Since intuitively speaking, a Fourier transform decomposes a signal into its frequencies or, in other words, into sinusoids, I'm confused about the exact definition which seems to decompose a signal into real and imaginary sinusoids.
For a real signal, how can it be decomposed into imaginary sinusoids? Are the imaginary portion supposed to all cancel out eventually?
Best Answer
For real signals:
a. the even part of the real signal gets transformed to the real part of the real signal's Fourier Transform, which will be even; and
b. the odd part of the real signal gets transformed to the imaginary part of the real signal's Fourier Transform, which will be odd.
If $f(t)$ is your real signal, then it's even and odd parts before being transformed, are:
$$f_e(t) = \dfrac{f(t) + f(-t)}{2}$$ $$f_o(t) = \dfrac{f(t) - f(-t)}{2}$$
So, if $f(t)$ is real and
$$F(s) = \mathscr{F}\left\{f(t)\right\}$$
then
$$\Re[F(s)] = \mathscr{F}\left\{f_e(t)\right\}$$ $$i\Im[F(s)] = \mathscr{F}\left\{f_o(t)\right\}$$