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From: ken@turtlevax.UUCP (Ken Turkowski)
Newsgroups: net.math
Subject: Re: Re: Non-linear systems: discontinuous functions
Message-ID: <662@turtlevax.UUCP>
Date: Wed, 13-Feb-85 00:55:27 EST
Article-I.D.: turtleva.662
Posted: Wed Feb 13 00:55:27 1985
Date-Received: Thu, 14-Feb-85 07:27:48 EST
References: <209@talcott.UUCP> <328@rlgvax.UUCP> <384@hou2g.UUCP> <1027@sunybcs.UUCP> <386@hou2g.UUCP> <115@redwood.UUCP> <2619@umcp-cs.UUC Wed, 13-Feb-85 00:55:27 EST
Reply-To: ken@turtlevax.UUCP (Ken Turkowski)
Organization: CADLINC, Inc. @ Menlo Park, CA
Lines: 66

In article <3188@umcp-cs.UUCP> chris@umcp-cs.UUCP (Chris Torek) writes:
>If this Dirac delta thing (I don't want to call it a function) is defined as
>
>	f(x) = 0,  x <> 0
>
>	integral from -infinity to infinity f(x) = 1
>
>then in order to make any sense at all, its integral must also be one over
>any interval containing the point zero, and zero for all others.  *Most*
>peculiar...

Great deduction, Sherlock.

>Whatever does one do with this beast, anyway?  (I have a vague idea that
>Dirac integrals are useful in quantum mechanics or something like that.)

I thought that everyone knew what the Dirac delta function was.
I guess not everyone on the net is an electrical engineer...

The Dirac delta distribution (formally not a function) is the formal
derivative of the unit step function (er, distribution), u(x), which is
0 for x less than zero, and 1 for x greater than zero.  Still not
satisfied because we haven't found a function?  Well, integrate it one
more time, and you have the unit ramp function, defined as:

ramp(x) = 0, x <= 0
	  x, x >= 0

Take the formal derivative of the unit ramp, and you have a unit step.
One more derivative gives a unit impulse or delta, and another
derivative gives a unit doublet.

The delta function is a unit under the operation of convolution:

		infinity
f(x)*g(x) = integral	   f(t)g(x-t)dt
		-infinity

where * denotes convolution, not multiplication.  We have

f(x)*delta(x) = f(x)

A linear differential system can be described by a convolution, where
f(x) is called the impulse response, and g(x) is the driving function
or input.  Such differential equations can be solved simply by purely
algebraic techniques.

Many of the netlanders are probably familiar with the concept of frequency.
The delta function is related to frequency through the Fourier transform:

		-infinity
F(t) = integral		  f(x)exp{-itx}dx
		infinity

When f(x) is a delta function,

F(t) = exp{-itx} = cos(xt) - i sin(xt),

a sine wave of frequency x, so that a sine wave in time corresponds to
a delta function in frequency.

-- 

Ken Turkowski @ CADLINC, Menlo Park, CA
UUCP: {amd,decwrl,nsc,seismo,spar}!turtlevax!ken
ARPA: turtlevax!ken@DECWRL.ARPA