im confused on how these folks seems to like convert a frequency into a time function, and a time function into a frequency function. i know that time function uses amplitude that varies over time, but i dont understand the mystery of frequency domain. if time has amplitude then frequency has what?
here is a sample image
i still cant get the relevance or the gist between transitioning between time and frequency. thanks
On
A simple way to think about the relationship between signals in the time domain versus frequency domain is to consider the expansion of a signal as a linear combination of its projections onto basis vectors, each of which represents a specific frequency:
$$ x(t) = \int X(\omega) \phi(\omega, t)\, d\omega $$
where the functions $\{\phi(\omega, \cdot)\}$ form a basis for the signal space.
To gain intuition, think of this as a weighted sum of the "frequency vectors" $\phi(\omega, \cdot)$, with weights $X(\omega)$.
Often one uses the exponential function $\phi(\omega, t) = e^{-i2\pi \omega t}$ for the basis vector, which represents a (complex) sinusoidal wave of frequency $\omega$.
Now, for each $\omega$, the number $X(\omega)$ is just the projection of your signal $x$ onto the $\omega$-th frequency vector:
$$ X(\omega) = \int_{\mathbb R} x(t) e^{-i2\pi \omega t} \, dt $$
and this number can be interpreted as telling you how much of the $\omega$-the frequency is present in your signal.
You can plot the (magnitude of the) function $X(\omega)$ as $\omega$ varies along the horizontal axis, as in the bottom pane of your figure, and you have a representation of your function in the frequency domain. For each $\omega$, the height of the function $|X(\omega)|$ tells you the weight or "importance" of the frequency $\omega$ in the linear combination of your signal (the first integral above).
(Of course, I'm being very informal and non-rigorous here, but rigor is what your textbook is for. I'm just trying to give you a sense of how, given a signal $f(t)$ in the time domain, the Fourier coefficients $X(\omega)$ represent the signal in the frequency domain.)
In electronics, for example, if you have a filter and you want to know its behaviour vs. frequency, you have to transform the response of the circuit vs time to a function of the frequency. This means you must use the Fourier transform. Whenever you want to know how a dynamic system behaves vs frequency, you have to use it. For example: $$\ddot{x}(t)+\omega^2x(t)=f(t)$$ is the harmonic oscillator differential equation. If you want to know all the frequencies of the oscillator forced by an external force you have to use the Fourier transform of the $x(t)$ and so you obtain $X(\omega)$. You can plot $X(\omega)$ vs. $\omega$ and this gives you, for every frequency ($\omega=2\pi f)$ the 'power' associated to that frequency.