Blind equalization of digital communication channels using high-order moments

522 zyxwv zyxw zyxwvutsrqponmlkjih zyxwvutsrq zyxwvutsrqp zyxwvutsrqp zyxwvutsrqp IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 39, NO. 2 . FEBRUARY 1991 in a variety of white additive input noise distributions, namely, for uniform, Gaussian, and mixed noise. The mixed noise that has been used in the simulations consists of zero mean unit variance white Gaussian noise and 10% impulsive noise taking values + 10 and - 10. Such an input noisy signal is shown in Fig. 2(a). The size of the signal vector is 256 samples in all cases. The optimal filter for the mixed Gaussian and impulsive noise is not known exactly. However, it is close to the median filter, as it has been found by using (5) and by evaluating R numerically. The adaptive L-filter has been chosen to have n = 5 coefficients. In all experiments the initial filter coefficients have been chosen a, = 1 / 5 , i = 1, . . . , 5, so that the initial L-filter is equivalent to the moving average filter. The value of the step size chosen for the LMS L-filter in the case of the mixed noise is p = This step is smaller than that given by (20) and guarantees stability. The update of the filter coefficients was done according to (15). However, since the noise distribution is symmetric about the mean of the distribution, the unbiasedness condition [l 11 n has been enforced at each step of the algorithm by scaling the coefficients obtained by (15). The output of the adaptive LMS L-filter is shown in Fig. 2(b). It is clearly seen that its performance is very good after a short adaptation period. The plot of the filter coefficients a , , a 3 , a5 corresponding to minimum, median, and maximum, is shown in Fig. 3(a). As expected, it is seen that the filter coefficient u l , u5 decrease and that u3 increases. This fact indicates that the adaptive L-filter tends to the median in this case. The coefficient estimation error J ( a, i ) is defined as follows: (39) where a;p‘, j = 1, . . . , n is the optimal set of coefficients. This error function is plotted in Fig. 3(b). It clearly decreases with time. However, the convergence speed of the LMS algorithm is relatively slow. The performance of the adaptive RLS L-filter in the presence of mixed impulsive and Gaussian noise is shown in Fig. 3(b), (c). By observing Fig. 3(b) it is seen that it has much faster convergence than the LMS filter. It also converges to the median filter, as is indicated in Fig. 3(c), because the coefficients a , , u5 tend to zero and the coefficient a3 tends to 1. Both RLS and LMS adaptive L-filters have also been tested for the cases of white additive uniform and Gaussian noise. As expected, they converge to the arithmetic mean and to the midpoint filter, respectively. An open question which is still under investigtion is the theoretical study of the convergence properties of the algorithms that have been presented in this correspondence. We feel that tighter bounds on the convergence rate can be imposed. REFERENCES T. Alexander, Adaptive Signal Processing. Berlin: Springer, 1986. M. Bellanger, Adaptive Digital Filters and Signal Analysis. Marcel Dekker, 1987. J. D. Proakis and D. G. Manolakis, Znfroduction to Digifal Signal Processing. New York: Macmillan, 1988. G. L. Sicuranza and G. Ramboni, “Adaptive nonlinear digital filters using distributed arithmetic,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, no. 3, pp. 518-526, June 1986. J . C. Slapeton and S . C. Bass, “Adaptive noise cancellation for a class of nonlinear dynamic reference signals,” in Proc. Int. Symp. Circuits Sysr., 1984, pp. 268-271. I. Pitas and A. N. Venetsanopoulos, “Nonlinear order statistic filters for image filtering and edge detection,” Signal Processing, vol. IO, pp. 395-413, 1986. R. Bernstein, “Adaptive nonlinear filters for simultaneous removal of different kinds of noise in images,” IEEE Trans. Circuits Syst., vol. CAS-34, no. 11, pp. 1275-1291, Nov. 1987. X. Z. Sun and A. N. Venetsanopoulos, “Adaptive schemes for noise filtering and edge detection by use of local statistics,” IEEE Trans. Circuits Syst., vol. CAS-35, no. 1, pp. 57-69, Jan. 1988. H. A David. Order Statistics. New York: Wiley. 1981. [lo] 1. Pitas and A. N. Venetsanopoulos, Nonlinear Digital Filters: Principles and Applications. Kluwer Academic, 1990. [ I I ] A. C. Bovik, f.S . Huang, and D. C. Munson, “A generalization of median filtering using combinations of order statistics,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, no. 6, pp. 13421349, Dec. 1983. [12] A. S. Householder, The Theory ofMarrices in Numerical Analysis. Waltham, MA: Blaisdell, 1964. ~~ zyxwvu Blind Equalization of Digital Communication Channels Using High-Order Moments Boaz Porat and Benjamin Friedlander Abstract-This correspondence describes new algorithms for blind equalization of digital communication channels of the QAM type. The algorithms use the fourth-order statistical moments of the symbol sequence to explicitly estimate the channel impulse response. The estimated impulse response is used, in turn, to construct a linear meansquare error equalizer. V. CONCLUSIONS A nonlinear adaptive L-filter is presented that can easily adapt to the noise probability distributions. It can be used for the filtering of both long-tailed and short-tailed distributions. It performs well in the cases where linear adaptive filters fail, e.g., in the case of impulsive noise. Two different algorithms to update the filter coefficients have been presented. They have a close resemblance to the LMS and RLS adaptive algorithms used in the adaptive linear FIR filters. The LMS algorithm can be easily implemented, it has low computational complexity, but it converges slowly to the optimum. Faster convergence rates can be obtained by using the RLS adaptation algorithm. However, this algorithm has a much higher computational complexity. This fact restricts its use in real-time applications and in image processing. In general, the adaptive L-filters have definite advantages over their linear counterparts. The extra computational complexity needed involves only the calculation of running ordering. However, this computational load is not large if special running ordering algorithms or structures are used. I. INTRODUCTION The most common approach to adaptive equalization of digital communication channels is by using known training sequences. The equalizer is typically a linear transversal (FIR) filter, the coefficients of which are adjusted to minimize the mean-square error. After the initial training, the equalizer is normally switched to a decision-directed mode, where the detected symbols are considered Manuscript received May 7, 1989; revised April 16, 1990. This work was supported by the National Science Foundation under Grant ISI-87600 95 and by the Army Research Office under Contract DAAL 03-89-C-0007. B. Porat is with the Department of Electrical Engineering, TechnionIsrael Institute of Technology, Haifa 32000, Israel. B. Friedlander is with Signal Processing Technology, Ltd., Palo Alto, CA 94303. IEEE Log Number 9041 114. zyxwvutsr 1053-587X/91/0200-0522$01.OO 0 1991 IEEE I zyxwvutsrq IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 39, NO. 2, FEBRUARY 1991 as the true ones, and the error is used to adjust the equalizer gains as in the training mode. The situation where initial training sequence is not available at the receiver is known as the blind equalization problem. In cases where the intersymbol interference and the noise are sufficiently small so that “the eye is open,” decision-directed equalization is possible even without an initial training sequence. However, in the presence of severe intersymbol interference (“when the eye is closed”), other approaches are necessary. Most of the existing works on blind equalization use nonquadratic cost functions, and recursively minimize these functions with respect to the equalizer’s parameters. Typically, the adaptation is done by gradient-type algorithms, e.g., the LMS or its numerous variations. Some important works on the blind equalization problem are [1]-[8] In this correspondence we propose two algorithms for blind channel equalization for general QAM signals. These algorithms are based on the fourth-order statistical moments of the received data sequence. The first of the two is the linear least squares type algorithm, which uses ideas put forward by Giannakis and Mendel [9]. The second algorithm is the nonlinear least squares type, and is based on the previous works of the present authors, mainly [lo]- Wl. 523 channel parameters { hk} from the output sequence { y,, 1 5 t 5 T }. The estimated parameters are then used to reconstruct (detect) the symbol stream {U,} by means of a linear equalizer. 111. LINEARLEASTSQUARES ESTIMATION ALGORITHM In this section we derive a simple linear least squares algorithm for estimating the channel response { hk} from the second- and fourth-order moments of the output sequence. The algorithm is based on the approach of Giannakis and Mendel [9], with proper modifications necessary for the complex noncausal model at hand. The algorithm is not optimal in any sense, but its performance can be quite satisfactory in cases where the intersymbol interference is not severe, or when the time variation of the channel is very slow, so that a large number of data points can be accumulated for a single estimate of the channel parameters. This algorithm is also essential to the nonlinear least squares algorithm described in the next section, where it is used for initialization purposes. Let us denote r(m) d(m) = ~ { x ~ x ~ + ~ x f + , } ; = ~{x:x,+,}; c ( n ) = d(m) - 2r(O)r(m). (4) The { r ( m ) } are the covariances of the process. The { d ( m ) } are the so-called diagonal fourth-order moments, while the { c ( m ) } are the corresponding diagonal fourth-order cumulants. It is easy to show, using standard techniques, that { r ( m ) , c ( m ) } are given by the following expressions: zyxwvu zyxwvutsrqp zyxwvutsrqponm 11. PROBLEMFORMULATION The communication signals considered in this paper are of the quadrature amplitude modulation (QAM) type. A QAM symbol can be described as a complex number, belonging to a discrete set in the complex plane, called the symbol constellation. Let { U,} denote the symbol stream. The U, are assumed to be independent identically distributed random variables. Each U, can take one of M possible values with probability 1 /M.The symbol constellation is assumed to possess sufficient symmetry such that all odd-ordered moments are zero. The even-order moments will be denoted by The communication channel is assumed to be linear. In reality, it is usually time varying. However, we make the common assumption that the time variation is sufficiently slow compared to the rate at which the channel parameters are to be estimated. In other words, the channel is assumed to be time-invariant during the observation interval. The equivalent discrete-time complex impulse response of the channel is denoted by { hk } . This includes the transmit and receive filters, the channel itself and the sampler (one sample per symbol is assumed). Thus, in the absence of‘noise, the output symbol sequence is given by r ( m ) = 71.1 t; hk*hk+rn; c(m)= c(2.2 hk*hk*+rnh:+rn where P2,2 = 7 2 . 2 - 2 d . I . Let C ( z ) be the z transform of { c ( m ) } , i.e., (6) zyx zyxwvutsrqpon where H ( z ) is the transfer function of the channel, and H 3 ( z ) is the z transform of {hTh?; -ql 5 I 5 q 2 } . Let R ( z ) be the z transform of { r ( m ) } , i.e., R(z) = 71.1 c hk*hk+mZ-rn= y l , l H * ( Z - ’ ) H ( z ) . rn (7) We get from (6) and (7) C(z)H(z) + EH3(Z)R(Z) = 0 . (8) where E = - c L 2 , 2 / Y I . I . Since the impulse response { h,} is finite, we can write (8) explicitly as zyxwvutsrqpo zyxwvutsrqpon zyxwvutsrqp The channel impulse response is infinite in general. However, as commonly seen in the communication literature, we approximate it by a finite impulse response. We also assume an additive complex Gaussian white noise with variance a:, so the actual sampler’s output is (5 (5 ( r n $ q ~ ( m ) ~ - r+n )E h,z-*) k = -41 k=-ql h*h2k z - k ) 42 Yr = k = -41 hkU,-k + U,. (9) (3) The order of the impulse response will be denoted by q = ql + q2. Noncasuality is introduced into the model in order to enable us to assume that the impulse response is effectively centered at ho, i.e., that the ideal response is n, = U,. Alternatively, we could have used a causal model and assumed that the impulse response is centered at some hL ( L > 0). There is no loss of generality in assuming that ho = 1, and scaling the symbol sequence { U ,} accordingly. This scaling can be complex in general, i.e., it can include both amplitude scaling and phase shift. The problem we address in this paper is the estimation of the Let us define qk = Eh,*h:; -41 5 k 5 t 10) q2. We treat the { q k } as free parameters, even though in reality they depend on the impulse response parameters { hk } . Using our assumption that ho = 1, we can rewrite (9) in the following form: = -c(I - k ) ; -(2q, + q2) 5 1 5 (2q2 + ql). (11) 524 zyxw zyxwvutsrqponmlkjih zyxwvutsrq zyxwvutsrq zy zyxwvutsrq zyxwvuts zyxwvutsrq IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 39, NO. 2. FEBRUARY 1991 This is an overdetermined set of equations, with 3q + 1 equations 1 unknowns (regarding the qk as independent of the h k ) . and 2q The coefficient matrix and the right-hand side vector of this set dependonthemoments{r(m),O Im 5 q } (sinter(-rn) = r * ( m ) ) and { c ( m ) , - q i m 5 q } . If { c ( m ) , r ( r n ) } were known, it could be solved exactly. The channel estimation algorithm uses, instead, estimates of these quantities, computed from the measurements as follows: + channel parameters 8 is very difficult to express analytically. Therefore, estimation based on global minimization of (17) cannot be regarded as a practical algorithm. Suppose, however, that we can find a consistent estimate of E ( e ) , computed directly from the measurements { y r , 0 5 t 5 T } , and denote this estimate by 2. Let 8 be the global minimizer of the cost function +(e) = [s(e) - i]"~-l[s(e)- $1. (18) In [12] it was shown that, subject to some regularity conditions (formally defined in [12]), this estimate is also asymptotically minimum variance, i.e., it achieves the same asymptotic variance as the global minimizer of (17). Thus, nonlinear minimization of (18) can lead to practical estimation algorithm, provided we can find a consistent estimate, E, of C ( e ) . Similarly to [12], the entries of the matrix can be computed as follows: c When the sample moments { i ( m ) , i ? ( m ) } are substituted in (1 l ) , we obtain a set of equations of the form AX = -b. T . cov { i ( k ) , P * ( l ) } (13) zyxwvutsrqponm The individual entries of the matrix A and the vectors x and b are obvious from (11). This set is solved in the least squares sense, i.e., x = -(AHA)-'A"~. T . cov { i(k),d*(I)} (14) The estimates { h,, G k } are finally extracted from the components of x. The values of 7 , . p 2 , 2can be estimated, if desired, by IV. NONLINEAR LEASTSQUARESESTIMATION ALGORITHM In this section we describe a nonlinear least squares algorithm that is asymptotically minimum variance in a sense defined below. The algorithm is based on ideas developed in [12]. Because of its improved accuracy, the algorithm is useful in cases where the channel has severe intersymbol interference, and/or the number of data points available for a single estimate is relatively small. On the other hand, the nonlinear algorithm requires considerably more computations than the linear one, which may limit its use in some real-time applications. Let 0 denote the vector of unknown parameters-the channel impulse response { hk } and possibly the moments of the symbol constellation y I ,I , p 2 , 2 .Let us denote by s a vector consisting of some subset of the moments { r ( m ) ; 0 Im Iq } and { d ( m ) ;- q 5 m 5 q } . Similarly, we will denote by the corresponding vector of estimated moments, defined in (12a)-(12c). The dimension of s is required to be equal or larger than the dimension of 0. In the extreme case, s consists of the entire set of moments, and then the entries of 4 are exactly the moments used in the linear least squares algorithm described in Section 111. Let C (e) b e the asymptotic normalized covariance matrix of the vector 4, i.e., q e ) = Tlim -m T . E { ( $ - s)(i - s)"] (16) where ( . ) " denotes complex conjugate transposition. Let V ( 0 ) be the nonlinear cost function v(e) = [?(e) - i]"~-l(e)[qe) - 41. (17) In [lo] it was shown that the global minimizer of V (e ) is an asymptotically minimum variance estimate of 8 in the class of estimates which are "well behaved" functions of j . (in a sense formally defined in [lo]). However, the dependence of the matrix C ( e ) on the where i, ,i = min (0, k ) - max (0, I ) - q = max (0, k ) - min (0, I ) + q. (19d) The nonlinear least square estimation algorithm can now be described as follows. i) Compute the estimated moments from (12a)-(12c). ii) Compute the estimated, covariances of the estimate! moments from (19a)-( 19c) and use them to construct the matrix E. iii) Run the linear least squaJes algorithm described in Section I11 and obtain the estimates { h,, ql, I , fi2, } . Build the vector Bo from these estimates. Bo serves as an initial condition to the nonlinear minimization that follows. iv) Use some nonlinear minimization procedure to minimize the cost function $ ( e ) with respect to 8 , starting with the above initial condition. V. THE EQUALIZER Equalization schemes can be broadly classified into two categories: linear equalizers, and decision-directed equalizers [ 131. Blind equalization is characterized by relatively inaccurate estimation of the channel (compared to equalization based on a training sequence), especially during the initial phase. Under such circumstances, decision-directed equalization tends to yield relatively high error rates, and may even lead to divergence. For this reason, we have chosen to test the channel estimators described in the previous sections with a linear equalizer. Of course, after initial convergence (when the "eye" becomes sufficiently open), it may be desirable to switch to decision-directed equalization. Switching to decision-directed mode is considered "safe" when the intersymbol I zyxwvut zyxwvuts zyxw zyxwvutsrq 525 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 39, NO. 2, FEBRUARY 1991 t: \ -181 0 1 " 02 04 1 06 ' 08 " 1 12 ' 14 " 16 N 18 ' -15' -15 2 5 -10 0 5 I 15 IO REAL PART x10' Fig. 2. The symbol constellation before equalization. Fig. 1. Companson between the experimental and the theoretical ISI. zyxwvuts 2 interference (ISI) gets below a certain threshold.' This threshold is in the range of - 15 dB [8] for quaternary QAM signals, and may be lower for multilevel QAM signals. The equalizer is chosen as a linear transversal (FIR) filter whose coefficients are computed so as to minimize the mean-square error (MSE). Let { gk; -K 5 k 5 K } denote the impulse response of the equalizer corresponding to the estimated channelresponse { i k } . We denote the column vector of the { & } by g. Let e be a column vector of dimension 2 K q 1 with 1 in the ( K + q1 + 1 )th position and zeros elsewhere. Let P be the ( 2 K q 1 ) X (2K 1 ) matrix whose ( i ,j ) t h entry is ki-j-ql (or zero, if the subscript i - j - q1 is outside the range [ - q l , q 2 ] ) . It is well known [13, pp. 205-2171 that the minimum mean-square error equalizer g is given by zyxwvutsrqp + + + + + zyxwvutsrq REAL PART Fig. 3. The symbol constellation after equalizatlon. VI. AN EXAMPLE In this section we illustrate the performance of the blind equalization method proposed in this paper by an example. The channel impulse response was taken as { 2 - 0.4j, 1.5 + 1.8j, 1 , 1.2 1.3j, 0.8 1.6j }. This channel has an initial IS1 of + 12 dB, i.e., it represents a case of extremely severe distortion. The signal was quaternary QAM, and the SNR was 40 dB. The length of the equalizer was taken as 65. Fig. 1 shows the mean residual IS1 (after equalization), obtained from 100 Monte Carlo runs, for ten values of data length, up to 20 000 (marked by the circles in the graph). Also shown in this figure is the corresponding theoretical IS1 (the computation of which is not discussed in this correspondence). As we see, the actual performance matches the theoretical one very well. Figs. 2 and 3 show the received and the equalized symbol constellations, respectively, of a single simulation. Here we used 15 000 data points to estimate the channel, of which 2000 are shown in the figures. It is interesting to observe the apparent rotation of the reconstructed constellation, due to the imperfect estimation of the channel. Fig. 4 shows the frequency responses of the channel, before and after equalization. As we see, the response of the equalized channel is almost flat, except for the notch at frequency 0.47. + VII. CONCLUSIONS We have presented two methods for blind equalization of QAM channels, based on the fourth-order moments computed from the 'The IS1 is defined as the sum of square magnitudes of all nonzero terms of the total impulse response (channel plus equalizer), divided by the square magnitude of the zero term. m p -10 0'1 0'2 0:3 04 05 0'6 fKquency 07 0'8 09 07 08 09 1 1 zyxwvu zyxwvutsrqpo PHASE 200 100 3E 8 O -100 m0 01 02 03 04 05 06 f7equency 1 Fig. 4. The frequency responses of the channel before and after equali. zation. received symbol stream. Both methods estimate the channel parameters explicitly, and use the estimated parameters to construct a linear MSE equalizer. The first equalizer is based on the least squares solution of a linear set of equations. The performance of this equalizer is not optimal in any sense, but it is adequate for channels with mild intersymbol interference, or when the number of data points available for estimating the channel response is very large. Another application of the linear least squares algorithm is to provide initial estimates to the second, nonlinear least squares algorithm. 526 zyxwvutsrqponmlkjihgf zyxw IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 39, NO. 2. FEBRUARY 1991 The second equalizer is based on nonlinear minimization of a certain cost function, which takes into account the second-order statistical properties of -the estimated moments. This algorithm is asymptotically minimum variance in a sense defined in Section IV. Its convergence rate is very fast, but its computational complexity is considerably higher than that of the first equalizer. ing, based on energy. As such, it is ah almost immediately usable, reasonably robust, and easily implemented algorithm. However, the necessary parameter setting and adjusting detracts from its ease of implementation. This correspondence will suggest an improved method of determining the probability density function of the energy in the recording, from which energy level based threshold parameters can be automatically determined. The second part will clarify and correct the flowcharts presented in the original paper. The notation will be that of Lamel’s flowchart. zyxwvutsrqpo zyxwvutsrqpon zyxwvutsrqp REFERENCES [l] Y. Sato, “A method of self-recovering equalization for multilevel amplitude-modulation systems,” ZEEE Trans. Commun., vol. COM23, no. 6, pp. 679-682, June 1975. [2] A. Benveniste, M. Goursat, and G. Ruget, “Robust identification of a nonminimum phase system: Blind adjustment of a linear equalizer in data communications,” ZEEE Trans. Automat. Contr., vol. AC25, no. 3, pp. 385-399, June 1980. [3] A. Benveniste and M. Goursat, “Blind equalizers,’’ IEEE Trans. Commun., vol., COM-32, no. 8, pp. 871-883, Aug. 1984. [4] D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems, IEEE Trans. Commun., vol. COM-28, no. 11, pp. 1867-1875, Nov. 1980. [5] G. B. Foschini, “Equalizing without altering or detecting data,” AT&TTech. J . , vol. 64, no. 8, pp. 1885-1911, Oct. 1985. [6] J. C. Treichler and B. G. Agee, “A new approach to multipath correction of constant modulus signals,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, no. 2, pp. 459-412, Apr. 1983. [7] J. R. Treichler and M. G. Larimore, “New processing techniques based on the constant-modulus adaptive algorithm,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, no. 2, pp. 420431, Apr. 1985. [SI Z. Pritzker and A. Feuer, “The variable length stochastic gradient algorithm,” submitted for publication. [9] G. B. Giannakis and J. M. Mendel, “Identification of nonminimum phase systems using higher order statistics,” ZEEE Trans. Acousr., Speech, Signal Processing, vol. 37, no. 3, pp. 360-377, Mar. 1989. [IO] B. Porat and B. Friedlander, “Performance analysis of parameter estimation based on high-order moments,” J . Adaprive Confr., Signal Processing, vol. 3, pp. 191-229, 1989. [ l l ] B. Friedlander and B. Porat, “Adaptive IIR filtering based on highorder statistics,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, no. 4, pp. 485-495, Apr. 1989. 1121 B. Friedlander and B . Porat, “Asymptotically optimal estimation of MA and ARMA parameters of non-Gaussian processes from highorder moments,” ZEEE Trans. Automat. Contr., vol. 35, no. 1, pp. 27-35, Jan. 1990. [13] A. P. Clark, Equalizers f o r Digital Modems. London: Pentech, 1985. ” 11. AUTOMATIC DETERMINATION OF THRESHOLDS The level-based parameters depend on the ambient noise and can be automatically set by examining the energy level distribution in a single recording. For recordings with noise quieter than 35 dB below the speech, the energy level thresholds K I throu h K,, which are required for the algorithm in Lamel et al.’s paper to run, can be determined from the distribution of energy. Lamel suggests generating a histogram to estimate the noise level, but a procedure given in [l] can also estimate the voicing signal level and is simpler, faster, and more precise. The procedure is as follows. First, sort the frames’ energies so that R ( i ) < R ( i + 1 ). Then, the probability of the energy R being between R ( i ) and R ( i + J ) is inversely proportional t o R ( i J ) - R ( i ) , i.e., 4 + PDF[i + 3/21 = constant R(i + J) - R(i) where PDF is an estimate of the probability density of the energy in the entire recording. The interval J should be chosen such that the resulting distribution shows 2 modes-one for noise, and one for voiced speech. A reasonable value for J is the number of frames corresponding to 0.25 s. From the distribution, Lamel’s K,, the “averaged” background noise level, can be estimated as the mode (the local maximum of PDF from (1)) of the distribution among the lowest 10 dB, and the typical voicing level, U of Table I, can be assumed to be the mode of the highest 15 dB. The rest of the K thresholds can be estimated as shown in Table I. Care should be taken to ensure that K2 is greater than K3.This is no problem if the signal-to-noise ratio of the recording is over 30 dB. zyxwvutsrqpon zyxwvu Comments on “An Improved Endpoint Detector for Isolated Word Recognition” Ben Reaves Abstract-Robust word boundary detection remains an unsolved problem. This correspondence presents an automatic threshold setting algorithm and corrects a paper on single utterance word boundary detection in a quiet environment, promising great improvement in the accuracy of fully automatic word recognition and assistance in the hand labeling of endpoints. I. INTRODUCTION In their paper,’ Lamel et al. published a detailed flowchart for determining the boundaries of a single utterance within a recordManuscript received October 5, 1989; revised April 30, 1990. The author is employed by the Speech Technology Laboratory (a Division of Panasonic Technologies Inc.), Santa Barbara, CA, and is working at Matsushita Electrical Industrial Company, Central Research Laboratory, Moriguchi, 570, Japan. IEEE Log Number 9041 120. ‘L. F. Lamel, L. R. Rabiner, A. E. Rosenberg, and J. G. Wilpon, ZEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. 777-785, Aug. 1981. 111. CLARIFICATIONS AND CORRECTIONS Because of some inconsistencies between the notation of the text and that of the flowchart in the original paper, it was not clear how to set some of the time-based parameters. Table I1 will be of assistance here. The following corrections are necessary for obtaining the results published in Lamel e t al.’s paper.’ They refer to Figs. 1 through 5. 1) In Fig. 1, on the path from the “YES” output of “ K , < & ( l ) 5 K2,” a box labeled “CHECK l,” with L and 1 as arguments, is inserted. 2) The “CHECK 1” subroutine, Fig. 2, has an “I = I - 1” box inserted between the “YES”output of the “ 1 > L” decision and the “2.” 3) The left side decision of Fig. 3 is j ‘ j - 1 = I. The prime is removed from the corresponding figure in Lamel et al.’s work. 4) The lower “LEVEL CHECK” subroutine of Fig. 4 shows K 1 as indices for PB and P E , not K as in Lamel et al.’ 5) Fig. 5 shows a “NO” decision from “a # 1 & b # L” which was omitted in Lamel et al. 6) The in the first section, labeled “Pulse Reordering,” of Fig. 5(b) of Lamel et al.’ represents a zero. Only here was zero represented by 4; elsewhere it was represented by 0. + ’ + + ’ zyxwvutsr 1053-587X/91/0200-0526$01.OO 0 1991 IEEE