time-frequency domain 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
 A: 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.
A: 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.)
A: This is a bit old, but here are some things to think about.
First, the title is a bit misleading. The title is "time-frequency domain" which is, really, something else. (more on that later).
The whole point of representing a signal in the frequency domain is to be able to see things with more clarity, and to gain more info. The frequency domain also has amplitude.
Fourier transform is about decomposing a signal into sinusoïd of different frequencies. Each sinusoïd with a different frequency 'contributes' its own way so that the 'whole' gives you back your original signal.
This is useful in two ways: Deconstruction, and reconstruction.
Deconstruction is obvious, you want to see which frequencies are in a signal.
Reconstruction is something you don't think about on the spot (as it is presented in courses): It allows you to go the other way. You can "build" a non sinusoidal signal using simple sine waves of different frequencies.
So for example: You can build a square wave using sine waves, which isn't something one thinks about, really.
Another thing is bandwidth. By decomposing a signal, you answer the question: Which frequencies bear the most power or information.
So if the signal you break down is concentrated in the first three harmonics, you don't need to transmit the other ones and save a lot on bandwidth. So instead of transmitting the whole signal, you transmit just the frequencies that are the most relevant.
Also, it enables you to see clearly.
For example, let's say you have a signal that is modulated by another (let's assume they're both sine waves). In the time domain (i.e amplitude vs. time), the result of such modulation is a mess, but in the frequency domain it's so clear, it jumps at you: The frequencies present in the signal are represented by delta functions. You couldn't tell in the time domain.
And finally, I said earlier that the title is misleading.
There is time representation.
There is frequency representation.
There is also time-frequency representation. That is, the signal is both represented in the time-domain and in the frequency domain.
This was needed because Fourier transform tells you about the spectral characteristics of a signal, but doesn't tell you how these change over time.
In other words: It tells you that this frequency exists in that signal, but it doesn't tell you when it appears. Just that it's there.
This is why there's what is called TFRs (Time-Frequency Representations) Joint Distributions, etc..
Have a look at Wigner-Ville Distribution, look up Leon Cohen, too.
If you want to dig deeper, take a look at "Time Frequency Signal Analysis and Processing, A Comprehensive Reference". Elsevier. Edited by Boualem Boashash.
There's also the excellent "Time Frequency Distributions, A Review" by Leon Cohen.
