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Modified Model and Algorithm of LMS Adaptive Filter for Noise Cancellation

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Abstract

The present research investigates the innovative concept of LMS adaptive noise cancellation by means of a modified algorithm using an LMS adaptive filter along with their detailed analysis. Three types of equations viz. output, error, and weight update are used in the LMS algorithm. The error equation of traditional LMS algorithm is modified which gives better results as compared to traditional LMS algorithm and their types. The comparison parameters used in the present analysis are signal-to-noise ratio, mean square error, maximum absolute error, and energy ratio between signals of error and output signals. The proposed modified model was found to have a higher signal-to-noise ratio with respect to the traditional model of LMS adaptive noise cancellation. The signal-to-noise ratio of the proposed model has also been compared with some other types of LMS algorithms and was found better in most of the cases.

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Authors and Affiliations

  1. Department of Electronics and Communication Engineering, College of Engineering and Technology, IILM Academy of Higher Learning, Greater Noida, Gautam Budh Nagar, Uttar Pradesh, 201306, India

    Awadhesh Kumar Maurya & Priyanka Agrawal

  2. Amity University, Noida, Uttar Pradesh, 201313, India

    Shubhra Dixit

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  1. Awadhesh Kumar Maurya
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  2. Priyanka Agrawal
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Correspondence to Awadhesh Kumar Maurya.

Appendix

Appendix

1.1 LMS Algorithms for Adaptive Filters

The output signal y(n) and error signal e(n) for the adaptive filter is given in Eqs. (1) and (2), respectively. The different types of LMS algorithms are known by their weight updated algorithms.

1.1.1 Weight Updated Algorithms

The different types of weight updated algorithms are described as follows:

1.1.1.1 Traditional LMS Algorithm
$$ W\left( {n + 1} \right) = W\left( n \right) + X\left( n \right)\mu e\left( n \right) $$
(30)

where W(n) = [w0(n) w1(n) … wL−1(n)]T is the weight coefficient vector, X(n) = [x(n) x(n − 1) … x(n − L + 1)]T is the input signal vector, e(n) is the error signal, μ is the step size and L is the filter length of LMS adaptive filter and 0 < µ < 1/λmax.

The λmax is the maximum eigenvalue of correlation matrix Rx. Here,

$$ R_{X} = E\left[ {X\left( n \right)X^{\text{T}} \left( n \right)} \right] $$
(31)
1.1.1.2 Normalized LMS (N-LMS) Algorithm
$$ W\left( {n + 1} \right) = W\left( n \right) + \beta \frac{X\left( n \right)}{{\varepsilon + \left\|{X\left( n \right)^{2} }\right\|}}\mu e\left( n \right) $$
(32)

where 0 < β < 2, \( {\left\|{X\left( n \right)^{2} }\right\|} = X^{\text{T}} \left( n \right) {X\left( n \right)} \) and \( \mu = \frac{\beta }{{\left\|{X\left( n \right)^{2} }\right\|}} \)

1.1.1.3 Modified Normalized LMS (MN-LMS) Algorithm
$$ W\left( {n + 1} \right) = W\left( n \right) + \beta \frac{X\left( n \right)}{{\varepsilon + \left\|{X\left( n \right)^{2} }\right\| }}\mu e\left( n \right) $$
(33)

where ε = small positive number.

1.1.1.4 Leaky LMS (L-LMS) Algorithm
$$ W\left( {n + 1} \right) = \left( {1 - \mu \gamma } \right)W\left( n \right) + X\left( n \right)\mu e\left( n \right) $$
(34)

where leaky coefficient γ, 0 < γ ≪ 1 and 0 < µ < (γ + λmax).

1.1.1.5 Block LMS (B-LMS) Algorithm

It is also known as block-updating LMS algorithm. The filter coefficients are updated only once for each block of L-samples. The output signal of a kth block of LMS filter,

$$ y\left( {kL + l} \right) = W^{\text{T}} \left( {kL} \right)X\left( {kL + l} \right) $$
(35a)

The error signal of kth block of LMS filter,

$$ e\left( {kL + l} \right) = d\left( {kL + l} \right) - y\left( {kL + l} \right) $$
(35b)

The weight updated algorithm of kth block of LMS filter,

$$ W\left( {\left( {k + 1} \right)L} \right) = W\left( {kL} \right) + \mu \frac{1}{L}\mathop \sum \limits_{l = 0}^{L - 1} e\left( {kL + l} \right)X\left( {kL + l} \right) $$
(35c)

where l = 0, 1, 2,…, L − 1.

1.1.1.6 Sign-Error LMS (SE-LMS) Algorithm
$$ W\left( {n + 1} \right) = W\left( n \right) + X\left( n \right)\mu {\text{sgn}} \left[ {e\left( n \right)} \right] $$
(36)

where sgn(.) = signum function

$$ {\text{sgn}}[e(n)] = \left\{ {\begin{array}{*{20}l} {1\quad {\text{for}}\;e\left( n \right) > 0} \hfill \\ { 0\quad {\text{for}}\;e\left( n \right) = 0} \hfill \\ { - 1\quad {\text{for}}\;e\left( n \right) < 0} \hfill \\ \end{array} } \right. $$
1.1.1.7 Sign-Data LMS (SD-LMS) Algorithm
$$ W\left( {n + 1} \right) = W\left( n \right) + {\text{sgn}} \left[ {X\left( n \right)} \right]\mu e\left( n \right) $$
(37)
1.1.1.8 Sign-Data-Normalized LMS (SDN-LMS) Algorithm
$$ w_{k} \left( {n + 1} \right) = w_{k} \left( n \right) + \frac{\mu }{{\left| {x\left( {n - k} \right)} \right|}}e\left( n \right)x\left( {n - k} \right) $$
(38)
1.1.1.9 Sign-Sign LMS (SS-LMS) Algorithm
$$ W\left( {n + 1} \right) = W\left( n \right) + {\text{sgn}} \left[ {X\left( n \right)} \right]\mu {\text{sgn}} \left[ {e\left( n \right)} \right] $$
(39)
1.1.1.10 Sign-Sign LMS Algorithm with Leakage Term (SS-LMS-LT)
$$ W\left( {n + 1} \right) = \left( {1 - \mu \gamma } \right)W\left( n \right) + {\text{sgn}} \left[ {X\left( n \right)\left] {\mu {\text{sgn}} } \right[e\left( n \right)} \right] $$
(40)
1.1.1.11 Variable Step-Size LMS (VS-LMS) Algorithm
$$ w_{k} \left( {n + 1} \right) = w_{k} \left( n \right) + \mu_{k} e\left( n \right)x\left( {n - k} \right) $$
(41)

where µmin < µ < µmax and step size adjusted for each coefficient.

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Maurya, A.K., Agrawal, P. & Dixit, S. Modified Model and Algorithm of LMS Adaptive Filter for Noise Cancellation. Circuits Syst Signal Process 38, 2351–2368 (2019). https://doi.org/10.1007/s00034-018-0952-z

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