I am having a discussion with another person about some data that I am looking at. The data can be roughly modeled as a ramp with an additive sinusoid. My friend is arguing that we can model this as:

$\displaystyle f(t) = a t+sin(2 \pi \omega t)$

where $\displaystyle a,t,\omega \in \Re^+ $

He then argues to use the Fourier transform to find the spectrum.

However, I am arguing the following points:

1) The function $\displaystyle f $ is not periodic, as required by the Fourier transform

2) For the Fourier transform to exist, a sufficient condition is that the function is absolutely integrable. The function, as defined above, including the integration limits as defined by the support of $\displaystyle t $, is not absolutely integrable.

3) When my friend is computing the above Fourier transform above, the result is actually the form for a 'sawtooth' plus sinusoid. He is choosing (limiting) the limits of integration, in this case, to $\displaystyle (0, T) $, so by computing the Fourier transform, he is implicitly making the signal periodic with period $\displaystyle T $.

Are my arguments correct?

Assuming what I have said is correct, I expect the discussion will then go as follows: real world signals (such as the output of an amplifier or A/D converter) saturate at some value. So the signal doesn't increase indefinitely. So is it reasonable to say that the "form" of the Fourier transform is still correct? That is, can anything be said about the form of the Fourier transform whose signal is 1) not periodic and 2) violates the existence condition? That is, could limiting arguments be made?

Thank you