figure 4.7a with diamonds. The same considerations hold true for the samples in the range χ_R, considering the scale parameter, instead of the standard deviation. In this case, the new subset is labelled with ${\text{χ}}_R^{*}$ and its elements are shown in Figure 4.7b with diamonds. Finally, for the range χ_c, all the time samples have been randomly generated, that is, ${\text{χ}}_C^{*}={\text{χ}}_C$. Using the earlier definitions, a realisation of the SISO PL channel is obtained with the following procedure:

•  For each time instant in ${\text{χ}}_L^{*}$, generate a random variable accordingly to the Weibull distribution with scaling factor given by Equation 4.8 and shape parameter equal to 3/4. Then, randomly change the sign of each random variable.

•  For each time instant in ${\text{χ}}_C^{*}$, generate a random variable accordingly to the Gaussian distribution with mean and variance given in Table 4.1. Then, randomly change the sign of each random variable.

•  For each time instant in ${\text{χ}}_R^{*}$, generate a random variable accordingly to the Gaussian distribution with zero mean and variance given in Equation 4.9.

•  The generated channel must have its maximum absolute value at the time instant MaxPos. Let it be ξ all the samples that have an absolute value higher than ξ are removed from the subsets ${\text{χ}}_L^{*},{\text{χ}}_C^{*}$ or ${\text{χ}}_R^{*}$.

•  Interpolate, with a linear function, the channel impulse response on all the time instants χ_L, χ_C and χ_R (the removed samples need to be obtained by interpolation).

•  Evaluate the RMS-DS of the generated SISO PL channel.

•  Introduce an average attenuation using the relation between the RMS-DS and the average attenuation proposed in [19] and shown in Figure 4.8.

Let H_SISO,k be the channel frequency response. An example is reported in Figure 4.3b with the dashed line.

4.4.2    MIMO Channel Model

Let ${\tilde{H}}_{MIMO,k}$ be a N_r × N_t matrix that denotes the MIMO channel coefficients on carrier k. The three receiving signals are not linearly dependent; hence, the coefficients ${\tilde{H}}_{MIMO,k}^{\left(r,t\right)}$ could be generated as follows:

${\tilde{H}}_{MIMO,k}^{\left(r,t\right)}=H_{SISO,k},$

(4.19)

where each (r,t) refers to a different SISO channel realisation. As a consequence, by construction, the coefficients of ${\tilde{H}}_{MIMO,k}$ are statistically uncorrelated. However, as shown in Figure 4.9, the coefficients of ${\tilde{H}}_{MIMO,k}$ must be correlated.

Due to this observation, the channel correlation is modelled based on the assumption that the phenomena correlating the channel taps act only at the two link extremities (as it happens in wireless channels [20,21]), that is, for the generic kth carrier, the correlated MIMO channel coefficients are as follows:

$H=G\cdot R_r^{1/2}{H}^{\prime }{R}_t^{1/2},$

(4.20)

where the N_tN_r channel taps $h_u^{\prime r\left(r,t\right)}\in N\left(0,1\right)$ are independent and identically distributed (i.i.d.) variables. Because of G, the two correlation matrices R_t and R_r can be arbitrarily scaled.

Therefore, the problem is to derive a synthetic model to generate such matrices as close as possible to the observed ones.

In the following sections, it will be analysed how to introduce a statistical correlation.

4.4.2.1    Spatial Correlation Model

Now, both R_t and R_r must be modelled. Being interested not only in the covariance matrices but also in their eigenvalues, the considered eigenvector–eigenvalue decompositions are:

$R_t=U_t\cdot D_t\cdot U_t^H,$

(4.21)

$R_r=U_r\cdot D_r\cdot U_r^H,$

(4.22)

and their eigenvectors and eigenvalues are drawn separately, respecting the channel properties (as estimated in Section 4.3.4.1).

The eigenvalues are modelled as uniform random variables (as suggested by Figure 4.12, dashed lines), with average and standard deviation values optimised evaluating the measurements. However, one should remember that covariance matrices have been normalised. Since the trace of any square matrix equals the sum of its eigenvalues, this constraint is inherited by eigenvalues, too, and can be satisfied, for example, eventually normalising them. For the time being, consider a trace normalisation to 1. In order to model the distribution of the eigenvalues of R_t, two variables uniformly (U(⋅)) distributed x₁ and x₂ are introduced such that${\overline{x}}_1$ is obtained as

${\overline{x}}_1=\frac{x_1}{x_1+x_2}$

(4.23)

with $x_i\in U\left(x_{i,min},x_{i,max}\right)$. Similarly, for R_r,

${\overline{x}}_1=\frac{x_1}{x_1+x_2+x_3}.$

(4.24)

First, notice that the earlier two cases can be considered as special cases of

$\overline{x}=\frac{x}{x+y}$

(4.25)

with $x\in U\left(x_{min},x_{max}\right)$ and y having any PDF f_Y(y). In the former case, y = x₂ has uniform distribution as well. In the latter, y = x₂ + x₃ will exhibit a triangular PDF. In the same way, higher-order convolutions of uniform variables are easy to manage and integrate. The PDF of the normalised variable (Equation 4.25) is computed as follows:

$\begin{array}{l}{f}_{\overline{X}}\left(\overline{x}\right)={\displaystyle \underset{x=x_{min}}{\overset{x_{max}}{\int }}\frac{1}{{\text{Δ}}_x}}{\displaystyle \underset{y_{min}}{\overset{y_{max}}{\int }}{f}_Y\left(y\right)\text{δ}\left(\overline{x}=\frac{x}{x+y}\right)dydx,}\\ =\frac{1}{{\text{Δ}}_x}\max \left({\displaystyle \underset{\max \left(x_{min},\frac{\overline{x}}{1-\overline{x}}{y}_{min}\right)}{\overset{\min \left(x_{max},\frac{\overline{x}}{1-\overline{x}}{y}_{max}\right)}{\int }}\frac{x}{{\overline{x}}^2}{f}_Y\underset{@z}{\underbrace{\left(x=\frac{1-\overline{x}}{\overline{x}}\right)}}dx,0}\right),\\ =\frac{1}{{\text{Δ}}_x{\left(1-\overline{x}\right)}^2}\max \left({\displaystyle \underset{\max \left(x_{min},\frac{1-\overline{x}}{\overline{x}},y_{min}\right)}{\overset{\min \left(x_{max},\frac{1-\overline{x}}{\overline{x}},y_{max}\right)}{\int }}z{f}_Y\left(z\right)dz,0}\right),\end{array}$

(4.6)

Δ_x = x_max – x_min and δ(⋅) being the impulsive (Dirac) generalised function. Notice that the integral in Equation 4.26 can be simply interpreted as a truncated expectation of y. Finally, if the rescaled version of $\overline{x}$ is considered, that is, the actual synthetic eigenvalues, the following equations are obtained:

$\text{λ=α}\overline{x}$

(4.27)

$f_{\text{Λ}}\left(\text{λ}\right)=\frac{f_{\overline{X}}\left(\text{λ}/\text{α}\right)}{\text{α}}$

(4.28)

with α = N_t or N_r, respectively. By means of the earlier PDF, the mean and variance of the N_t = 2 and N_r = 3 uniform eigenvalues have been iteratively optimised, so that the mean and variance of their normalised versions are as close as possible to the measured eigenvalues, along all the experiments. Table 4.2 reports these optimisation results.

Figure 4.12a and b compares measured eigenvalues with those of the earlier model. Dashed lines are the actual eigenvalues; solid lines are the synthetic eigenvalues, obtained with the optimised parameters; as already stated, horizontal lines represent their average values over the N = 92 channel measurements. We expect this to match very well.

TABLE 4.2

Upper and Lower Bounds, Mean and Standard Deviation for the Uniform Random Variables x_i of the Eigenvalues Model

Source:  Tomasoni, A, Riva, R, and Bellini, S, Spatial correlation analysis and model for in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on March 2012. Copyright © 2012 IEEE. With permission.

The eigenvector choice is now considered. Our focus is on the main diagonal elements of R_t and R_r that must be as close as possible to the actual ones. If the eigenvectors were circularly distributed (i.e. with uniform PDF on the unitary hypersphere), one could choose among many strategies to draw them, for example, singular value decomposition, eigenvalue–eigenvector decomposition or QR decomposition of a square matrix of i.i.d., zero-mean Gaussian samples. Actually, this is not the case, since Figure 4.11 shows that covariance matrices have pronounced diagonal elements, compared to off-diagonal ones. Anyway, the QR decomposition of a square matrix with Gaussian samples can still be exploited, yet biasing the mean of the elements on its main diagonal, to emphasise them:

$QT=aI+W,$

(4.29)

where

$w^{\left(i,j\right)}\in N\left(0,1\right)$

Q is unitary (QQ^H = Q^HQ = I)

T is upper triangular

In the limit, when a → 0, Q is circularly distributed; conversely, when a ↔ ∞, Q ↔ I; therefore, also covariance matrices would have a diagonal structure.

To avoid dependency of the eigenvectors on the column position w.r.t.Q (the chosen QR is based on a successive Gram–Schmidt orthonormalisation process), the columns and rows of Q are randomly permuted by a permutation matrix Π, obtaining

$U=\text{Π}Q{\text{Π}}^H.$

(4.30)

Also the biasing parameters a_t and a_r have been optimised numerically, to find the best match with R_t and R_r, respectively. The optimisation ended with a_t = 1.25 and a_r = 8.Figure 4.13 compares measured diagonal elements with the synthetic ones. Again, dashed lines represent the actual values, while solid lines are the diagonal entries of Equations 4.21 and 4.22, assuming the earlier two models for the eigenvalues and the eigenvectors. Horizontal lines represent their average values over the N = 92 channel measurements. Again, the match between the model and the actual measurements is very good.^*

FIGURE 4.14
Rank statistics of the diagonal elements of R_t and R_r. (a) Transmitter ports 1 and 2 correspond to the couples L-N and L-PE, respectively. (b) Receiver ports 1, 2 and 3 correspond to the couples L-N, L-PE and N-PE, respectively. (From Tomasoni, A., Riva, R., and Bellini, S., Spatial correlation analysis and model for in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, March 2012. Copyright © 2012 IEEE. With permission.)

Finally, the rank statistics of the covariance matrices diagonal elements is studied. Indeed, the mapping between the elements of Figure 4.13 and the transmitter and receiver ports is not uniform. Actually, some ports are likely to have gains larger than others and must most frequently be associated to the highest eigenvalues. These statistics are reported in Figure 4.14.

4.4.2.2    Simpler Correlation Model

In this section, a simpler spatial correlation model is introduced A spatial correlation between each SISO link can be introduced for each carrier based on an approach documented in several publications on MIMO channel modelling (e.g.[22]). The transmitter and receiver correlation matrices, based on the definition in [23], are explicitly stated as

$R_t=\left[\begin{array}{cc}1& {\text{ρ}}_t\\ {\text{ρ}}_t& 1\end{array}\right],R_r=\left[\begin{array}{ccc}1& {\text{ρ}}_r& {\text{ρ}}_r^2\\ {\text{ρ}}_r& 1& {\text{ρ}}_r\\ {\text{ρ}}_r^2& {\text{ρ}}_r& 1\end{array}\right].$

(4.31)

Successively, the correlated channel matrix H for each carrier is obtained using Equation 4.13.

The eigenvalues of the proposed matrices in Equation 4.31 obtained resolving the characteristic equation are

$\{\begin{array}{cc}{\text{λ}}_{R_t}^{\left(1\right)}=& 1+{\text{ρ}}_t\\ {\text{λ}}_{R_t}^{\left(2\right)}=& 1-{\text{ρ}}_t\end{array}$

(4.32)

and

$\{\begin{array}{ccc}{\text{λ}}_{R_r}^{\left(1\right)}& =& 1-{\text{ρ}}_r^2\\ {\text{λ}}_{R_r}^{\left(2\right)}& =& \frac{1}{2}\left(2+{\text{ρ}}_r^2+\sqrt{8+{\text{ρ}}_r^2}\right),\\ {\text{λ}}_{R_r}^{\left(3\right)}& =& \frac{1}{2}\left(2+{\text{ρ}}_r^2-\sqrt{8+{\text{ρ}}_r^2}\right),\end{array}$

(4.33)

Considering the results obtained in Section 4.3.4.1 on the average values of measured eigenvalues of R_t and R_r, the values of ρ_t = 0.4 and ρ_r = 0.6 guarantee similar behaviour for the eigenvalues of the models in Equation 4.31. Thus, it is recommended to adopt these last values for the model parameters, in order to obtain simulated channel models close to the field measurements in terms of channel capacity. The elements on the diagonal of R_t and R_r still remain equal to 1, thus losing insight on some characteristics observed in Figures 4.11 and 4.14. Nevertheless, this simpler model allows to study MIMO PL channels either in terms of spatial correlation or in terms of channel capacity.

4.5    Characterisation of Noise in MIMO PL Channels

In contrast to MIMO channel modelling on the PL, the analysis of MIMO PL noise is relatively unexplored to date. In many previous treatments of the PL channel, noise is either not addressed or often simply assumed to have an independent and Gaussian distribution. Pagani et al. [15] uses the ETSI STF410 measurements focusing on the characterisation of noise power with respect to frequency and different geographies. Hashmat et al. [16,17] provide two statistical models for the background noise found in MIMO PL channels. In addition to [15,16,17], which deal specifically with PL noise, a number of contributions also treat the case of correlated noise in MIMO communication systems in [24,25,26,27].

FIGURE 4.15
Variation of noise power across frequency (average over all measured paths). (From Rende, D., Nayagam, A., Afkhamie, K., Yonge, L., Riva, R., Veronesi, D., Osnato, F., and Bisaglia, P., Noise correlation and its effects on capacity of inhome MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

Noise characteristics are derived using the captured samples in all three receiver ports during the silent period shown in Figure 4.2. Noise samples are used to calculate the noise covariance matrix R_n,k, defined in Section 4.2, for every carrier. Noise power on all three receiver ports as well as the correlation between them is derived from the estimated noise covariance matrix.

The noise power on each receiver port corresponds to the diagonal elements of R_n,k. The variation of noise power across frequency is shown in Figure 4.15. The noise power is averaged across all measured paths for each frequency. It is seen that noise tends to be higher in the 1.8-30 MHz frequency band as compared to frequencies greater than 30 MHz. It is also seen that the average noise power is similar across all three of the receiver ports. Thus, from a noise power perspective, a certain receiver port is not better than any of the others.

For every carrier k, the correlation coefficient between noise in receiver ports i and j is defined in terms of elements of R_n,k as

$C_k^{\left(i,j\right)}=\frac{\left|R_{n,k}^{\left(i,j\right)}\right|}{\sqrt{R_{n,k}^{\left(i,i\right)}{R}_{n,k}^{\left(j,j\right)}}},i,j=1,\dots ,N_r.$

(4.34)

Note that the off-diagonal elements of R_n,k are complex since the noise n_k is complex, and hence, the correlation is defined in terms of the magnitude of $R_{n,k}^{\left(i,j\right)}$ in Equation 4.34. When the noise is identical on ports i and j, then $C_k^{\left(i,j\right)}$. When the noise is completely uncorrelated, then the correlation coefficient becomes 0. Correlation coefficients for MIMO PL noise for all carriers from all measured paths are collected to obtain the PDF and CDF that are shown in Figure 4.16. Note that the density function is heavy tailed, that is, noise with high correlation is not unlikely on the PL. Also, the CDF indicates that there is a higher correlation on L-PE and N-PE wire pairs. For instance, there is a 20% chance that $C_k^{\left(2,3\right)}$ is greater than 0.7 and only a 15% chance that $C_k^{\left(1,2\right)}$ or $C_k^{\left(1,3\right)}$ is greater than 0.7.

FIGURE 4.16
Density and distribution functions of the correlation coefficients of MIMO PL noise. (From Rende, D., Nayagam, A., Afkhamie, K., Yonge, L., Riva, R., Veronesi, D., Osnato, F., and Bisaglia, P., Noise correlation and its effects on capacity of inhome MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

The frequency dependence of the correlation coefficients of PL noise is shown in Figure 4.17. It is seen that noise is more correlated on the lower frequencies as compared to the higher frequencies. On the lower frequencies, it is also seen that the correlation is higher between the L-PE and N-PE wire pairs as compared to the other pairs of receiver ports.

4.5.1    Eigenspread Analysis of MIMO PL Channels

In this section, results on eigenspread of the MIMO PL channel matrix are presented. For a given channel matrix H, the eigenspread is defined as the ratio of the square root of the largest eigenvalue of H^HH to the square root of the lowest eigenvalue [28]. For square H matrices, the eigenspread is the ratio of the eigenvalues of H, and for rectangular H matrices, it is the ratio of the singular values of H. The eigenspread is also referred to as the condition number of a matrix. In our case, the channel is a 3 × 2 matrix and hence the condition number is defined as $\text{ψ=}{\text{λ}}_H^{\left(1\right)}/{\text{λ}}_H^{\left(2\right)}$, where ${\text{λ}}_H^{\left(1\right)}$ is the largest singular value of H and ${\text{λ}}_H^{\left(2\right)}$ is the smallest singular value. A large eigenspread means that there is more correlation in the channel (the channel is more ill conditioned), whereas a small eigenspread means that the channels are less correlated. The signal to noise ratios (SNRs) on the two transmit streams are proportional to ${\left({\text{λ}}_H^{\left(1\right)}\right)}^2$ and ${\left({\text{λ}}_H^{\left(2\right)}\right)}^2$ [9].A large eigenspread implies that the SNR of the first stream that corresponds to the largest eigenvalue is significantly better than the other stream. Since the SNRs are proportional to the eigenvalues, the capacity of the MIMO channel is also dependent on the eigenvalues [9]. Thus, in order to understand how the noise correlation affects capacity, it is also important to understand how the noise correlation affects the eigenvalues and eigenspread.

FIGURE 4.17
Variation of noise correlation coefficients across frequency. (From Rende, D., Nayagam, A., Afkhamie, K., Yonge, L., Riva, R., Veronesi, D., Osnato, F., and Bisaglia, P., Noise correlation and its effects on capacity of inhome MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

To isolate the effect of the noise correlation, the eigenspread is calculated for three different cases:

1.  For the composite channel, $H_w={\left(R_{n,k}\right)}^{-1/2}H$. The eigenspread in this case affects the capacity of a PL channel with correlated noise.

2.  For the raw PL channel, H. The eigenspread in this case affects the capacity of a PL channel with i.i.d. noise.

3.  For a fictitious channel that has independent but nonidentically distributed (i.n.i.d.) noise, $H_D={\left(R_{ndiag}\right)}^{-1/2}H$. In this case, R_ndiag is a diagonal matrix with different entries along the diagonal elements. This fictitious channel does not have any correlation between the noise received in different MIMO parts. It provides a mechanism to study channel and noise conditions from the noise correlation perspective and enables direct comparison between correlated and independent noise cases. The eigenspread in this case affects the capacity of a PL channel with i.n.i.d.noise. In order to keep the comparison to the eigenspread of the composite channel fair, the elements of R_ndiag are generated such that the Frobenius norms of R_ndiag and R_n are equal, that is, $\Vert R_{ndiag}\Vert =\Vert R_n\Vert$. This is accomplished by computing each diagonal entry of R_ndiag as

$R_{ndiag}^{\left(\upsilon ,\upsilon \right)}=\sqrt{{\displaystyle \sum }_{r=1}^{N_r}{\left|R_n^{\left(\upsilon ,r\right)}\right|}^2},\upsilon =\text{1,}\dots \text{,}{N}_r.$

(4.35)

The normalisation earlier preserves the noise power on each receiver port in R_n and R_ndiag. Thus, a comparison of capacities or eigenspreads on H_w and H_D is fair because it compares uncorrelated and correlated noise with equal effective powers.

The variation of the eigenspread of the MIMO PL channel across frequency is shown in Figure 4.18. Eigenspread for a given carrier is obtained by averaging the eigenspread for that carrier across all the paths that were measured. It is seen that eigenspread for the raw PL channel and the fictitious channel with i.n.i.d. noise is nearly constant across frequency. Eigenspread of the composite MIMO channel H_w shows more variation across frequency with higher eigenspread in low frequencies.

FIGURE 4.18
Frequency dependence of eigenspread of the MIMO PL channel when averaged over all homes. (From Rende, D., Nayagam, A., Afkhamie, K., Yonge, L., Riva, R., Veronesi, D., Osnato, F., and Bisaglia, P., Noise correlation and its effects on capacity of inhome MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

FIGURE 4.19
PDF of the eigenspread of the composite MIMO PL channel under different noise assumptions. (From Rende, D., Nayagam, A., Afkhamie, K., Yonge, L., Riva, R., Veronesi, D., Osnato, F., and Bisaglia, P., Noise correlation and its effects on capacity of inhome MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

The PDF of the eigenspread of the MIMO PL channel is shown in Figure 4.19. The PDF is obtained by looking the corresponding channel matrices over all the carriers on all paths that were measured. It is observed that eigenspread is in general not very high for the three different cases.

The PDF can be approximated by the Rayleigh distribution and this fact has also been observed for the eigenspread of random matrices in [29].

It is also seen that eigenspreads for the raw PL channel and the fictitious channel with i.n.i.d noise have almost identical distributions. Thus, as long as the noise is independent, the actual noise powers on the three different ports do not have a significant influence on the eigenspread. In this case, the eigenspreads are dominated by the correlation in the channel only. The eigenspread for the composite channel is higher compared to the independent noise cases. This suggests that noise correlation increases the eigenspread of the composite channel. Since eigenspread is an indication of increased channel correlation, this in turn implies that noise correlation causes a reduction in capacity compared to the case of having uncorrelated noise.

However, channel correlation or eigenspread is not the only factor that determines capacity. In fact, the capacity is determined by the individual eigenvalues. The two eigenvalues for the composite PL MIMO channel for R_n and R_ndiag are shown in Figure 4.20. Note that the two eigenvalues are closer together when the noise is i.n.i.d.and the spread in eigenvalues is larger when noise is correlated. When the noise is correlated, each of the eigenvalues increases. Since SNRs are proportional to eigenvalues, the capacity will actually increase when the noise is correlated. So, even though the noise correlation increases the spread in eigenvalues (as seen in Figure 4.20), this is more than compensated for by the increase in the individual eigenvalues. The fact that capacity increases when the noise is correlated is also observed in [24] and will be illustrated for MIMO PL channels in the next section.

FIGURE 4.20
Variation of eigenvalues with frequency for the MIMO PL channel (averaged over all measured paths). (From Rende, D., Nayagam, A., Afkhamie, K., Yonge, L., Riva, R., Veronesi, D., Osnato, F., and Bisaglia, P., Noise correlation and its effects on capacity of inhome MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

4.5.2    Impact of Noise Correlation on the MIMO PL System

The effect of the noise correlation on the capacity of a MIMO PL system with two or three receiver ports is studied in this section. The channel capacity for a single carrier in a MIMO system with correlated noise is given in [24]:

$C=BW\log \left(\left(I+R_n^{-1}HF{H}^H\right)\right).$

(4.36)

The system capacity is obtained by summing the capacity over all the carriers. It is easy to derive the expression for capacity in Equation 4.36 by starting with the pre-whitened linear model given in Equation 4.3. In Equation 4.36, BW is the carrier bandwidth.F is found through the water-filling solution after applying singular value decomposition on the composite MIMO channel $H_w=R_n^{-1/2}H$.

It was shown that noise in MIMO PL channels is correlated. In order to study the effect of noise correlation on capacity, the capacity is studied with three different noise assumptions as described in the beginning of Section 4.5, namely, correlated noise, i.i.d. noise and i.n.i.d.noise. The capacity of 2 × 2 and 2 × 3 MIMO PL channels with correlated and independent noise is shown in Figure 4.21. The results illustrate that noise correlation actually improves channel capacity. This observation can also be found in [24,25,26,27]. It is seen that 2 × 2 and 2 × 3 MIMO capacities with correlated noise are higher than the capacities with independent noise.

FIGURE 4.21
Capacity of MIMO PL channels with correlated and independent noise. (From Rende, D., Nayagam, A., Afkhamie, K., Yonge, L., Riva, R., Veronesi, D., Osnato, F., and Bisaglia, P., Noise correlation and its effects on capacity of inhome MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

Note also that 2 × 3 MIMO shows a larger difference between capacities with correlated and independent noise as compared to 2 × 2 MIMO. The additional receiver port in the 2 × 3 system when compared to the 2 × 2 MIMO system is the N-PE wire pair. It is mentioned that the correlation between L-PE and N-PE is higher than the other wire pairs (see Figure 4.16). Thus, by including the N-PE wire pair, the correlation among the noise samples goes up thereby increasing the capacity more for the 2 × 3 case when compared to the 2 × 2 case.

The results on capacity are obtained by summing the capacity of each individual carrier. It is possible that the capacity could be dominated by a few carriers that exhibit low noise correlation, and this is clouding the results. To establish that this is not the case, capacity as a function of frequency for R_n and R_ndiag is plotted in Figure 4.22.

The capacity for a single carrier is obtained as the average of the capacity on that carrier in each path that is tested. It can be seen that noise correlation improves capacity on all frequencies. The benefit is higher on lower frequencies because the noise correlation is actually higher on lower frequencies as shown in Figure 4.17.

FIGURE 4.22
MIMO capacity as a function of frequency for correlated and independent noise (averaged over all measured paths). (From Rende, D., Nayagam, A., Afkhamie, K., Yonge, L., Riva, R., Veronesi, D., Osnato, F., and Bisaglia, P., Noise correlation and its effects on capacity of inhome MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, April 2011. Copyright © 2011 IEEE. With permission.)

4.6    Conclusions

In this chapter, a statistical description of PL channels were provided and the noise in the 1.8–88 MHz band was analysed from a MIMO perspective. Channel and noise samples collected on 92 paths in five different homes in North America formed the basis of the analysis.

This chapter began reporting properties that provide insight into important physical characteristics of MIMO PL channels and compares them to previous studies on SISO PL channels finding a good match. A MIMO PL channel model was proposed, based on field measurements and on the extracted statistics related to meaningful physical parameters. As a first step, each single communication link within the MIMO channel matrix was completely characterised through the collected statistics. Then, the MIMO PL channel model is defined, introducing a representation of the correlation that is verified on the field. The channel model proposed here was suited to perform system-level simulations in a realistic scenario.

The last portion of this chapter was devoted to MIMO PL noise characterisation; in particular, it has been shown that the noise is correlated on the L-N, L-PE and N-PE receiver ports. The strongest correlation is measured between the L-PE and N-PE receiver ports. Moreover, the correlation is stronger on lower frequencies when compared to higher frequencies. The effect of the noise correlation on the capacity of a MIMO PL system with two transmit ports and three receiver ports is studied, and it is observed that noise correlation indeed helps to increase the MIMO channel capacity.

The analysis and the models described so far can be conveniently leveraged to simulate and compare the performance of different MIMO PL communication systems.

References

1.  D. Veronesi, R. Riva, P. Bisaglia, F. Osnato, K. Afkhamie, A. Nayagam, D. Rende and L. Yonge, Characterization of in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, Udine, Italy, April 2011, pp. 42–47.

2.  D. Rende, A. Nayagam, K. Afkhamie, L. Yonge, R. Riva, D. Veronesi, F. Osnato and P. Bisaglia, Noise correlation and its effects on capacity of inhome MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, Udine, Italy, April 2011, pp. 60–65.

3.  A. Tomasoni, R. Riva and S. Bellini, Spatial correlation analysis and model for in-home MIMO power line channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, Beijing, China, March 2012, pp. 286–291.

4.  HomePlug AV Specification, Version 1.1, 21 May 2007, HomePlug PowerLine Alliance. http://www.homeplug.org.

5.  Standard for Local and Metropolitan Area Networks – Part 11: Wireless LAN MAC and PHY Specifications. Amendment 5: Enhancements for Higher Throughput, http://standards.ieee.org/findstds/standard/802.11n-2009.html, IEEE 802.11n.

6.  3G Release 8. See version 8 of Technical Specifications and Technical Reports for a UTRAN-based 3GPP system, 3GPP TR 21.101, http://www.3gpp.org/article/lte, accessed 17 October 2013.

7.  R. Hashmat, P. Pagani and T. Chonavel, MIMO capacity of inhome PLC links up to 100 MHz, in Workshop on Power Line Communications (WSPLC), Udine, Italy, October 2009, pp. 4–6.

8.  R. Hashmat, P. Pagani, A. Zeddam and T. Chonavel, MIMO communications for inhome PLC networks: Measurements and results up to 100 MHz, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, Rio de Janerio, Brazil, March 2010, pp. 120–124.

9.  L. Stadelmeier, D. Schill, A. Schwager, D. Schneider and J. Speidel, MIMO for inhome power line communications, in International ITG Conference on Source and Channel Coding (SCC), Ulm, Germany, January 2008, pp. 1–6.

10.  C. L. Giovaneli, B. Honary and P. G. Farrell, Space–time coding for power line communications, in Communication Theory and Applications (ISCTA), International Symposium on, Ambleside, Lake Districk, UK, July 2003, pp. 162–169.

11.  R. Hashmat, P. Pagani, A. Zeddam and T. Chonavel, A channel model for multiple input multiple output in-home power line networks, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, Udine, Italy, April 2011, pp. 35–41.

12.  F. Versolatto and A. M. Tonello, A MIMO PLC random channel generator and capacity analysis, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, Udine, Italy, April 2011, pp. 66–71.

13.  A. Schwager, W. Bäschlin, H. Hirsch, P. Pagani, N. Weling, J. L. González Moreno and H. Milleret, European MIMO PLT field measurements: Overview of the ETSI STF410 Campaign & EMI analysis, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, Beijing, China, April 2012, pp. 298–303.

14.  D. M. Schneider, A. Schwager, W. Bäschlin and P. Pagani, European MIMO PLC field measurements: Channel analysis, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, Beijing, China, April 2012, pp. 304–309.

15.   P. Pagani, R. Hashmat, A. Schwager, D. M. Schneider and W. Bäschlin, European MIMO PLC field measurements: Noise analysis, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, Beijing, China, April 2012, pp. 310–315.

16.  R. Hashmat, P. Pagani, T. Chonavel and A. Zeddam, Analysis and modeling of background noise for inhome MIMO PLC channels, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, Beijing, China, April 2012, pp. 316–321.

17.  R. Hashmat, P. Pagani, T. Chonavel and A. Zeddam, A time domain model of background noise for inhome MIMO PLC networks, IEEE Trans. Power Deliv., 27(4), 2082–2089, October 2012.

18.  F. Versolatto and A. M. Tonello, An MTL theory approach for the simulation of MIMO power-line communication channels, IEEE Trans. Power Deliv., 26(3), 1710–1717, July 2011.

19.  S. Galli, A simplified model for the indoor power line channel, in Power Line Communications and Its Applications (ISPLC), IEEE International Symposium on, Dresden, Germany, March 2009, pp.13–19.

20.  D. McNamara, M. Beach and P. Fletcher, Spatial correlation in indoor MIMO channels, in Proceedings of IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, Pavilho Atlantico, Lisbon, Portugal, September 2002, pp. 290–294.

21.  K. Yu, M. Bengtsson, B. Ottersten, D. McNamara, P. Karlsson and M. Beach, Second order statistics of NLOS indoor MIMO channels based on 5.2 GHz measurements, in Proceedings of IEEE Global Telecommunications ConferenceRapajic and R. A. Kennedy, Channel capacity estimation, San Antonio, TX, November 2001, pp. 156–160.

22.  J. P. Kermoal, L. Schumacher, K. I. Pedersen, P. E. Mogensen and F. Frederiksen, A stochastic MIMO radio channel model with experimental validation, IEEE J. Sel. Areas Commun., 20(6), 1211–1226, August 2002.

23.  S. L. Loyka, Channel capacity of MIMO architecture using the exponential correlation matrix, IEEE Commun. Lett., 5(9), 369–371, September 2001.

24.  S. Krusevac, P. Rapajic and R. A. Kennedy, Channel capacity estimation for MIMO systems with correlated noise, Proceedings of IEEE GLOBECOM ‘05, St. Louis, MO, December 2005, pp. 2812–2816.

25.  Y. Dong, C. P. Domizioli and B. L. Hughes, Effects of mutual coupling and noise correlation on downlink coordinated beamforming with limited feedback, EURASIP J. Adv. Signal Process., 2009, Article ID 807830, 2009.

26.  C. P. Domizioli, B. L. Hughes, K. G. Gard and G. Lazzi, Receive diversity revisited: Correlation, coupling and noise, in Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM ‘07), Washington, DC, November 2007, pp. 3601–3606.

27.  J. W. Wallace and M. A. Jensen, Mutual coupling in MIMO wireless systems: A rigorous network theory analysis, IEEE Trans. Wireless Commun., 3(4), 1317–1325, 2004.

28.  V. Madisetti and D. B. Williams, The Digital Signal Processing: Handbook, CRC Press, Boca Raton, FL, 1997.

29.  A. Edelman, Eigenvalues and condition numbers of random matrices, PhD dissertation, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 1989.

*  Permission to re-use part of the material in [1,1,3] in this and in all subsequent editions, revisions and derivative works in English and in foreign translations, in all formats, including electronic media, has been granted to the authors by the Institute of Electrical and Electronics Engineers, Incorporated (the ‘IEEE’).

*  Of course, the eigenvectors model can be further refined to find a better match between synthetic and measured off-diagonal covariance matrix samples. For example, one could run a multivariate optimisation of a biasing matrix A rather than of the diagonal one aI that we have chosen for simplicity. In our opinion, the measurement campaign at the moment is too small to justify such a precision.