Performance of DMT Systems Under Impulsive Noise Gokhan Pay and Mehmet Safak Hacettepe University Department of Electrical and Electronics Engineering 06532 Beytepe, Ankara, Turkey ee.truva.hacetteue.edu.tr,
[email protected] Keywords: Discrete Multitone (DMT) modulation, QAM, SER (Symbol Error Rate), Middleton's Class A Distribution, Impulsive Noise, Central Limit Theorem
Abstract The performance of DMT systems is studied in impulsive noise channels by using Middleton's Class A noise distribution. A statistical model is developed to generate Class A distributed impulsive noise samples and this model is used for the performance analysis. An exact expression is derived for the subchannel SER. By employing the central limit theorem, the subchannel SER is further simplified. The results are presented as a function of the effective subchannel SNR for different values of impulsive noise index.
1. Introduction Multicanier modulation is not only convenient in dispersive channels due to one tap equalization, but it is also effective against the impulse noise due to its longer symbol duration, as stated in [I]. Nevertheless, the performance of DMT system over dispersive, impulsive noise channels has not been investigated adequately, most probably due to the lack of a realistic model for the channel noise. Class A distributed impulsive noise was used by several researchers in single carrier systems [2,3] and multicanier modulation was extensively investigated under Gaussian background noise [4]. This paper addresses the performance of DMT systems in dispersive channels characterised by Middleton's class A noise model.
2. Statistical Model for Impulsive Noise Consider the random variable n which has Class A pdf [2,5,6] given by
where
A
: impulsive index, characterizes the impulse "traffic",
P
:total mean power of the channel noise,
P = bG2 + 0,2 ,
2
bG :mean power of the Gaussian component of the channel noise, 2
6, :mean power of the Impulsive component of the channel noise, 0,'
= ((j~~)+r)/(i+r),
Note that the Class A pdf given by (1) may be expressed as
-00
where
40
X
:Poisson distributed random variable with pdf [7] " A'
P X ( ~=)e - " C - - ; - ~ ( x - j ) , j=o
A ~ ( x= x )
J.
: Gaussian distributed random variable with zero mean and variance
Ii
pox2:
In view of the last equation, a class A noise sample n(m),may be expressed as
n(m) =
(m)+
ZY (m)
where
x, ( m ) :White Gaussian background noise sequence with zero mean and variance 0 ,2 , y(m)
:White Gaussian sequence with ra,mean and variance C12/A,
K,
: Statistically independent, Poisson distributed random sequence whose pdf characterized by
A,
m :Time index of the noise samples. All random sequences in the model, are statistically independent fiom each other. To assess the validity of (3) for generating class A distributed noise samples, the pdf obtained by using the noise samples generated by (3) are compared with the theoretical class A pdf, given by (1). It is important to note that the noise generated is essentially flat since no correlations are introduced in simulation. The results are shown in Figure 1, where the solid lines denote the theoretical class A pdf and the crosses denote the results of simulation produced by 100,000 computer-generated statistically independent samples. Average power of the impulsive noise component was taken as 1 W for both cases. Figure 1 shows the perfect match for all values of A and even over the tails of the pdf.
r
3. DMT Subchannel SER FEQ (Frequency Domain Equalizer) output of a DMT receiver [1,8,9] for the 1 'th subcarrier :
where
4
:message point from a square QAM constellation in which the min. distance is 1 ,
dl
:decision variable for l t h subcarrier,
N
:half the DFT length,
HI = gz
n(m)
IH,
leJyt:DFT of sampled channel impulse response, evaluated at i t h subcarrier frequency , :real-valued scaling applied to the unity minimum distance constellation , :additive channel noise samples, given by (3)
noise amplitude lo0
.
Pdf I
I
noise amplitude Figure 1 Comparison between theoretical (solid lines) and empricial (crosses) Class A pdf s. The probability of symbol error on 2 th subchannel (or subcanier) :
where
4
: number of bits allocated to the 2 th subchannel, assumed to be even ,
M I = 2*1:number of message points in the 2 th subchannel constellation ,
P,,
:half the probability of correct decision along in-phase axis of 2 th subchannel QAh4 constellation:
:half the probability of correct decision along quadrature axis of 2 th subchannel QAh4 constellation:
In order to take expectation E{
) of (5) and (7), the pdf of the argument is needed. Since the main source of
randomness is K, sequence in the argument, using the pdf of (6) and (8) and making the necessary variable transformations, one gets the final expressions for (5) and (7) as
m, =O
where
SNR, =
IH, I
~ower,
M , -1
I+-
TAN
is the signal to background noise power ratio at the input of the receiver,
0,
Power, =
g; (2" - 1) is the power allocated to 1th subcarrier , 6
The expression for
is obtained fiom (9) if
xRis replaced by X I :
Thus, (4), (9)-(11) provide the exact subchannel SER for DMT over dispersive channels suffering fiom the impulsive noise, modeled by Middleton's Class A distribution. The expression (9) can be further simplified further by averaging over the long-term pdf of y, .
4. Subchannel SER by employing Central Limit Theorem In the previous section, an exact pdf of (6) and (8) is derived for any N with no assumption on the phase of the subchannel T l .However, this method leads to a subchannel SER expression that is too complex. Note that, for large values o f 2 N , each term in the summations of (6) and (8) may be considered as statistically independent fiom each other, since
K, 's are assumed to be statistically independent in (3); All cosine and sine terms in (6) and (8) contain the same random phase
Y,,
which is assumed to be
uniformly distributed over the interval [o,&). Thus, considering the relation between any two terms of the summations in (6) or (8), cosine and sine terms appear as simple deterministic multiplicative constants. The randomness of Y, will be important in the pdf s of (6) and (8). Note that each term of the summations in (6) and (8) can take any non-negative real values depending on
K,
and y,. Additionally, the pdf of each term in the summation of (6) or of (8) is the same since sine and cosine functions are periodic over any
2n ml is not critical. 2N
2~ interval. Consequently, from a statistical point of view, the argument term
Then, Central Limit Theorem [lo]states that for large number of terms 2 N , the pdf of summations in (6)and (8) approaches Gaussian pdf. Consequently, the pdf of (6)and (8)also approaches Gaussian pdf . Note that for DMT systems 2N is typically 512 or more. can then be shown to become equal to each other and the subchannel SER expression (4) reduces
PRcand to
with
Here, X is a Gaussian random variable with its mean and variance given by
It can be shown that (13)can be expressed in integral form:
where
The subchannel SER, which is given by (12)and (14)for large values of N ,is characterised by two parameters, namely, TXSNR, and @ .Figure 2 shows the subchannel SER versus eflective SNR of the i t h subchannel,
r
defined by x S W for N = 256, MI = 64 and A = 0.2 and A = 1.Note that the SER has a floor for the values of effective subchannel SNR, exceeding about 28 dB. Note that when the effective subchannel SNR is sufficiently high, erf{]in (14) becomes approximately equal to 1, at values of u for which exp&u2} is significant. Then, (12)represents the irreducible SER, which is dependent on parameter, @ given by (15).
5. Conclusions This paper presents an exact analysis of the subchannel SER in DMT systems using M-ary QAM and a simple expression for the subchannel SER obtained by using the central limit theorem. The simple expression provides a clear view of the SER is influenced by which the parameters, A and r, characterising the class A noise. The subchannel SNR was observed to have a floor of lo4 for the values of the effective subchannel SNR values exceeding about 28 dB.
Effective subchannel SNR
Figure 2 Subchannel SER versus
Tx SNRl for A=l and
A=0.2.
Acknowledgment This study is supported by Turkish Scientific and Technical Research Council ( T ~ I T A K )under the project number EEEAG-198E023.
References [I] John A. C. Bingham, Multicarrier Modulation for Data Transmission :An idea Whose Time Has Come, IEEE Communications Magazine, pp. 5-14, May 1990. : [2] Jong-Soo Seo, Sung-Joon Cho and Kamilo Feher, Impact of Non-Gaussian Impulsive Noise on the Performance of High-Level QAM, IEEE Trans. Electromagnetic compatibility, pp. 177-180, May 1989.
[3] Arthur D. Spauldiig, David Middleton, Optimum Reception in an Impulsive ~ntekerenceEnviroment-Part I: Coherent Detection, IEEE Trans. Communications, pp. 910-923, September 1977. [4] Irving Kalet, The Multitone Channel, IEEE Trans. Communications, pp. 119-124, February 1989. [5] David Middleton, Statistical-Physical Models of Electromagnetic Interference, IEEE Trans. Electromagnetic Compatibility, pp. 106-127, August 1977.
[6]Leslie A. Beny, Understanding Middleton's Canonical Formula for Class A Noise, IEEE Trans. Electromagnetic Compatibility, pp. 337-344, November 1981. [7] Peyton Z. Peebles, Jr. Probability, Random Variables and Random Signal Principles, pp. 381-382, McGrawHill 1993. [8] John M. Cioffi, Asymmetric Digital Subscriber Lines, in The CommunicationsHandbook by Jeny D. Gibson, pp. 450-479, CRC Press LLC, 1997. [9] Simon Haykin, Communication Systems, pp. 431-448, John Wiley & Sons, 2001. [lo] John G. Proakis, Digital Communications, pp. 61-62, McGraw-Hill, 1995.
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