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

[Artist]

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.

 

[Tim Wescott]

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.

Parseval's relation says that the time integral of the _square_ of the
impulse response (i.e. the total energy) is equal to the frequency
integral of the _square_ of the frequency response* (i.e. the total energy).

You have added to your difficulties by trying to look at the area under
the frequency response in a Bode plot, which is a log-log plot, not a
linear plot.

* times a factor of 2 pi that I always have to look up.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google? See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

 

[Rene Tschaggelar]

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.

Beside signal parts being passed through a fourpole,
there are also reflected signal parts.

Rene

 

[kkrish]

Laplace transform has an atteneuation
constant but fourier shows the frequency domain represtation of a
signal as such.Fourier transform of a signal should give the exact
time domain representation when inverse fourier transform is taken.

 

[John O'Flaherty]

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.

I think the output of a filter fed by a true step function has infinite
energy, so you can't really sum it up. You have to use an input, like
an impulse, that has finite energy.

{snipped}

 

[Artist]

Parseval's relation says that the time integral of the _square_ of the
impulse response (i.e. the total energy) is equal to the frequency
integral of the _square_ of the frequency response* (i.e. the total
energy).

You have added to your difficulties by trying to look at the area under
the frequency response in a Bode plot, which is a log-log plot, not a
linear plot.

* times a factor of 2 pi that I always have to look up.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google? See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

I am not trying to calculatet the area under a Bode Plot. I use it only to
make the observation the area under it changes.

I had not considered the squaring. I understand it now. The filter with the
higher bandwidth will have an impulse response with more of its area higher
off the baseline than one with a lower bandwidth. The frequency response
curve never gets higher off baseline than its DC gain. So with squaring the
result will be higher energy for the filter with higher bandwidth and more
area under its frequency response curve.

Thanks