# Impulse Response, Fourier and Laplace Transforms, and Parseval's Relation

*From*: "Artist" <artist@xxxxxxxxxxxxxxxx>*Date*: Wed, 6 Sep 2006 19:45:15 -0700

I need help resolving a contradiction relating the area under an impulse

response curve of a system and the system's frequency response.

According to Parseval's Relation the area under the impulse response must

equal the area under the impulse response's Fourier Transform. Also, this

Fourier Transform is the frequency response of the system having that

impulse response. This is a consequence of both having to have the same

energy. I have thought about this and it makes perfect sense to me. But when

looking at Bode plots of a single pole low pass filter I see the area under

the frequency response changing with changes in the time constant of the

filter. If two such filters having different bandwidths have their outputs

normalized to the same DC gain, the area under their impulse responses must

also be the same because when the impulse response of each is convolved with

a step function the final values they settle on must be the same, the DC

gain value.

So what I do not understand is how a Bode Plot of a filter's Laplace

Transform can show an area under the frequency reponse curve that obviously

changes with the time constant of the exponential decay of their impulse

response, yet the area under the Fourier Transform does not change with

change with time constants but at the same time the Fourier Transform must

also be the frequency response.

Are these two different kinds of bandwidths? I got puzzled by this apparent

contradiction when I started thinking about how the time constant of the

exponential decay of an impulse response relates to the square root of the

area under a frequency response curve and resulting system output noise in

response to white input noise.

.

**Follow-Ups**:**Re: Impulse Response, Fourier and Laplace Transforms, and Parseval's Relation***From:*John O'Flaherty

**Re: Impulse Response, Fourier and Laplace Transforms, and Parseval's Relation***From:*Rene Tschaggelar

**Re: Impulse Response, Fourier and Laplace Transforms, and Parseval's Relation***From:*Tim Wescott

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