Low Pass Filters and Laplace Transform

[jendrix]

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

 

[Hesch]

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.

 

[milesyoung]

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.

 

[jendrix]

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?

 

[Hesch]

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.

 

[milesyoung]

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.

 

[rude man]

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).