2704 Proof of Green's Formula

Green's Formula: For the equation $$P(D)y=f(t),y(t)=0,y'(t)=0 \text{ for } t<0\tag{1}$$ the solution for $t > 0$ is given by $$y(t)=(f*w)(t)=\int_{0^-}^{t+} f(\tau)w(t-\tau)d\tau\tag{2}$$ where $w(t)$ is the weight function (unit impulse response) for the system.

Proof: The proof of Green's formula is surpisingly direct. We will use the linear time invariance of the system combined with superposition and the definition of the integral as a limit of Riemann sums.
To avoid worrying about $0^-$ and $t^+$ we will assume that $f(t)$ is continuous. With appropriate care, the proof will work for an $f(t)$ that has jump discontinuities or contains delta functions.
As we saw in the session on Linear Operators in the last unit, linear time invariance means that $$y(t) \text{ solves } P(D)y=f(t) \rArr y(t-a) \text{ solves } P(D)y=f(t-a)\tag{3}$$ Or, in the language of input-response, if $y(t)$ is the response to input $f(t)$ then $y(t - a)$ is the response to input $f(t - a)$.
First we will partition time into intervals of width $\Delta t$. So, $t_0 = 0, t_1 = \Delta t, t_2 = 2\Delta t$, etc.

Next we decompose the input signal $f(t)$ into packets over each interval. The $k$th signal packet, $f_k(t)$ coincides with $f(t)$ between $t_k$ and $t_{k+1}$ and is 0 elsewhere $$f(t_k)=\begin{cases} f(t)& \text{for }t_k\leq t \leq t_{k+1}\\ 0& \text{elsewhere} \end{cases}$$
It is clear that for $t > 0$ we have $f(t)$ is the sum of the packets $$f(t)=f_0(t)+f_1(t)+f_2(t)+\ldots+f_k(t)+\ldots$$ A single packet $f_k(t)$ is concentrated entirely in a small neighborhood of $t_k$ so it is approximately an impulse with the same size as the area under $f_k(t)$. The area under $f_k(t) \approx f(t_k)\Delta t$. Hence, $$f_k(t)\approx (f_k(t)\Delta t)\delta(t-t_k)$$ The weight function $w(t)$ is response to $\delta(t)$. So, by linear time invariance the response to $f_k(t)$ is $$y_k(t)\approx (f_k(t)\Delta t)w(t-t_k)$$ We want to find the response at a fixed time. Since $t$ is already in use, we will let $T$ be our fixed time and find $y(T)$. Since $f$ is the sum of $f_k$, superposition gives $y$ is the sum of $y_k$. That is, at time $T$ $$\begin{aligned} y_(T)&\approx y_0(T)+y_1(T)+\ldots\\ &=(f(t_0)w(T-t_0)+f(t_1)w(T-t_1)+\ldots)\Delta t\tag{4} \end{aligned}$$ We can ignore all the terms where $t_k > T$. (Because then $w(T - t_k) = 0$, since $T - t_k < 0$.) If $n$ is the last index where $t_k < T$ we have $$y_(T)\approx (f(t_0)w(T-t_0)+f(t_1)w(T-t_1)+\ldots+f(t_n)w(T-t_n))\Delta t$$ This is a Riemann sum and as $\Delta t \rarr 0$ it goes to an integral $$y(T)=\int_0^T f(t)w(T-t)dt$$ Except for the change in notation this is Green's formula $(2)$.

Note on Causality: Causality is the principle that the future does not affect the past. Green's theorem shows that the system $(1)$ is causal. That is, $y(t)$ only depends on the input up to time $t$. Real physical systems are causal.
There are non-causal systems. For example, an audio compressor that gathers information after time $t$ before deciding how to compress the signal at time $t$ is non-causal. Another example is the system with input $f(t)$ and output $y(t)$ where $y$ is the solution to $y' = f(t + 1)$.