If I have a periodic signal such as a cosine wave, the application of the Fourier Transform will produce two pulses: one to show the frequency in the positive x-axis, and one of the same value in the negative x-axis.
Will this always be the case for all periodic signals, and if so, then technically can the idea that if I apply the inverse Fourier Transform on a series of pulses of certain values that are the same on both the positive and negative x-axis, i.e. the pulses of certain frequencies on the positive x-axis are mirrored onto the negative x-axis across the y-axis will be the able to produce a periodic signal?
Thanks!
Fourier Transform does not decompose signal into cosine and sine but into complex exponential i.e. $e^{i\omega t}$. For real signals, the magnitude in frequency domain is symmetric w.r.t. $y$ - axis because the imaginary parts need to cancel each others.