Impulse response of linear time-varying system

Question

I am confused about linear time-varying system. For a time varying system, the output is given by \begin{align} y(t)=\int x(\tau) h_{\tau}(t) d\tau, \end{align} where $ h_{\tau}(t)$ is the output of the system given the input $\delta(t-\tau)$. By manipulating the integration above, we have \begin{align} y(t)=\int x(t-\tau) h_{t-\tau}(t) d\tau \end{align} I found that in some refeneces, we can write above as $y(t)=\int x(t-\tau) h(t,\tau) d\tau$, where $h(t,\tau)=h_{t-\tau}(t)$. To avoid confusion, here I let $g(t,\tau)=h_{t-\tau}(t)$. Thus, \begin{align} y(t)=\int x(t-\tau) g(t,\tau) d\tau \end{align}

From my understanding, the $g(t,\tau)$ is just obtained by substituting the variables in $ h_{t-\tau}(t)$. However, I am not sure whether we can treat $g(t,\tau)$ as the system output of some input signal.

Based on the definition of $h_{\tau}(t)$, though $ h_{\tau}(t)$ is the output of the system given the input $\delta(t-\tau)$, $h_{\tau}(t)$ in fact defines a function over $(t,\tau), \forall t,\tau$. Therefore, the value of $h_{t-\tau}(t)$ can be obtained by substituting the pair $(t,t-\tau)$ in to $h_{\tau}(t)$.

However, can we treat $h_{t-\tau}(t)$ as the output of $\delta(t-(t-\tau))$, i.e., $\delta(\tau)$?

Here, how can we understand the input $\delta(\tau)$? In my view, I think the input of a system should be a function over $t$, and $\tau$ is just a parameter. Does this mean that the input is a constant?

Answer 1

As you've correctly noted, there are two common ways to describe the relation between input and output of a linear time-varying system. The first is

$$y(t)=\int_{-\infty}^{\infty}x(\tau)h(t,\tau)d\tau\tag{1}$$

In this case the kernel $h(t,\tau)$ is the response at time $t$ to an impulse at time $\tau$. I've always found it instructive to look at a few special cases:

• LTI: the system is time-invariant if $h(t,\tau)$ only depends on the difference $t-\tau$, i.e., $h(t,\tau)=h'(t-\tau)$.
• causality: the system is causal if $h(t,\tau)=0$ for $t<\tau$.
• modulation: according to definition $(1)$, a modulator with modulation function $m(t)$ is described by $h(t,\tau)=m(t)\cdot\delta(t-\tau)$.

The second common description of the input-output relation of a linear time-varying system is

$$y(t)=\int_{-\infty}^{\infty}x(t-\tau)h(t,\tau)d\tau\tag{2}$$

In this case, the response to an input $\delta(t-t_0)$ is given by $h(t,t-t_0)$. This means that $h(t,\tau)$ is the response at time $t$ to an impulse at time $t-\tau$. Let's take a look at the same special cases as above:

• LTI: the system is time-invariant if $h(t,\tau)$ is independent of $t$, i.e., $h(t,\tau)=h'(\tau)$.
• causality: the system is causal if $h(t,\tau)=0$ for $\tau<0$.
• modulation: according to definition $(2)$, a modulator with modulation function $m(t)$ is described by $h(t,\tau)=m(t)\cdot\delta(\tau)$.