14.6
Deconvolution
Deconvolution, also known as inverse filtering, is the process of extracting the impulse response from known input and output signals. This technique…
The system's impulse response can be utilized to determine the output response through input signal and impulse response convolution.
Acquiring this impulse response, given an input signal and output, is called deconvolution or inverse filtering. It is the process of obtaining one of the constituent signals in the convolution sum.
Given an input signal and an output response, deconvolution can be performed using polynomial division or recursive algorithm methods to yield the impulse response.
In the polynomial division approach, sequences are seen as coefficients of descending-order polynomials. Long division is then executed to obtain the impulse response.
In the recursive algorithm method, the output response is initially defined as the convolution sum, which can be formulated as a recursive algorithm. The equation is simplified by setting the variable n to zero, allowing the impulse response for positive values of n to be obtained.
The number of evaluations needed for the impulse response is determined by substituting signal lengths into the given relation. The final impulse response value is calculated for the obtained number.
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Q1: What is deconvolution and why is it used in signal processing?
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.
Q2: How does the polynomial division method work for deconvolution?
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.
Q3: What are the advantages of using the recursive algorithm method for deconvolution?
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.
Q4: How do you determine the number of evaluations needed in deconvolution?
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.
Q5: What is the relationship between convolution and deconvolution?
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.
Q6: When would you use deconvolution in practical engineering applications?
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.
Q7: What mathematical techniques are available for performing deconvolution?
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.