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



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.


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