Deconvolution

Deconvolution, also known as inverse filtering, is the process of extracting the impulse response from known input and output signals. This technique is vital in scenarios where the system's characteristics are unknown, and they must be inferred from the observable signals.

Deconvolution involves several mathematical techniques to derive the impulse response. One common approach is polynomial division. In this method, the input and output sequences are treated as coefficients of descending-order polynomials. By performing long division on these polynomials, the impulse response can be obtained. This method is straightforward and provides an efficient means to determine the impulse response when the system's input-output relationship is expressed in polynomial form.

Another effective technique for deconvolution is the recursive algorithm method. Here, the output response is represented as a convolution sum, which can be transformed into a recursive algorithm. The recursive nature of this method allows for the systematic simplification of the convolution sum. By setting the variable n to zero, the equation is simplified, and the impulse response for positive values of n can be determined. This method is particularly useful when dealing with long sequences, as it reduces the computational complexity involved in the deconvolution process.

The number of evaluations required to determine the impulse response depends on the lengths of the input and output signals. This can be calculated by substituting the signal lengths into a given relation. Once the necessary number of evaluations is determined, the final value of the impulse response can be calculated accurately. This step is crucial for ensuring that the derived impulse response is precise and reliable for predicting the system's behavior under various input conditions.