Time-varying "impulse response" According to many references [1,2], the time-varying "impulse response" can compute wireless channel output $y(t)$ at time $t$ using the following expression:
$$
y(t) = \int h(\tau, t) x(t - \tau) d\tau
$$

In both references, they state that this represents the response of the channel at time $t$ to an impulse applied at time $t-\tau$. 

It seems reasonable to assume that there is some version of x(t) that involves a delta function that we can apply as an input that returns $h(\tau,t)$ as the output.  

Trying:
$$x(t) = \delta(t)$$
$$\implies y(t) = \int h(\tau, t) \delta(t - \tau) d\tau = h(t,t) \qquad \text{nope}
$$

Trying:
$$x(t) = \delta(t-\tau')$$
$$\implies y(t) = \int h(\tau, t) \delta(t - \tau - \tau') d\tau = h(t-\tau',t) \qquad \text{nope, but closer}
$$

Is there a way to generate something resembling $h(\tau,t)$ as the output? 

$$
$$

References:

[1] Proakis, Digital Communications, 5th ed, p.832

[2] Goldsmith, Wireless Communications, 1st ed, p.67
In writing $h(\tau,t)$ with $t$ is time and $\tau$ is delay, we are in the model that $\tau$ varies "differently" from $t$. Or in other words, they are different notions in spite of the fact that they are both time unit.

Similarly, in this equation
$$y(t) = \int h(\tau, t) x(t - \tau) d\tau$$

The time index of $y(t)$ varies "differently" from the time index of $h(\tau,t)$.

It is easier to write $h(\tau, t')$ and fix the measuring moment $t'=t_0$ for $h(\tau,t_0)$ and applying an input $\delta(t)$

$$y(t) = \int h(\tau, t_0) \delta(t - \tau) d\tau = h(t,t_0)$$

The index $t$ of $h(t,t_0)$ plays the role of delay $\tau$ that is exactly the same expression $h(\tau,t_0)$.

Redo the experiment at $t'=t_1, t_2, ..., $ we have the two dimensional time varying impulse reponse $h(\tau,t)$.