# Why the delta at the end of the approximation?

The equation to approximate an input signal with a unit impulse in Continuous Time, is shown below, before we take the limit $\hat{x}(t)=\frac{lim}{\Delta\rightarrow0}\sum^{\infty}_{-\infty}x(k\Delta)\delta_\Delta(t-k\Delta)\Delta$ <-- why is there a final $\Delta$ multiplying the $\delta_\Delta(t-k\Delta)$?

• try defining formally $\delta_\Delta(t)$. what is the height of that function, given a specific $\Delta$? May 21, 2014 at 19:58
• Could you clarify what do you mean? Is it the discretation process you're after?
– Royi
May 21, 2014 at 21:10
• This is to do with the convolution integral and dissertation sounds like it has something to do with representing a CT signal as as series of discrete signals. i think the $\delta_\Delta(t-k\Delta)\Delta$ part does this?
– Sach
May 22, 2014 at 7:19
• the height of the function at every $\Delta$ is given by $x(k\Delta)$ for the relevant value of k.
– Sach
May 22, 2014 at 7:20
• yes @Princ3Sach, the height of the function, $\hat{x}(t)$ when $k\Delta \le t < (k+1)\Delta$ is $x(k\Delta)$, but the height of $\delta_\Delta(t-k\Delta)$ is $\frac{1}{\Delta}$ because the width is $\Delta$ and the area has to be 1. so now do you see why you need that factor of $\Delta$ in there? May 23, 2014 at 3:00

Here the signal is in continuous time domain. We can approximate any signal with weighted integral of unit impulse. As the signal is in continuous time domain integration is used instead of summation.

The actual equation is , $$x(t)=\int_{-\infty}^{\infty}{x(t_0)\delta(t-t_0)dt}$$

In your equation ${lim}_{\Delta\rightarrow0}\sum^{\infty}_{-\infty}$ stands for integration so $\Delta$ is required at the end, which stands for dt in integral.

Hope you clear that.

• GREAT! I think i get it. Appreciate it!
– Sach
May 22, 2014 at 7:21

$\delta_\Delta(t-k\Delta)$ is a spike with value 1. $\Delta$ is the width. It is like you are approximating the area under $x(t)$ with a bunch of rectangles of height $x(k\Delta)$ and width $\Delta$. So area is width by height, hence the multiplication. $x(k\Delta)\delta_\Delta(t-k\Delta)$ is a just a way of writing the discrete set of values at $x(k\Delta)$.