complex-analysisfourier analysisfourier series
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
Well, since the wavelet transform is not precisely a time-frequency analysis but more a time-scale analysis one can not really display the wavelet coefficients in the time frequency plane. (However, even for the STFT the time-frequency plane is only a crutch for illustration purposes.)
Maybe it helps to think of $N=2^n$. Then you can analyze your signal into $n$ scales with the wavelet transform. See, e.g., this image with $n=3$:
Then the wavelet-transform stores the $c_{31}$ and all the $d_{ij}$'s. If you want to put the coefficients in the time-frequency plane you would have to stretch the tiles as you indicated in your figures. There would be $n$ vertical colums (in $\omega$), the rightmost having $2^{n/2}$ boxes and the number of boxes halving in each column (the last two containing just one box). This adds up to $2^n$.
However, I suggest to read Mallat's "A wavelet tour of signal processing - the sparse way" Chapters 1.3, 4.3 and 8.1. It's quite hard to produce illustrating figures for the wavelet transform here...
Considering the efficiency: To be a bit more precise, the discrete wavelet transform (DWT) of a signal of length $2^n$ and a filter with length $k$ takes $O(k2^n)$, while FFT is $O(n2^n)$ with comparable constants. Hence, DWT is faster it the filter is not too long.
To your first question: Yes, $Y_0$ is the amplitude of the component with zero frequency, which is a constant: $Y_0 \mathrm{e}^{2\pi\mathrm{i}m0/N}=Y_0 \mathrm{e}^0=Y_0$. You can see that you need a component like this because all the non-constant components have an average of $0$, so $Y_0$ is the only component that can give the sum a non-zero average, and in fact it simply measures the average of your signal.
To your second question: I'm not sure I understand it correctly. First, we don't obtain amplitudes for $N-1$ frequency components, but for $N$ frequency components -- this might be related to your first question. Second, why do you write $N/2$ for $N$ even but $N\pm1$ for $N$ odd? Did you mean $(N\pm1)/2$? The reason that $Y_{N/2-n}=\overline{Y_{N/2+n}}$ is that $\mathrm{e}^{2\pi\mathrm{i}m(N/2-n)/N}=\mathrm{e}^{2\pi\mathrm{i}m(N/2-n)/N}\mathrm{e}^{-2\pi\mathrm{i}m}=\mathrm{e}^{2\pi\mathrm{i}m(-N/2-n)/N}=\overline{\mathrm{e}^{2\pi\mathrm{i}m(N/2+n)/N}}$.
Best Answer
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.