If I have the following model, $$\int_x\int_\tau F(x)\cdot\delta(t -d(x) - \tau)\: d\tau \: dx$$ where $t$ represents time in nanoseconds (ns), $d(x)$ is round trip time in ns and $\tau$ is total time from the starting time in ns. How can I explicitly write out $d(x)$ to represent time if the $x$ value I pass in is measured in meters.
I originally thought $d(x)$ should be linear function in the form: $$d(x) = m \cdot x + b $$ where $b$ is the default offset time in nanoseconds (when $x = 0$, we get $b$), $m$ is slope, and $x$ is a measurement in meters that affects nanoseconds. But since $x$ and $b$ are in different units, then this cannot output a round trip time in nanoseconds, ($x$ alone can't add with $b$).
I'm currently using values for $m$ that I don't think convert $x$ into a unit of time. I noticed it did not do what was intended when working this out numerically in a computer program. I am using real data for the $m$ values. I need to make sure that $m$ does some sort of conversion to with $x$ before I do the integration right? It should convert this value into a value of nanoseconds.