Digital Signal Processing Reference
In-Depth Information
Many other important works and approaches have been presented in the
vast literature on the subject. A nice and helpful scan of the literature can be
found in [99].
4.1 The Unsupervised Deconvolution Problem
The importance of convolution is notorious, as it models the input-output
mapping of a linear time-invariant system. Deconvolution is the inverse
mathematical operation that allows the recovery of the input signal from
the output signal. If the system response is available, it is relatively straight-
forward to develop both time-domain and frequency-domain algorithms to
perform deconvolution.
An early application of deconvolution has been in seismic signal process-
ing. Robinson developed a pioneer work on the subject in his PhD thesis at
MIT [253]. His research attracted the attention of Wiener and Levinson, both
working at MIT at that time. In fact, Robinson's work represented the first
successful application of the recently developed Wiener theory on predic-
tion and filtering. His aim was to obtain information about the structure of
the Earth by the estimation of the impulse response of a layered earth model,
i.e., an FIR model.
Theproblembecameintricatethough, sinceanestimateoftheinputsignal,
the so-called seismic wavelet, was not available. Such lack of information
characterizes the problem as unsupervised. To solve it, Robinson derived the
predictivedeconvolutionprocedurebyconsideringtwosimplifyinghypothe-
ses [139]: (1) the seismic wavelet is the impulse response of an all-pole system,
sothatitisnecessarilyminimumphase; (2)theimpulseresponseofthelayered
earth model behaves like a white noise, so that it has a flat spectral shape.
Indeed, if we compare the above hypotheses with the properties of the
PEF given in Sections 3.8.1 and 3.8.2, we verify that the desired impulse
response of the model may be recovered as a prediction-error signal when
the output of the model (i.e., the measured signal that is recorded in a
seismograph) is applied to the input of a PEF.
Robinson's approach can be generalized as follows. If a given signal
x
(
n
)
obeys a convolution relationship,
x
(
n
)
=
s
(
n
)
∗
w
(
n
)
(4.1)
where
s
(
n
)
is an uncorrelated (white) signal and
w
(
n
)
is a minimum-phase
impulse response, we can recover
s
(
n
)
and
w
(
n
)
in an unsupervised way by
obtaining the prediction-error signal
K
e
f
(
n
)
=
x
(
n
)
−
a
k
x
(
n
−
k
)
(4.2)
k
=
1
