I'm studying electrical engineering and use correlation, Fourier transform and Laplace transform a lot. I know how and when to use them, however, the interpretation I've seen in the lectures still leave me a bit hungry.
Fourier series is easily interpreted as how much a harmonic contributes to a periodic signal. However, I feel that the way Fourier transform were taught, fails to give a similar simple interpretation.
The focus of these transformations in my lectures is mainly on the transformation of known functions or using the properties (e.g. $d/dt => s$), without much interpretation for transformation a randomly chosen signal. In labs however we focus on FT of measures signals of sensors. We do not know the function beforehand.
Let's assume well behaving (so realistic) real signals $f(t)$ and $g(t)$ here (which we for instance measure). Now I understand that the cross-correlation
$R(\tau)=\int_{-\infty}^{\infty} f(t)g(t+\tau)dt$
is a measure of similarity of functions $f(t)$ and $g(t)$ over a time shift $\tau$.
Can you then say that
$F(s)=\int_{0}^{\infty} f(t)e^{-st}dt = \int_{0}^{\infty} f(t)e^{-\sigma t}[cos(\omega t)+isin(\omega t)]dt $
Is then a measure for similarity between $f(t)$ and the decaying exponential $e^{-st} = e^{-(\sigma +i\omega)t} = e^{-\sigma t}[cos(\omega t)) +i sin(\omega t)]$ that starts at $t=0$? The real part of $F(s)$ expresses the similarity to a decaying cosine and the imaginary part to a decaying sine.
Similarly
$F(\omega)=\int_{-\infty}^{\infty} f(t)e^{-\omega t}dt = \int_{-\infty}^{\infty} f(t)(cos(\omega t)+isin(\omega t))dt $
expresses in its real part the similarity of $cos(\omega t)$ and in the imaginary part $sin(\omega t)$. This then easily explains that some function that is highly similar (but not exactly) to $cos(\omega_{0} t)$ has relative high magnitude for F($\omega_{0}$) in $\omega_{0}$ but also for a band around $\omega_{0}$, as the integral is not exactly zero (which I see in my measurement labs)
Is this interpretation of measures of similarity correct to some degree (especially in the context of measuring unknown signals)?