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In some applications, the system or environment where processing is happening is unknown or time-dependant. In such applications, adaptive filters are required because they possess self-adjusting capabilities. An adaptive filter is mainly a digital filter that will adjust its filter parameters (coefficients) to converge the filter to the optimal solution of defined cost function using input data from the environment. The main target of these filters is to estimate the unknown entity of an input signal. They are used to reshape certain input signals in such a way that the filter output is a good estimate of the given desired signal.
The two steps involved in adaptive filtering process are: –
Filtering process,which produces an output signal in response to the given input signal using the updated filter coefficients, which later helps to adjust filter parameters in the adaptation process.
Adaptation process,to adjust the filter parameters according to the time varying environment. In this step, filter parameters (coefficients) are updated so that the cost function converges to its optimal solution by finding the best match between the desired signal and filter output.
The major applications of adaptive filters include noise cancellation, acoustic echo cancellation, bio-medical signal enhancement, equalizations of communication channels, active noise control, system identification, speech coding, multi-channel noise reduction and adaptive control systems. It works generally for the adaptation of signal-changing environments, and unknown or time-varying noise. For example: in echo cancellation, a dynamic mathematical model of channel that creates echo is generated by continuously monitoring the received signal. This model is used to create an estimate of echo path which is then subtracted from the signal to remove the effect of echo from desired signal.
2. Adaptive filter algorithms
Different kinds of adaptive filter algorithms includes
Least Mean Square Algorithm (LMS)
Variable Least Mean Square Algorithm (VLMS)
Normalized Least Mean Square Algorithm (NLMS)
Recursive Least Square Algorithm (RLS)
Affine Projection Algorithm (APA)
Kalman Algorithm
Among these algorithms, LMS, NLMS, VLMS, and RLS are generally considered as conventional adaptive filtering techniques.
The following outline the three basic steps for all algorithms:
Computing the output of the digital filter with a set of filter coefficients
Generation of an estimated error by comparing the filter output and desired signal
Adjusting filter coefficients based on the estimated error
Choosing the right kind of algorithm depends mainly on applications.
2.1.1 Least Mean Square Algorithm (LMS)
LMS Algorithm is a linear adaptive filtering algorithm and its a member of a stochastic gradient-based algorithm. An important feature of this algorithm is its robustness and low computational complexity. This is a fixed step-size algorithm and can be used in a wide range of applications such as channel equalization and echo cancellation, but it is not considered useful when a long echo duration is present as in case of teleconferencing. In teleconferencing, long impulse response or long memory is required to cope with the long duration of echo. LMS algorithm in time domain does not have a long memory to cope with the long-duration echo therefore it causes the problem of increased computational complexity. It mainly aims to reduce the mean square error between the signals. This algorithm uses a step size parameter to control immediate change in updating factor. As the value of step size decreases, the convergence speed to optimal values is slower and for large values, the filters will diverge and become unstable. So, we have to select the step size accordingly. It requires (2N+1) additions and (2N+1) multiplications, where N is length of adaptive filter.
Filter coefficients updating equation in LMS,
where,
μ – Step size, 0 < μ < 1
x(n) – Input signal
e(n) – Error signal
2.1.2 Normalized Least Mean Square Algorithm (NLMS)
By using the normalized step size parameter in LMS algorithm, it becomes NLMS algorithm. Normalized step size improves the convergence behavior in NLMS algorithm. So, it becomes more powerful in applications like speech recognition. This algorithm is an equally simple but more robust variant of LMS algorithm, and also keeps a better balance between simplicity and performance than LMS algorithm. In this algorithm, the step size parameter is chosen based on the current input values. As human speech has more energy in low frequencies, this algorithm gives good echo cancellation for low frequencies and poor for high frequencies. Step size varies adaptively by following the changes in input signal level which prevents filter weights from diverging and makes the algorithm more stable and faster converging compared with a fixed step size algorithm. NLMS algorithm requires (3N+1) multiplications and 1 division.
Weight vector updating equation becomes,
Step size for computing the weight updating factor is,
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