State-space representation of gammachirp filter

Question

Let $h$ be the impulse response of the gammachirp filter:

$$ h(t) = \exp(c_1 + c_2 t + c_3 \ln t) \cos(c_4 + c_5 t + c_6 \ln t) $$

The output $y$ of the filter is given by the convolution of the input $x$ and $h$:

$$ y = x \ast h $$

What is the state-space representation of this filter?

Answer 1

The impulse response of a LTI system with a finite dimensional state space can be computed with,

$$ h(t) = \left\{\begin{array}{l l} C\, e^{A\, t} B + D\, \delta(t) & \text{if } t \geq 0 \\ 0 & \text{else} \end{array} \right. $$

which should always consist out of a finite sum of exponentials (potentially complex conjugate pairs and/or multiplied by polynomials when $A$ has Jordan blocks of larger than size one).

When $c_3$ and $c_6$ are not zero, then the impulse response you provided can not be written as a finite sum of such terms. Therefore there is no state space model, with a state space of finite size, which behaves exactly like such filter.

Answer 2

A state-space representation can either indicate a LTI, LTV or nonlinear system. As @fibonatic said, your system surely does not have a LTI state-space representation. Assume that the system is described in s-s form as: $$\begin{align}\dot{x}&=f(x,t)+u\\y&=x\end{align}$$ Then we can write $$\begin{align}\dot{h}&=f(h,t)+\delta(t)\\ &=\left\{\left(c_2+\frac{c_3}t\right)-\left(c_5+\frac{c_6}t\right)\tan(c_4 + c_5 t + c_6 \ln t)\right\}h(t)\\&\stackrel{\Delta}= g(t)h(t)\end{align}$$ This might give us the idea that we can write $f(x,t)$ as $$f\left(x,t\right)=g(t)x-\delta(t)$$ but as you see, this $f$ will be dependent on the input which contradicts our initial assumption that $f(x,t)$ is independent of $u$. So there can't be any affine state space representation of this filter.