Convolution
Convolution is often presented in a manner that emphasizes efficient calculation over comprehension of the convolution itself. To calculate in a pointwise fashion, we're told: "flip one of the input signals, and perform shift+multiply+add operations until the signals no longer overlap." This is numerically valid, but you could in fact calculate the convolution without flipping either signals. We'll perform the latter here for illustration. Consider the convolution of the following constant input and causal impulse reponse:
$ y[n] \,=\, x[n] \ast h[n] $
$ x[n] \,=\, 1 \;\;\;\;\; \forall \, n $
$ h[n] = \left\{ \begin{array}{lr} \mathrm{e}^{-n} & : n \geq 0\\ 0 & : n < 0 \end{array} \right. $
Our input function x is a constant 1. You can think of this as a (Kronecker) delta function occurring at every discrete time step, and therefore causing the impulse reponse to 'go off' at every time step (see graphs below for better visualization). What we term the 'convolution' is just the summation of all these time-shifted impulse reponses that have non-zero outputs at the time we want to find the convolution for (ie time n for y[n]). To be more specific we can consider an impulse that was generated at time n=0. We know that output from that singular impulse response should be $ (1)*(e^{0}) = 1 $ at n=0, $ (1)*(e^{-1}) = \frac{1}{e} $ at n=1, $ (1)*(e^{-2}) = \frac{1}{e^{2}} $ at n=2, and so on. Without even using to the convolution formula given to us in class, we can intuitively grasp what the convolution should be for any time n (doesn't matter what n we choose since input has always been and will always be the same). It should be the
$ \int_{-\infty}^{\infty} h(\tau)\,\mathrm{d}\tau \,=\, \int_{0}^{\infty} h(\tau)\,\mathrm{d}\tau \;\;\;\;\; \because h(t)=0 \;\;\; \forall \, t<0 $
$ \Rightarrow \int_{0}^{\infty} \mathrm{e}^{-\tau}\,\mathrm{d}\tau \,=\, \left.-\mathrm{e}^{-\tau}\right|_{0}^{\infty} \,=\, -(\mathrm{e}^{-\infty} - \mathrm{e}^{0}) \,=\, -(0 - 1) \,=\, 1 $
$ \int_{-\infty}^{\infty} h(t-\tau)\,\mathrm{d}\tau \,=\, \int_{-\infty}^{t} h(t-\tau)\,\mathrm{d}\tau \;\;\;\;\; \because h(t)=0 \;\;\; \forall \, t<0 $
$ \Rightarrow \int_{-\infty}^{t} \mathrm{e}^{-(t-\tau)}\,\mathrm{d}\tau \,=\, \int_{-\infty}^{t} \mathrm{e}^{\tau-t}\,\mathrm{d}\tau \,=\, \left.\mathrm{e}^{\tau-t}\right|_{-\infty}^{t} \,=\, \mathrm{e}^{0} - \mathrm{e}^{-\infty-t} \;\; (\forall \, t>0) \,=\, 1 - 0 \,=\, 1 $
