Delta/Heaviside Functions

  • Thread starter Ted123
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  • #1
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Homework Statement



Compute the integral

[tex]F(x) = \int^x_{-\infty} f(t) \;dt[/tex]

of the linear combination of Dirac delta-functions

[tex]f(t) = -2\delta (t) + \delta (t-1) + \delta (t-2)[/tex].

Express the result analytically (piecewise on a set of intervals) and draw a sketch of the function [tex]F(x)[/tex].

The Attempt at a Solution



Does [tex]F(x) = -2H(x) + H(x-1) + H (x-2)[/tex] where H is the Heaviside function?

I know how to express the Heaviside/Delta functions in terms of 'jumps' in a graph but the actual values could be anything couldn't they? For instance:

[tex]\begin{displaymath} F(x) = \left\{ \begin{array}{lr}
0, & \;x \leq 0\\
-2, & \;0 < x \leq 1\\
-1, & \;1<x\leq 2\\
0, & \;x > 2
\end{array}
\right.[/tex]

and

[tex]\begin{displaymath} F(x) = \left\{ \begin{array}{lr}
1, & \;x \leq 0\\
-1, & \;0 < x \leq 1\\
0, & \;1<x\leq 2\\
1, & \;x > 2
\end{array}
\right.
\end{displaymath}[/tex]

both respresent that linear combination of Heaviside functions don't they?
 
Last edited by a moderator:

Answers and Replies

  • #2
L-x
66
0

Homework Statement



Compute the integral

[tex]F(x) = \int^x_{-\infty} f(t) \;dt[/tex]

of the linear combination of Dirac delta-functions

[tex]f(t) = -2\delta (t) + \delta (t-1) + \delta (t-2)[/tex].

Express the result analytically (piecewise on a set of intervals) and draw a sketch of the function [itex]F(x)[/itex].

The Attempt at a Solution



Does [tex]F(x) = -2H(x) + H(x-1) + H (x-2)[/tex] where H is the Heaviside function?

I know how to express the Heaviside/Delta functions in terms of 'jumps' in a graph but the actual values could be anything couldn't they? For instance:

[tex]\begin{displaymath} F(x) = \left\{ \begin{array}{lr}
0, & \;x \leq 0\\
-2, & \;0 < x \leq 1\\
-1, & \;1<x\leq 2\\
0, & \;x > 2
\end{array}
\right.[/tex]

and

[tex]\begin{displaymath} F(x) = \left\{ \begin{array}{lr}
1, & \;x \leq 0\\
-1, & \;0 < x \leq 1\\
0, & \;1<x\leq 2\\
1, & \;x > 2
\end{array}
\right.
\end{displaymath}[/tex]

both respresent that linear combination of Heaviside functions don't they?
should make it a bit easier to read, can't get your array to work though.
 
  • #3
446
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should make it a bit easier to read, can't get your array to work though.
This should make it easier!

[PLAIN]http://img824.imageshack.us/img824/9868/heaviside.png [Broken]
 
Last edited by a moderator:
  • #4
L-x
66
0
You have the correct shape, however you may define a value at x= 0, 1, 2 for the heavside step function. Personally I think 1/2 is the most sensible, because it means that the function is odd and the approximation H(x)={1+tanh(kx)}/2 holds exactly in the limit k->infinity.

0, 1 and 0.5 are all valid choices to use, and will often depend on what exactly you are using H for.

http://www.wolframalpha.com/input/?i=0.5(1+tanh+(x))
 
  • #5
446
0
You have the correct shape, however you may define a value at x= 0, 1, 2 for the heavside step function. Personally I think 1/2 is the most sensible, because it means that the function is odd and the approximation H(x)={1+tanh(kx)}/2 holds exactly in the limit k->infinity.

0, 1 and 0.5 are all valid choices to use, and will often depend on what exactly you are using H for.

http://www.wolframalpha.com/input/?i=0.5(1+tanh+(x))
In my definition of H, H(0) is undefined.
 

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