Deconvolution

Deconvolution, also called inverse filtering, extracts the impulse response from known input and output signals. It is essential when system characteristics are unknown and must be inferred from observable signals. This technique reverses the convolution process to recover one of the constituent signals in the convolution sum.

In polynomial division, input and output sequences are treated as coefficients of descending-order polynomials. Long division is performed on these polynomials to obtain the impulse response. This straightforward approach provides an efficient means to determine the impulse response when the system's input-output relationship is expressed in polynomial form.

The recursive algorithm method represents the output response as a convolution sum, which is transformed into a recursive algorithm. By setting the variable n to zero, the equation simplifies and the impulse response for positive values of n is determined. This method reduces computational complexity, making it particularly useful for long sequences.

The number of evaluations required to determine the impulse response depends on the lengths of the input and output signals. This value is calculated by substituting the signal lengths into a given relation. Once determined, the final impulse response value can be calculated accurately for predicting system behavior.

Convolution uses the impulse response and input signal to determine the output response. Deconvolution reverses this process: given the input signal and output response, it recovers the impulse response. Deconvolution is the inverse operation of convolution, enabling system identification when the system's characteristics are unknown.

Deconvolution is used when system characteristics are unknown and must be inferred from observable input and output signals. Engineers apply it to identify system behavior, reverse signal distortion, and recover original signals that have been filtered or modified by unknown systems. It is fundamental to system identification and signal recovery tasks.

Two primary deconvolution techniques are polynomial division and recursive algorithms. Polynomial division treats sequences as polynomial coefficients and uses long division to find the impulse response. Recursive algorithms formulate the output as a convolution sum and simplify it systematically. Both methods yield the impulse response but differ in computational efficiency and applicability.