Generalized channel impulse response shortening for ... - CiteSeerX
GENERALIZED CHANNEL IMPULSE RESPONSE SHORTENING FOR DISCRETE MULTITONE TRANSCEIVERS Bo Wangt, Tulay Adalzt, Qingli L i d , and Milan Vlajnid
t Department of Computer Science and Electrical Engineering University of Maryland, Baltimore County, Baltimore, MD 21250 SNortel Networks, Gaithersburg, MD 20878 ABSTRACT In Discrete Multitone (DMT) transceivers, a cyclic prefix, whose length is longer than that of the channel impulse response, is inserted between modulated symbols to avoid the intersymbol interference (ISI). To reduce the inefficiency due to the use of a long cyclic prefix, a finite impulse response filter, known as a time domain equalizer (TEQ), is used to shorten the effective channel impulse response. In addition, the TEQ can be used to suppress the stopband-noise and passband interference such as radio frequency interference (RFI) in DMT systems by changing the TEQ’s spectral shape. In this paper, we generalize two time-domain TEQ training methods by defining composite squared cost functions and derive the new algorithms for the TEQ which can jointly shorten the channel impulse response and suppress the stopband noise and passband interference. 1. INTRODUCTION
The Discrete Multitone (DMT) system has become an attractive technology for enabling delivery of a variety of multimedia services over the existing telephone networks. It has been chosen as the industry modulation standard for Asymmetrical ,Digital Subscriber Loop (ADSL) modems and is also a candidate modulation scheme for very-high-speed digital subscriber line (VDSL) systems which can provide much higher bit rates over shorter loops [l]. In the DMT system, the cyclic prefix provides a guard time between blocks. If the length of the channel response is not longer than that of the cyclic prefix, y, DMT symbols can be transmitted free of intersymbol interference (ISI). Using large y values to compensate for the length of the channel response, however, decreases the efficiency (introduces an overhead of y / ( N + y)) and increases latency. A time domain equalizer to shorten the effective channel impulse response has been the most popular equalization approach for DMT systems [2] - [6]. A major advantage of the DMT system is that its modulation and demodulation can be implemented by an efficient algorithm, the IFFT/FFT. However, due to its finite length, the FFT vector at the receiver establishes a spectral overlapped filter bank [7], [8]. Due t o the poor subchannel Research supported in part by Maryland Industrial Partnerships and Nortel Networks under grant number 2218.12.
isolation of this FFT filter bank, narrowband noise or interference can affect multiple neighboring subchannels. In the DMT systems for ADSL, there are two operation modes: echo-cancellation (EC) DMT and frequency division multiplexing (FDM) DMT. In FDM-based DMT system, upstream and downstream signals use different frequency bands. Because of the leakage effect of the FFT operation, the energy of the stopband noise and interference can be spread into passband subchannels and results in the decreasing of the SNRs for bins in the passband. In DMT-based VDSL system, even with relatively modest levels, narrowband interference such as radio frequency interference (RFI) can seriously degrade the performance of basic DMT transceiver as a result of the leakage effect 111. In [7], the authors propose the idea of suppressing the stopband noise and interference for FDM-based DMT-ADSL system by reducing the energy of the TEQ in the stopband. In addition, the TEQ can be used to effectively suppress the narrowband interference in DMT-based VDSL system by realizing a notch in the transfer function of the TEQ [l]. In this paper, we generalize the optimal and least-squares (LS) impulse response shortening methods in [5] by defining composite cost functions. We then derive the corresponding algorithms by minimizing the cost functions such that the channel impulse response can be shortened and the noise and interference can be suppressed. In the simulation part, we apply these two generalized response shortening algorithms to the FDM-based DMT-ADSL system to suppress the stopband noise and interference. It has been shown by simulations that these two generalized algorithms can effectively serve the two purposes of shortening the channel impulse response and suppressing the stopband noise and interference,
2. GENERALIZED OPTIMAL IMPULSE RESPONSE SHORTENING Assume the impulse response of the channel h,, and that of the TEQ a,, then the equalized impulse response can be represented as:
heqn= h,
(1) where * denotes the linear convolution. Suppose the lengths of h, and a, are N and M respectively, then the length of heqnis N M - 1. We would like t o force as much of the equalized channel response as possible t o lie in the y 1
+
276 0-7803-5700-0/99/$10.0001999IEEE
* a,
+
consecutive samples. Usually, the channel impulse response can not be perfectly shortened by the TEQ such that some residual energy of the shortened impulse response will lie outside the 7 + 1 consecutive taps with the highest total energy. We can use the shortening signal to noise ratio (SSNR) [5] to evaluate the performance of the TEQ. This SSNR is defined as the ratio of the energy in the largest consecutive 7 1 taps to the energy in the remaining taps of the equalized response. In [5), the authors propose the optimal TEQ training method in terms of SSNR ignoring the frequency-domain characteristics of the TEQ such as its spectral shape. However, the shape of the TEQ is also important to improve the overall performance such as the channel capacity [7], [9].We generalize the optimal impulse response shortening method in [5] such that both channel response shortening and selective band noise and interference suppression can be realized. For continuance, we adopt most of the notation in [5].We can rewrite Eqn. (1) as: -
+
hepnz H a = H
[71
Optimal shortening in terms of SSNR can be expressed as choosing a to minimize hZallhwoll while satisfying the constraint: hEinhwin= 1[5]. This constraint is used t o prevent the trivial solution, a = 0. The expression for the energy outside and inside the window can be expressed as:
T
T
T
ao
hEinhwin = a HwinHwina = a Ba (6) where D and B are symmetric and positive semidefinite matrices. Assume A ( k ) is the N-point DFT of the TEQ such that A ( k ) = aie-'2"ki/N where k is the index
aM-1
of frequency bins. To achieve both the channel response shortening and TEQ suppression at desired frequency bins, we define a composite cost function:
where H cad be expressed as:
zzil
k=O
= aTDa + aTSa = aT(D+ S)a
aTGa
(7)
where
...
0
0
hN-i
+
Let hwinrepresent a window of y 1 consecutive samples of heqnstarting from sample d , and assume hwollrepresent the remaining N M - 7 - 2 samples of &,,. With these defkiitions, we have:
Nl2
+
Pi
= CCOS(iWk)W(k), k=O
Wk
=
27rk N
-
hwinG Hwina where H,,
can be written as: hd
hd-i
hd+i
hd
hd+y
hd+y-l
... ... ..
*
and hwaii
where Hwollcan be expressed as:
hd-M+1 hd-M+2
hd+-r-M+l
Because D and F are both symmetric and positive semidefinite, so is matrix G. Then the generalized optimal TEQ problem can be posed as choosing a to minimize &,,t while satisfying the constraint aTBa = 1. Here, SC is the suppression coefficient, and Q is the normalized quantity which balances the channel response shortening and TEQ suppression at some desired frequency subchannels. W ( k ) is the weighting function which assumes the value 0 for unsuppressed bins and a positive value for suppressed subchannels. In [5],the length of the TEQ is assumed t o be less than that of the cyclic prefix (for long TEQ case, see [lo]). The rows of Hwjnis composed of shifted windows of the channel impulse response and in practical applications it has been observed that Hwin is full-rank [5). Hence the matrix B
277
is positive definite and can be decomposed using Cholesky Factorization [ll]into:
B
+
y 1, M is the length of the TEQ filter, and d is the delay of the target response. The ARMA model representation can be written as the standard difference equation:
= QAQ~ =(QA+)(QA~)~ z
F F ~
+
(10)
where A is a diagonal matrix with the eigenvalues of B as the diagonal entries, and the columns of Q are the orthonormal eigenvectors. Because B is of full rank, the matrix F-' exists. Define
y = FTa
(11)
yTy = aTFFTa= aTBa = 1
(12)
=
+ +
y(n) a l y ( n - 1) . . . a M - l y ( n - M boz(n - d ) blz(n - d - 1) + . , . +bL-lz(n - d - L 1) e ( n )
+
+ 1)
+ +
(17)
or from Fig. (l),the error sequence between the outputs of the target channel and the TEQ can be represented as:
then
Solving a in Eqn. (11) yields: a=
(
where ~
~
1
-
l
~
(13)
We find that
aTGa = yTF-'G(FT)-'y = yTCy (14) where We define C G F-'G(FT)-'. So, generalizedoptimal shortening c p be considered as choosing y to minimize yTCy with constraint yTy = 1. Then generalized optimal TEQ can be computed as [5] :
In order to suppress the noise and interference at certain frequency subchannels in the DMT system, we define the following composite cost function:
aopt= (FT)-'lmin (15) where lmin is the unit-length eigenvector corresponding to the minimum eigenvalue Amin of matrix C.
3. GENERALIZED LEAST-SQUARES
IMPULSE RESPONSE SHORTENING
N/2
&lS
w(~)IA(~)I'
= ~ [ e ' ( n ) l +0
where a is the normalized suppression coefficient which can balance the channel shortening and TEQ suppression at the selected frequency bins. Here we choose W ( k )as 0 for unsuppressed frequency bins and positive values for suppressed frequency subchannels. If we differentiate El, with respect to 0 and set the results to zero, we obtain the following set of linear equations:
+e=p where
* =[
Figure 1: LS method for channel response shortening
911
9 1 2
9 2 1
9 2 2
]
The least-squares (LS) approach in [5] uses the fact that the channel transfer function in the DMT system can be approximated by an autoregressive moving average (ARMA) model:
A TEQ whose transfer function is equal to A ( z - ' ) can be introduced a t the receiver side as shown in Fig. (l), which forces the effective channel to a sufficiently short target channel B ( z - ' ) . In Eqn. (16), L is the length of the target response where L should be less than or equal to
(21)
k=O
912
=-
R,,(d - M
. ..
+ 1)
*21
278
...
Rvz(d-1) R,,(d - 2)
R,=(d+L-2) R,,(d L - 3)
. . . R,,(d T
=912
+
+L -M )
Figure 2: Channel impulse response
Here, SC is the suppression coefficient. We assume that we have an estimate of the channel impulse response h, before the TEQ training, which can be identified directly by sending a pseudo-random sequence repeatedly 1121. If we assume
+
R z z ( k ) = E { s ( n k ) z ( n ) } = S,S(k) we can thenodirectly compute the components of
R:
R , z ( k ) = E { y ( n + k ) z ( n ) } = Szhk
(28)
N-1
&(k)
= E { y ( n + k ) y ( n ) } = Sz
Figure 3: Channel transfer function
(27)
hihi+&
(29)
i=O
Hence solution of the set of linear equations shown in Eqn. (22) yields the parameters of the TEQ such that the channel response is shortened and the energy of the TEQ at the selected frequency bins is suppressed. In our simulations, we use Gaussian elimination method to solve the set of linear equations. 4. SIMULATION RESULTS
In this section, we apply the two generalized impulse responseshartening algorithms to jointly shorten the channel impulse response and suppress the energy of the TEQ in the stopband in the FDM-based DMT system. The channel impulse and its squared transfer function shown in Figs. 2 and 3 are used for the two generalized shortening algorithms. For both cases, N equals to 512, the length of cylic prefix, 7 = 32, d = 18, L = 33 and M = 20. The starting and ending indices for stopband are 0 and 45,and the frequency bins 46 to 256 are considered as the passband. The starting and ending indices for suppression region (SR) are 0 and 39. In our case, we choose W ( k )= 1 for k = 0,. . . ,39, and 0 for the remaining frequency bins. Usually, it is impossible to completely suppress the TEQ's energy in the stopband. Some residual energy of the TEQ will lie outside the passband. In this paper, we define the passband energy to stopband energy ratio (PESER) as the measure for the stopband noise and interference suppression by the TEQ. Because we can remove the noise effect during the start-up training phase by averaging multiple received signal blocks,
we don't consider additive noise effect in the channel in our simulations. First, we run the generalized optimal response shortening algorithm to shorten the channel impulse response with suppression coefficient SC equal to 0.0 and 0.1 respectively. Note that SC equal to 0.0 case corresponds to the design for channel shortening only IS]. Fig. 4 shows the squared transfer functions of the two resultant TEQs by using the generalized optimal response shortening algorithm for SC equal to 0.0 and 0.1 respectively. We can see in Fig. 4 that when we introduce the noise suppression (SC equal to 0.1 in this case), the residual energy of the TEQ in the stopband decreases substantially. Table 1 shows the resultant SSNR and PESER values for the two different suppression coefficient values. As seen in Table 1, the joint shortening and suppression causes few dB loss in the SSNR but provides very significant energy suppression of the TEQ in the stopband.
Figure 4: TEQ spectral responses for generalized optimal shortening In the second part of the simulations, we apply the gen-
279
SC
SSNR (dB)
0.0 0.1
63.02 59.70
PESER (dB) 12.39 57.99
eralized LS response shortening algorithm to the same channel response shown in Fig. 2 with suppression coefficient equal to 0.0 and 0.1 respectively. Fig. 5 shows the squared transfer functions of the two resultant TEQs by using the generalized LS response shortening algorithm for SC equal to 0.0 and 0.1 respectively. We can see in Fig. 5 that when we introduce the noise suppression (SC equal to 0.1 in this case), the residual energy of the TEQ in the stopband decreases substantially. Table 2 shows the resultant SSNR and PESER values for the two different supression coefficient values. As seen in Table 2, the joint shortening and TEQ suppression at stopband gives up very little in the channel response shortening but provides a large advantage in the energy suppression of the TEQ in the stopband. It may be true that for the two generalized impulse response shortening algorithms, a little shorter TEQ can be used-if only shortening of the channel impulse response is needed. Hoyever, the stopband noise and interference rejection will significantly improve the performance of the receiver (71 and can relax the strict requirement of the analog bandpass filters at the transmitter and receiver.
Figure 5: TEQ spectral responses for generalized LS shortening
ing the stopband noise and interference. Compared to the generalized LS shortening, the generalized optimal shortening algorithm has higher computational complexity because of eigenvector calculation. We observe that the SSNR and PESER values depend on the delay of the target channel and the TEQ length for both algorithms. From Tables 1 and 2, we can see that when the stopband noise suppression is performed for the two algorithms with same delay of the target channel and length of the TEQ, the generalized LS method obtains better PESER but worse SSNR values compared to generalized optimal shortening algorithm. Also note in Fig. 4 that the resultant TEQs have deep nulls at certain frequency bins though the method is optimal in terms of SSNR, resulting in frequency response not desirable [9]. 6. REFERENCES [l] J. M. Cioffi, V. Oksman, J. J. Werner, T. Pollet, P. M. P. Spruyt, J. S. Chow and K. S. Jacobsen, “Very-high-speeddigi-
tal subscriber lines,“ in IEEE Communications Magazine, pp. 72-79,Apr. 1999. [2] J. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “Equalizer training algorithms for multicarrier modulation systems,” in Proc. Int. Conf. on Communications, pp. 761-765,(Geneva, Switzerland), May 1993. [3] N. Al-Dhahir, and J. M. Cioffi, “Efficiently computed reduced-parameter input-aided MMSE equalizers for ML detection: A unified approach,” IEEE %ns. Inform. Theory, vol. 42,no. 3, pp. 903-915,May 1996. [4]M. N&e, and A. Gather, “Time-domain equalizer training for ADSL”, in Proc. Int. Conf. on Communications, pp. 10851089, (Montreal, Canada), June 1997. [5] P.J. W. Melsa, R. C. Younce, and C. E. Rohrs, “Joint impulse response shortening for discrete multitone transceivers,” IEEE Thana. Communications, vol. 44,no. 12,pp. 1662-1672, Dec. 1996. [6] B. Wang, and T. Adali, “Joint impulse responses shortening for discrete multitone systems,” to appear in Proc. IEEE Globecom Symposium on Communication Theory, Rio de Janeiro, Brazil, Dec. 1999. [7] J. V. Kerckhove, and P. Spruyt, “Adapted optimization criterion for FDM-based DMT-ADSL equalization,” in Proc. Int. Conf. on Communications, pp. 1328-1334,(Dallas, TX), June 1996. [8] M. Webster, and R. Roberts, “Finite length equalization for FFT-based multicarrier systems - an error-whitening view-
SC 0.0
0.1
SSNR(dB) 59.61 56.23
PESER(dB) 12.31 60.49
~
5. CONCLUSIONS AND DISCUSSIONS
In this paper, we propose two generalized time-domain impulse response shortening algorithms to estimate the coefficients of the time domain equalizer of the DMT system such that the channel impulse response is shortened and the noise and interference at the selected frequency bins is suppressed. It has been shown by simulations that these
point,” in Proc. Asilomar Conf. on Signals, Systems and Computers, pp. 555-559,Nov. 1997. [9] B. Farhang-Boroujeny, and M. Ding, “An eigen-approach to the design of near-optimum time domain equalizer for DMT transceivers,” in Proc. Int. Conf. on Communications, (Vancouver, Canada), June 1999. [lo] C. Yin, and G. Yue, “Optimal impulse response shortening for discrete multitone transceivers,” Electronic Letters, vol. 34,no. 1, pp. 35-36,Jan. 1998. [ll] L.N. ’Itefethen, and D. Bau, 111, Numerical Linear Algebra. Siam, Philadelphia, PA, 1997. [12] J. A. C.Bingham, and F. van der Putten, Ti.413 Issue 2: Standards Project for Interfaces Relating to Carrier to Customer Connection of Asymmetrical Digital Subscriber Line (ADSL) Equipment, ANSI Document, No. TlE1.4/97-007R6, Sept. 1997.
280
t Department of Computer Science and Electrical Engineering University of Maryland, Baltimore County, Baltimore, MD 21250 SNortel Networks, Gaithersburg, MD 20878 ABSTRACT In Discrete Multitone (DMT) transceivers, a cyclic prefix, whose length is longer than that of the channel impulse response, is inserted between modulated symbols to avoid the intersymbol interference (ISI). To reduce the inefficiency due to the use of a long cyclic prefix, a finite impulse response filter, known as a time domain equalizer (TEQ), is used to shorten the effective channel impulse response. In addition, the TEQ can be used to suppress the stopband-noise and passband interference such as radio frequency interference (RFI) in DMT systems by changing the TEQ’s spectral shape. In this paper, we generalize two time-domain TEQ training methods by defining composite squared cost functions and derive the new algorithms for the TEQ which can jointly shorten the channel impulse response and suppress the stopband noise and passband interference. 1. INTRODUCTION
The Discrete Multitone (DMT) system has become an attractive technology for enabling delivery of a variety of multimedia services over the existing telephone networks. It has been chosen as the industry modulation standard for Asymmetrical ,Digital Subscriber Loop (ADSL) modems and is also a candidate modulation scheme for very-high-speed digital subscriber line (VDSL) systems which can provide much higher bit rates over shorter loops [l]. In the DMT system, the cyclic prefix provides a guard time between blocks. If the length of the channel response is not longer than that of the cyclic prefix, y, DMT symbols can be transmitted free of intersymbol interference (ISI). Using large y values to compensate for the length of the channel response, however, decreases the efficiency (introduces an overhead of y / ( N + y)) and increases latency. A time domain equalizer to shorten the effective channel impulse response has been the most popular equalization approach for DMT systems [2] - [6]. A major advantage of the DMT system is that its modulation and demodulation can be implemented by an efficient algorithm, the IFFT/FFT. However, due to its finite length, the FFT vector at the receiver establishes a spectral overlapped filter bank [7], [8]. Due t o the poor subchannel Research supported in part by Maryland Industrial Partnerships and Nortel Networks under grant number 2218.12.
isolation of this FFT filter bank, narrowband noise or interference can affect multiple neighboring subchannels. In the DMT systems for ADSL, there are two operation modes: echo-cancellation (EC) DMT and frequency division multiplexing (FDM) DMT. In FDM-based DMT system, upstream and downstream signals use different frequency bands. Because of the leakage effect of the FFT operation, the energy of the stopband noise and interference can be spread into passband subchannels and results in the decreasing of the SNRs for bins in the passband. In DMT-based VDSL system, even with relatively modest levels, narrowband interference such as radio frequency interference (RFI) can seriously degrade the performance of basic DMT transceiver as a result of the leakage effect 111. In [7], the authors propose the idea of suppressing the stopband noise and interference for FDM-based DMT-ADSL system by reducing the energy of the TEQ in the stopband. In addition, the TEQ can be used to effectively suppress the narrowband interference in DMT-based VDSL system by realizing a notch in the transfer function of the TEQ [l]. In this paper, we generalize the optimal and least-squares (LS) impulse response shortening methods in [5] by defining composite cost functions. We then derive the corresponding algorithms by minimizing the cost functions such that the channel impulse response can be shortened and the noise and interference can be suppressed. In the simulation part, we apply these two generalized response shortening algorithms to the FDM-based DMT-ADSL system to suppress the stopband noise and interference. It has been shown by simulations that these two generalized algorithms can effectively serve the two purposes of shortening the channel impulse response and suppressing the stopband noise and interference,
2. GENERALIZED OPTIMAL IMPULSE RESPONSE SHORTENING Assume the impulse response of the channel h,, and that of the TEQ a,, then the equalized impulse response can be represented as:
heqn= h,
(1) where * denotes the linear convolution. Suppose the lengths of h, and a, are N and M respectively, then the length of heqnis N M - 1. We would like t o force as much of the equalized channel response as possible t o lie in the y 1
+
276 0-7803-5700-0/99/$10.0001999IEEE
* a,
+
consecutive samples. Usually, the channel impulse response can not be perfectly shortened by the TEQ such that some residual energy of the shortened impulse response will lie outside the 7 + 1 consecutive taps with the highest total energy. We can use the shortening signal to noise ratio (SSNR) [5] to evaluate the performance of the TEQ. This SSNR is defined as the ratio of the energy in the largest consecutive 7 1 taps to the energy in the remaining taps of the equalized response. In [5), the authors propose the optimal TEQ training method in terms of SSNR ignoring the frequency-domain characteristics of the TEQ such as its spectral shape. However, the shape of the TEQ is also important to improve the overall performance such as the channel capacity [7], [9].We generalize the optimal impulse response shortening method in [5] such that both channel response shortening and selective band noise and interference suppression can be realized. For continuance, we adopt most of the notation in [5].We can rewrite Eqn. (1) as: -
+
hepnz H a = H
[71
Optimal shortening in terms of SSNR can be expressed as choosing a to minimize hZallhwoll while satisfying the constraint: hEinhwin= 1[5]. This constraint is used t o prevent the trivial solution, a = 0. The expression for the energy outside and inside the window can be expressed as:
T
T
T
ao
hEinhwin = a HwinHwina = a Ba (6) where D and B are symmetric and positive semidefinite matrices. Assume A ( k ) is the N-point DFT of the TEQ such that A ( k ) = aie-'2"ki/N where k is the index
aM-1
of frequency bins. To achieve both the channel response shortening and TEQ suppression at desired frequency bins, we define a composite cost function:
where H cad be expressed as:
zzil
k=O
= aTDa + aTSa = aT(D+ S)a
aTGa
(7)
where
...
0
0
hN-i
+
Let hwinrepresent a window of y 1 consecutive samples of heqnstarting from sample d , and assume hwollrepresent the remaining N M - 7 - 2 samples of &,,. With these defkiitions, we have:
Nl2
+
Pi
= CCOS(iWk)W(k), k=O
Wk
=
27rk N
-
hwinG Hwina where H,,
can be written as: hd
hd-i
hd+i
hd
hd+y
hd+y-l
... ... ..
*
and hwaii
where Hwollcan be expressed as:
hd-M+1 hd-M+2
hd+-r-M+l
Because D and F are both symmetric and positive semidefinite, so is matrix G. Then the generalized optimal TEQ problem can be posed as choosing a to minimize &,,t while satisfying the constraint aTBa = 1. Here, SC is the suppression coefficient, and Q is the normalized quantity which balances the channel response shortening and TEQ suppression at some desired frequency subchannels. W ( k ) is the weighting function which assumes the value 0 for unsuppressed bins and a positive value for suppressed subchannels. In [5],the length of the TEQ is assumed t o be less than that of the cyclic prefix (for long TEQ case, see [lo]). The rows of Hwjnis composed of shifted windows of the channel impulse response and in practical applications it has been observed that Hwin is full-rank [5). Hence the matrix B
277
is positive definite and can be decomposed using Cholesky Factorization [ll]into:
B
+
y 1, M is the length of the TEQ filter, and d is the delay of the target response. The ARMA model representation can be written as the standard difference equation:
= QAQ~ =(QA+)(QA~)~ z
F F ~
+
(10)
where A is a diagonal matrix with the eigenvalues of B as the diagonal entries, and the columns of Q are the orthonormal eigenvectors. Because B is of full rank, the matrix F-' exists. Define
y = FTa
(11)
yTy = aTFFTa= aTBa = 1
(12)
=
+ +
y(n) a l y ( n - 1) . . . a M - l y ( n - M boz(n - d ) blz(n - d - 1) + . , . +bL-lz(n - d - L 1) e ( n )
+
+ 1)
+ +
(17)
or from Fig. (l),the error sequence between the outputs of the target channel and the TEQ can be represented as:
then
Solving a in Eqn. (11) yields: a=
(
where ~
~
1
-
l
~
(13)
We find that
aTGa = yTF-'G(FT)-'y = yTCy (14) where We define C G F-'G(FT)-'. So, generalizedoptimal shortening c p be considered as choosing y to minimize yTCy with constraint yTy = 1. Then generalized optimal TEQ can be computed as [5] :
In order to suppress the noise and interference at certain frequency subchannels in the DMT system, we define the following composite cost function:
aopt= (FT)-'lmin (15) where lmin is the unit-length eigenvector corresponding to the minimum eigenvalue Amin of matrix C.
3. GENERALIZED LEAST-SQUARES
IMPULSE RESPONSE SHORTENING
N/2
&lS
w(~)IA(~)I'
= ~ [ e ' ( n ) l +0
where a is the normalized suppression coefficient which can balance the channel shortening and TEQ suppression at the selected frequency bins. Here we choose W ( k )as 0 for unsuppressed frequency bins and positive values for suppressed frequency subchannels. If we differentiate El, with respect to 0 and set the results to zero, we obtain the following set of linear equations:
+e=p where
* =[
Figure 1: LS method for channel response shortening
911
9 1 2
9 2 1
9 2 2
]
The least-squares (LS) approach in [5] uses the fact that the channel transfer function in the DMT system can be approximated by an autoregressive moving average (ARMA) model:
A TEQ whose transfer function is equal to A ( z - ' ) can be introduced a t the receiver side as shown in Fig. (l), which forces the effective channel to a sufficiently short target channel B ( z - ' ) . In Eqn. (16), L is the length of the target response where L should be less than or equal to
(21)
k=O
912
=-
R,,(d - M
. ..
+ 1)
*21
278
...
Rvz(d-1) R,,(d - 2)
R,=(d+L-2) R,,(d L - 3)
. . . R,,(d T
=912
+
+L -M )
Figure 2: Channel impulse response
Here, SC is the suppression coefficient. We assume that we have an estimate of the channel impulse response h, before the TEQ training, which can be identified directly by sending a pseudo-random sequence repeatedly 1121. If we assume
+
R z z ( k ) = E { s ( n k ) z ( n ) } = S,S(k) we can thenodirectly compute the components of
R:
R , z ( k ) = E { y ( n + k ) z ( n ) } = Szhk
(28)
N-1
&(k)
= E { y ( n + k ) y ( n ) } = Sz
Figure 3: Channel transfer function
(27)
hihi+&
(29)
i=O
Hence solution of the set of linear equations shown in Eqn. (22) yields the parameters of the TEQ such that the channel response is shortened and the energy of the TEQ at the selected frequency bins is suppressed. In our simulations, we use Gaussian elimination method to solve the set of linear equations. 4. SIMULATION RESULTS
In this section, we apply the two generalized impulse responseshartening algorithms to jointly shorten the channel impulse response and suppress the energy of the TEQ in the stopband in the FDM-based DMT system. The channel impulse and its squared transfer function shown in Figs. 2 and 3 are used for the two generalized shortening algorithms. For both cases, N equals to 512, the length of cylic prefix, 7 = 32, d = 18, L = 33 and M = 20. The starting and ending indices for stopband are 0 and 45,and the frequency bins 46 to 256 are considered as the passband. The starting and ending indices for suppression region (SR) are 0 and 39. In our case, we choose W ( k )= 1 for k = 0,. . . ,39, and 0 for the remaining frequency bins. Usually, it is impossible to completely suppress the TEQ's energy in the stopband. Some residual energy of the TEQ will lie outside the passband. In this paper, we define the passband energy to stopband energy ratio (PESER) as the measure for the stopband noise and interference suppression by the TEQ. Because we can remove the noise effect during the start-up training phase by averaging multiple received signal blocks,
we don't consider additive noise effect in the channel in our simulations. First, we run the generalized optimal response shortening algorithm to shorten the channel impulse response with suppression coefficient SC equal to 0.0 and 0.1 respectively. Note that SC equal to 0.0 case corresponds to the design for channel shortening only IS]. Fig. 4 shows the squared transfer functions of the two resultant TEQs by using the generalized optimal response shortening algorithm for SC equal to 0.0 and 0.1 respectively. We can see in Fig. 4 that when we introduce the noise suppression (SC equal to 0.1 in this case), the residual energy of the TEQ in the stopband decreases substantially. Table 1 shows the resultant SSNR and PESER values for the two different suppression coefficient values. As seen in Table 1, the joint shortening and suppression causes few dB loss in the SSNR but provides very significant energy suppression of the TEQ in the stopband.
Figure 4: TEQ spectral responses for generalized optimal shortening In the second part of the simulations, we apply the gen-
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SC
SSNR (dB)
0.0 0.1
63.02 59.70
PESER (dB) 12.39 57.99
eralized LS response shortening algorithm to the same channel response shown in Fig. 2 with suppression coefficient equal to 0.0 and 0.1 respectively. Fig. 5 shows the squared transfer functions of the two resultant TEQs by using the generalized LS response shortening algorithm for SC equal to 0.0 and 0.1 respectively. We can see in Fig. 5 that when we introduce the noise suppression (SC equal to 0.1 in this case), the residual energy of the TEQ in the stopband decreases substantially. Table 2 shows the resultant SSNR and PESER values for the two different supression coefficient values. As seen in Table 2, the joint shortening and TEQ suppression at stopband gives up very little in the channel response shortening but provides a large advantage in the energy suppression of the TEQ in the stopband. It may be true that for the two generalized impulse response shortening algorithms, a little shorter TEQ can be used-if only shortening of the channel impulse response is needed. Hoyever, the stopband noise and interference rejection will significantly improve the performance of the receiver (71 and can relax the strict requirement of the analog bandpass filters at the transmitter and receiver.
Figure 5: TEQ spectral responses for generalized LS shortening
ing the stopband noise and interference. Compared to the generalized LS shortening, the generalized optimal shortening algorithm has higher computational complexity because of eigenvector calculation. We observe that the SSNR and PESER values depend on the delay of the target channel and the TEQ length for both algorithms. From Tables 1 and 2, we can see that when the stopband noise suppression is performed for the two algorithms with same delay of the target channel and length of the TEQ, the generalized LS method obtains better PESER but worse SSNR values compared to generalized optimal shortening algorithm. Also note in Fig. 4 that the resultant TEQs have deep nulls at certain frequency bins though the method is optimal in terms of SSNR, resulting in frequency response not desirable [9]. 6. REFERENCES [l] J. M. Cioffi, V. Oksman, J. J. Werner, T. Pollet, P. M. P. Spruyt, J. S. Chow and K. S. Jacobsen, “Very-high-speeddigi-
tal subscriber lines,“ in IEEE Communications Magazine, pp. 72-79,Apr. 1999. [2] J. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “Equalizer training algorithms for multicarrier modulation systems,” in Proc. Int. Conf. on Communications, pp. 761-765,(Geneva, Switzerland), May 1993. [3] N. Al-Dhahir, and J. M. Cioffi, “Efficiently computed reduced-parameter input-aided MMSE equalizers for ML detection: A unified approach,” IEEE %ns. Inform. Theory, vol. 42,no. 3, pp. 903-915,May 1996. [4]M. N&e, and A. Gather, “Time-domain equalizer training for ADSL”, in Proc. Int. Conf. on Communications, pp. 10851089, (Montreal, Canada), June 1997. [5] P.J. W. Melsa, R. C. Younce, and C. E. Rohrs, “Joint impulse response shortening for discrete multitone transceivers,” IEEE Thana. Communications, vol. 44,no. 12,pp. 1662-1672, Dec. 1996. [6] B. Wang, and T. Adali, “Joint impulse responses shortening for discrete multitone systems,” to appear in Proc. IEEE Globecom Symposium on Communication Theory, Rio de Janeiro, Brazil, Dec. 1999. [7] J. V. Kerckhove, and P. Spruyt, “Adapted optimization criterion for FDM-based DMT-ADSL equalization,” in Proc. Int. Conf. on Communications, pp. 1328-1334,(Dallas, TX), June 1996. [8] M. Webster, and R. Roberts, “Finite length equalization for FFT-based multicarrier systems - an error-whitening view-
SC 0.0
0.1
SSNR(dB) 59.61 56.23
PESER(dB) 12.31 60.49
~
5. CONCLUSIONS AND DISCUSSIONS
In this paper, we propose two generalized time-domain impulse response shortening algorithms to estimate the coefficients of the time domain equalizer of the DMT system such that the channel impulse response is shortened and the noise and interference at the selected frequency bins is suppressed. It has been shown by simulations that these
point,” in Proc. Asilomar Conf. on Signals, Systems and Computers, pp. 555-559,Nov. 1997. [9] B. Farhang-Boroujeny, and M. Ding, “An eigen-approach to the design of near-optimum time domain equalizer for DMT transceivers,” in Proc. Int. Conf. on Communications, (Vancouver, Canada), June 1999. [lo] C. Yin, and G. Yue, “Optimal impulse response shortening for discrete multitone transceivers,” Electronic Letters, vol. 34,no. 1, pp. 35-36,Jan. 1998. [ll] L.N. ’Itefethen, and D. Bau, 111, Numerical Linear Algebra. Siam, Philadelphia, PA, 1997. [12] J. A. C.Bingham, and F. van der Putten, Ti.413 Issue 2: Standards Project for Interfaces Relating to Carrier to Customer Connection of Asymmetrical Digital Subscriber Line (ADSL) Equipment, ANSI Document, No. TlE1.4/97-007R6, Sept. 1997.
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