Statistical Analysis of the effect of Impulse noise in Multi Antenna System

Statistical Analysis of the Effect of Impulse Noisefor Multi Antenna System

MOHAMMAD ISMAIL HOSSAIN

February 25, 2012

Statistical Analysis of the Effect of Impulse Noise for MultiAntenna System

Report Submitted By

Mohammad Ismail Hossain

Communications, Systems and Electronics

School of Engineering and Science

Jacobs University Bremen

[email protected]

Supervisor: Prof. Dr. Werner Henkel

2

Abstract

In this report we address the noise which is impulsive, non-Gaussian prevail-ing in wireless environments and followed by a heavy-tailed distribution. Mostcommonly used heavy-tailed distribution Middleton’s Class-A noise has been in-troduced in our study. In wireless communication, we may undergo several kindsof indoor and outdoor sources for impulsive noise. In order to detect and mit-igate the effect of impulsive noise, a model over multiple antenna system wasadopted while this is an accurate model for the thermal noise present at thereceiver and measured noise from different noise sources. For the detection andmitigation of impulsive noise, separate models are used to analyze the effect ofimpulse noise in wireless environments. In addition, an ignition circuit was im-plemented which gives similar type of impulse noise generates from car ignitionand impulse segments have been measured from the ignition circuit. Using mea-sured data, analytical models for the statistics of impulse noise including voltagehistogram, probability density functions of voltages, Gaussian-Gaussian modelhas been developed, where Gaussian-Gaussian model provides an accurate ap-proximation for class-A model. Levenberg–Marquardt algorithm has been usedto demonstrate the approximation routine to fit with our measured data withvarious model functions and to estimate required parameters. Finally, Matlabsimulations were performed to examine the effects of these non-linearities forboth Middleton’s Class-A and HK models were used.

Chapter 1

Introduction

Wireless technology is the foundation for the much anticipated omnipresent com-munication networks that will allow people and machines to transfer and receiveinformation on the move, anytime and anywhere. This technology will enablean endless array of applications such as wireless phones, wireless Internet access,wireless local area networks, smart homes and appliances, automated highways,distance learning, sensor networks, video conferencing, and remote medicine.There are many technical challenges that must be overcome in order to makethis vision a reality and most usual problem is noise which arises from wirelessnetworks.

In wireless communications systems, various kinds of noise are experienced.In most of the cases, thermal noise is the common one which is basically anAdditive White Gaussian Noise (AWGN) and can be illustrated by a Gaussianmodel. Nevertheless, very rarely wireless communications systems are interferedby only white Gaussian noise. The human-made electromagnetic (EM) environ-ment, and much of the natural one, is fundamentally impulsive and it cannotbe assumed to be Gaussian. The impulsive noise has a highly structured formcharacterized by significant probabilities of large interference levels and shortduration [1]. The impulsive noise or electromagnetic interference (EMI) can befound in many indoor and outdoor environments [2]. Since wireless networks donot use expensive signal and control cables for data transmission, they are eas-ier to install and use, and provide cost-effective solutions for these applications.However, the impulsive character can drastically degrade the performance andthe reliability of wireless communications systems even in case of high signal tonoise ratios (SNR). In order to guard against unacceptable performance, the truecharacteristics of the noise must be taken into account. For this reasons we needto implement appropriate model of impulse noise.

Impulsive noise is most frequent in many wireless communication applica-tions. For instance, automotive ignition noise, power transmission lines, coronaeffect and arc generating circuit components are examples of impulsive noisesources which are encountered mainly in metropolitan areas [3]. In indoor wire-

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less communications, devices with electromechanical switches such as electricalmotors in elevators, refrigerator units, photocopy machines, and printers areconsidered as impulsive noise sources. Furthermore, microwave ovens, drill ma-chines, trains, trams, cash register receipt printers, gas-powered engines, carignitions, welding and compressor motors produce impulsive noise on frequencybands which fall together with the operating frequencies of current wireless net-works [4, 5]. Due to electromagnetic interference from independent sources, re-ceivers are affected by disruption of radio frequency Interference (RFI). There arethree sources of RFI, namely: natural, inherent, and man-made. The first caseis happened by natural phenomena such as lightning and radiation from the sunand galactic sources. This type of interference is commonly called atmosphericnoise. The second type is the noise within a piece of electronic equipment,caused by thermal excitement of electrons flowing through circuit resistance.The man-made noise is produced by a number of different classes of electricaland electronic equipment and systems. These sources of RFI include high powerbroadcast systems and a multitude of other communications systems [6].

RFI is a combination of independent radiation events prevailing non-Gaussianstatistics. Middleton’s Class A, B and C noise models (statistical-physical) [7]and Symmetric Alpha-Stable models (statistical) [8] are used for RFI model-ing. They are well-suited for modeling the predominantly non-Gaussian randomprocesses that arise from the nonlinear phenomena that govern electromagneticinterference. Symmetric Alpha-Stable processes are included due to their mathe-matically tractable form for parameter estimators and communication detectors[9]. In this project, we restrict our attention to combating Middleton’s Class Amodel.

In the sense of error free communication we need to have sufficient knowl-edge of the sources that might be responsible for causing disturbance in ourentire wireless communication systems. So we need to have knowledge of variousmodels commonly used for different noise analysis since different sources producedifferent kinds of characteristics.

In this project, for measuring the effect of impulse noise in multi-input-multi-output (MIMO) wireless system, we implemented a multi-antenna wireless re-ceiving environment that receives the bandwidth of WLAN system from varioussources. With different spacing and distance we analyzed the effect of impulsenoise from different sources and we also produced an ignition circuit that pro-vides same noise compared with car ignition. Finally, these effects of impulsenoise were analyzed with calibrated devices.

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Chapter 2

Background and Methodology

2.1 Impulsive NoiseImpulse noise is an additive source that is only active for very short intervalsin time. Because of its small duty cycle, the average power of impulse noise ismuch lower than its instantaneous power during active intervals. This results ina large peak-to-average ratio (PAR) - the salient feature of this type of noise [7].Impulse noise is a more problematic source of error when it activates frequentlywith impulse power larger than background noise. In this case, the error eventsare dominated by the impulse noise. On the other way, we can say impulsivenoise is usually described as a process characterized by explodes of one or moreshort pulses whose amplitude, duration and time of occurrence are random [10].The inter-arrival time of these pulses is generally assumed to be greater than thetime constants of the measuring system. This does not introduce any restrictionsand simply means that individual pulses can be resolved by the system [10].

2.2 Sources of Impulsive NoiseThere are many potential sources of impulsive noise in our entire world. Asdescribed earlier, due to electromagnetic interference from independent sources,receivers are affected by disruption of radio frequency Interference (RFI). Thereare mainly three sources of RFI, which are [6][4]

1. Natural Source: These type of noise are happened by natural phenomenasuch as lightning and radiation from the sun , cosmic, extraterrestrial solarand galactic sources etc.. This type of interference is commonly calledatmospheric noise.

2. Inherent Source: Inherent noise is the noise within a piece of electronicequipment, caused by thermal excitement of electrons flowing through cir-cuit resistance.

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3. Man-made source: There are many sources of impulse noise which are pro-duced by a number of different classes of electrical and electronic equipmentand systems. These sources of RFI include high power broadcast systemsand a multitude of other communications systems [6]. We can define man-made sources with indoor source and outdoor source in our environments.

→ Indoor source: In our daily life we are consecutively using many domesticappliances which produce most frequent impulse noise. For example, houseappliances such as washing machines, dish washers, refrigerator units, pho-tocopy machines, elevator, food mixers, irons, ovens, kettles, electric razors,drills, central heating thermostats and light switches are common sourcesof impulsive noise [12].

→ Outdoor source: In outdoor environments we may experience various kindof impulsive noise sources, e.g. , gas-powered engines, car ignitions, weld-ing, compressor motors, ignition systems (traffic, lawn mower, etc), highgrid power lines, corona effects and medical equipment are remarkablesources [3].

2.3 MIMO System

Multiple antenna systems are systems where multiple transmitting and/or multi-ple receiving antennas are deployed in a wireless network, and are often referredto as multiple input multiple output (MIMO) or vector systems. Huge gains indata rate for point-to point systems by employing multiple antennas at both thetransmitter and receiver, multiple-input multiple-output (MIMO) techniques areused to improve the robustness and performance of wireless links. Here, the termmultiple-input multiple-output refers to the use of an array of antennas for bothtransmitting and receiving. MIMO approaches the enabling of better wirelesscommunications because they mitigate problems inherent in ground-to-groundlinks, which are the most common links used by wireless devices, including cellphones and WiFi [13]. Typically, ground-to-ground links are not line of sight.The electromagnetic waves transmitted from the antennas bounce around theenvironment in a complicated fashion and end up at the receiver coming frommultiple directions and with varying delays. The effect produced by the di-rection/delay interactions is referred to as multipath, a condition that must beaccommodated by ground-to-ground systems [13].

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2.4 Noise Modeling for Radio frequency inter-ference (RFI)

Two general approaches for modeling electromagnetic interference (EMI) arethrough physical modeling and through statistical-physical modeling. In phys-ical modeling, each source of EMI would require a different circuit model. Onthe other hand, statistical-physical models, provide universal models for accu-rately modeling EMI from natural and human-made sources. The key statistical-physical models are the Middleton’s Class A, B and C noise models [7]. Mid-dleton model are the most widely accepted model for RFI primarily since thesemodels are canonical, i.e. their mathematical form is independent of the phys-ical environment. Middleton models are classified with respect to the receiverbandwidth:

Class A: It’s called narrowband noise model, because of the interference spec-trum is narrower than the receiver bandwidth.

Class B: This is known as broadband noise model.In this model interferencespectrum is wider than the receiver bandwidth. The Class B interference model[7] is analytically more complex since two characteristic functions are now neededto approximate the exact characteristic function. Hence we have two expressionsfor the envelope density, one for small and intermediate envelope values , theother for the larger values . The examples of such interference included atmo-spheric noise, automotive ignition noise, and arc welders.

Class C: Mixed Case: Since Class C interference is a sum of the Class Aand Class B interference models, no specific derivations for Class C are required.Furthermore, Middleton proved in [7] that in Class C can be approximated toClass B models in most cases.

Impulsive noise models for systems operating at low frequencies have beenproposed by the ITU [10]. These models are based on measurements of medianlevels of interfering noise.

Mathematically, impulsive interference is usually modeled as a train of pulses[11][12]

n (t) = ∑iAiPwi(t− τi)

where the amplitude Ai and arrival time τi of each pulse is a random variablewhose distribution is a priori unknown. The shape of the pulses is determinedby the impulse response of the actual receiving or measuring system.

Hence, we limit our discussion to Class A Middleton Models for the remainderof the report.

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2.4.1 Middleton’s Class-A ModelRecent work development of a physical-statistical model for radio noise interfer-ence done by Middleton, thus he classified this interference in the following threebroad categories: A, B, and C [7]. The spectral bandwidth of the noise enteringthe receiver is comparable to or less than the receiver bandwidth at the receiver’sfront-end stages. Specifically, Class A interference produces negligible transientsin the typical receiver, as shown this model has been used to develop optimumdetection algorithms for a wide range of communications problems [14]. Mid-dleton’s Class A model is the first model which treats narrowband interferenceprocesses. An advantage of Middleton’s model is that it can be expressed in acanonical form, so that noise from many different specific interference scenarioscan all be represented by the same model but with a different set of coefficients..A second critical feature of this model is that it is also analytically tractable, aswell as computationally manageable [14].

The Class A probability density function (PDF) for the the nature of thenoise source an be expressed by

Pclass−A = ∑∞m=0 e

−A Am

m!√

2πσ2m

e− x2

2σ2m

Where, σ2m =

mA

+Γ1+Γ

Hence the Class A model [7] is uniquely determined by the following twoparameters:

→ A is the overlap or impulsive index . It is the product of the average numberof emissions events impinging on the receiver per second and mean durationof a typical interfering source emission, and A ∈ [10−2, 1] in general. Thesmaller A, the more "structured" (in time) is the interference. Conversely,the larger A the more Gaussian and less structured is the noise. When Ais ∞, the noise is Gaussian [7].

→ Γ is called the Gaussian factor and it is the ratio σ2G

Ω2A, where σ2

G is theintensity of the independent Gaussian component, Ω2Ais intensity of theimpulsive non-Gaussian component, and Γ ∈ [10−6, 1] in general.

2.4.2 Symmetric Alpha stable (Sα S) ModelWhile Middleton Class A and Class B models [7] are known to accurately modelRFI sources, their practical applications are limited due to the intractable formof their distributions. In particular, Class B interference model is difficult to

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use due to the existence of five parameters and also an empirically determinedinflection point (εB) [3]. Hence many authors have considered Symmetric Al-pha Stable (SαS) model [10] as an approximation to Middleton Class B model.This approximation is particularly accurate for the case of narrowband receptionwithout a Gaussian component, as well as the case of a symmetric pdf withouta Gaussian component.

Symmetric Alpha Stable (SαS) models [10] are used to model the statisticalproperties of an “impulsive” signal. A random variable is said to have (SαS)distribution if its characteristic function is of the form,

φ (ω) = ejδω−γ|ω|α

Hence the following three parameters uniquely identify a (SαS) distribution[10],

• α is the characteristic exponent and is a measure of the ”thickness” of thetail of the distribution, where α ∈ [0, 2] in general.

• δ is the localization parameter. It is the mean when 1 ≤ α ≤ 2 and themedian when 0 ≤ α ≤ 1, where δ ∈ [−∞, ∞] in general.

• γ is scale parameter or the dispersion and is similar to the variance of theGaussian distribution, where γ ≥ 0 in general.

2.4.3 Extension of Middleton’s Class-A model for MultiAntenna System

In Middleton’s Class-A model he assumed single antenna system is present forreception purpose. McDonald extends the Class A model to spatial diversityschemes and produces a multi-dimensional version of Class A model by devel-oping mathematical expressions for the signals simultaneously received at twoantennas after match filtering. In order to limit complexity and to promoteclarity, only the two antenna array case is considered. The normalized randomvariable (x, y) PDF for antenna 1 and antenna 2 is [21]:

fR(x, y) = e−A

2πc0c′0

√1−k2

0exp(−

x2c20

+ y2

c′20

+ 2xyk0c0c′0

2(1−k20) ) + (1−e−A)

2πc1c′1

√1−k2

1exp(−

x2c21

+ y2

c′21

+ 2xyk1c1c′1

2(1−k21) )

c0 =√

Γ11+Γ1

c′0 =

√Γ2

1+Γ2

c1 =√

A−1+Γ11+Γ1

c′1 =

√A−1+Γ2

1+Γ2

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Where A is the overlap index, Γ1,Γ2 is the ratio of Gaussian power to thenon-Gaussian power for antenna l and antenna 2. k0 is correlation coefficient oftwo antennas for the Gaussian component. k1 is correlation coefficient of twoantennas for the non-Gaussian component [21].

Using multivariate case we can modify Middleton’s Class - A for multi- an-tenna systems. A bivariate Middleton’s Class-A model for 2 antenna system hasbeen considered in [15]. An extension fornr ≥ 2 can be expressed as[16],

f (x) = ∑∞m=0 amg(x, µ, σ2

m)

Where, am = e−AAm

m! , µ = 0 and g (x, σ2m) = e

− x22σ2m√

2πσ2m

Middleton’s Class-A model for multi antenna system can be representedby[17]

f (x) = ∑∞m=0

am

(2π)nr2 Km

12e−

xTK−1m x

2

where Km is an nrx nr covariance matrix and nr is random number variableand detail derivations can be found in [17].

2.4.4 Gaussian-Gaussian ModelA Gaussian Mixture Model is a parametric probability density function repre-sented as a weighted sum of Gaussian component densities. Gaussian Mixturesmodel are commonly used as a parametricmodel of the probability distribution ofcontinuous measurements. This model can be used as an approximation of a widevariety of symmetric zero-mean random variable, e.g. , the Laplace distributionand the SaS distribution [6].

The Gaussian-Gaussian model offers an accurate approximation to the Class-A model, The Gaussian-Gaussian model is as follows:

Pclass−A = e−A√2πσ2

0e− x2

2σ20 + 1−e−A√

2πσ21e− x2

2σ21

Where, A is the impulsive index, σ0 and σ1 are Standard Deviation[6].

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2.4.5 The HK (Henkel/Kessler) ModelUsing the generalized Gaussian distribution as an impulsive noise model wasproposed in [18][6] for modeling the voltage histogram impulsive noise on twistedpairs and resulted from a measurement campaign of Deutsche Telekom, and ithas the following form:

A suitable approximation can be achieved from this paper [18] for the pureimpulsive noise density, which can be expressed as

fi (u) = e− u

u0

15

10Γ(5)u0= e

− uu0

15

240u0

Which has the same distribution of a generalized Gaussian with parameters:µ = 0, u0= standard deviation, u= voltages of measured data. This is thequite good approximation for pure impulsive noise density. Now if we includebackground noise which is the only disturbance between impulses, we obtain theequation in following form,

ftot (u) = Pfn(u) + (1− P )fn(u)∗fi(u)

Where, fn (u) = 1√2πσ2 e

− x22σ2

Where, fn (u)is Gaussian noise density, σis the standard deviation and P∈[0, 1] is its relative portion. For ftot(u) during impulse background noise alsohere and we can also see the effect disturbance with convolution of both densities(gaussian and impulsive) [18].

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Chapter 3

Experimental Analysis

3.1 Instrument set up for measurementBased on our multiple antenna systems we constructed number of four anten-nas resonating at 2.45 GHz center-frequency for taking measurement. For themeasurement 190 MHz bandwidth filters, 300- 4300 MHz frequency mixers and1700- 4200 MHz power splitter were used. To implement overall set up antennas,band pass filter, down-converting mixer, oscillator, and a digital oscilloscope foracquiring the sampled and down-sampled data at an intermediate frequency wererequired.

OSCILLOSCOPE MIXER

BAND-PASS FILTER

ANTENNAS

LOCAL OSCILLATOR

𝑓𝑟 − 𝑓𝑐 𝑓𝑟

𝑓𝑐

Figure-1: Block diagram of multi antenna receiving system

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For taking impulse noise measurement we had to consider following steps:

→ Monopole antennas were designed at Lamda/4 where lamda is the wave-length at 2.45 GHz RF center frequency and checked for the resonatingfrequency using the scattering-parameter, i.e., S-parameters-S11. We usedthese antennas only for receiving purpose and they have Omni- directionalreceiving capability and antennas were positioned vertically to the base(ground-level). Antennas were placed with equidistance forming a Uni-form Linear Array (ULA).

→ The antennas were connected to the band pass filter with coaxial cablewhich filtered the RF signal with a bandwidth ranges from 2530 MHz to2340 MHz. The outputs of the band-pass filters were linked up with mixerswhich were down-converting according to the frequency of local oscillatorsignal that was applied to the local-oscillator (LO) input port of the mixer.Signals from both ports are mixed to yield the Intermediate Frequency (IF)of interest. The IF was finally filtered by the oscilloscope.

→ All of the mixers are connected with power splitter which takes carrierfrequency from signal generator and gives number of four outputs.

→ In order to supply 2.1 GHz carrier frequency , a signal generator was con-sidered.

→ Digital oscilloscope was used for some executions such as sampling, filteringand down-sampling of the IF signal for further use of data.

Measurement-frame was considered to maintain the same orientation, structure,environment and surroundings.

3.2 Car Ignition Circuit

In the project work, an ignition circuit which provides same impulse noise likecar ignition was implemented using Ignition coil, Spark plug, capacitor, relaydriver circuit, relay and voltage controlled oscillator for supplying rectangularpulse which can be controlled by changing capacitor.

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Ignition coil

Spark Plug

Voltage source (+12v) Switching Circuit

Capacitor

Figure-2: Block diagram of car ignition circuit

This relay driver circuit drives relay coil from a low power output, usuallyfrom VCO. It is used to switch high loads or a load that needs AC current tooperate. The relay will be actuated when the input of the circuit goes high.The protection diode is used to protect the transistor from the reverse currentgenerated from the coil of the relay during the switch off time. The values forresistor and transistor vary accordingly. The way to calculate them is:

First we calculate the load current:

IL = VSRL

Then we calculate the transistor hFE. It must be at least 5 times the loadcurrent IL divided by the maximum output current from the Input to the baseof the transistor.

hFE (min) > 5× ILIinput

Now the transistor Qs according to its current gain hFE has been selected.Then the calculated the base resistor RB was in the form

RB = 0.2×RL × hFE

The considered protective diode was 1N4001 or it can be any general purposediode. A voltage control oscillator as rectangular pulse generator which is 16-pinpackage has been utilized. It has typical maximum frequency 85 MHz, frequencyspectrum range 1 Hz to 60 MHz and power dissipation 525 MW. The outputfrequency of the oscillator can be controlled by changing external capacitor andapproximated as follows:

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f0 = (5×10−4

Cext)

Where, frequency in Hz and capacitance in farads [t].

Figure-3: schematic diagram of relay driver.

The following are the components to implement this circuit:

→ Voltage Controlled oscillator (similar type of SN54s124)

→ Resistor (1.5k, 2.2k)

→ Diode (1N4001 or it can be any general purpose diode)

→ Transistor (NPN)

→ Relay Coil (172 Ohms)

→ Capacitors (2.2µf , 4.7 µf , 10µf , 20µf , 100µf)

→ Ignition Coil

→ Spark plug

→ Capacitor for Ignition coil

→ Connecting wire

→ Voltage source

→ Voltage regulator (7805): for converting voltage from 12v to 5 volt.

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3.3 Measurements TechniqueWe analyzed impulse noise from different sources like drilling machine, car igni-tion circuit etc.. For taking measurement the antennas were placed at suitableplace and noted inter-antenna distances. All coaxial cable connections were madeby us. By using Vector Network Analyzer connection cables were calibrated andtransfer function of filter-mixer pairs were measured, then resulting data weresaved for further analysis.

Using data-acquiring Matlab code, impulse segments from different sourcesthrough oscilloscope were taken. Also 1000 blocks and 500 samples have beenconsidered for taking measurement from drilling machine . Whenever for carignition circuit 1000 blocks and 250 samples are considered where each blockcontains 5 mv threshold. The signal was sampled at 5 GHz and saved after down-sampling with 2.5 GHz and segment size of impulse was 0.2µs. The measurementswere taken from 50 cm and 100 cm from both drilling machine and car ignitioncircuit. The inter-antenna distances were taken of λ/4 and 3λ/4.

During measurement time for car ignition circuit, sparks inside the relay andspark plug were observed. When noise was generated, oscilloscope was triggeredand noise segments were counted. As the data from Matlab code through os-cilloscope are in time domain, for further analysis it should be converted intofrequency domain.

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Chapter 4

Parameter Estimation

In this chapter, we discuss various parameter estimation procedure for both ClassA and the Henkel/Kessler (generalized Gaussian distribution) noise models.

4.1 The HK (Henkel/ Kessler) ModelIn this report, we implemented an approximation routine by Matlab to estimateunknown parameters using Levenberg–Marquardt method. A good approxima-tion for pure impulsive noise density of voltages was achieved by [18]-

fi (u) = e− u

u0

15

10Γ(5)u0= e

− uu0

15

240u0

If we include background noise then we can obtain,

ftot (u) = Pfn(u) + (1− P )fn(u)∗fi(u)

Where, fn (u) = 1√2πσ2 e

− x22σ2

From our measured data we implemented voltage histogram then we tried tofit curve for outer area with pure impulsive noise density equation and middlepart for Gaussian noise density. From outer area we estimated u0 and frominner part which is Gaussian we estimated σ. Finally, we estimated P for overallequation which is mixed with impulsive noise and background noise. Using thisapproximation method we can arbitrary choose areas and can get correspondingparameters.

Table-1: Parameters approximation for voltage-frequency distribution.

Parameters P σ (mV) u0 (nV)Values 0.06594 0.891 81.395

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4.2 Middleton’s Class-A ModelThe Middleton Class A model [7] can be expressed as

Pclass−A = ∑∞m=0 e

−A Am

m!√

2πσ2m

e− x2

2σ2m

Where

σ2m =

mA

+Γ1+Γ

Since the Gaussian-Gaussian model offers an accurate approximation to theClass-A model, estimation aproximation can be done for this model. The Gaussian-Gaussian model is as follows:

Pclass−A = e−A√2πσ2

0e− x2

2σ20 + 1−e−A√

2πσ21e− x2

2σ21

Parameters under estimation are A, σ0 and σ1. Estimation procedure can bedone in two ways, one is based on method of moments [19,20] and another isusing our approximation routine.

Estimation procedure based on method of moments (MOM): Let, a, b and care denoted for the following moments computed over the received signal v (datavectors).

a =√

π2E|v| = e−Aσ0 + (1− e−A)σ1

b = Ev2 = e−Aσ20 + (1− e−A)σ2

1

c =√

2π4 E|v|

3 = e−Aσ30 + (1− e−A)σ3

1

Where, we can find [20]

Standard Deviation, σ0 = ab−c+√

(ab−c)2−4(a2−b)(b2−ac)2(a2−b)

Standard Deviation, σ1 = ab−c−√

(ab−c)2−4(a2−b)(b2−ac)2(a2−b)

Actually, a, b, and c can simply be obtained from the empirical moments over1×mn ( length of data vectors) observations:

a =√

2π

(1mn

∑mnk=1 |vk|

)

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b = 1mn

∑mnk=1 v

2k

c =√

42π

(1mn

∑mnk=1 |v2

k|)

The estimated parameters can be obtained as

A = −ln(1− σ0−a

σ0−σ1

)Using approximation routine: Our implemented approximation routine gives

quite accurate estimation for parameters. So, required parameters estimated byour approximation routine.

Table-2: Parameters approximation for Gaussian - Gaussian Model.

Parameters A σ0 (mV) σ1 (mV)Values 0.04 0.912 20.63

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Chapter 5

Simulations and Results onMeasurement

In this chapter, we provide simulation experiments to illustrate the performanceof our experiement and the approximation routine to estimate parameters. Mat-lab operations were performed to visualize the effects of these non-linearities.After observing these, simulation results show that our routine and measure-ment have good performance.

5.1 Impulse SegmentsAfter taking measurement we saved data and visualized data with different ap-proaches. The measured data saved in time domain, figure-1 shows the data withrespect to segment time for different channels from drilling machine source.

0 0.5 1 1.5 2

x 10−7

−0.02

−0.01

0

0.01

0.02

0.03Channel−1,Segment−1,Antenna−1

Time in seconds

Am

plitu

de

0 0.5 1 1.5 2

x 10−7

−0.02

−0.01

0

0.01

0.02

0.03


Time in seconds

Am

plitu

de

0 0.5 1 1.5 2

x 10−7

−0.02

−0.01

0

0.01

0.02


Time in seconds

Am

plitu

de

0 0.5 1 1.5 2

x 10−7

−0.02

−0.01

0

0.01

0.02


Time in seconds

Am

plitu

de

Figure- 4: Impulse segment for inter-antenna distances 3L4 from drillingmachine at 50 cm distance.

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If we observe these above plots we can see slightly difference between differentchannels. Figure-5 shows the data with respect to segment time for differentchannels from car ignition circuit source. From figure-5 some changes betweenchannels with amplitudes are observed and it’s different from segments aquiredfrom drilling machine.

0 0.5 1 1.5 2

x 10−7

−4

−2

0

2

4

6x 10

−3Channel−1,Segment−1,Antenna−1

Time in seconds

Am

plitu

de

0 0.5 1 1.5 2

x 10−7

−0.01

−0.005

0

0.005


Time in seconds

Am

plitu

de

0 0.5 1 1.5 2

x 10−7

−4

−2

0

2

4

6

8x 10


Time in seconds

Am

plitu

de

0 0.5 1 1.5 2

x 10−7

−5

0

5

10x 10


Time in seconds

Am

plitu

de

Figure-5: Impulse segment for inter-antenna distances L4 from car ignitioncircuit at 50 cm distance.

5.2 Probability Density Function (PDF)

We estimated mean (µ) and variance (σ2) for every channels and finally plottedGaussian density for all channels, figure-6 shows the Gaussian pdf for number of4 channels and see the effect from channel to channel with pdf.

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−0.1 −0.05 0 0.05 0.1

10−100

100

Estimated Gaussian Density for channel−1

Voltages

pdf

−0.1 −0.05 0 0.05 0.1

10−100

10−50

100


Voltages

pdf

−0.1 −0.05 0 0.05 0.1

10−100

10−50

100


Voltages

pdf

−0.1 −0.05 0 0.05 0.1

10−50

100


Voltages

pdf

Figure-6: Estimated Gaussian distribution from 4 channels

5.2.1 Voltage Histogram

−0.05 0 0.05−14

−12

−10

−8

−6

−4

−2

Voltage Histogram for channel−1

Voltages(u) in volt

log

h(u)

−0.05 0 0.05−14

−12

−10

−8

−6

−4

−2


Voltages(u) in volt

log

h(u)

−0.05 0 0.05−14

−12

−10

−8

−6

−4

−2


Voltages(u) in volt

log

h(u)

−0.05 0 0.05−14

−12

−10

−8

−6

−4

−2


Voltages(u) in volt

log

h(u)

Figure-7: voltage histogram for four channels

Figure-8 shows the estimated approximation for pure impulsive noise densitywith measured histogram and we take this approximation for outer area whichis fitted with measured data.

21

−0.06 −0.04 −0.02 0 0.02 0.04 0.06−14

−12

−10

−8

−6

−4

−2

Voltages(u)in volt

log

h(u)

Histogram (Impulsive)

measured dataApproximated

Figure-8: impulse noise density histogram with fitted curve

Figure-9 shows the estimated approximation for Gaussian noise density withmeasured histogram and we take this approximation for inner area which is fittedwith measured data.

−0.06 −0.04 −0.02 0 0.02 0.04 0.06−14

−12

−10

−8

−6

−4

−2

Voltages(u)in volt

log

h(u)

Histogram (Gaussian)

measured datafitted

Figure-9: Gaussian density histogram with fitted curve

We can see the effect of background noise from Gaussian noise density his-togram. After convolving both pure impulse density and gaussian density we cansee the overall effect in the following figure-10.

22

−0.06 −0.04 −0.02 0 0.02 0.04 0.06−14

−12

−10

−8

−6

−4

−2

Voltages(u) in volt

log

h(u)

Histogram (Overall)

measured datafitted

Figure-10: Overall density

We can arbitrary select an area what we have needed for example we want tohave voltage ranges -0.06 to -0.02 and 0.02 to 0.06 for outer area, -0.004 to -0.004inner area which is background noise and overall from -0.075 to 0.075. Figure-11 shows the plots for different selected areas.

−0.06 −0.04 −0.02 0 0.02 0.04 0.06−14

−12

−10

−8

−6

−4

−2

Histogram (measured)

Voltages(u) in volt

log(

u)

−0.06 −0.04 −0.02 0 0.02 0.04 0.06−14

−12

−10

−8

−6

−4

−2

Voltages(u) in volt

log(

u)

Histogram (Impulsive)

selectedotherfitted

−0.06 −0.04 −0.02 0 0.02 0.04 0.06−14

−12

−10

−8

−6

−4

−2

Voltages(u) in volt

log(

u)

Histogram (Gaussian)

selectedotherfitted

−0.06 −0.04 −0.02 0 0.02 0.04 0.06−14

−12

−10

−8

−6

−4

−2

Voltages(u) in volt

log(

u)

Histogram (Overall)

selectedotherfitted

Figure-11: Histogram with fitted curve for different selected areas

23

5.2.2 Gaussian-Gaussian ModelWe also applied this approximation method for Gaussian - Gaussian distributionand from here we can get figures 12 and13.

−0.06 −0.04 −0.02 0 0.02 0.04 0.06−14

−12

−10

−8

−6

−4

−2

Voltages(u) in volt

log

h(u)

Histogram (Outer)

f vs. Bfit 1

Figure-12: Histogram with outer fitted curve

−0.06 −0.04 −0.02 0 0.02 0.04 0.06−14

−12

−10

−8

−6

−4

−2

Voltages(u) in volt

log

h(u)

Histogram (Overall)

fff vs. BBfit 1

Figure-12: Histogram for overall fitted curve

24

5.3 Frequency Response for Channels

We calibrate channels with filters, mixers, carrier generator and coaxial cablesfor finding transfer function and frequency responses. We have plotted frequencyresponses for different channels in the following figures.

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8−40

−35

−30

−25

−20Magnitude (dB) for channel−1

Frequency (GHz)

Mag

nitu

de (

dB)

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8−4

−2

0

2

4Phase for channel−1

Frequency (GHz)

Pha

se

Figure-14: Frequency response for channel 1

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8−45

−40

−35

−30


Frequency (GHz)

Mag

nitu

de (

dB)

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8−4

−2

0

2


Frequency (GHz)

Pha

se


25

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8−45

−40

−35

−30


Frequency (GHz)M

agni

tude

(dB

)

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8−4

−2

0

2


Frequency (GHz)

Pha

se


2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8−40

−35

−30

−25


Frequency (GHz)

Mag

nitu

de (

dB)

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8−4

−2

0

2


Frequency (GHz)

Pha

se


26

Chapter 6

Conclusion

In our project we mainly focused on generation of impulse noise, detection ofnoise, receiving and analyzing impulse noise. We discussed different types RFInoise for indoor and outdoor environment noise sources. We shown the effect ofimpulse noise for multi antenna system and have seen the different characteristicfor indoor and outdoor noise sources.We implemented a car ignition circuit forgetting noise like from car ignition. Then it provides a mathematical descriptionof the noise statistics by providing various models that are being used to modelthis time of noise. After that, the paper gives various algorithms that can be usedto estimate the parameters of the mentioned models and goes on to evaluatingthe accuracy of the given models in fitting real data samples.We modeled ourmeasured data with various model function, e.g. , Gaussian - Gaussian model,The HK model for pure impulsive density and impulsive density with backgroundnoise. It is noticed that the effects of the impulsive noise can be mitigated by theapplication of the appropriate method for both Class A and HK model. In thisreport, we proposed approximation routine with using Levenberg–Marquardt al-gorithm to estimate parameters and this approximation routine can be used forvarious wireless environment. We visualized the effect how it looks like. In con-clusion, this report described computationally tractable methods for mitigatingimpulsive noise which can have a substantial impact on current wireless environ-ment. In this way our entire communication systems would be more sufficient,reliable and secure.

AcknowledgementI would like to express my heartiest gratitude Professor Dr. Werner Henkel for hiskind and scholastic supervision with stimulating suggestions and encouragementto accomplish this project. I would also like to thank Saaifan Khodr for his allkinds support during this work.

27

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Documents

Statistical Analysis of the effect of Impulse noise in Multi Antenna System