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**Chapter 3 Impulsive Noise Model**

In this chapter, two models for non-Gaussian impulsive noise from the literature will be introduced and compared to measured results. In this thesis, simpli ed versions of Mid-dleton’s canonical statistical-physical model are used [22]. This chapter begins with a brief introduction of Middleton’s model, followed by two simpli ed versions. The analytical ex-pressions for the APD graph of the components of this model are described. The APD graph was explained in further detail in Section 2.1.3. This chapter concludes with an explanation of estimating the parameters for these simpli ed models and a few representative examples comparing computer simulations using these models to actual measurements.

** Introduction**

Many modern communication systems have been designed to work in a channel corrupted by additive white Gaussian noise. In the presence of non-Gaussian impulsive noise, the perfor-mance may be signi cantly degraded. In order to analyze the performance of such systems in impulsive noise, an appropriate noise model is needed. The simpli ed models in this chapter are based on the estimated amplitude probability distribution of noise measurements taken in Chapter 2. These models do have some limitations. The simpli ed models ignore the underlying physics of the impulsive noise and instead attempt to match measured data to known probability distributions. Since the models are matched to the output of the mea-surement system, the statistics of the impulsive noise prior to the antenna are not known. Transferring these results to receivers of di erent bandwidths is further complicated by the fact that the power of Gaussian noise is proportional to the received bandwidth while the power of impulsive noise is proportional to the square of the received bandwidth. Further-more, most measurements are necessarily done over a nite period of time, and the statistics of the impulsive noise may change for di erent times of day. Finally, basing the model on the amplitude distribution alone ignores the time correlation of the noise.

**Middleton’s Model**

The noise model developed by Middleton in [22] is a detailed statistical-physical model of man-made noise. In particular, it models the rst order statistics of the amplitude of man-made noise after the nal IF lter of the measurement system. Therefore, the physical nature of the underlying waveforms of man-made noise are not described. The parameters for the model would need to be adjusted for di erent receivers and scenarios. Noise is de ned as any undesired signal at the receiver. It is assumed that there are an in nite number of possible noise emission sources. The amplitudes, frequencies, and durations of the emission pulses are considered to be random. The locations of the sources as well as the emission times are assumed to be Poisson-distributed. There are three basic classes of models used based on the e ect of man-made noise on the receiver: Class A, B, and C. Class A models the amplitude of the noise when the response of the receiver is dominated by the amplitude of the pulse itself. This means that the duration of a given pulse must be much longer than the inverse of the bandwidth of the receiver used. Put another way, the bandwidth of a given pulse must be much smaller than the receiver bandwidth so that it passes through the receiver largely unchanged. Class B models noise pulses when the response of the receiver is dominated by the transients, or « ringing » in the receiver [22]. This means that the duration of the pulse must be much shorter than the inverse of the bandwidth of the receiver. Therefore, the bandwidth of a given pulse will be much larger than the receivers lter. The Class C model is a combination of both Class A and B models. These are summarized in Table 3.1.

Since noise modeled by Class A has a bandwidth narrower than the receiver, it will pass largely unchanged through the nal IF lter, and its transient response will be negligible compared to the pulse amplitude. This causes the APD graph to have a sharp increase at low probabilities. This sharp increase usually appears as a step function. Example sources of noise, which have been modeled using Middleton’s Class A, include an ore crushing machine in a mine using a receive bandwidth of 1-1.2 kHz and power lines [22], [23]. It is important to note that the class of the model is determined by the ratio of the receiver bandwidth and the bandwidth of the noise impulses. Noise modeled with Class A for one receiver may be better modeled with Class B for another narrower bandwidth receiver. Middleton models the CCDF of the Class A model for impulsive noise, which is the probability that the normalized instantaneous amplitude will exceed a given threshold with a Gaussian mixture model

In contrast to Class A, the Class B model represents the situation where the amplitude is dominated by the transient response of the receiver. Therefore, instead of a step function, there will be an increasing slope in the APD graph at low probabilities. Figure 3.1 shows the di erence between Class A and B using the simpli ed models to be discussed in Section 3.3. The slope of the Class A model has a sharp increase, almost like a step function at low probabilities, while the Class B model has a more gentle change in slope. The APD for Class B noise is broken into two parts: one for small and intermediate values (0 < _{0} < _{B}), and another for large amplitudes ( _{B} < _{0} < 1). The APD for larger values is of the same form as Class A. The APD for small and intermediate values is more complicated.

**Simpli cation of Middleton’s Models**

While Middleton’s model for impulsive noise has been shown to have excellent results [24], it can be too complicated to easily use in practice. Middleton’s Class A model uses a Gaussian mixture with an in nite number of terms. It has been shown that the Class A model can be approximated with only two mixture terms [26]. The authors of [14], [15] have introduced simpli ed versions of Middleton’s Class B model, which uses a train of impulses and a lter to mimic the properties of noise impulses with a bandwidth greater than the receiver. This section will begin with the simpli cation of the Class B model and analytical descriptions of the APD curves. Following this there will be a discussion of the simpli cation of the Class A model.

**Simpli ed Class B Model**

For the model developed by [15], it was desired to not only simplify Middleton’s Class B model but also to model the noise prior to the IF lter instead of after it. It was desired to have a model for the noise prior to the IF lter so that results would be transferable to receivers of di ering bandwidths. The noise was modeled as the sum of two components, one Gaussian and the other non-Gaussian. The non-Gaussian component was represented with a Weibull distribution. The model for the complex baseband representation of the noise prior to the nal IF lter is

n^_{k} = (xe_{k}b_{k} + g_{k})e^{j} ;

where n^_{k} is the complex baseband representation of the noise, is a uniformly-distributed random variable on the interval (0 u < 2 ), g_{k} is the Rayleigh-distributed amplitude of the Gaussian noise component, b_{k} is the Weibull-distributed amplitude of the impulsive noise component, and xe_{k} is a binary random variable determining whether a pulse is present. The author of [15] used a common phase parameter shared by the nominal and impulsive components of the noises while the author of [14] used independent phase values for each component. In practice, there is no real di erence when the amplitudes for the impulsive component of the noise are much larger than the nominal component.

Pulses were assumed to arrive randomly with a Poisson distribution in time. The parameter is the mean pulse arrival rate, and t is the sampling period. Therefore, the quantity t is the probability that a pulse will be present in a given sample and can be related to Middleton’s impulsive index A_{A}. It is assumed that prior to the IF lter the bandwidth of the pulse is much larger than the bandwidth of the IF lter. Therefore, the pulses are represented as impulses with a duration much shorter than the sampling period. In order to get the complex baseband representation of the noise after the nal IF lter, n^_{k} must be ltered using a lter that represents the receiver to be modeled. In the rest of this section, the statistics of Gaussian noise and Weibull-distributed impulsive noise as well as some observations of their APD will be examined.

**Statistics for Gaussian Noise**

The Gaussian component of the noise, g_{k}, is described by a single parameter, the aver-age power w_{og}. The APD graph is based on the CCDF of the instantaneous power. The probability density function for the instantaneous power of Gaussian noise is given in [15

where p(w) is the probability density function of the instantaneous power, w, and w_{og} is the average power of the signal. The CCDF of the instantaneous power can be derived from the probability density function

There are two parameters that describe the Weibull distribution, w_{ow} and . The parameter w_{o} shifts the plot up and down on the y-axis. When F_{W}^{0} (w) = 0:368 then 10 log(w) = 10 log(w_{ow}) so that just as in the case of Gaussian noise, w_{o} can be read from the intersection of the Weibull distribution and the 37% mark on the APD graph. Equation(3.12) shows that the slope of a Weibull distribution is 10 .

An example comparing the APD graph of Weibull and Gaussian noise can be found in Figure 3.2. The Gaussian distributed noise has an average power, w_{og} = 0 dB. Two di erent distributions of Weibull noise are plotted. All three have average power, w_{ow} = 0 dB. The parameter for is set to 2 and 3. The average power of all three distributions can be read directly from the 37% mark. It can be seen from the gure that the Gaussian noise has a slope of -10 dB for a unity distance on the APD graph. Similarly, the slopes for the two Weibull noise distributions are -20 dB and -30 dB.

1 Introduction

1.1 Motivation

1.2 Contributions

1.3 Thesis Summary

2 Measurement Campaign

2.1 Single Antenna

2.2 Multi Antenna

2.3 Conclusions

3 Impulsive Noise Model

3.1 Introduction

3.2 Middleton’s Model

3.3 Simplication of Middleton’s Models

3.4 Estimating the Parameters for the Two Models

3.5 Conclusion

4 Eects of Impulsive Noise on the Energy Detector

4.1 System Model and Detector Statistics

4.2 SNR Walls for an Energy Detector

4.3 Eects of Impulsive Noise on the Sample Complexity of an Energy Detector

4.4 Eect on the Central Limit Theorem Convergence

4.5 Conclusions

5 Eects of Spatially Correlated Impulsive Noise on the Energy Detector

5.1 System Model

5.2 Correlated Noise Measurements

5.3 Performance of an Energy Detector in Uncorrelated Impulsive Noise

5.4 Performance of an Energy Detector in Spatially Correlated Impulsive Noise

5.5 Numerical Simulations

5.6 Conclusions

6 Receiver Design

6.1 Overview of Project

6.2 Design Methodology

6.3 System Architecture

6.4 Results

6.5 Conclusions

7 Conclusions

7.1 Summary

7.2 Future Work

Appendices

Bibliography

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Measuring Noise in the VHF Band and Its E ect On Low SNR Signal Detection