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Performance Comparison of Three Different Estimators for the Nakagami m Parameter Using Monte Carlo Simulation

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Ali Abdi at New Jersey Institute of Technology
Ali Abdi
Mostafa Kaveh
Mostafa Kaveh
Mostafa Kaveh
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Abstract

Nakagami distribution has proven useful for modeling the multipath faded envelope in wireless channels. The shape parameter of the Nakagami distribution, known as the m parameter, can be estimated in different ways. In this contribution, the performance of the inverse normalized variance, Tolparev-Polyakov, and the Lorenz estimators have been compared through Monte Carlo simulation, and it has been observed that the inverse normalized variance estimator is superior to the others over a broad range of m values.
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PerformancecomparisonofthreedifferentestimatorsfortheNakagamim
parameterusingMonteCarlosimulation
AliAbdiandMostafaKaveh
ABSTRACT
Nakagamidistributionhasprovenusefulformodelingmultipathfadedenvelopeinwirelesschannels.Theshape
parameteroftheNakagamidistribution,knownasthemparameter,canbeestimatedindifferentways.Inthis
contribution,theperformanceoftheinversenormalizedvariance,Tolparev-Polyakov,andtheLorenzestimators
havebeencompared throughMonteCarlo simulation,andit hasbeenobservedthat theinversenormalized
varianceestimatorissuperiortotheothersoverabroadrangeofmvalues.
I.INTRODUCTION
The Nakagami distribution is a good candidate for modeling the fluctuations of the received signal
envelope )(tR ,propagatedthroughmultipathfadingwirelesschannels[1].TheNakagamiprobabilitydensity
function(PDF)hasthefollowingform:
0,
2
1
,0,exp
)(
2
)( 212 ≥Ω≥≥
Ω
−
ΩΓ
=−mr
mr
mrm
rf m
mm
R, (1)
where (.)
Γ
isthegammafunctionandmand
Ω
aretheshapeandscaleparametersgivenby[2,p.4]:
(
)
][,
][ ][ 2
2
2
2RE
RVar
RE
m=Ω= . (2)
Intheaboveformulas,E[.]andVar[.]denoteexpectationandvariance,respectively.Notethatinasense,mis
theinverseofthenormalizedvarianceof 2
R
[2,p.4].
Asisgenerallywellaccepted,ausefulandreasonableestimatorfor
Ω
is =
−
=Ω N
ii
RN 1
2
1
ˆ,whereNisthe
number of available samples i
R of the envelope. On the other hand, several different methods have been
proposedintheliteratureforestimatingm.Inthispaperwestudytheperformanceofthoseestimatorsusing
MonteCarlosimulationandshowthatonlyoneofthemhasthebestperformanceoverabroadrangeofvalues
ofm.
II.THREEESTIMATORSFORTHENAKAGAMImPARAMETER
Letusrepresentthekthsamplemomentby k
µ,i.e. =
−
=µ N
i
k
ik RN 1
1.Withthisnotation 2
ˆµ=Ω .Forthe
kthordermomentoftheNakagamirandomvariableRwehave[2,p.10]:
2
2
)(
)2(
][ k
k
kmm
km
RE Ω
Γ
+
Γ
=. (3)
So,ageneralmoment-basedestimator[3,pp.272-274], k
m
,formmaybewrittenas:
2
2
2
~
)
~
(
)2
~
(
k
k
k
kk
k
mm
km µ
µ
=
Γ
+
Γ
. (4)
Foroddk,wemustsolveatranscendentalequation(involvinggammafunction)toobtain k
m
,whileforevenk
wehaveanalgebricequationwhichisgenerallypreferred.Theoretically, k
m
~
mustbeclosetothetruevalueof
mforanyk.Butsinceintheprocessofmodelingempiricaldata(whichinevitablyhavefiniterange)usingan
infiniterangePDF,higherordersamplemomentsdeviatefromthetheoreticalmomentssignificantly[4],i.e. k
µ
differsfrom ][ k
RE drasticallyforlargekwhenRisarandomvariablewithinfiniterange,itisbettertousethe
lowestpossibleevenordersamplemoment.For 2
=
k,formula(4)reducestotheidentity
1
1
=
,whilefor 4
=
k
weobtaintheinversenormalizedvariance(INV)estimator, INV
m
ˆ:
2
24
2
2
ˆµ−µ µ
=
INV
m. (5)
Thenamecomesfromthefact thatbyreplacingthemomentsinthedefinitionofmin (2)withthesample
moments,wearriveat INV
m
ˆ.
TheTolparev-Polyakov(TP)estimator, TP
m
ˆ,hasbeenproposedin[5]:
)ln(4
)ln()3/4(11
ˆ
2
2
B
B
mTP µµ++
=, (6)
whereln(.)isthenaturallogarithmand
(
)
N
N
ii
RB 1
1
2
∏=
=.
ThethirdestimatoristheLorenz(L)estimator, L
m
ˆ[6]:
29.12
12
2
12 ])([ 4.17
)(
4.4
ˆdBdB
dBdB
L
mµ−µ
+
µ−µ
=, (7)
where dB
k
µisthekthsampledBmoment,i.e. =
−
=µ N
i
k
i
dB
kRN 1
1)log20( ,and log(.) isthelogarithmtothebase
10(noteaminortypoin[1, p. 154,eqn.(5.81)]whereformula(7)abovehasbeenreportedwith1.29asa
coefficientandnotintheexponent).
III.COMPARINGTHEPERFORMANCEOFESTIMATORSVIAMONTECARLOSIMULATION
InordertogenerateNakagamidistributedsamples,it is usefultonotethatifR isaNakagamirandom
variable,then 2
R
P
=
isagammarandomvariable[7,p.49]:
0,exp
)(
)( 1≥
Ω
−
Γ
Ω
=−p
mp
m
pm
pf m
m
P. (8)
Forgeneratinggamma distributedsamples,we haveusedthe StatisticsToolboxof Matlab.Withoutloss of
generality,throughoutthesimulationswehavegenerateddatawith
1
=
Ω
.Foranyfixedmfromtheset{0.5,1,
1.5,…,19,19.5,20},broadenoughtocoverthepracticalrangeofmfordifferentpropagationenvironments,
andanyfixedNfromtheset{100,1000,10000},500sequencesoflengthNweregenerated.Letm
ˆdenoteany
ofthethreepreviouslydefinedestimators.Thesamplemeanof m
ˆ, =
−500
1
1ˆ
500 jj
m,isplottedinFigs.1-3versus
m,togetherwiththesampleconfidenceregion,definedby
×
±
2
(samplestandarddeviationof m
ˆ),wherethe
samplestandarddeviationof m
ˆwascalculatedaccordingto 2
500
1
1
500
1
2
1)
ˆ
500(
ˆ
500 =
−
=
−−jj
jjmm .Thesample
confidenceregiondefinedhereisusefulforexaminingthevariationsofdifferentestimatorsintermsofmandN.
AccordingtoFig.1,forsmallN( 100
=
N), INV
m
ˆisunbiased,while L
m
ˆhasapositivebiaswhichincreases
asmincreases.Thesampleconfidenceregionsofboth INV
m
ˆand L
m
ˆaresimilar,whilethesampleconfidence
regionof TP
m
ˆistoobroadtomakethisestimatorusefulandreliableforsmallN.FormoderateN( 1000
=
N)in
Fig. 2, L
m
ˆ still has the same amount of increasing bias. The sample confidence regions of all the three
estimatorshavetightened.However,thesampleconfidenceregionfor TP
m
ˆismuchwiderthanthoseof INV
m
ˆ
and L
m
ˆ.ThesamebehaviorcanbeobservedinFig.3forlargeN( 10000
=
N).
IV.CONCLUSION
Accordingtotheabovediscussion,itisobviousthatforestimatingtheNakagamimparameter,theinverse
normalizedvarianceestimatorissuperiortotheothertwoestimators,becausecontrarytotheLorenzestimatorit
doesnotshowapositiveincreasingbiasversusm,andincontrastwiththeTolparev-Polyakovestimator,hasa
narrowsampleconfidenceregion.
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  • Feb 2023
  • ULTRASOUND MED BIOL
Objective: Modelling ultrasound speckle to characterise tissue properties has generated considerable interest. As speckle is dependent on the underlying tissue architecture, modelling it may aid in tasks such as segmentation or disease detection. For the transplanted kidney, where ultrasound is used to investigate dysfunction, it is unknown which statistical distribution best characterises such speckle. This applies to the regions of the transplanted kidney: the cortex, the medulla and the central echogenic complex. Furthermore, it is unclear how these distributions vary by patient variables such as age, sex, body mass index, primary disease or donor type. These traits may influence speckle modelling given their influence on kidney anatomy. We investigate these two aims. Methods: B-mode images from n = 821 kidney transplant recipients (one image per recipient) were automatically segmented into the cortex, medulla and central echogenic complex using a neural network. Seven distinct probability distributions were fitted to each region's histogram, and statistical analysis was performed. Discussion: The Rayleigh and Nakagami distributions had model parameters that differed significantly between the three regions (p ≤ 0.05). Although both had excellent goodness of fit, the Nakagami had higher Kullbeck-Leibler divergence. Recipient age correlated weakly with scale in the cortex (Ω: ρ = 0.11, p = 0.004), while body mass index correlated weakly with shape in the medulla (m: ρ = 0.08, p = 0.04). Neither sex, primary disease nor donor type exhibited any correlation. Conclusion: We propose the Nakagami distribution be used to characterize transplanted kidneys regionally independent of disease etiology and most patient characteristics.
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Automatic segmentation of ultrasound images using SegNet and local Nakagami distribution fitting model
Article
  • Mar 2023
  • BIOMED SIGNAL PROCES
Automatic and accurate segmentation of ultrasound (US) images plays a vital role in computer-aided diagnosis of many diseases. However, multiple degradations within US images, such as speckle noise, intensity inhomogeneity, shadows, low contrast, and low signal-to-noise ratio, often cover up the details of imaging tissues and blur the edges of lesions, resulting in limited accuracy for existing segmentation algorithms. To tackle these issues, a coarse-to-fine segmentation method combining deep learning with an active contour model (ACM) is proposed in this paper. At the first stage, a SegNet is trained and employed to predict the segmentation mask because it not only has a superior boundary delineation capability, but also has high efficiency in terms of memory and computational time during inference. Due to the multiple degradations of US images, the unavailability of large-scale US image datasets, and the loss of spatial information during downsampling, the SegNet may only infer the coarse object boundaries, especially for the tissue with blurred edges. At the second stage, the segmentation mask of the SegNet is refined to locate the accurate object boundaries by a novel ACM termed the local Nakagami distribution fitting (LNDF) model, which takes advantage of discrepancies between Nakagami distributions of different tissues around the boundaries within US images. Experimental results on two US image datasets demonstrate that, compared with some state-of-the-art segmentation methods, our proposed method achieves the highest segmentation accuracy at the cost of acceptable running time, and thereby adapts to the automatic segmentation of US images in clinical applications.
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Model selection
Chapter
  • Jan 2022
This initial chapter shows how wireless channel modeling can be done when epistemic uncertainty may play a major role in the prediction of system performance. Although most of the tools described here can be adapted to different situations, we focus our attention on the derivation, based on incomplete knowledge available, of the probability density function of the random fading attenuation process affecting wireless communications. We proceed as follows: We first describe how a probability density function of a given family, whose form is known but whose parameters are unspecified, can be fitted to experimental data: Rayleigh, Rice, and Nakagami-m fading models. If only incomplete information about a random variable is available, and in particular its probability density function has an unknown form, we describe how the maximum-entropy method can be used to determine a probability density function consistent with what is known about the random variable. Assuming a given class of fading models in which the true model, or at least a good approximation to it, is believed to lie, we show how the Akaike Information Criterion can be used to make the best model choice.
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Applying Nakagami Distribution for BER Analysis of Equal Gain Transmission in MU-MIMO Systems
Article
  • May 2021
  • AEU-INT J ELECTRON C
In this paper, the bit-error rate (BER) performance of equal gain transmission (EGT) has been analyzed in multiuser multiple-input multiple-output (MU-MIMO) systems. The multiuser interference (MUI), conditioned by the EGT precoding, is found highly correlated with the desired signal when the number of transmit antennas M is in moderate-scale (M⩽64). For such case, the common independent Gaussian assumption for MUI loses precision. In this paper, a translated Nakagami approximation is proposed to characterize the equivalent channel effect, including the channel fading of the desired user and the correlated MUI. The resulted analytical BER equation shows a satisfactory accuracy for both moderate-scale and large-scale MU-MIMO systems.
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Theoretical distribution functions of multipath propagation and their parameters for mobile radio communication in quasi-smooth terrain
Article
  • Dec 1979
Due to the superposition of many partial waves arriving from different directions, the field strength received by a mobile antenna is an intensely fluctuating function of space. For the planning of radio service areas, the statistical distribution of the amplitudes should be known to improve frequency efficiency. Three statistical distributions were considered: the Weibull distribution, the Nakagami-m-distribution and the mixture of Rayleigh and log-normal distributions. Formulas are given for the determination of the distribution parameters from measurements. For the choice of the best distribution, three different methods are discussed. Results from measurements in quasi-smooth terrain in the 450-mHz-range are presented.
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Estimation of the parameters of the Nakagami probability distribution in a detector with stabilization of the false-alarm probability
Article
  • Jun 1988
An expression for estimating the Nakagami-distribution parameters is obtained which simplifies the determination of the decision threshold stabilizing the given false-alarm probability under conditions of an unknown and changing noise background. An error analysis indicates that the proposed approach is quite accurate.
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The Mobile Radio Propagation Channel
Article
  • Jan 2000
Fundamentals of VHF and UHF propagation propagation over irregular terrain propagation in built-up areas area coverage and planning tools characterisation of multipath phenomena wideband channel characterisation other mobile radio channels and methods of characterisation sounding sampling and simulation man-made noise and interference multipath mitigation techniques.
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The m-Distribution—A General Formula of Intensity Distribution of Rapid Fading
Article
  • Dec 1960
This paper summarizes the principal results of a series of statistical studies in the last seven years on the intensity distributions due to rapid fading.
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Principles of Mobile Communication
Book
  • Jan 1996
This chapter begins with a brief history of wireless systems and standards, including cellular systems, cordless telephone systems, and wireless local and personal area networks (LANs and PANs). The discussion of cellular standards includes GSM, IS-54/136, IS-95, PDC, cdma2000, UMTS, and WiMAX. The discussion of cordless phones includes DECT and PHS, and the discussion of IEEE802.11a/b/g, IEEE802.15 and Bluetooth. The chapter then introduces frequency reuse and the cellular concept, and discusses the propagation phenomenon that are found in land mobile radio environments along with additive impairments such as co-channel interference (CCI) and noise. Afterwards, the cellular land mobile radio link budget is considered, including the effects of interference loading, shadow margin and handoff gain that are peculiar to cellular frequency reuse systems. The chapter concludes with a discussion of coverage and capacity issues for cellular radio systems.
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Radar clutter modelling using finite PDF tail
Article
  • Mar 1996
The authors propose the truncation of the tail of theoretical PDFs used in clutter modelling. It is shown that the experimental moments deviate from their theoretical counterparts if this truncation is not incorporated into the theoretical PDFs. This is caused by the tail of experimental PDFs being finite as opposed to the theoretical ones. Spiky Weibull statistics are used to demonstrate the effectiveness of the proposed technique and validate our conclusion
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  • Jan 1996
  • 256-258
  • V A Anastassopoulosandg
  • Lampropoulos
V.AnastassopoulosandG.A.Lampropoulos, " RadarcluttermodellingusingfinitePDFtail, " Electron. Lett.,vol.32,pp.256-258,1996.
Estimation of the Nakagami probability stabilization
  • R G Tolparev
  • V A Polyakov
Estimation of the Nakagami probability density parameters in a detector employing false-alarm probability stabilization
  • Jan 1988
  • 113-115
  • R G Tolparev
  • V A Polyakov
R. G. Tolparev and V. A. Polyakov, "Estimation of the Nakagami probability density parameters in a detector employing false-alarm probability stabilization," Telecommun. and Radio Eng., vol. 43, pp. 113-115, 1988.