Summary: How to isolate two convolved signals.
Deconvolution is exactly what it sounds like: the undoing of convolution. This means that instead
of
mixing two signals like in convolution, we are isolating them. This is useful for analyzing the
characteristics of the input signal and the impulse response when only given the output of the
system. For example, when given a convolved signal
| Ideal Deconvolution System |
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Instead of producing one system that outputs both the convolved signals, it will be much easier for our purposes to consider separate systems that output one of the signals at a time. Thus, we desire the following systems:
| Separate Systems |
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What it looks like each of these systems is doing is annihilating the undesired signal. This is, in fact, exactly correct. This system is a homomorphic filter.
A frequently applied method is to convert the convolution of two signals into a sum, and then implement a homomorphic filter to remove one of the signal components. This is the basis for cepstral analysis, so we will cover this later. A diagram of this method follows:
| A Possible Deconvolution Method |
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The isolation of two convolved signals depends greatly on the characteristics of both signals. Thus, a wide variety of deconvolution methods exist. Since this is a study on speech analysis, we will cover only the deconvolution methods which focus the signals of the source filter model: the excitation signal and the impulse response of the vocal tract filter.
A few deconvolution methods that we will use in speech analysis are:
We study the first of these in the next area covering the cepstrum.Rabiner, Lawrence R, and Schafer, Ronald W. Digital Processing of Speech Signals. Bell Laboratories, 1978.