Noise Distribution

4.8 Conclussion

5.1.3 Noise Distribution

The instantaneous noise power envelope such as that shown in Figure 5-2 is captured by the measuring instrument and can be defined by:

power=𝐼²+𝑄² (5.1)

where I and Q are the in-phase and quadrature components of the waveform. Unfortunately, the power line noise signal is difficult to characterize due to its intrinsic random behaviour. In order to extract meaningful information from such a random process, a statistical description of the PLC noise power level is required. A statistical analysis of the PLC signal is provided

Figure 5-3: Time-Frequency spectrum of PLC noise captured in an office

Figure 5-4: Time-Domain spectrum of PLC noise captured in an office

through the complimentary cumulative distribution function (CCDF). A CCDF curve shows how much time the signal spends at or above a given power level. In this work, we observe the signal level relative to its average power. Thus, the peak-to-average ratio is actually being measured as opposed to the absolute power level. This deviation will be expressed in dB. The percentage of time the signal spends at or above a certain power level defines the probability for that particular power level. A CCDF curve is thus a plot of relative power levels vs. probability. Figure 5-6(a) shows a CCDF of PLC noise registered in an office. In

Figure 5-5: Spectrogram of PLC noise captured in an office

the same plot, a Gaussian noise CCDF is plotted for reference. The cumulative distribution (CDF) and the probability distribution functions are also shown in Figure 5-6(b) and 5-6(c) respectively. We derive the CDF from the CCDF in the following manner:

𝐹(𝑥) = 1−𝐹¯(𝑥), (5.2)

where 𝐹(𝑥) is the CDF and 𝐹¯(𝑥) is the CCDF. The cumulative distribution function is ideally the opposite of its compliment and is defined as follows:

𝐹(𝑥) =𝑃(𝑋≤𝑥) = ∑︁

𝑦:𝑦≤𝑥

𝑝(𝑦) (5.3)

The cumulative distribution describes the probability that a random variable 𝑋 with a given probability distribution will exhibit values less than or equal to 𝑥. Once the CDF is obtained, the probability density function can also be determined. The relationship between the CDF and the PDF is such that the PDF is the derivative of the CDF. It can thus be determined as follows, provided 𝑓_𝑥 is continuous at 𝑥:

𝑓_𝑋(𝑥) = 𝑑

𝑑𝑥𝐹_𝑋(𝑥) (5.4)

where 𝑓_𝑋(𝑥) is the PDF and 𝐹_𝑋(𝑥) is the CDF. The probability distribution can be represented as follows:

(a) CCDF of PLC noise registered in an office

(b) CDF of PLC noise registered in an office

(c) PDF of PLC noise registered in an office

Figure 5-6: Statistical distribution of impulsive noise measured at a wall plug

𝑓𝑋(𝑥) =𝑃(𝑋 =𝑥) (5.5)

Which is the probability that a certain power level above the average power occurs.

This work has presented a detailed measurement procedure for noise measurement for broadband power line communication systems. The measurements are acquired using a high

resolution signal analyser from Tektronix that is capable of providing high resolution noise spectrums both in time and frequency domain. Some characteristics which are normally difficult to observe with conventional digital oscilloscopes or vector spectrum analyzers are shown with adequate clarity. The time-frequency characteristics of PLC noise shows that there is a strong presence of high frequency cyclostationary noise well beyond that which is synchronous with the mains frequency. Through measurements, we are able to determine the complementary cumulative distribution function (CCDF) of the noise powers that equals or exceed the average power of the noise. Though, this is not a measure of absolute noise power, it provides valuable information as to how often certain exceedance power levels are attained. We can see from Figure 5-6(a) that the most stressful noise (further right of the graph) occurs less often compared to that which is closer to the average power in magnitude.

The same conclusion can be drawn from the DPX spectrum shown in Figure 5-2. The blue coloured signals occurring above the yellow/red signals are of higher magnitude and appears less frequent.

The distribution of the power levels above the average power level represents the dis- tribution of impulsive noise occurrences. It is evident in Figure 5-6(a) through 5-6(c) that this noise definitely does not follow the classical additive white Gaussian noise (AWGN).

Clearly, there is a large presence of high power impulsive noise components in the PLC environment than suggested by the AWGN assumption. As an example: the Gaussian dis- tribution suggests that the probability of a noise component being 7.5 dB above the average power is 0.5, whereas according to measurements for the same power exceedance of 7.5 dB, the probability is 0.15. On the other hand, the AWGN assumption exaggerates the presence of noise components with a power level within the vicinity of the average power level. As an example: The Gaussian distribution predicts that the probability of a noise component being 2.5 dB above the average power is 0.7. According to the measurements, the probability of a noise component being 2.5 dB more that the average noise power is approximately 0.3.

This power statistics gives us a perspective of how often there is presence of impulsive noise. In that case, impulsive noise is not classified as whether cyclostationary, synchronous or asynchronous. It is rather presented as seen at the receiver. Since the distribution of these impulsive components does not follow any standard probability distribution, it is most likely that it could be modelled by a combination of known probability distributions or a completely new model. It is also of note that this distribution will vary from location to

location as well as time to time. However, with intensive measurements a unified model or class of models could possibly be developed and defined for typical indoor scenarios.