Proof of the scaling property of an impulse function

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SUMMARY

The discussion focuses on proving the scaling property of the impulse function, specifically the equation delta(a(t-to)) = 1/abs(a)*delta(t-to). The user explores the definition of the delta function, particularly delta(a, x), and substitutes k(t-to) for x to demonstrate that the area under the function scales by a factor of 1/k while the function itself remains unshifted. This method effectively validates the scaling property, although the user questions the mathematical rigor of their approach.

PREREQUISITES
  • Understanding of impulse functions in signal processing
  • Familiarity with the properties of the Dirac delta function
  • Basic knowledge of limits and mathematical proofs
  • Concept of area under curves in calculus
NEXT STEPS
  • Study the properties of the Dirac delta function in detail
  • Learn about the implications of scaling properties in signal processing
  • Explore mathematical rigor in proofs related to impulse functions
  • Investigate applications of impulse functions in systems analysis
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Students and professionals in signal processing, mathematicians focusing on functional analysis, and anyone seeking to understand the properties and applications of impulse functions in systems theory.

amolraf
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I am presently taking my first course in signals and systems and I have been charged with proving the scaling property of the impulse function; that:

delta(a(t-to)) = 1/abs(a)*delta(t-to)

I am seriously miffed and need some help.
 
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Well I messed around with it for a bit and found one way to do it- take the definition of the delta function before the limit- like this:

0 for x < -a
delta(a, x) = 1/2a for -a < x < a
0 for x > a

and substitude k(t-to) for x...you'll find after messing around with it that a gets pulled in by a factor of 1/k. The area becomes 1/k and but the function is not translated, and hence your property is proved, although how rigorous this would be considered by mathematical standards I don't know.
 

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