A New tatistical Model of the Noise Power Density pectrum for Powerline Communication Dirk Benyoucef Institute of Digital Communications, University of aarland D 66041 aarbruecken, Germany E-mail: Dirk.Benyoucef@LNT.uni-saarland.de Abstract In this paper a model for the disturbances on power line chaels is represented. The model is based on extensive long-term measurements in office and home environments. The parameters of the model are based on random variables with space-dependent statistical distributions. A substantial advantage of this concept is its simple adaptation to new types of environment. This adaptation can be realized with only a few additional measurements and, in this coection, with a reconfiguration of the model parameters. This opens a simple possibility to calculate the system capacities of the new environment. The small number of necessary model parameters predestined this model for simulations. 1. Introduction In the literature different models of the disturbances on power line chaels are suggested. Most of these models [3, 2, 4] are based on the approximation of the power density spectrum of a small number of measurements. These models do not consider statistical aspects of the disturbances and are less suitable for simulations. Philipps [3] was the first to indicate a model which considered the statistical characteristics of the disturbances. This model is based on the decomposition of the power density spectrum into an exponentially decreasing background noise and a number of narrow band interferers as well as W-radio signals. The statistical characteristics of the narrow band disturbers were examined at a small number of measurements associated with two different environments. The background noise was not analyzed on its statistical characteristics. The approximation of the narrow band disturbers was implemented by ideal band-pass filters and is not adapted to the actual shape of the disturbers. In this paper a new statistical model for the interference at power line chaels is presented, which is based on extensive long-term measurements. All model parameters are modelled by statistical random variables and here density function. This choice guarantee the statistical character of the disturbance in the model. A other boundary condition of the new model is the few numbers of parameters. This advantage is important for simulations. The statistical characteristics of the noise were analyzed on the basis of extensive time domain measurements (2600 measurements per day) at many different network access points (plug sockets) in an office environment. The results are that the probability density of the noise was approximately Gauss-distributed. The effect of temporal impulse disturbers is neglected due to their small occurrence. Because of these boundary conditions it is sufficient for the modelling to consider only the power density spectrum and the second order statistics respectively. For the modelling of the power density spectrum daily measurements were executed. 56 different network access points (plug sockets) inside a office and home environment were considered. Each daily measurement covered 2500 individual measurements with 401 samples in each case. The considered frequency range was about 0-30 MHz. It turned out that the power density spectrum was subjected to strong fluctuations at different places. 2. Mathematic Description of the Model From the analysis of the disturbances [1] it is known that the distribution of the amplitude of the disturbance is nearly Gaussian. Therefore it is sufficient to consider the power density spectrum for the modelling. The basis of the modelling is the superposition of background noise and the narrow band disturbances. We do not make a difference between W radios and other narrow band disturbances in the form of spectral lines, because normally we find spectral lines in bundled form. For the modelling, these bundles of disturbers are approximated by their envelope. The modelling of the impulse noises is not done in this model. The reason is that the impulse noises rarely occur with really high amplitudes [1] and therefore have only little influence on the transmission.
The power density spectrum of noise { (mes) (f, t) } will be time-averaged for the modelling, E (mes) (f, t) = (f). The dependence of the power density spectrum of the day time can be modelled independently with the knowledge of the standard deviation. These average power density spectrums (mes) (f) will be approximated by the superposition of the background noise and narrow band disturbances, eq. (1) describes this. (mes) (mes) (f) (M) (f) = (h) (f) + N k=1 (s,k) (f) (1) (M) (f) is the modelled power density spectrum, (h) (f) describes the background noise and (s,k) (f) specify the narrow band interferers. As function for the background noise we use a first-order exponential function. (h) (f) = N 0 + N 1 e f f 1 (2) N 0 is the constant noise power density, N 1 and f 1 are the parameters of the exponential function. For the approximation of the narrow band interferers we used a parametric gaussian function. The main advantage of this function are the few numbers of parameters. Furthermore, the parameters can be individually found out from the measurements. Considering many measurements, a small variance is shown. (s,k) (f) = A k e (f f 0,k ) 2 2 B k 2 (3) The parameters of the function are A k for the amplitude, f 0,k is the center frequency and B k is the bandwidth of the gaussian function. Therefore we get for the eq. (1) (M) (f) = N 0 + N 1 e f f 1 + N k=1 A k e (f f0,k) 2 2 B k 2. (4) The model consists of three parameters (N 0, N 1, f 1 ) for the background noise and four parameters (A k, f 0,k, B k, N) for the narrow band interferers. Every one of these parameters should be described by probability density functions. This reflects the statistical character of the model. In order to determinate the parameters and their distribution we analyzed the measured power density functions. Figure 1 shows this analysis scheme. The measured power density spectrum (mes) (f) is split into the background noise (h) (rel) (f) (upper branch) and into the narrow band interferers (f) (bottom branch). The three parameters of the exponential function are extracted from the background noise curve (upper branch, left and right figure). These are used to extract the narrow band interferers (bottom branch, left figure). This curve builds the basis for the extraction of the number of gaussian functions and their parameters. The t WGN ource w ( t ) ww ( ) G = M Coloring Filter Figure 2: Noise model n ( t ) ( M ) criterion of optimization for the determination of the numbers is the area coverage with the measured power density spectrum. By joining the modelled background noise with the modelled narrow band interferers we get the modelled power density spectrum (M) (f) (right figure). The synthesis of the noise is based on the filtering of Gaussian white noise. As filter function we use the synthetic power density spectrum. The main idea is the transformation of the power density function. We consider a linear time-invariant system (filter) that is characterized by its frequency response G(f). We assume that w(t) is the random input signal to the system. This signal is characterized by its power density function ww (f). And we also assume that n(t) denotes the output signal that is characterized by its power density function (f). Between these signals the following relation exists [5]: With white noise we get (f) = G(f) 2 ww (f) (5) This gives us the design of the filter (f) = σ 2 w G(f) 2. (6) G(f) = σ w (f) with (f) 0 f. (7) Figure 2 shows the synthesis of the color noise n(t) from white noise w(t). 3. Analyzing the Background Noise The goal of this section is the investigation of the model parameters and her probability density functions. The background noise is this part of measured noise without narrow band interferer. At fist it is important to eliminant the narrow band interferer from the measured power density function. As a favourably method it turned out a two step procedure. In the first step we used a median filter and the seconde step used a filter thats build the minimum of a window. In the left part of the figure 3 the basis line (wide line) is shown. It is quit plain to show the good fitting at the measured power density function (thin line). In the following we determine the model parameters. Exponential functions up to the third order were regarded (M = 3 corresponds to
background noise ( h) power density spectrum -125-125 synthesis model ( mes) -130-135 -130-135 ( M ) -80-80 -100-100 40 40 30 30 20 20 10 10 0 0 ( rel) narrow band interferer Figure 1: cheme of the parameter determination 80 80 [dbm/hz] smoothing curve [dbm/hz] approximation Figure 3: measured power density function and the approximately background noise
seven parameters). (h) (f) = M k=1 N k e f f k (8) Higher orders however were not regarded because of the higher number of parameters and the associated increasing complexity. The third order term showed only a very small improvement on the modelling of the background noise and therefore had not to be further considered. In contrast to this the term of second order has a clear influence up to 500kHz. However, it was shown that the parameters, which were determined for the individual power density functions, can only very badly be described by standardized probability densities. ince this frequency range to 500kHz is less important for the high rate data communication, an exponential function of first order was used for the modelling. (h) (f; N 0, N 1, f 1 ) = N 0 + N 1 e f f 1 (9) N 0 is the constant noise power density, N 1 and f 1 are the parameter of the exponential function. The determination of the parameters from the smoothed power density spectrum leads to the problem of the analytic curve fitting, which is nonlinear as a rule. For the solution of this optimization task a large scale algorithm was consulted. As optimization criterion the algorithm uses the square error. This iterative working algorithm needs an estimation of the initial values for the parameters, which were empirically determined. 3.1. tatistic Characteristics of the Parameters The goal of the modelling of the background noise exists in the description of the parameters of the model function by probability densities and/or distribution functions, in order to consider the statistic conditions. For the estimation of the probability densities and/or distribution functions extensive measurements of power density spectra in both environment types were examined and modelled. The parameters determined from this form the basis for the relative frequentnesses, and/or for the cumulative frequentnesses of these parameters. From these an estimation is determined for the probability densities and/or distribution functions which can be accepted, which are then used in the model. In [1] the evaluation was described in detail. Table 1 shows in summary the determined parameters of the background noise as well as the used probability densities with their values for the regarded environment types. 4. Analysis of the narrow band interferer This section describes how to determinate the model parameters of the individual narrow band disturbers and how to investigate the statistical distributions of the model parameters. The basis for the parameter extractions is the relative power density spectrum, equation (10), which corresponds to the difference between the measured and the approximated power density spectrum of the background noise. (rel) (f) = (mes) (f) (h) (f) (10) The determination of the model parameters is done in three stages. At first the relative power density spectrum is submitted to a smoothing through a maximum value filter. The following stage determines the maxima of the smoothed relative power density spectrum and intends a first estimation of the parameters of the model function. The last stage optimizes the determined parameters in such a way that the different cross-sectional areas between the relative and the modelled power density spectrum become minimal. The exact proceeding is described in [1]. 4.1. tatistic Characteristics of the Parameters As in section 3, also here the goal consists of describing the certain characteristic values by probability densities in order to model the statistical character of the narrow band disturbers. From the measured power density spectra the characteristic values of the narrow band disturbers were determined in the before described procedure. We build with these values the relative or cumulative frequencies and estimate from these frequencies the parameters for the probability density functions. In table 2 the determined distributions and their parameters are summarized. 5. ummary of the Analysis Results The analysis of the background noise has given for every modelled parameter a description by a probability density function, see table 1. It also showed that the background noise for the upper frequency band can be seen as constant. Considering the results of the analysis of the narrow band disturbers, we realize that probability density functions have been determined for the number of disturbers, the bandwidth of the disturbers and for the center frequency of the disturbers, sees table 2. When we determined the characteristic values of the probability densities it turned out that a splitting of the regarded frequency range into individual frequency bands is favourable. In this way, on one hand we ensure the reality proximity, on the other hand the complexity of the model decreases if only individual frequency bands for system investigations are needed. That means that the model in this case is easy to scale. 6. ynthesis of the Disturbance
Table 1: Characteristic values of the probability densities of the parameters of the background noise N 0 N 1 f 1 in dbm/hz in dbm/hz in MHz Distribution Normal distribution Uniform distribution shifted Exponential distribution Office building σ N0 = 1, 29 a N0 = 23, 06 λ f1 = 1, 300 µ N0 = 135, 00 b N0 = 74, 97 f min = 0, 096 Residential building σ N0 = 4, 14 a N0 = 30, 83 λ f1 = 0, 840 µ N0 = 137, 20 b N0 = 70, 96 f min = 0, 100 Table 2: Characteristic values of the probability densities of the parameters of the narrow band disturbers Frequency band N B k A k Office building in MHz in dbm/hz Normal distribution modified Normal distribution (N) Exponential distribution logarithmic Normal distribution (LN) Uniform distribution (G) 0MHz-30MHz σ N = 12, 0 λ B = 0, 33 µ N = 1, 40 B min = 0, 19 0MHz-10MHz σ N = 6, 94 λ B = 0, 30 σ A = 19, 20 (N) µ N = 1, 00 B min = 0, 19 µ A = 6, 50 (N) 10MHz-30MHz σ N = 4, 98 λ B = 0, 39 µ N = 0, 91 B min = 0, 19 10MHz-20MHz σ N = 3, 64 λ B = 0, 45 σ A = 16, 60 (N) µ N = 0, 69 B min = 0, 19 µ A = 4, 80 (N) 20MHz-30MHz σ N = 1, 34 λ B = 0, 20 σ L,A = 1, 50 (LN) µ N = 0, 52 B min = 0, 19 µ L,A = 0, 70 (LN) Residential building 0MHz-30MHz σ N = 12, 0 λ B = 0, 17 µ N = 1, 20 B min = 0, 23 0MHz-10MHz σ N = 5, 47 λ B = 0, 20 a A = 0, 97 (G) µ N = 0, 88 B min = 0, 23 b A = 54, 40 (G) 10MHz-20MHz σ N = 6, 47 λ B = 0, 15 µ N = 0.80 B min = 0, 23 10MHz-20MHz σ N = 3, 94 λ B = 0, 18 σ A = 23, 20 (N) µ N = 0, 35 B min = 0, 23 µ A = 9, 60 (N) 20MHz-30MHz σ N = 2, 53 λ B = 0, 09 σ L,A = 1, 40 (LN) µ N = 0, 72 B min = 0, 23 µ L,A = 0, 96 (LN)
80 80 (M) n n [dbm/hz] (M) [dbm/hz] n n (a) Office building (b) Residential building Figure 4: Two examples of the synthesizing of the power density spectrum The synthesis of the disturbance takes place in two steps. At first a power density spectrum is produced from the determined model parameters. This spectrum is then used for the filtering of Gaussian white noise (see Fig. 2). The power density spectrum is produced by the superposition of the background noise and the narrow band disturbers, eq. (4). The parameters of the equation (4) were determined in the preceding chapters and are defined by its distributions (table 1 and 2). In figure 4, synthesized power density spectra for the two regarded environments are represented. In both figures we see the exponential decrease of the background noise up to 5MHz. Afterwards the background noise remains on constant level. The number and the bandwidth of the narrow band disturbers are represented more strongly in this realization for the office building. In contrast to this the magnitude of the narrow band disturbers is higher in the realization of the residential building, which was determined also in reality. 7. Conclusion Universität des aarlandes, Lehrstuhl für Nachrichtentechnik, 2002. [2] Holger Philipps. Modelling of powerline communication chaels. In International ymposium on Powerline Communications and its Applications, pages 14 21, Lancaster, UK, 30. März - 1. April 1999. haon Foundation. [3] Holger Philipps. Development of a statistical model for powerline communication chaels. In International ymposium on Powerline Communications, page?, Limerick, Irland, 5. April - 7. April 2000. haon Foundation. [4] Holger Philipps. Hausinterne tromversorgungsnetze als Übertragungswege für hochratige digitale ignale. haker Verlag, Aachen, 1. auflage edition, 2002. Dissertation. [5] G. Proakis, John. Digital Communications. McGraw- Hill, dritte auflage edition, 1995. In this work a new model for the disturbances on power line chaels was given. The model consists only of seven parameters, whereby every parameter is a random variable and thus is described by its distribution density function. The efficiency of the model lies in the few numbers of parameters. We needed only seven random experiments to build a power density spectrum, which we used to generate a synthesis filter for the noise. References [1] Dirk Benyoucef. Codierte Multiträgerverfahren für Powerline und Mubilfunkanwendungen. Dissertation,