Help converting dirac delta function

[jwang34]

Homework Statement

SO I'm given a dirac delta function, also known as a unit impulse function.
d(t-t'_=(1/P) sum of e^[in(t-t')], for n from negative to positive infinity.
I need to graph this.

Homework Equations

I understand that at t', there is a force made upon the system which results in an impulse function. I need to convert the e^[in(t-t')] to get rid of the imaginary component. For this, Eulers equation is e^(it)=cost+isint

The Attempt at a Solution

So just working with the inside of the summation, e^[in(t-t')]=cos[n(t-t')]+isin[n(t-t')]. But this is treating (t-t') as the independent variable when t is the independent variable. So I'm really not sure if this is right. Any help is greatly appreciated. Thanks.

 

[CompuChip]

Is it an assignment to graph it, or you want to draw it for yourself? And are you supposed to use the equation you gave?

Because, I'm not sure if you can graph it from that definition. Basically, what you wrote down is the Fourier series of the delta function (if you don't know what it is, forget I even said that) but I don't think it will help you draw it.

Basically, you would have to do the following. Start by drawing a square around t = t' with corners (t - 1/2, 0), (t + 1/2, 0), (t + 1/2, 1), (t - 1/2, 1). So you have a square with area one centered at t'. Now halve its width, while doubling it's height (such that the area of the rectangle remains the same, and the base line of the rectangle is still on the axis and it's centered at t'). Now repeat this, making the rectangle smaller and higher all the time, keeping it centered at t' --- making it infinitely thin and infinitely high such that the are remains one. You will not be able to draw this, at some point, it will just be a very high thin half-line from (t', 0) upwards to (t', infinity).

By the way, it is possible to draw it in the space where not t is along the axis, but n is.

 

[jwang34]

Ok, thanks for the reply and I understand what you mean, but basically I'm supposed to show that as n increases, the graph does a better job of approximating the dirac delta function. Also, I just checked the equation again and d(t-t') has a period of 2pi, so P=2pi. So if I interpret your reply correctly, if the width goes to zero, the graph becomes a better representation of the delta function.