Time Domain Filter Estimation

*System identification* is the subject of identifying filter
coefficients given measurements of the input and output signals
[46,78]. For example, one application is
*amplifier modeling*, in which we measure (1) the normal
output of an electric guitar (provided by the pick-ups), and (2) the
output of a microphone placed in front of the amplifier we wish to
model. The guitar may be played in a variety of ways to create a
collection of input/output data to use in identifying a model of the
amplifier's ``sound.'' There are many commercial products which offer
``virtual amplifier'' presets developed partly in such a
way.^{F.6} One
can similarly model electric guitars themselves by measuring the pick
signal delivered to the string (as the input) and the normal
pick-up-mix output signal. A separate
identification is needed for each switch and tone-control position.
After identifying a sampling of models, ways can be found to
interpolate among the sampled settings, thereby providing ``virtual''
tone-control knobs that respond like the real ones
[101].

In the notation of the §F.1, assume we know and and wish to solve for the filter impulse response . We now outline a simple yet practical method for doing this, which follows readily from the discussion of the previous section.

Recall that convolution is *commutative*. In terms of the matrix
representation of §F.3, this implies that the input signal and
the filter can switch places to give

or

Here we have indicated the general case for a length causal FIR filter, with input and output signals that go on forever. While is not invertible because it is not square, we can solve for under general conditions by taking the

The *Moore-Penrose pseudoinverse* is easy to
derive.^{F.7} First multiply Eq.
(F.7) on the left by the
transpose of
in order to obtain a ``square'' system of
equations:

Since is a square matrix, it is invertible under general conditions, and we obtain the following formula for :

Thus, is the

If the input signal
is an *impulse*
(a 1 at time
zero and 0 at all other times), then
is simply the identity
matrix, which is its own inverse, and we obtain
. We expect
this by definition of the impulse response. More generally,
is invertible whenever the input signal is ``sufficiently
exciting'' at all frequencies. An LTI filter frequency response can
be identified only at frequencies that are excited by the input, and
the accuracy of the estimate at any given frequency can be improved by
increasing the input signal power at that frequency.
[46].

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