Adaptive Linear Transversal Filter Frequency Domain Equalizer for Telephone Line Using the Widrow LMS Algorithm Atli Lemma 1 , Mesfin Belachew1 Summary
A local loop of a public switched telephone network (PSTN) has 3kHz bandwidth, which is insufficient for high-speed data transmission. Nevertheless, for traditional network operators, it has become of great interest to research on copper loops in view of aggressive competition with cable operators. Adaptive line equalization in time and frequency domains, are focus of research in developing digital subscriber loop (DSL) modems that effectively use existing copper. It is known that inter - symbol interference (ISI), inter - carrier interference (ICI), and additive white Gaussian noise (AWGN) are major problems that limit the performance of fast DSL, such as very high - speed DSL (VDSL), modems [3]. As a research on this problem, we present an adaptive linear transversal filter frequency domain equalizer for the audio band using the Widrow least mean square (LMS) algorithm (cf. [4][5]). We developed model for a typical local loop from transmission line characteristics and derived the adaptive equalizer that undo the effect of ISI and AWGN. The work is found to have laid the foundation for further endeavors on combating the effect of ISI, ICI, and AWGN in multi - tone signaling in copper loops. Introduction
A telephone local loop requires adaptive equalization to undo the severe effect of inter - symbol interference (ISI) due to long channel dispersion for fast data transmission [1][2]. Such a channel is characterized as a band-limited low pass filter [1]. A maximum likelihood sequence estimation (MLSE) equalizer is optimum, but, with complexity exponential in channel dispersion length [1][2]. If for instance, the symbol alphabet is ‘M’ and the number of overlapped symbols (ISI) is ‘L’, then the Viterbi detector computes ML+1 metrics for each received symbol [1]. Apart from that, design of optimum equalizer requires the a priori knowledge of spectral response (both magnitude and phase) of the channel with sufficient precision [1]. This, however, is difficult to assume or find as practical channels, like the telephone line, in general, have time - varying response particularly when high - speed communication is required [1]. There are sub-optimum methods, with linear transversal filter equalizer being most often used and considered in this work. Here, complexity is linear in dispersion length; hence becomes more feasible [1]. 1 Electrical Engineering Department, Faculty of Engineering, Bahir Dar University, Bahir Dar, Ethiopia, Email: fatli
[email protected];
[email protected]
A lot of research has been made on the criterion for optimizing the filter coefficients. It is known that the average probability of error is the most appropriate measure of performance to optimize the coefficients in a digital communication system. Nevertheless, the probability of error is a highly non - linear function of the equalizer coefficients [1]. There are two widely used optimizing criteria; peak distortion criterion and mean square error (MSE) criterion [1][2]. The peak distortion criterion minimizes the worst - case ISI at the equalizer output, while in the MSE criterion the equalizer coefficients are varied to minimize the mean square value of the error between successive computations. The mean square error (MSE) is the preferred optimizing criterion as it takes care of the effect of both ISI and AWGN [1]. An adaptive equalizer is needed to adjust time variations of channel characteristics. The Widrow LMS algorithm is an implementation of the MSE criterion and hence is ideal to track a slowly time varying channel, such as the telephone line. Telephone Channel Model
A local loop part of a telephone network of distributed resistance of R Ω/km and attenuation of a dB/km is considered. Also, series line inductance and shunt conductance is neglected to simplify the model computation. As a result of this assumption, the channel can be safely modeled as a resistance-capacitance transmission line filter. The attenuation for of the model is [2]: A( f ) =
pπ f RC
(1)
It is convenient to develop discrete - time models (from analog response) in the design of equalizers and other signal processors for band - limited channels that result ISI, such as the local loop of a public data network. Nevertheless, evaluating the performance of an equalizer for a discrete - time channel becomes difficult. The digital frequency (z) domain transfer function of the filter model of the local loop is given by: H (Z ) =
1 1 + τ ln(Z )
(2)
Where τ is the time constant of the resistance - capacitance (RC) low pass filter transmission line model of the channel. Now, the power series expansion of ln(z) in the neighborhood of the center Zo =1 + j0 can be found to be: ∞
ln(Z ) =
∑(
n=1
1)n
1)n
(Z
n
(3)
Then, replacing this power series in the channel model function above, we get an infinite impulse response (IIR) form of the channel model transfer function to be: H (Z ) =
1 1 + ∑∞ n=1 (
1)n τ (Z n1)
(4)
n
Channel response infinite length is not useful in many practical cases, and hence the first N terms are considered to get an N th order FIR filter form as: N
H (Z ) =
∑ h(n)Z
n
(5)
n=0
The random change in the channel response, h, can be simulated by introducing a random variable ξ through the following transformation: h¯ = h + ξ
(6)
ξ is a random variable that reflects random change in the channel response. It can take a probability distribution similar to that of the channel noise, for example. In this paper, we used pseudo random number of zero mean, Gaussian distributed with different variance values. Adaptive Linear Transversal Equalizer
Consider the following channel-equalizer arrangement n
v
r
K(Z)
H E(Z)
v’
Figure 1: Channel-equalizer Arrangement The use of Widrow LMS algorithm gives the adaptive linear transversal filter equalizer transfer function to be: HE (Z ) =
σ2ν H (Z 1 ) H (Z )H (Z 1)σ2ν + σ2n
=
∑Ni=0 di Z ∑2N j=0 c j Z
i j
(7)
Let us define an equivalent noise-to-signal power ratio as: ∆ [SNR℄
1
σσ2n 2
(8)
ν
Now, substituting pertinent variables in the equalizer function, we find the set of coefficients: ¯ di = h;
i = 0; 1; 2; ; N
8 j h >< ∑k 0 cj = >: ∑c2Nj j ;h k 0 =
( k +δk )(hN j+k +δN j+k )
SNR
( k +δk )(hk +δk )+1
=
(9)
SNR
;
;
j = 0; 1; 2; ; N 1 j = N + 1; ; 2N j=N
(10)
Figure 2: Spectral Response of a 4km Telephone Line: RC Module of Diameter 0.4mm, Resistance 2.28ohm/km, and Attenuation of 1.45dB/km at f=800Hz
Conclusion
Adaptive channel equalizer design for high - speed telephone line data transmission is a prominent research problem [1][2][3]. A linear equalizer yields good performance for channel with no spectral nulls as in the telephone channel [1]. However, they are not effective for channels with spectral zeros, like radio channels
Figure 3: Spectral Plot of Adaptive Equalizer at SNR=40dB
[1]. In this work, an N th order discrete-time linear filter model of a telephone copper based local loop is developed. An adaptive linear transversal filter equalizer that undo the severe effect time spread and noise on channels, is developed. The work produced a satisfactory performance as Figure 2 and 3 show. Robust model of the copper loop (channel) may be found when series inductance and shunt conductance of the copper line are considered. Finally, the basic limitation of linear equalizers to combat severe inter - symbol interference as in radio and band - limited channels used for high speed data traffic necessitates the investigation and development of nonlinear equalizers with low computational complexity [1]. Reference
1. John G. Proakis Digital Communications, 3rd McGraw-Hill International, Electrical Engineering Series, New York, U.S.A. 1995. 2. Atli Lemma, Adaptive Telephone Line Equalizer for Digital Transmission, M.Sc. Thesis, Addis Ababa University, Addis Ababa, Ethiopia, 2000. 3. Thierry Pollet and Miguel Peeters,Equalization for DMT - Based Broadband Modem, Alcatel, Marc Moonen, ESAT, Katholiek Universiteit Leuven; Luc Vandendorpe, TELE, Universite’ Catholique de Louvain, IEEE Communications Magazine, May 2000 Vol. 38 No. 5. 4. B. Widrow et al., Stationary and nonstationary learning characteristics of the LMS adaptive filter, Proc. IEEE, vol. 64, pp. 1151–1162, 1976. 5. S. C. Douglas and W. Pan. Exact expectation analysis of the LMS adaptive filter. IEEE Transactions on Signal Processing, 43(12):2863–2871, Dec.1995.
