# Proper convolution notation

What would be the correct way to write down the convolution in "star" notation for these two functions; $h(t)$ and $\delta(t-x)$. $\delta$ is the Dirac delta function. The integral notation should be

$$\int_{-\infty}^{\infty}h(t-\tau)\delta(\tau-x)d\tau$$

It feels a little awkward with the delta function in there.

• As far as I see, this evaluates to $h(t - x)$, and this is the simplest way you can write it. Am I missing something? Apr 29, 2014 at 16:48
• That is true but I just wanted to know how to write down the notation for the convolution with a dirac delta function not centered at $0$. Apr 29, 2014 at 16:49
• Perhaps $h \star (\delta \circ T_x)$, where $T_x$ is the translation by $x$ (that is, $T_x : \mathbb R \to \mathbb R$, defined by $\tau \mapsto \tau - x$)? Apr 29, 2014 at 16:50
• Yea that makes sense. I guess there isn't a real convention then. Thanks Apr 29, 2014 at 16:53

How about $h* \delta(\cdot - x)$?
Maybe not as rigorous as the previous comments, but probably a bit nicer in notation: You could define an ensemble of functions $\delta_\tau(t):= \delta(t-\tau)$ and then write the convolution as $h*\delta_\tau$.