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Re: plot dirac delta function? [message #49359 is a reply to message #49355]

Wed, 19 July 2006 09:28

Benjamin Hornberger is currently offline

Benjamin Hornberger
Messages: 258
Registered: March 2004

Senior Member

kuyper@wizard.net wrote:
>
> The Dirac delta function is defined by the following equations:
>
> delta(x) = 0 if x ne 0
>
> integral of delta(x) dx from x0 to x1 = 1, if x0 < 0 and x1 > 0
>
> Notice that this definition fails to identify explicitly the value of
> delta(0). That's because there's no meaningful value that can be
> assigned to delta(0). The best you can do is to call it infinity, but
> even that's not quite right, for reasons that I don't remember right
> now.

I think that (the last paragraph) applies to the continuous case. In the
discrete case, it follows quite clearly from the two equations above
that the delta function must be an array which is 1 for the center (more
on that below) and zero otherwise. Two examples:

1. Convolution of a function (array) with a delta function must
reproduce the function (or shift if the delta function is not centered).
This is clearly fulfilled by an array which has a 1 in one element and
zeroes otherwise.

2. The Fourier Transform of a (centered) delta function is a constant.
The value of the constant depends somewhat on the definition of the FT
(in IDL, the forward FFT of a delta function as defined above is a
constant 1/N, N being the number of array elements, and the reverse FFT
is a constant 1).

You have to keep in mind that for frequency analysis, IDL assumes the
zero frequency ("center") to be at the zero element in real AND Fourier
space. I, like many others, prefer to shift all arrays by minus/plus N/2
before and after an FT and work on "centered" arrays, which makes the
plots look more intuitive.

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[Message index]

		Re: plot dirac delta function? By: James Kuyper on Wed, 19 July 2006 12:20
		Re: plot dirac delta function? By: Benjamin Hornberger on Wed, 19 July 2006 09:28
		Re: plot dirac delta function? By: James Kuyper on Wed, 19 July 2006 08:31
		Re: plot dirac delta function? By: Paolo Grigis on Wed, 19 July 2006 04:52
		Re: plot dirac delta function? By: swingnut on Sun, 23 July 2006 17:54

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