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 .
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.
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 pseudoinverse of . Doing this provides a least-squares system identification method .
Thus, is the Moore-Penrose pseudoinverse of .
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. .
It is also straightforward to introduce a weighting function in the least-squares estimate for by replacing in the derivations above by , where is any positive definite matrix (often taken to be diagonal and positive). In the present time-domain formulation, it is difficult to choose a weighting function that corresponds well to audio perception. Therefore, in audio applications, frequency-domain formulations are generally more powerful for linear-time-invariant system identification. A practical example is the frequency-domain equation-error method described in §I.4.4 .
delete('sid.log'); diary('sid.log'); % Log session echo('on'); % Show commands as well as responses N = 4; % Input signal length %x = rand(N,1) % Random input signal - snapshot: x = [0.056961, 0.081938, 0.063272, 0.672761]' h = [1 2 3]'; % FIR filter y = filter(h,1,x) % Filter output xb = toeplitz(x,[x(1),zeros(1,N-1)]) % Input matrix hhat = inv(xb' * xb) * xb' * y % Least squares estimate % hhat = pinv(xb) * y % Numerically robust pseudoinverse hhat2 = xb\y % Numerically superior (and faster) estimate diary('off'); % Close log fileOne fine point is the use of the syntax `` '', which has been a matlab language feature from the very beginning . It is usually more accurate (and faster) than multiplying by the explicit pseudoinverse. It uses the QR decomposition to convert the system of linear equations into upper-triangular form (typically using Householder reflections), determine the effective rank of , and backsolve the reduced triangular system (starting at the bottom, which goes very fast) [29, §6.2].F.8
+ echo('on'); % Show commands as well as responses + N = 4; % Input signal length + %x = rand(N,1) % Random input signal - snapshot: + x = [0.056961, 0.081938, 0.063272, 0.672761]' x = 0.056961 0.081938 0.063272 0.672761 + h = [1 2 3]'; % FIR filter + y = filter(h,1,x) % Filter output y = 0.056961 0.195860 0.398031 1.045119 + xb = toeplitz(x,[x(1),zeros(1,N-1)]) % Input matrix xb = 0.05696 0.00000 0.00000 0.00000 0.08194 0.05696 0.00000 0.00000 0.06327 0.08194 0.05696 0.00000 0.67276 0.06327 0.08194 0.05696 + hhat = inv(xb' * xb) * xb' * y % Least squares estimate hhat = 1.0000 2.0000 3.0000 3.7060e-13 + % hhat = pinv(xb) * y % Numerically robust pseudoinverse + hhat2 = xb\y % Numerically superior (and faster) estimate hhat2 = 1.0000 2.0000 3.0000 3.6492e-16
State Space Realization