# How do I make sense of the cosine wave having Fourier Transform coefficients which have infinite magnitude?

To illustrate my question better, consider the Fourier Transform of an aperiodic (as a periodic cosine wave has a Fourier Transform not Fourier Series) cosine wave

$$f(x) = \begin{cases} \cos(2\pi f_0x), & \text{ 0 \le x \le 2\pi} \\ 0, & \text{else} \end{cases}$$

$$F[f(x)] = \frac{1}{2}(\delta(f - f_0) + \delta(f + f_0))$$

This above implies an aperiodic cosine wave consists of two Fourier Transform coefficients which have infinite magnitude (as a delta function has infinite magnitude). Therefore, does this imply that a perfect cosine wave can not be produced in real life?

• (1) I don't think your FT is correct. (2) Regardless, a pure cosine wave cannot be produced in real life: you'd have to have turned your source on before the universe even began! Not to mention the need for an infinite amount of energy. – MBaz Sep 17 '19 at 23:57
• @MBaz I often see questions stating "The Fourier Transform of a cosine wave", I thought they couldn't possibly be referring to a periodic cosine wave, so it shouldn't it be an aperiodic wave like I've shown? – vecohah Sep 18 '19 at 0:03
• Please see this question and answers for a hint: dsp.stackexchange.com/q/6038/41790 – Ed V Sep 18 '19 at 0:21
• @vecohah Mathematically it is possible to find the FT of a periodic cosine wave, but you get Dirac deltas. The FT of the truncated cosine doesn't have any deltas. – MBaz Sep 18 '19 at 2:00

Your function is both integrable, and square integrable, because it is continuous on the finite support $$[0\;2\pi]$$. In this context ($$L_1 \cap L_2$$), its Fourier transform is pretty well defined as a classical function (not a distribution like the Dirac), and continuous. So, your formula is incorrect.

So there is no infinite magnitude. Moreover, one cannot simply say that Dirac have "infinite magnitude".

Yet, this would not imply that a perfect cosine wave can not be produced in real life; produced by whom? However, there is little chance that somebody could observe, or capture, a perfect cosine wave in real life (caveat: if real life can be described by a continuous time variable, and amplitude could be continuous as well).