Adaptive Filter - Block Diagram

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 and interfering noise

• 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 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.