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 ${\underline{x}}$ and $\underline{y}$ and wish to solve for the filter impulse response $\underline{h}^T=[h_0,h_1,\ldots,h_{N_h-1}]$ . 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

$$\underbrace{% \left[\begin{array}{c} y_0 \\ [2pt] y_1 \\ [2pt] y_2 \\ [2pt] y_3 \\ [2pt] y_4 \\ [2pt] y_5 \\ [2pt] y_6 \\ \vdots \end{array}\right]}_{\displaystyle\underline{y}} = \underbrace{% \left[\begin{array}{ccccccc} x_0 & 0 & 0 & 0 & 0 \\ [2pt] x_1 & x_0 & 0 & 0 & 0 \\ [2pt] x_2 & x_1 & x_0 & 0 & 0 \\ [2pt] x_3 & x_2 & x_1 & x_0 & 0 \\ [2pt] x_4 & x_3 & x_2 & x_1 & x_0 \\ [2pt] x_5 & x_4 & x_3 & x_2 & x_1 \\ [2pt] x_6 & x_5 & x_4 & x_3 & x_2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \end{array}\right]}_{\displaystyle\mathbf{x}} \underbrace{% \left[\begin{array}{c} h_0 \\ [2pt] h_1 \\ [2pt] h_2 \\ [2pt] h_3\\ [2pt] h_4\end{array}\right]}_{\displaystyle\underline{h}},$$

or

$$\underline{y}= \mathbf{x}\underline{h}. \protect$$ (F.7)

Here we have indicated the general case for a length $N_h=5$ causal FIR filter, with input and output signals that go on forever. While $\mathbf{x}$ is not invertible because it is not square, we can solve for $\underline{h}$ under general conditions by taking the pseudoinverse of $\mathbf{x}$ . Doing this provides a least-squares system identification method [46].

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

$$\mathbf{x}^T\underline{y}= \mathbf{x}^T\mathbf{x}\underline{h}$$

Since $\mathbf{x}^T\mathbf{x}$ is a square $N_h\times N_h$ matrix, it is invertible under general conditions, and we obtain the following formula for $\underline{h}$ :

$$\zbox {\underline{h}= \left(\mathbf{x}^T\mathbf{x}\right)^{-1}\mathbf{x}^T\underline{y}} \protect$$ (F.8)

Thus, $\left(\mathbf{x}^T\mathbf{x}\right)^{-1}\mathbf{x}^T$ is the Moore-Penrose pseudoinverse of $\mathbf{x}$ .

If the input signal $x$ is an impulse $\delta (n)$ (a 1 at time zero and 0 at all other times), then $\mathbf{x}^T\mathbf{x}$ is simply the identity matrix, which is its own inverse, and we obtain $\underline{h}=\underline{y}$ . We expect this by definition of the impulse response. More generally, $\mathbf{x}^T\mathbf{x}$ is invertible whenever the input signal is ``sufficiently exciting'' at enough 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].