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A Time-Varying Convergence Parameter for the LMS Algorithm in the Presence of White Gaussian Noise

ABSTRACT
A novel approach for the least-mean-square (LMS) estimation algorithm is proposed. Rather than using a fixed convergence parameter μ, this approach utilizes a time-varying LMS parameter μ_n. This technique leads to faster convergence and provides reduced mean-squared error compared to the conventional fixed parameter LMS algorithm. The algorithm has been tested for noise reduction and estimation in narrow-band FM signals corrupted by additive white Gaussian noise.
For the LMS algorithm in a white Gaussian noise environment. A general power decaying law has been studied, however, other time-varying laws could also be applicable. The main idea is to set the convergence parameter to a large value in the initial state in order to speed up the algorithm convergence. The modified algorithm has been tested for noise reduction and estimation in linear frequency-modulated (LFM) narrowband signals corrupted by additive white Gaussian noise.
INTRODUCTION
The LMS algorithm is a well-known adaptive estimation and prediction technique. It has been extensively studied in the literature and widely used in a variety of applications. The performance of the LMS algorithm is highly dependent on the selected convergence parameter μ and the signal condition. A larger convergence parameter value leads to faster convergence of the LMS algorithm, i.e., convergence of the filter coefficients to their optimal values. After coefficients converge to their optimal value, the convergence parameter should be small for better estimation accuracy .
In this project, we propose a time-varying convergence parameter μ_n. for the LMS algorithm in a white Gaussian noise environment. A general power decaying law has been studied, however, other time-varying laws could also be applicable. The main idea is to set the convergence parameter to a large value in the initial state in order to speed up the algorithm convergence. As time passes, the parameter will be adjusted to a smaller value so that the adaptive filter will have a smaller mean-squared error. The modified algorithm has been tested for noise reduction and estimation in linear frequency-modulated (LFM) narrowband signals corrupted by additive white Gaussian noise. Simulation results have shown that the modified LMS algorithm has better performance in terms of convergence speed than the conventional LMS algorithm with a constant convergence parameter and the least-mean-squares error close is to the optimal value.
An adaptive filter is a filter that self-adjusts its transfer function according to an optimizing algorithm. Because of the complexity of the optimizing algorithms, most adaptive filters are digital filters that perform digital signal processing and adapt their performance based on the input signal. By way of contrast, a non-adaptive filter has static filter coefficients (which collectively form the transfer function).
For some applications, adaptive coefficients are required since some parameters of the desired processing operation (for instance, the properties of some noise signal) are not known in advance. In these situations it is common to employ an adaptive filter, which uses feedback to refine the values of the filter coefficients and hence its frequency response.
Generally speaking, the adapting process involves the use of a cost function, which is a criterion for optimum performance of the filter (for example, minimizing the noise component of the input), to feed an algorithm, which determines how to modify the filter coefficients to minimize the cost on the next iteration.
As the power of digital signal processors has increased, adaptive filters have become much more common and are now routinely used in devices such as mobile phones and other communication devices, camcorders and digital cameras, and medical monitoring equipment.
EXAMPLE
Suppose a hospital is recording a heart beat (an ECG), which is being corrupted by a 50 Hz noise (the frequency coming from the power supply in many countries).
One way to remove the noise is to filter the signal with a notch filter at 50 Hz. However, due to slight variations in the power supply to the hospital, the exact frequency of the power supply might (hypothetically) wander between 47 Hz and 53 Hz. A static filter would need to remove all the frequencies between 47 and 53 Hz, which could excessively degrade the quality of the ECG since the heart beat would also likely have frequency components in the rejected range.
To circumvent this potential loss of information, an adaptive filter could be used. The adaptive filter would take input both from the patient and from the power supply directly and would thus be able to track the actual frequency of the noise as it fluctuates. Such an adaptive technique generally allows for a filter with a smaller rejection range, which means, in our case, that the quality of the output signal is more accurate for medical diagnoses.
2.1 BLOCK DIAGRAM
The block diagram, shown in the following figure, serves as a foundation for particular adaptive filter realisations, such as Least Mean Squares (LMS) and Recursive Least Squares (RLS). The idea behind the block diagram is that a variable filter extracts an estimate of the desired signal
To start the discussion of the block diagram we take the following assumptions:
The input signal is the sum of a desired signal d(n) and interfering noise v(n)
x(n) = d(n) + v(n)
The variable filter has a Finite Impulse Response (FIR) structure. For such structures the impulse response is equal to the filter coefficients. The coefficients for a filter of order p are defined as
The error signal or cost function is the difference between the desired and the estimated signal
The variable filter estimates the desired signal by convolving the input signal with the impulse response. In vector notation this is expressed as
where
is an input signal vector. Moreover, the variable filter updates the filter coefficients
at every time instant
where is a correction factor for the filter coefficients. The adaptive algorithm generates this correction factor based on the input and error signals. LMS and RLS define two different coefficient update algorithms.
2.2 APPLICATIONS OF ADAPTIVE FILTERS
Noise cancellation
Signal prediction
Adaptive feedback cancellation
LEAST MEAN SQUARES (LMS)
Least mean squares (LMS) algorithms is a type of adaptive filter used to mimic a desired filter by finding the filter coefficients that relate to producing the least mean squares of the error signal (difference between the desired and the actual signal). It is a stochastic gradient descent method in that the filter is only adapted based on the error at the current time. It was invented in 1960 by Stanford University professor Bernard Widrow and his first Ph.D. student, Ted Hoff.