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?
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References:
[1] Proakis, Digital Communications, 5th ed, p.832
[2] Goldsmith, Wireless Communications, 1st ed, p.67