N. J. Bershad and J. C. M. Bermudez, "A nonlinear analytical model for the quantized LMS algorithm -- the power-oftwo step size case," IEEE Transactions on Signal Processing, vol. SP-44, no. 11, pp. 2895--2900, Nov. 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....does not account for the slowdown phenomenon which occurs when the LMS weights do not get regularly updated due to excessive roundoff error. For the special case of quantized LMS, which is LMS implemented with finite precision coefficients and infinite precision data, Bermudez and Bershad [17] [18], 19] presented a recursive learning curve approximation which gives more accurate predictions of the steady state MSE than the standard analysis, the standard analysis giving increasingly poor predictions as coefficient resolution decreases. In this paper we use the standard analysis ....

....of the vertical plane in Figure 3 (right of the thick line in Figure 4) while the other region is the slowdown region. 11 B. 4 Predicting Onset of Slowdown The slowdown phenomenon occurs when one or more components of the input to the quantizer Q c in (2) falls below the LSB of Q c [12] 17] [18], 23] The corresponding onset of slowdown can be defined as the minimum integer k 0 such that ; or fi Imfx (11) for some i 2 f0; p Gamma 1g. Here Ref Deltag and Imf Deltag denote real and imaginary parts, and Delta c = 2 is the granularity of the coefficient ....

[Article contains additional citation context not shown here]

N. J. Bershad and J. C. M. Bermudez, "A nonlinear analytical model for the quantized LMS algorithm -- the power-oftwo step size case," IEEE Transactions on Signal Processing, vol. SP-44, no. 11, pp. 2895--2900, Nov. 1996.

....analysis does not account for the slowdown phenomenon which occurs when the LMS weights do not get regularly updated due to excessive roundoff error. For the special case of quantized LMS, which is LMS implemented with finite precision coefficients and infinite precision data, Bermudez and Bershad [17], 18] 19] presented a recursive learning curve approximation which gives more accurate predictions of the steady state MSE than the standard analysis, the standard analysis giving increasingly poor predictions as coefficient resolution decreases. In this paper we use the standard analysis ....

....the left of the vertical plane in Figure 3 (right of the thick line in Figure 4) while the other region is the slowdown region. 11 B. 4 Predicting Onset of Slowdown The slowdown phenomenon occurs when one or more components of the input to the quantizer Q c in (2) falls below the LSB of Q c [12] [17], 18] 23] The corresponding onset of slowdown can be defined as the minimum integer k 0 such that ; or fi Imfx (11) for some i 2 f0; p Gamma 1g. Here Ref Deltag and Imf Deltag denote real and imaginary parts, and Delta c = 2 is the granularity of the coefficient ....

[Article contains additional citation context not shown here]

J. C. M. Bermudez and N. J. Bershad, "A nonlinear analytical model for the quantized LMS algorithm -- the arbitrary step size case," IEEE Transactions on Signal Processing, vol. SP-44, no. 5, pp. 1175--1183, May 1996.

....a feature unique to some nite precision implementations that oc12 curs when the LMS weights do not get regularly updated due to excessive roundo error. For the special case of the LMS algorithm implemented with nite precision coecients and in nite precision data, Bermudez and Bershad [6, 7, 8] presented a recursive learning curve approximation that gives more accurate predictions of the steady state MSE than the standard analysis, the standard analysis giving increasingly poor predictions as coecient resolution decreases. 2.1.3 Overview of Contribution In this chapter, we use the ....

....to the left of the vertical plane in Figure 2.3 (right of the thick line in Figure 2.4) while the other region is the slowdown region. Predicting Onset of Slowdown The slowdown phenomenon occurs when one or more components of the input to the quantizer Q # in (2. 2) falls below the LSB of Q # [6, 7, 15, 28]. The corresponding onset of slowdown can be de ned as the minimum integer k 0 such that # # Re# x # ### e ## # # # # # 2 or # # Im# x # ### e ## # # # # # 2 (2.11) for some i # #0; p# 1#. Here Re### and Im### denote real and imaginary parts and # = 2 ### is ....

[Article contains additional citation context not shown here]

N. J. Bershad and J. C. M. Bermudez, \A Nonlinear Analytical Model for the Quantized LMS Algorithm { the Power-of-Two Step Size Case," IEEE Trans. on Signal Processing,vol. SP-44, no. 11, pp. 2895-2900, Nov. 1996.

....a feature unique to some nite precision implementations that oc12 curs when the LMS weights do not get regularly updated due to excessive roundo error. For the special case of the LMS algorithm implemented with nite precision coecients and in nite precision data, Bermudez and Bershad [6, 7, 8] presented a recursive learning curve approximation that gives more accurate predictions of the steady state MSE than the standard analysis, the standard analysis giving increasingly poor predictions as coecient resolution decreases. 2.1.3 Overview of Contribution In this chapter, we use the ....

....to the left of the vertical plane in Figure 2.3 (right of the thick line in Figure 2.4) while the other region is the slowdown region. Predicting Onset of Slowdown The slowdown phenomenon occurs when one or more components of the input to the quantizer Q # in (2. 2) falls below the LSB of Q # [6, 7, 15, 28]. The corresponding onset of slowdown can be de ned as the minimum integer k 0 such that # # Re# x # ### e ## # # # # # 2 or # # Im# x # ### e ## # # # # # 2 (2.11) for some i # #0; p# 1#. Here Re### and Im### denote real and imaginary parts and # = 2 ### is ....

[Article contains additional citation context not shown here]

J. C. M. Bermudez and N. J. Bershad, \A Nonlinear Analytical Model for the Quantized LMS Algorithm { the Arbitrary Step Size Case," IEEE Trans. on Signal Processing, vol. SP-44, no. 5, pp. 1175-1183, May 1996.

....standard one in the literature whenever finite precision arithmetic effects are being studied, although it is often implicit in the assumptions. For example, the assumption that all variables are suitably scaled so that overflow never occurs in fact requires that all variables be bounded (see [9] [14] [16] There are several algorithms that can be used to compute estimates w k for w . In this paper, we focus on the following adaptive schemes of the LMS class. LMS. In the standard LMS algorithm, the estimates w k are computed via [17] 18] w k 1 = w k x k e(k) with initial ....

.... unbiased estimates, since the estimates computed by LMS have this 6 We should note that the results obtained with this linear model for the quantization error are valid if the so called stopping phenomenon does not occur (i.e. when the step size is large enough, see [26] 15] Reference [14] considers an alternative nonlinear model for finite precision errors, albeit under the more restrictive assumption of iid Gaussian input variables with R = oe 2 x I see further the comments immediately before the concluding remarks of [27] 7 To simplify the notation, in this section we ....

J. C. M. Bermudez and N. J. Bershad. A nonlinear analytical model for the quantized LMS algorithm --- the arbitrary step size case. IEEE Transactions on Signal Processing, 44(5):1175--1183, May 1996.

....to minimize the mean square error (MSE) between the filter s output and a primary signal. Two important performance measures of the algorithm are its steady state MSE and convergence rate [1] In practice, the LMS algorithm is implemented in finite precision, and often with fixed point arithmetic [2, 3, 4]. The performance penalty incurred as a result of finite precision implementation has been analyzed in [2] and [3, 4] In [2] Caraiscos and Liu derived an expression for the increase in steady state MSE due to quantization of data and coefficients using an additive white noise model for the ....

.... of the algorithm are its steady state MSE and convergence rate [1] In practice, the LMS algorithm is implemented in finite precision, and often with fixed point arithmetic [2, 3, 4] The performance penalty incurred as a result of finite precision implementation has been analyzed in [2] and [3, 4]. In [2] Caraiscos and Liu derived an expression for the increase in steady state MSE due to quantization of data and coefficients using an additive white noise model for the quantizers. Bermudez and Bershad [3, 4] considered an implementation of the LMS algorithm with infinite precision data and ....

[Article contains additional citation context not shown here]

N. J. Bershad and J. C. M. Bermudez, "A nonlinear analytical model for the quantized LMS algorithm -- the power-of-two step size case," IEEE Transactions on Signal Processing, vol. SP-44, no. 11, pp. 2895--2900, Nov. 1996.

....to minimize the mean square error (MSE) between the filter s output and a primary signal. Two important performance measures of the algorithm are its steady state MSE and convergence rate [1] In practice, the LMS algorithm is implemented in finite precision, and often with fixed point arithmetic [2, 3, 4]. The performance penalty incurred as a result of finite precision implementation has been analyzed in [2] and [3, 4] In [2] Caraiscos and Liu derived an expression for the increase in steady state MSE due to quantization of data and coefficients using an additive white noise model for the ....

.... of the algorithm are its steady state MSE and convergence rate [1] In practice, the LMS algorithm is implemented in finite precision, and often with fixed point arithmetic [2, 3, 4] The performance penalty incurred as a result of finite precision implementation has been analyzed in [2] and [3, 4]. In [2] Caraiscos and Liu derived an expression for the increase in steady state MSE due to quantization of data and coefficients using an additive white noise model for the quantizers. Bermudez and Bershad [3, 4] considered an implementation of the LMS algorithm with infinite precision data and ....

[Article contains additional citation context not shown here]

J. C. M. Bermudez and N. J. Bershad, "A nonlinear analytical model for the quantized LMS algorithm -- the arbitrary step size case," IEEE Transactions on Signal Processing, vol. SP-44, no. 5, pp. 1175--1183, May 1996.

....LMS weight update falls below the least significant bit (LSB) of the coefficient quantizer. Bermudez and Bershad [14] showed that the stopping phenomenon is actually a slowdown phenomenon which does not stop the algorithm s updating, but causes a severe increase in convergence time. In [15] [16] they bypassed the linear white noise assumption and derived an accurate recursive approximation for an LMS algorithm with quantized weight update and infinite precision data, which they called quantized LMS. Their work showed that the white noise assumption is invalid for this type of algorithm, ....

....effects of coefficient resolution and data resolution on MSE. Fortunately, in many operating regimes of practical interest, it has been demonstrated that this approach gives useful and accurate performance predictions [11] 13] 17] In fact, unlike the quantized LMS algorithm of [15] [16], the finite precision LMS algorithm can operate without experiencing the stopping phenomenon. It is in this operating region that the model used in [12] gives accurate results. Accordingly, we adopt the white noise approach for our analysis of finite precision LMS. While our results are directly ....

[Article contains additional citation context not shown here]

N. J. Bershad and J. C. M. Bermudez, "A nonlinear analytical model for the quantized LMS algorithm -- the power-oftwo step size case," IEEE Transactions on Signal Processing, vol. SP-44, no. 11, pp. 2895--2900, Nov. 1996.

....precision LMS weight update falls below the least significant bit (LSB) of the coefficient quantizer. Bermudez and Bershad [14] showed that the stopping phenomenon is actually a slowdown phenomenon which does not stop the algorithm s updating, but causes a severe increase in convergence time. In [15], 16] they bypassed the linear white noise assumption and derived an accurate recursive approximation for an LMS algorithm with quantized weight update and infinite precision data, which they called quantized LMS. Their work showed that the white noise assumption is invalid for this type of ....

....the effects of coefficient resolution and data resolution on MSE. Fortunately, in many operating regimes of practical interest, it has been demonstrated that this approach gives useful and accurate performance predictions [11] 13] 17] In fact, unlike the quantized LMS algorithm of [15], 16] the finite precision LMS algorithm can operate without experiencing the stopping phenomenon. It is in this operating region that the model used in [12] gives accurate results. Accordingly, we adopt the white noise approach for our analysis of finite precision LMS. While our results are ....

[Article contains additional citation context not shown here]

J. C. M. Bermudez and N. J. Bershad, "A nonlinear analytical model for the quantized LMS algorithm -- the arbitrary step size case," IEEE Transactions on Signal Processing, vol. SP-44, no. 5, pp. 1175--1183, May 1996.

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