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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 R (t ) , propagated through multipath fading wireless channels [1]. The Nakagami probability density function (PDF) has the following form: f R (r ) =
mr 2 1 2m m r 2 m −1 exp , r ≥ 0, m ≥ , Ω ≥ 0 , − m 2 Ω Γ ( m )Ω
(1)
where Γ(.) is the gamma function and m and Ω are the shape and scale parameters given by [2, p. 4]:
(E[ R ]) m= 2
2
Var[ R 2 ]
, Ω = E[ R 2 ] .
(2)
In the above formulas, E[.] and Var[.] denote expectation and variance, respectively. Note that in a sense, m is the inverse of the normalized variance of R 2 [2, p. 4]. ˆ = N −1 As is generally well accepted, a useful and reasonable estimator for Ω is Ω
N i =1
2
Ri , where N is the
number of available samples Ri 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 m PARAMETER Let us represent the kth sample moment by µ k , i.e. µ k = N −1
N i =1
k ˆ = µ . For the Ri . With this notation Ω 2
kth order moment of the Nakagami random variable R we have [2, p. 10]: E[ R k ] =
Γ ( m + k 2) k 2 Ω . Γ( m) m k 2
~ , for m may be written as: So, a general moment-based estimator [3, pp. 272-274], m k ~ Γ ( m k + k 2) µk = . k 2 k 2 ~ )m ~ µ2 Γ(m k k
(3)
(4)
~ , while for even k For odd k, we must solve a transcendental equation (involving gamma function) to obtain m k ~ must be close to the true value of we have an algebric equation which is generally preferred. Theoretically, m k
m for 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. µ k
differs from E[ R k ] 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 k = 2 , formula (4) reduces to the identity 1 = 1 , while for k = 4 we obtain the inverse normalized variance (INV) estimator, mˆ INV : mˆ INV =
µ2
2
µ4 − µ2
2
.
(5)
The name comes from the fact that by replacing the moments in the definition of m in (2) with the sample moments, we arrive at mˆ INV . The Tolparev-Polyakov (TP) estimator, mˆ TP , has been proposed in [5]: mˆ TP =
1 + 1 + (4 / 3) ln(µ 2 B) 4 ln(µ 2 B )
,
where ln(.) is the natural logarithm and B =
(6)
(∏
N
i =1
Ri
2
)
1N
.
The third estimator is the Lorenz (L) estimator, mˆ L [6]: mˆ L =
4 .4 µ
dB 2
− (µ ) dB 1
2
+
[µ
dB 2
17.4 , − (µ 1dB ) 2 ]1.29
−1 dB where µ dB k is the kth sample dB moment, i.e. µ k = N
(7) N i =1
(20 log Ri ) k , 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 is a Nakagami random variable, then P = R 2 is a gamma random variable [7, p. 49]: f P ( p) =
m Ω
m
p m −1 mp exp − , p≥0. Γ ( m) Ω
(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 Ω = 1 . For any fixed m from the set {0.5, 1, 1.5, …, 19, 19.5, 20}, broad enough to cover the practical range of m for different propagation environments, and any fixed N from the set {100, 1000, 10000}, 500 sequences of length N were generated. Let mˆ denote any of the three previously defined estimators. The sample mean of mˆ , 500 −1
500 j =1
mˆ j , is plotted in Figs. 1-3 versus
m, together with the sample confidence region, defined by ± 2 × (sample standard deviation of mˆ ), where the sample standard deviation of mˆ was calculated according to
500 −1
500 j =1
2 mˆ j − (500 −1
500 j =1
mˆ j ) 2 . 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 ( N = 100 ), mˆ INV is unbiased, while mˆ L has a positive bias which increases as m increases. The sample confidence regions of both mˆ INV and mˆ L are similar, while the sample confidence region of mˆ TP is too broad to make this estimator useful and reliable for small N. For moderate N ( N = 1000 ) in Fig. 2, mˆ L 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 mˆ TP is much wider than those of mˆ INV and mˆ L . The same behavior can be observed in Fig. 3 for large N ( N = 10000 ). 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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[2]
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[3]
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[4]
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[5]
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[6]
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[7]
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