20 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 38, NO. 1, JANUARY 1991 Reduced Complexity Echo Cancellation Using Orthonormal Functions Gordon W. Davidson and David D. Falconer, Fellow, ZEEE Absfruct -To reduce the echo canceler complexity associated with the cancellations of subscriber loop echo responses with long tails, we propose and investigate a hvo-stage digital echo canceler, in which the second stage is an adaptively weighted combination of orthonormal IIR responses. Satisfactory performance (70-dB cancellation) was found for a variety of loops at 80 and 320 kbaud symbol rates with 15 such IIR responses, based on Laguerre functions. The first echo canceler stage is a conventional transversal filter with 20-40 tap coefficients. Substantial complexity reduction was achieved by the new structure. Adaptation convergence was also analyzed and simulated. I. INTRODUCTION CHO cancellation is a technique used to achieve full E duplex transmission on twisted pair subscriber loops. The echo canceler is an adaptive filter that attempts to match the impulse response of the echo path, so that a replica of the echo can be subtracted from the received signal [ll. Because the far end data can be attenuated up to about 45 dB by the channel, a reduction in echo power of as much as 60-70 dB may be required. The simplest form of the echo canceler is a transversal filter, with tap spacing equal to the symbol interval. Accord- ing to the North American standard for ISDN basic access, the line rate is 160 kbaud, with four-level PAM signaling (2BlQ) at an 80-kbaud symbol rate. However, there is also the possibility of very high bit rates, such as one of 800 kbaud that has recently been proposed [2]. To achieve the required accuracy in cancellation without prefiltering, the transversal filter impulse response must span several hundred symbol intervals, for 80 kbaud signaling, and over a thousand inter- vals for the higher rates. This amounts to a large computa- tional complexity for the echo canceler, making implementa- tion difficult. Previously, l-D or high-pass filtering has been used to shorten the echo impulse response, but this increases the effect of noise and crosstalk, especially for high bit rates. This paper considers a method of reducing the complexity of a digital echo canceler by breaking it into two stages. The first stage is a transversal filter that spans the first few symbol intervals of the echo impulse response, and the Manuscript received December 18, 1989; revised May 30, 1990. This work was supported by a grant from TRIO (Telecommunications Re- search Institute of Canada). This paper was recommended by Associate Editor Y. F. Huang. G. W. Davidson was with Ottawa-Carleton Center for Communica- tions Research, Carleton University, Ottawa, Ont., Canada. He is now with the Department of Electrical Engineering, University of British Columbia, Vancouver, B.C., Canada. D. D. Falconer is with the Department of Systems and Computer Engineering, Ottawa-Carleton Center for Communications Research, Carleton University, Ottawa, Ont., Canada K1S 5B6. IEEE Log Number 9039218. second stage approximates the remainder or tail, of the response by a linear combination of orthonormal functions. These functions are chosen so that they can be implemented by simple recursive filters. Thus, if the number of orthonor- mal functions required in the approximation is not large, a significant reduction in computational complexity can be achieved. Our choice of a fixed set of orthonormal functions differs from approaches using IIR filters with adaptice poles [3], [4], and results in a stable, simpler adaptation algorithm. Our approach is also more general and should yield a better performance/complexity tradeoff than previously reported IIR echo cancelers with a single fixed pole [51-[81. The reason for the two-stage approach is the shape of a typical transhybrid echo response such as that of Fig. 1. The first part of the response has a rapid time variation, while the tail of the response is slowly decaying toward zero. Thus the tail may be well approximated by combining a few simple IIR responses, whereas a conventional FIR (transversal) filter representation would involve far more parameters. Fig. 2 shows an example of a fully digital echo canceler consisting of two stages, the first being a conventional transversal and/or memory canceler spanning say 10-15 baud intervals, and the second stage being a linear combination of digital IIR filter sections. The two-stage strategy can also be employed in an analog data echo canceler placed before the digital receiver’s A/D converter. In this case the analog “ pre-canceler” removes most of the interfering echo to reduce the resolution re- quired of the A/D converter [9]. 11. DIGITAL TWO-STAGE CANCELER Consider a sampled received signal of the form CO y, = hjdi-, + ni j=O where di near end data symbols, yi received signal, hi echo impulse response, ni noise and far end data. Symbol rate sampling is assumed; the extension to sampling at a multiple of the symbol rate is done through interleaving [l]. Ideally, the response of the echo canceler should match the actual echo response, {hi}, and the output of the echo canceler would then equal the summation in (1). n- 0098-4094/91/0100-0020$01.00 0 1991 IEEE
Transcript
20 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 38, NO. 1, JANUARY 1991
Reduced Complexity Echo Cancellation Using Orthonormal Functions
Gordon W. Davidson and David D. Falconer, Fellow, ZEEE
Absfruct -To reduce the echo canceler complexity associated with the cancellations of subscriber loop echo responses with long tails, we propose and investigate a hvo-stage digital echo canceler, in which the second stage is an adaptively weighted combination of orthonormal IIR responses. Satisfactory performance (70-dB cancellation) was found for a variety of loops at 80 and 320 kbaud symbol rates with 15 such IIR responses, based on Laguerre functions. The first echo canceler stage is a conventional transversal filter with 20-40 tap coefficients. Substantial complexity reduction was achieved by the new structure. Adaptation convergence was also analyzed and simulated.
I . INTRODUCTION CHO cancellation is a technique used to achieve full E duplex transmission on twisted pair subscriber loops.
The echo canceler is an adaptive filter that attempts to match the impulse response of the echo path, so that a replica of the echo can be subtracted from the received signal [ll. Because the far end data can be attenuated up to about 45 dB by the channel, a reduction in echo power of as much as 60-70 dB may be required.
The simplest form of the echo canceler is a transversal filter, with tap spacing equal to the symbol interval. Accord- ing to the North American standard for ISDN basic access, the line rate is 160 kbaud, with four-level PAM signaling (2BlQ) at an 80-kbaud symbol rate. However, there is also the possibility of very high bit rates, such as one of 800 kbaud that has recently been proposed [2]. To achieve the required accuracy in cancellation without prefiltering, the transversal filter impulse response must span several hundred symbol intervals, for 80 kbaud signaling, and over a thousand inter- vals for the higher rates. This amounts to a large computa- tional complexity for the echo canceler, making implementa- tion difficult. Previously, l -D or high-pass filtering has been used to shorten the echo impulse response, but this increases the effect of noise and crosstalk, especially for high bit rates.
This paper considers a method of reducing the complexity of a digital echo canceler by breaking it into two stages. The first stage is a transversal filter that spans the first few symbol intervals of the echo impulse response, and the
Manuscript received December 18, 1989; revised May 30, 1990. This work was supported by a grant from TRIO (Telecommunications Re- search Institute of Canada). This paper was recommended by Associate Editor Y. F. Huang.
G . W. Davidson was with Ottawa-Carleton Center for Communica- tions Research, Carleton University, Ottawa, Ont., Canada. He is now with the Department of Electrical Engineering, University of British Columbia, Vancouver, B.C., Canada.
D. D. Falconer is with the Department of Systems and Computer Engineering, Ottawa-Carleton Center for Communications Research, Carleton University, Ottawa, Ont., Canada K1S 5B6.
IEEE Log Number 9039218.
second stage approximates the remainder or tail, of the response by a linear combination of orthonormal functions. These functions are chosen so that they can be implemented by simple recursive filters. Thus, if the number of orthonor- mal functions required in the approximation is not large, a significant reduction in computational complexity can be achieved.
Our choice of a fixed set of orthonormal functions differs from approaches using IIR filters with adaptice poles [3], [4], and results in a stable, simpler adaptation algorithm. Our approach is also more general and should yield a better performance/complexity tradeoff than previously reported IIR echo cancelers with a single fixed pole [51-[81.
The reason for the two-stage approach is the shape of a typical transhybrid echo response such as that of Fig. 1. The first part of the response has a rapid time variation, while the tail of the response is slowly decaying toward zero. Thus the tail may be well approximated by combining a few simple IIR responses, whereas a conventional FIR (transversal) filter representation would involve far more parameters. Fig. 2 shows an example of a fully digital echo canceler consisting of two stages, the first being a conventional transversal and/or memory canceler spanning say 10-15 baud intervals, and the second stage being a linear combination of digital IIR filter sections.
The two-stage strategy can also be employed in an analog data echo canceler placed before the digital receiver’s A /D converter. In this case the analog “ pre-canceler” removes most of the interfering echo to reduce the resolution re- quired of the A/D converter [9].
11. DIGITAL TWO-STAGE CANCELER Consider a sampled received signal of the form
CO
y , = hjdi- , + ni j = O
where
di near end data symbols, y i received signal, hi echo impulse response, ni noise and far end data.
Symbol rate sampling is assumed; the extension to sampling at a multiple of the symbol rate is done through interleaving [l]. Ideally, the response of the echo canceler should match the actual echo response, {hi}, and the output of the echo canceler would then equal the summation in (1).
n-
0098-4094/91/0100-0020$01.00 0 1991 IEEE
. . I . DAVIDSON AND FALCONER: REDUCED COMPLEXITY ECHO CANCELLATION
xi05
r i (s) Fig. 1. Typical echo response (for Bellcore loop #9).
d i
I Transmit
I
bo?
Fig. 2. Two-stage echo canceler. { L ; ) are IIR filters.
We consider an echo canceler with a set of Nb parame- ters: {bj, j = 0,1,. . . , Nb - I}, approximating the first Nb of {hi}, and another set of Nu parameters tu j , j = 0,l; . . ,Nu - 1) which are coefficients of Nu orthonormal functions used to approximate the tail portion of the echo (following the first Nb symbol intervals). The canceler's output is
N,-l No - 1
t i= + u j x j i ( 2 ) j = O j = O
where m
X i ; = d i - k - N , L j ( k ) , J = o , l , " ' , N u - l ( 3 ) k = O
and {L j (k ) , k = 0,1, * . . m} is the impulse response of the jth IIR filter, with the orthonormal property
m
, f n . k = O (4)
The output zi can also be represented in an obvious way, using N,-dimensional vectors b and di and Nu-dimensional vectors a and x i :
21
While (2) and (3) involve only a set of Nu orthonormal filter responses {L j (k ) } , the echo signal in (1) can be repre- sented in the same way in general by extending the number of orthonormal responses to infinity:
Nb-1 m
y i = c j x j i + ni (6) j = O j = O
where m
c j = L j ( k ) h k + N , , * (7) k = O
Implicit in (6) and (7) is the representation of the echo response tail as a linear combination of orthonormal wave- forms [lo],
m
h k + N h = C j L j ( k ) , k = 0 , 1 , " ' , m . (8) j = O
Representing the first Nb samples of the echo response by a vector h, and the first Nu values of {c j } by a vector c, we can rewrite (6) as
yi = hTd; + CTX, + vi + n, (9)
where m
( 10) v i = C . X . .
is the portion of the echo excluded by the approximation of the tail by Nu terms in (2).
In the following analysis it is assumed that the data sym- bols Idi} are uncorrelated, with variance U:. Thus since the filter responses { L k } are orthonormal, their outputs { x k ; } are also uncorrelated and have the same variance. Furthermore, it is assumed that { d J , and hence { x k i } , are uncorrelated with the noise and far end data, {ni l .
The error after subtracting the echo canceler output from the received signal is
(11)
I 11 j =Na
e . 1 1 1 1 = y . - z . = dT(h - b ) + xT(c - a ) + vi + n i .
I l l I1
22
cancellation = - lolog
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 38, NO. 1 , JANUARY 1991
c ( h j + N h - h e , ) '
C hi' \ I
(17) j = O
w-1
j = O
This gives the following mean squared error (MSE) ex- pression:
E(.') = Ih - bI2u; + IC - u12u; +U: +U: (12)
where
ad' = E@?); u,z = E ( n ; ) (13)
u:=E ( y 2 i ) - u d - 2 C ck2. (14)
and m
k = N,
This is minimized with respect to U and b by setting U = c and b = h, and the resulting minimum MSE (MMSE) is
MMSE = + (15)
111. EVALUATION OF ORTHONORMAL FUNCTIONS FOR DIGITAL CANCELER
The purpose of approximating the echo tail using or- thonormal functions is a reduction in complexity, and this can only be accomplished if the number of terms required in the expansion (Nu) is not large. This depends on the set of orthonormal functions used, the echo impulse response it- self, and the number of symbol intervals before the start of the echo tail ( N J . In general, the echo impulse response depends on the length and gauge of the line, and the number of bridged taps, etc. In this paper, for a given line configura- tion, echo impulse response samples were generated numeri- cally and used to determine the effectiveness of possible sets of orthonormal functions. The echo responses were com- puted for "Bellcore" loops specified in [ll]. Each response included hybrids with 50 mH transformers 1-gF dc-blocking capacitors, and no balance network, and also a 100% raised cosine transmit/receive filter combination.
As mentioned earlier, a reduction in echo power of as much as 70 dB is required.' As a performance criterion, then, it would be desirable to obtain the total echo power before and after cancellation. Assuming perfect cancellation by the first stage, these would be defined as
m
respectively, where
N o - 1
h e j = C k L k ( j ) , j = o , l ; * * , w . (16) k = O
Note, however, that it was necessary to work with finite length sequences, so that the echo impulse response had to be truncated to, say, W symbol intervals. Thus, it was impor- tant to ensure that the neglected echo power due to trunca- tion, q = w h f , was very small compared to the residual echo power. For the generated responses, a 70-dB cancellation implied a residual of about 10-l2. It was observed that at W = 750 for 80 kbaud, h& was about so that it was reasonable to assume that the sum of terms following this
'This is a conservative requirement to allow margin for implementa- tion and system parameter uncertainties. A more realistic requirement for high bit rate systems may be on the order of 60 dB.
TABLE I CANCELLATION USING LAGUERRE
AND EXPONENTIALS BEFOR E
Nu = 8
Cancellation (dB)
LOOP #9 AT 80 KBUAD,
30 59.3 50.9 50 61.94 53.6
67.7 57.2 90 73.2 61.9
I I - 1
. . .
DAVIDSON AND FALCONER: REDUCED COMPLEXITY ECHO CANCELLATION
1 2 3 4 5 6 7 8 9 10 11 12
that of the exponentials. Therefore, in the remainder of this paper only the Laguerre functions are considered. Secondly, the cancellation improves as Nb increases. This is because the rate of change of the echo impulse response decreases with time, and the larger the rate of change, the greater the error resulting from using a finite number of terms in the orthonormal expansion. Finally, for Nu = 8, a cancellation of 70 dB was only achieved when Nb was greater than or equal to 90. Thus for a smaller first stage of the echo canceler, more orthonormal functions are needed in the approxima- tion of the echo tail.
In the next part of the investigation, it was desired to find the cancellation that was possible with Laguerre functions and larger values of Nu. Fortunately, these could be gener- ated using a recursive formula [lo] that did not suffer the numerical difficulties as did direct applications of Gram-Schmidt orthogonalization. Here the first and second orthogonal functions are
I,( k ) = e - k p (20a)
I , ( k ) = ( k - a 1 ) M k ) (20b)
c kl"(kI2
c l 0 (k l2
w-1
(21) k = O
a1= w-1
k = O
and for n 2 2,
M k ) = ( k - a n ) L l ( k ) - P n L z ( k ) (22) w-1 w-1
k = O k = O c kl,- l(k)2
c 4-m2 c k - * ( k I 2
c kin - 1( k )I, - Z(k ) . (23) a, = w-1 ; P,= w-1
k = O k = O
The orthonormal functions (Ln(k)} are then found by nor- malization.
How cancellation varies with N, for a given Nb, is shown in Table 11 for Nb = 20. In this case 11 orthonormal functions are needed to achieve 70 dB.
In order to get a sample of a wide range of subscriber loops, echo impulse responses were generated for the lines summarized in Table 111, and the impulse responses are shown collectively in Fig. 3.
Finally, in addition to the 80 kbaud symbol rate that has been assumed so far, cancellation results were found for a 320 kbaud symbol rate. This was done to determine how the size of the first or second stage of the echo canceler might have to change for higher rates, indicating the possible saving in complexity as a function of baud rate.
The results are presented in Table IV for 80 kbaud and Table V for 320 kbaud. For a given line and value of Nb, the value of N, required to achieve 70-dB cancellation is shown. Only values of Nu up to 15 were considered, so if more than this was required the entry is left blank. Also indicated is the value of p , the parameter in the Laguerre functions, which has been optimized for the particular line, Nb and Nu.
It is seen that for a given symbol rate and an Nb that is not too small, N, and p are not greatly affected by the presence or arrangement of bridged taps, yet seem to be functions of the length of the line. Since the lines considered way in
Fig. 3. Echo impulse responses for the lines considered. (Symbol rate = 80 kbaud.) The dashed response differing from the others below 20 symbol intervals is that of line 6 (1500 ft), while the solid response differing below 20 symbol intervals is that of line 5 (5000 ft).
80
70
s 'f 60 1 0
50
0 0.01 0.02 0.03 0.04 00s 0.06 0.07 0.08 0.09 0.1
P
30
Fig. 4. Cancellation versus p ; 80 kbaud, Nb = 20, N, = 13.
TABLE I1 CANCELLATION USING
VERSUS N, LINE 1, Nh = 20,80 KBAUD
N, I Cancellation (dB)
24 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 38, NO. 1, JANUARY 1991
Line 1
P Na 0.0724 15 0.055 11 0.0501 10 0.0437 8
LINE 1: Line 2: Line 3: Line 4: Line 5: Line 6:
Line 2 Line 3 Line 4 Line 5 Line 6 P Na P Na P Na P Na P Na
TABLE V REQUIRED No FOR 70 DB CANCELLATION 320 KBAUD
I Line 1 1 Line 2 I Line 3 1 Line 4 1 Line 5 1 Line 6
length from 1500 to 18000 ft the results should give a good indication of the Na and p required to achieve 70 dB cancellation on a wide range of subscriber loops.
Assuming p is optimized, the values of Nb and Na for an echo canceler could be chosen from the table to minimize complexity. As will be discussed later, the complexity, as measured by the number of equivalent 2-b multiplications per symbol interval, is about 2Nb +2ON,. Thus minimizing complexity in the case of 80 kbuad, avoiding the smallest value of Nb, would determine Nb = 20 and Nu= 13. Simi- larly, for 320 kbuad the values would be Nb = 40 and Nu = 14. However, another important factor is the sensitivity of the cancellation to p, and this will affect the required value of
Fig. 4 shows cancellation versus p for each of the six lines, at 80 kbuad and for Nb = 20, Na = 13. 70 dB cancellation is achieved for all lines when p is between 0.0425 and 0.045. Note that p is most restricted by the cancellation for line 6, the 1500 ft line. But a shorter line results in less attenuation of the far end signal, so that far less than 70 dB cancellation is required. Thus in practice the allowed variation in p would be greater than that indicated in the figures. To
Na .
for p is almost too small to be discerned from the graph. In Fig. 8, however, where Nb = 40 and Na = 15, 70-dB cancella- tion is obtained for values of p from 0.012 to 0.015. In general, higher symbol rates imply smaller values of p and greater required accuracy.
Finally, note that the optimum p at 80 kbaud is about four times the optimum p for the corresponding Nb and N, at 320 kbaud. Thus given the size of the echo canceler, if the baud rate is changed, the value of p may be adjusted by simple time scaling.
IV. ADAPTATION
In the echo canceler of Fig. 2, the coefficient vectors b and a still depend on the particular line, so for the echo canceler to be generally useful, it must be possible to adapt b and a, to their optimum values. Using the LMS algorithm, the coefficients are updated every symbol interval according to
obtain a greater tolerance for p, either Nb or Na can be increased, as illustrated in Figs. 5 and 6. In Fig. 5, Na is increased to 14, keeping Nb at 207 and the range Of for 70-dB cancellation is from 0.039 to 0.057. On the other hand,
p to vary from 0.033 to 0.053. The resulting tolerance for P is
It is important to establish convergence of the MSE to an acceptable steady-state value. The expression for MSE in (12) is for constant coefficients, so to obtain an expression for MSE involving the updated coefficients and bi, we in Fig* 6 increasing Nb to 40 keeping Nu at l3 take the expectation of the terms with these vectors:
- about the same in either case, but the least complex alterna- tive is to set Nh = 20 and N, = 14.
Fig. 8. Cancellation versus p ; 320 kbaud, Nb = 40, Nu = 15.
lated, and furthermore, that successive data vectors di are statistically independent. The latter assumption has been shown not to introduce major inaccuracies for small a and p
Considering first E(lh - biI2), using the expression for e; 1121.
in (11) in the update (24), we get
bi+l - h = ( I - Bdid')( bi - h )
+ pdixT( c - u i ) + Pdini + p d i ~ ; (26)
and
E [ Ibi+l- h12]
= E [ ( bi - h ) T ( I - 2 p d i d T + p2did,'did')(bi - h )
+ p2( c - ai)Txid,Tdix'( c - U i )
1 + p2d'din? + p2d'd;V']. ( 2 7 )
( N b - l ) U ; + E [ d : ] . ( 28)
P The matrix E[didFdidr], is a constant diagonal matrix with entries: Fig. 6. Cancellation versus p ; 80 kbaud, Nb = 40, N, = 13.
Now for the 2 B l Q (four-level PAM) signalling being con- sidered, E[ df] < ( E [ so that a conservative estimate of the mean squared error will result from approximating
80
70 E[didTdidT] by Nbu;I. l
I Thus 60 -
5 50 .E
8 4 0 3 - -
i Similarly, for la, - c12>:
E ( l q + 1 - cl') 30
20
= E [ ( u i - c ) ~ ( I - 2 a X i X ~ + a 2 X i X ~ X i X f ) ( u ; - C )
Fig. 7. Cancellation versus p ; 320 kbaud, Nb = 40, No = 14. (30)
1 1 1
I l l I 1
26 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. 38, NO. 1, JANUARY 1991
Again, with the stated assumptions and noting that x k , and v, are uncorrelated, all the expectations can be evalu- ated except for E[x ,xTx ,xT] . In this case, note that the result is a matrix whose m, nth element is
Because of the orthogonality of the x k r , this is a constant diagonal matrix with entries:
( N o - l)U," + E [ 4J.
~(4,) =ad" C L4k,(j) (33)
(32)
Now, m
] = o
where Lk is a function from the orthonormal set, and since L k has a norm of unity, it satisfies
m
0 < L i , ( j ) < 1. (34) j = o
Thus a conservative estimate of the mean squared error is obtained by approximating E[x ,x :x ,xT] by N,ajZ, and (30) becomes
E(la,+, - Cl2) = (1-2ffU; + U 2 N o o j ) E ( l a , - C l 2 )
+ a2N,ajE( Ib, - hi2) + a2N,a:a: + a2Nua&:. (35)
With this system of equations in E(lb, - hI2) and E(la, - cl'), the problem becomes the same as that solved in [13]. To write (29) and (35) in matrix form, let
and
so that
i - 1 X i + , = HX, + k or X i = H i & + H j k . (37)
j = O
In this notation the MSE is
Thus the MSE converges if all the eigenvalues of H are within the unit circle. In the steady state
i l i m X i = ( I - H ) - ' k -+a (39)
Fig. 9. Filter structure with No orthonormal impulse responses.
The fastest convergence is obtained by choosing a and p to minimize the maximum eigenvalue of H , and this results in
giving an MSE at 2(~: + a:). Lower MSE can be achieved at the cost of slower convergence by selecting smaller values of a and p.
V. IMPLEMENTATION AND COMPLEXITY In order to implement the tail canceler, a set of filters
must be found whose impulse responses are the discrete Laguerre orthonormal functions. This problem can be ap- proached in the z-domain since orthonormality of two se- quences implies orthonormality of their z-transforms. Two transforms X ( z ) and Y ( z ) are orthogonal if they satisfy [14]
where the contour of integration includes the unit circle. Also X(z) has a norm of unity if
l X ( Z ) X * ( f)2r;il=l. dz
Consider
(43)
(44)
which has an inverse transform of the form
for some constants a,,,; . . , ann. This is the same as (18), which was chosen to be of the same form as the discrete weighted Laguerre polynomials. It is straightforward to show that the (4 , (x>> satisfy (42) and (43), and so are the z transforms of the Laguerre orthonormal filters.
The set of filters with the orthonormal impulse responses can now be easily drawn, as in Fig. 9, and the echo canceler is shown in Fig. 10.
The filter structure of the tail canceler in Fig. 9 has some desirable properties with respect to using a finite number of bits to represent the coefficient r . Namely, because only the one parameter is used in all stages, the impulse responses
I l l I
DAVIDSON AND FALCONER: REDUCED COMPLEXITY ECHO CANCELLATION
~
27
4 Hybrid
I I I -
c i Receive 'i
Fig. 10. Implementation of 2-stage echo canceler.
will always be orthogonal for any value of r with 0 < r < 1. This is important for maintaining 70-dB cancellation with the given value of Nu, and for adaptation. Also, because only one normalizing constant is used in the entire structure, the impulse response will always have the same norms. However, inaccurate representation of the coefficient will cause the norms to differ from unity, requiring adjustments in the adaptation stepsize and affecting the MSE or the rate of convergence.
Finally, given the allowable variation in p , a value can be chosen that is a power of 2- ' , say 2-". In this case r becomes 1 - 2-" so that the multiplication by r is replaced by a shift and a subtraction.
In considering the complexity, the computation is com- pared to that of an echo canceler implemented by one long transversal filter. In this direct implementation, the data symbols, which require only a 2-b representation, are multi- plied by the coefficient values. In the tail canceler, on the other hand, the multiplications involve the outputs of the orthonormal filters, which require say b x / 2 bits. Thus a multiplication in the tail canceler is roughly equivalent to 6, /2 multiplications in the direct implementation.
To perform the filtering and adaptation in a transversal filter, the number of multiplications is about two times the number of taps. In the tail canceler, assuming r is chosen to avoid multiplications, 2(bx / 2 ) N u equivalent 2-b multiplica- tions are needed. This gives a total complexity for the two- stage canceler of 2 Nb + b, N, multiplications per symbol interval. Taking into account IIR filtering with a pole at ~ = r = e - ~ , with p on the order of 0.01, and also the requirement that the quantization error be 70 dB or more below the echo, we estimate a wordlength 6, on the order of 20 [15]. The complexity thus becomes 2Nb +20Nu.
To compare this to the complexity of the direct implemen- tation, it was found that at 80 kbaud, a 70-dB cancellation required a transversal filter of at least 330 taps, or a com- plexity of 660. On the other hand, the complexity of the two stage canceler with N b = 2 0 and N,=14 ws 320. At 320 kbaud the saving in complexity increased dramatically. In this case a 1300-tap transversal filter was needed for 70-dB cancellation, with a complexity of 2600, whereas the two stage canceler with Nb = 40 and Nu = 15 had a complexity of 380. This reduction in complexity at high rates is due to the fact that the number of orthonormal functions does not change as the symbol rate increases. Also, while the size of the first stage increases with the baud rate, it is still very small compared to a single stage transversal filter. Note that
104 l o 5 a 10.10 -
10" - '0"; 0:2 0:4 06 08 1:2 1:4 116 1:s
symbol intcrvrlr XlW
Convergence of MSE; four level PAM, 80 kbaud, Bellcore Fig. 11. loop #9; Nb = 20, N, = 14; p = 0.05; adaptation constant = 0.00599.
in actual u-interface receivers such as those described in [6], [7], [8], a high-pass filter is usually employed to reduce the echo and end-to-end response tail, at the expense of some crosstalk noise enhancement. Such a prefilter has not been considered here, but it would undoubtedly have reduced the required values of Nu and Nb further.
VI. SIMULATION The echo canceler was simulated for a four-level PAM
signal at 80 kbaud. The levels were - 3, - 1, 1, and 3 so that a: = 5. The echo path was simulated as a 750-tap transversal filter with coefficients equal to the baud rate sampled echo impulse response of Bellcore loop #9. White noise with variance 0,' = was added at the output. The two-stage canceler of Fig. 10 was simulated with N b = 20, Nu=14, p = 0.05, and an adaptation constant of l/u:(Nu + Nb) = 0.00599.
By calculating CFi'h?, and determining the amount of cancellation from the graph in Fig. 5, the theoretical value for U: was found to be a:=8.1X10-'3. For the 0,' and stepsize just mentioned, this gives a steady-state MSE of 3.6 X 10- 12.
Fig. 11 shows the graph of squared error versus symbol interval, and is the average of ten independent simulations. The theoretical steady-state MSE is also indicated, showing good agreement with the simulated result.
1 1 i
28 IEEE TRANSACTlONS ON CIRCUITS AND SYSTEMS, VOL. 38, NO. 1, JANUARY 1991
VII. CONCLUSION A technique for echo cancellation has been presented in
which the tail of the echo response is approximated by a linear combination of orthonormal functions. Of the two possible sets of functions considered-Laguerre and expo- nentials-the Laguerre functions offered much better per- formance. Also it was found that with at most 15 such functions, 70-dB cancellation could be achieved on a wide range of subscriber loops using a single fixed value of the time constant parameter p. The parameter varies with the symbol rate. For a less stringent requirement of 60-dB can- cellation, the range of tolerable values of p is even wider.
The functions were implemented by a set of filters whose impulse responses corresponded to the orthonormal set of Laguerre functions. The coefficients multiplying the outputs of these filters were found to be adaptable using the LMS algorithm, with convergence depending only on the variance of the data, the number of coefficients and the stepsize.
Since the set of recursive filters could be implemented simply, the result was a significant reduction in the computa- tional complexity of the echo canceler. This was especially true for higher symbol rates, because the transversal filter in the first stage is relatively small, and the number of or- thonormal functions in the second stage does not increase with the baud rate.
As mentioned in the introduction, the two-stage echo cancellation approach has also been investigated by G. Cook for an analog “pre-echo canceler” in front of a loop transceiver’s A/D converter [7]. In that application the first-stage canceler is a very short analog transversal filter, and continuous-time orthonormal Laguerre functions are used.
Finally, this technique is not necessarily limited to echo cancellation. For example, in decision feedback equalization, the impulse response of the channel can be approximated. In this case, since less than 70-dB cancellation is required, fewer orthonormal functions are needed, meaning an even greater reduction in complexity.
[7] M. Arai et al., “Design techniques and performance of a LSI-based 2B1Q transceiver,” Proc. Globecom ’88, pp. 25.2.1-25.2.5, Dec. 1988.
[8] J. Girardeau et al., “ISDU U-transceiver algorithm, develop- ment system and performance,” in Proc. Globecom ’89, pp.
[9] C. G. Cook, “Two stage data echo cancellation utilizing or- thonormal functions for echo tail cancellation,” M. Eng. thesis, Dep. of Electronics, Carleton Univ. Feb. 1989.
[ 101 P. Beckmann, Orthogonal Polynomials for Engineering and Physicists. Golem Press, 1973.
[ 111 American National Standards Institute, “Integrated services digital network (ISDN) basic access interface for use on metal- lic loops for application on the network side of the NT (layer 1 specification),” ANSI T1-601-1988, Sept. 1988.
[12] J. E. Mazo, “On the independence theory of equalizer conver- gence,” Bell Syst. Tech. J., May-June 1979, p. 963.
[ 131 D. D. Falconer, “Adaptive reference echo cancellation,” IEEE Trans. Commun., p. 2083, Sept. 1982.
[14] T. Y. Young and W. H. Huggins, “Discrete orthonormal expo- nentials,” in Proc. IVEC, Oct. 1962.
[15] A. Antoniou, Digital Filters: Analysis and Design. New York: McGraw-Hill, 1979.
54.5.1-54.5.9, NOV. 1989.
b
Gordon W. Davidson received the B.S. degree in electrical engineering from the University of Calgary, Canada, in 1984, and the M. Eng. de- gree from Carleton University, Ottawa, Ont., Canada. He is currently working toward the Ph.D. degree in image analysis at the University of British Columbia, Vancouver, Canada.
After two years at Bell Northern Research developing software for circuit simulation, he worked at Carleton University as a research assistant in the areas of echo cancellation and
equalization for digital mobile radio.
ACKNOWLEDGMENT Thanks are due to G. Cook for providing data on the echo
responses, and to him and Professor C. H. Chan for advice and discussions.
REFERENCES [l] D. G. Messerschmitt, “Echo cancellation in speech and data
transmission,” IEEE J. Select. Areas in Commun., vol. SAC-2, pp. 283-297, Mar. 1984.
[2] J. W. Lechleider, “Feasibility study of very high bit rate digital subscriber lines,” Contribution TlE1/88-012 to TlEl Commit- tee, Mar. 1988.
1 I. Korn, “Analogue adaptive hybrid for digital transmission on subscriber lines,” Proc. Inst. Elect. Eng., pt.F, vol. 131, no. 5, Aug. 1984. G. Long, D. Shwed, and D. D. Falconer, “Study of a pole-zero adaptive echo canceller,” IEEE Trans. Circuits Syst., vol. CAS- 34, pp, 765-769, July 1987.
David D. Falconer (S’62-M’68-SM’78-F86) re- ceived the B.A.Sc. degree in engineering physics from the University of Toronto, Ont., Canada, in 1962, and the S.M. and Ph.D. degrees in electrical engineering from M.I.T., Cambridge, in 1963 and 1967, respectively.
After a year as a Post-Doctoral Fellow at the Royal Institute of Technology, Stockholm, Swe- den, he joined Bell Laboratories, Holmdel, NJ, from 1967 to 1980, first as a Member of the Technical Staff and later as Group Supervisor.
During 1976 to 1977 he was visiting professor at Linkoping University, Linkoping, Sweden. Since 1980 he has been with Carleton University, Ottawa, Ont., Canada, where he is a Professor in the Department of Systems and Computer Engineering. His interests are in digital commu- nications, signal processing, and communication theory. He was Editor for digital communications for the IEEE TRANSACTIONS ON COMMUNI-
B. E. Dotter, A. De la Plaza, D. A. Hodges, and D. G. Messerschmitt, “Implementation of an adaptive balancing hy- brid,” IEEE Trans. Commun., vol. COM-28, pp. 1408-1416, Aug. 1980. c. Mogavero, G. Nervo, and G. Paschetta, “Mixed recursive echo canceller,” in Proc. Globecom ’86, pp. 2.2.1-2.2.5, Dec. 1988. and in 1986.
CATIONS, from 1981 to 1987, and was co-guest editor for the Special Issue on voiceband telephone transmission of IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, September 1984.
Dr. Falconer is a Registered Professional Engineer in the Province of Ontario, Canada. He was awarded the IEEE Communications Society Prize Paper Award in Communications Circuits and Techniques in 1983
