Adaptive Filters

adaptive filters are also Digital filters that change their coefficients with the intention of bringing the filter closer to its optimal state. A cost function, typically the mean square of the error signal between the adaptive filter’s output and the desired signal, serves as the optimization criterion. The mean square error (MSE) converges to its minimal value as the filter adjusts its coefficients. The coefficients have converged to a solution and the filter has been modified at this point. The desired signal, d(k), is said to be very closely matched by the filter output, y(k). The filter adjusts to the new environment by generating a new set of coefficients for the new data when the characteristics of the input data are altered, a process that is referred to as the “filter environment.”

Application of Adaptive Filter:

1. System Identification: Identifying Unknown Systems Using an Adaptive Filter- Identifying an unknown system, such as the response of an unknown communications channel or the frequency response of an auditorium, to select fairly distinct applications is one common application of adaptive filters. Channel identification and echo cancellation are two additional applications.
2. Using an Adaptive Filter to Remove Noise from an Unknown System: Noise or Interference Cancellation- In commotion crossing out, versatile channels let you eliminate clamor from a sign continuously. Noise and desired information are combined here into the desired signal, the one to be cleaned. Feed the adaptive filter a signal n'(k) that is correlated to the noise to be removed from the desired signal to get rid of the noise.
3. Identification of an Inverse Response to an Unknown System via Inverse System-  adaptive filter becomes the inverse of the unknown system as e(k) decreases when the unknown system is placed in series with it. In order to keep the data at the summation synchronized, the procedure necessitates the addition of a delay to the desired signal d(k) path, as depicted in the figure. The system remains causal when the delay is added.
4. A Periodic Signal’s Future Values Can Be Predicted  Through Prediction- In order to predict signals, you must make important assumptions. Assume that the signal is periodic and either steady or slowly changing over time.

On the degraded image, which contains both the original image and noise, an adaptive filter is applied. With a predetermined mxn window region, the mean and variance are the two statistical measures on which a locally adaptive filter depends.