Why is the ideal low pass filter not achievable?

Question

I find myself to be quite confused by some concepts of DFT. Suppose I have some spectrum-normalized signal (which I simply receive without any prior information) $$A\cos\left(0.5n\right)\ +\ B\cos\left(0.7n\right)\ +\ C\cos\left(1.2n\right)\ +\ D$$ with A,B,C,D being some constants

Now I find that the DFT of the same signal corresponds to a different set of basis functions given by $e^{\frac{2\pi jn}{N}}$ for N being the number of data points.

Now I can retrieve the signal by multiplying each "complex frequency component" with its "complex amplitudes" and adding them. Now my question is as follows:

If I apply an ideal low pass filter, I should get an infinite sequence of data points because the sinc function would have infinite points in it and convolution of it with an infinite set of data points would be again infinite and double-sided. (I hope this is a correct guess??).

But if I simply had a mechanism to be able to separate this unknown sequence into the above form, (say some alternate transform that would give me this decomposition instead of the exponential decomposition) then will I be able to design the "Ideal low pass filter" in that case just by knowing the above decomposition? I feel it can be proved (though the uniqueness I'm not sure) that such a decomposition will exist because it would be a set of N polynomials in the cosines of the found natural frequencies, like for the mth data point$$y\left(m\right)\ =\ \sum_{i=1}^{N}\cos\left(m\alpha_{i}\right)$$ where $\alpha_{i}$ is the ith frequency. Here the frequencies are normalized so the +$2\pi$ for each frequency idea cannot be used to refute the uniqueness Also if I knew that the signal had exactly N points, then the signal could be represented at most by N real frequencies. Then why is the ideal low pass filter not realizable? I sincerely apologize if the question was very naive, it could potentially be so.

Answer 1

General Background: Representing a signal numerically in a computer requires the signal to be discrete. That means it's periodic in the other domain. When you use something like a DFT you need the signal to be discrete in both domains: That means the signals are ALSO periodic in both domains.

If I apply an ideal low pass filter, I should get an infinite sequence of data points because the sinc function ...

If you want to do this numerically in a computer, it's NOT a sinc function. The lowpass filter MUST be periodic in both domains and so what you get in the time domain is a periodic repetition of a sinc function with a time domain aliasing.

It's still infinite it time. In fact it MUST be infinite in time since it's periodic. As long as all signals involved are actually periodic with a shared Period, than things work "as expected" .

At this point the term "low pass" filter becomes complicated. Since the spectrum is periodic, it will pass arbitrarily high frequencies. So it needs to be interpreted: it removes higher frequencies in the base band or something like this.

$y\left(m\right)\ =\ \displaystyle\sum_{i=1}^{N}\cos\left(m\alpha_{i}\right)$

Your missing phase or sine term here but you are re-inventing the DFT or cosine transform. You run into exactly the same problem. How to choose N ?

Also if I knew that the signal had exactly N points

That's a contradiction. The signal that you described in your original equation is infinitely long. As soon as you make it finite in time, you are windowing it and that changes the spectrum and the math behind considerably.

then the signal could be represented at most by N real frequencies

Correct. That's for example the Discrete Cosine Transform, which is a close cousin of the DFT. But it makes no material difference here.

Then why is the ideal low pass filter not realizable?

It is realizable as long as everything periodic "the right way". The periodicity helps dealing with the infinite length in time: you need only to compute what's happening in one period and you have all information that can be had since all other periods are the same.

However most real world signals are NOT periodic "the right way", so ideal low pass filtering is rarely feasible.