Convolutions (finite memory)

Both beta mixture approximations and Erlang mixture approximations can be used together with the linear chain trick to approximate convolutions in the form

\[ z(t) = \int_{t-h}^t \alpha(t - s) x(s) \, \mathrm ds \]

by a set of ordinary differential equations.

Beta mixture approximations

The beta basis functions for the domain \(\mathcal D = [0, h]\) are given by

\[ \ell_m(t) = b_m t^m (h - t)^{M-m}, b_m = \frac{1}{h^{M+1}} \frac{(M+1)!}{m! (M-m)!}. \]

Before substituting the beta mixture approximation, we introduce the auxiliary basis functions

\[ B_{m, k}(t) = b_{m, k} \frac{1}{h^{m+k+1}} \frac{(m+k+1)!}{m! k!}. \]

Note that \(\ell_m = B_{m, M-m}\). Next, we substitute the approximation into the convolution

\[ \hat z(t) = \int_{t-h}^t \hat \alpha(t - s) x(s) \, \mathrm ds = \sum_{m=0}^M c_m \int_{t-h}^t \ell_m(t - s) x(s) \, \mathrm ds = \sum_{m=0}^M c_m \int_{t-h}^t B_{m, M-m}(t - s) x(s) \, \mathrm ds = \sum_{m=0}^M c_m z_{m, k}(t), \]

where we have introduced

\[ z_{m, k}(t) = \int_{t-h}^t B_{m, k}(t - s) x(s) \, \mathrm ds. \]

We derive a recursion for the normalization factor in the auxiliary basis functions:

\[ \begin{align*} b_{m, k} = \frac{1}{h^{m+k+1}} \frac{(m+k+1)!}{m!}{k!} = \frac{m+k+1}{m h} \frac{(m+k)!}{(m-1)! k!} = \frac{m+k+1}{m h} b_{m-1, k}, \\ b_{m, k} = \frac{1}{h^{m+k+1}} \frac{(m+k+1)!}{m!}{k!} = \frac{m+k+1}{k h} \frac{(m+k)!}{m! (k-1)!} = \frac{m+k+1}{k h} b_{m, k-1}. \end{align*} \]

We use these recursions to obtain the derivatives of the auxiliary basis functions

\[ \begin{align*} \dot B_{m, k}(t) &= m b_{m, k} t^{m-1} (h - t)^k - k b_{m, k} t^m (h - t)^{k-1} = \frac{m+k+1}{h} b_{m-1, k} t^{m-1} (h - t)^k - \frac{m+k+1}{h} b_{m, k-1} t^m (h - t)^{k-1} = \frac{m+k+1}{h} (B_{m-1, k}(t) - B_{m, k-1}(t)), & m, k &\geq 1, \\ \dot B_{m, 0}(t) &= m b_{m, 0} t^{m-1} = \frac{m+1}{h} b_{m-1, 0} t^{m-1} = \frac{m+1}{h} B_{m-1, 0}(t), & m &\geq 1, \\ \dot B_{0, k}(t) &= -k b_{0, k} (h - t)^{k-1} = -\frac{k+1}{h} b_{0, k-1} (h - t)^{k-1} = -\frac{k+1}{h} B_{0, k-1}(t), & k &\geq 1, \\ \dot B_{0, 0}(t) &= 0. \end{align*} \]

Furthermore, we exploit that

\[ \begin{align*} B_{m, k}(0) &= \begin{cases} \frac{k+1}{h}, & m = 0, \\ 0, & \mathrm{otherwise}, \end{cases}, & B_{m, k}(h) &= \begin{cases} \frac{m+1}{h}, & k = 0, \\ 0, & \mathrm{otherwise}, \end{cases} \end{align*} \]

Finally, we use Leibniz’ integral rule to derive differential equations for the auxiliary memory states:

\[ \dot z_{m, k}(t) = B_{m, k}(0) x(t) - B_{m, k}(h) x(t - h) + \int_{t-h}^t \dot B_{m, k}(t - s) x(s) \, \mathrm ds \]

We specialize this depending on \(m\) and \(k\):

\[ \begin{align*} \dot z_{m, k}(t) &= \frac{m+k+1}{h} \left(\int_{t-h}^t B_{m-1, k}(t - s) x(s) \, \mathrm ds - \int_{t-h}^t B_{m, k-1}(t - s) x(s) \, \mathrm ds\right) = \frac{m+k+1}{h} (z_{m-1, k}(t) - z_{m, k-1}(t)), & m,k &\geq 1, \\ \dot z_{m, 0}(t) &= -\frac{m+1}{h} x(t - h) + \frac{m+1}{h} \int_{t-h}^t B_{m-1, 0}(t - s) x(s) \, \mathrm ds = \frac{m+1}{h}(z_{m-1, 0}(t) - x(t - h)), & m &\geq 1, \\ \dot z_{0, k}(t) &= \frac{k+1}{h} x(t) - \frac{k+1}{h} \int_{t - h} B_{0, k-1}(t - s) x(s) \, \mathrm ds = \frac{k+1}{h} (x(t) - z_{0, k-1}(t)), & k &\geq 1, \\ \dot z_{0, 0}(t) &= \frac{1}{h} (x(t) - x(t - h)). \end{align*} \]

To summarize, the convolution \(z(t_f)\) can be approximated by \(\hat z(t_f)\) given by

\[ \hat z(t_f) = \sum_{m=0}^M c_m z_{m, M-m}(t_f), \]

where the auxiliary memory states are obtained as the solution to the initial value problem

\[ \begin{align*} z_{m, k}(t_0) &= \int_{t_0-h}^{t_0} \ell_m(t_0 - s) x(s) \, \mathrm ds, \\ \dot z_{m, k}(t) &= \frac{m+k+1}{h} (z_{m-1, k}(t) - z_{m, k-1}(t)), & m,k &\geq 1, & t &\in [t_0, t_f], \\ \dot z_{m, 0}(t) &= \frac{m+1}{h}(z_{m-1, 0}(t) - x(t - h)), & m &\geq 1, & t &\in [t_0, t_f], \\ \dot z_{0, k}(t) &= \frac{k+1}{h} (x(t) - z_{0, k-1}(t)), & k &\geq 1, & t &\in [t_0, t_f], \\ \dot z_{0, 0}(t) &= \frac{1}{h} (x(t) - x(t - h)), &&& t &\in [t_0, t_f]. \end{align*} \]

Note that this system can be reduced by exploiting the relation between Bernstein polynomials and Legendre polynomials (Farouki, 2012).

References

Farouki, R.T., 2012. The Bernstein polynomial basis: A centennial retrospective. Computer Aided Geometric Design 29, pp. 379–419. DOI: https:doi.org10.1016j.cagd.2012.03.001.