Low Pass Filters and Laplace Transform

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Discussion Overview

The discussion revolves around the realization of an ideal "brick-wall" lowpass filter in practice, particularly focusing on the relationship between time-domain and frequency-domain responses using the Laplace transform. Participants explore the implications of the filter's characteristics and the nature of input signals.

Discussion Character

  • Homework-related
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question whether r(t) represents any input signal or a specific type, suggesting it could be a periodic signal or the impulse response of the filter.
  • There is a discussion about the need to consider r(t) as a sum of harmonics when discussing lowpass filters and the implications for the transfer function H(s).
  • One participant suggests that the inability to change amplification in steps for varying ω is a key reason the ideal filter is unrealizable.
  • Another participant expresses uncertainty about explaining the unrealizability of a "brick-wall" step in amplification due to language barriers.
  • It is proposed that finding the impulse response of the brick-wall filter could clarify its unrealizability.
  • One participant argues that proving the unrealizability via the Laplace transform is problematic, particularly with the single-sided Laplace transform, which assumes no activity before t = 0.
  • Another participant notes that the Fourier transform indicates that the inverse transform of the brick filter has a response for t < 0, complicating the realization for a causal network.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of proving the unrealizability of the brick-wall filter using the Laplace transform, with some suggesting it is not possible while others explore the implications of the filter's characteristics.

Contextual Notes

There are limitations regarding assumptions about the input signal and the definitions of the transforms being discussed, particularly the implications of using single-sided versus double-sided Laplace transforms.

jendrix
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Homework Statement



Given that r(t) = L^-1 (Inverse laplace) *H(S) and by making the link between the time-domain and frequency-domain responses of a network, explain in detail why the ideal “brick-wall” lowpass filter is not realisable in practice. [/B]

Homework Equations

The Attempt at a Solution



Is r(t) any input signal or is it a specific type? I am completely stumped by this.

Regards
 
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jendrix said:
Is r(t) any input signal or is it a specific type?
In principle r(t) could be any type of periodical signal, but speaking of a "low pass filter", lower frequencies are meant. So r(t) must be regarded as a sum of its harmonics, ( Fourier transform ).

Having a transfer function, H(s), you must substitute the "s", so that it becomes H(jω). Then you asked to explain why the amplification in the filter cannot be changed in steps, varying ω.
 
jendrix said:
Is r(t) any input signal or is it a specific type?
From the context of your post, I'd assume H(s) is the transfer function of the brick-wall lowpass filter and so r(t) is its time-domain response (impulse response).

If you determine r(t), it should be pretty clear why it's unrealizable.
 
Hesch said:
In principle r(t) could be any type of periodical signal, but speaking of a "low pass filter", lower frequencies are meant. So r(t) must be regarded as a sum of its harmonics, ( Fourier transform ).

Having a transfer function, H(s), you must substitute the "s", so that it becomes H(jω). Then you asked to explain why the amplification in the filter cannot be changed in steps, varying ω.
Is it because you can't vary w as it is a filter and therefore w will stop at a certain value?
 
ω is the frequency of the input. You may set it to any value.

I cannot explain why a filter that makes a "brick-wall" step in the amplification from say ω = 9.99999 to 10.00001 is unrealizable, since english is not my mothers tongue.
 
Find the impulse response of whatever brick-wall filter you're considering to see why it's unrealizable.

The Wikipedia page on Sinc filters should be a big help.
 
I don't think this can be proven via the Laplace transform, at least not the commonly understood single-sided (s-s) laplace transform. That's because the s-s laplace a priori assumes no activity before t = 0 (the double-sided (d-s) transform allows this, as does the Fourier transform.)

With the Fourier transform it can readily be shown that the inverse transform of the brick filter has response for t < 0 but this is impossible for a causal network output where the input is zero for t < 0, hence the filter realization is impossible too. But the problem asked to use the laplace to prove this & I don't think that's doable (unless as I say the d-s laplace is used).
 

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