Echo removal is the process of removing

echo In audio signal processing and acoustics, an echo is a reflection of sound that arrives at the listener with a delay after the direct sound. The delay is directly proportional to the distance of the reflecting surface from the source and the lis ...

and

reverberation In acoustics, reverberation (commonly shortened to reverb) is a persistence of sound after it is produced. It is often created when a sound is reflection (physics), reflected on surfaces, causing multiple reflections that build up and then de ...

artifacts from audio signals. The reverberation is typically modeled as the

convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...

of a (sometimes time-varying)

impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...

with a hypothetical clean input signal, where both the clean input signal (which is to be recovered) and the impulse response are unknown. This is an example of an

inverse problem An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, sound source reconstruction, source reconstruction in ac ...

. In almost all cases, there is insufficient information in the input signal to uniquely determine a plausible original image, making it an ill-posed problem. This is generally solved by the use of a

regularization Regularization may refer to: * Regularization (linguistics) * Regularization (mathematics) * Regularization (physics) * Regularization (solid modeling) * Regularization Law, an Israeli law intended to retroactively legalize settlements See also ...

term to attempt to eliminate implausible solutions. This problem is analogous to

deblurring Deblurring is the process of removing blurring artifacts from images. Deblurring recovers a sharp image ''S'' from a blurred image ''B'', where ''S'' is convolved with ''K'' (the blur kernel) to generate ''B''. Mathematically, this can be repre ...

in the image processing domain.