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I was studying the introduction to wavelets and its benefit over the frequency domain. I said that:

Fourier analysis can't localize signals both in time and frequency domain. Fourier analysis can localize signal in frequency domain very well, but not so much in time domain. While wavelet has the advantage of localizing signals both in time and frequency domain.

What does this mean? Could you give me any example or link that explains this scenario better?

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In the Fourier transform, the basis functions are complex exponentials. These functions are perfectly localized in the frequency domain, i.e., they exist at one frequency, but they have no time localization because of their infinite duration. The localization of a function depends on its spread in time and frequency. A complex exponential has zero spread in frequency, but it has infinite spread in the time domain. Consequently, any signal analyzed by the Fourier transform is only localized in frequency (e.g., frequencies of sinusoidal components can be identified), but not at all in time (e.g., temporal changes of signal properties cannot be localized).

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To localize here means: to find where the signal is mostly concentrated, and with what precision. This could be either in the time or the frequency domain. An answer could be: the signal's center of mass is at 1.5 s, plus or minus 0.3 s; its main frequency is at 100 Hz, plus or minus 10 Hz.

Behind this formulation lay indeterminacy principles: one cannot be arbitrary accurate in location both in time and frequency. If you only want time or frequency accuracy, time (signal) or frequency (Fourier) is fine. But then you loose almost every precision in the other domain (resp. frequency or time).

Wavelet mitigate this effects, and time-windowed Fourier , somehow could help you

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  • $\begingroup$ Thanks for explaining what it means to localise a wave function. $\endgroup$ – onurcanbektas Apr 25 '18 at 3:46
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A FT provides enough information to perfectly reconstruct a bandlimited input, including exact localization of everything. But the information about localized behavior is spread out in the FT phase across an infinite extent of the frequency domain.

However, in practice, FFT analysis is done using one fixed size of window, without reconstruction, and looking only at local magnitudes, which do not contain enough info about whether an impulse (tone burst, etc.) occurred in the left, right or middle (etc.) of the FFT aperture.

Whereas, wavelet analysis is more often done using various sizes and overlaps of basis functions. If an impulse (narrow-band burst, etc.) is located at an unknown position within one wavelet, another set of smaller and offset wavelets might identify in it just one, and thus better localize it.

One could also do something similar by using a bunch of sets of different sizes of windowed FFTs at varying overlaps per set the get a roughly equivalent quality of localization information, but at a likely much higher computational cost than using wavelets for analysis.

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