Suppose we have a real signal $x(t)$. Now, we know that $x(t)$ can be represented as a sum of sines and cosines. w be the angular frequency.
If $a(\omega)$ be the coefficients of the cosine terms, and $b(\omega)$ be the coefficients of the sine terms, then if we perform a transformation on this signal such that now $b(\omega)$ are the coefficients of not sines but cosines and similarly, $a(\omega)$ the coefficients of the sines, then how will the waveform transform$[x(t)]$ look like?
What will be the similarity between this transformed waveform and the original waveform?
What is the name for this transformation (is it Hilbert transformation? )
Please attach graphs for an example $x(t)$ to clarify.