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Okay, let's embark on a journey from Fourier analysis to wavelet transforms, with a special focus on the Haar wavelet. This will be a detailed guide, covering the theoretical foundations, motivations for moving beyond Fourier, and a practical implementation of the Haar wavelet transform in Python.

*I. The Fourier Transform: A Foundation*

The Fourier Transform (FT) is a cornerstone of signal processing. It decomposes a signal (a function of time, space, or some other variable) into its constituent frequencies. The core idea is that any reasonably well-behaved signal can be expressed as a sum (or integral) of sines and cosines.

*1.1 Key Concepts*

*Frequency Domain:* The FT represents a signal in the frequency domain, where the horizontal axis represents frequency and the vertical axis represents the amplitude or magnitude of each frequency component.

*Basis Functions:* The FT uses sines and cosines (or complex exponentials) as its basis functions. These functions are orthogonal to each other, meaning they don't interfere when you decompose a signal.

*Decomposition:* The FT decomposes the signal into a set of coefficients that indicate the contribution of each frequency component.

*Time-Frequency Localization:* The FT provides excellent frequency resolution (identifying precise frequencies), but it lacks time resolution. It tells you what frequencies are present in a signal, but not when they occur. This is because the sine and cosine waves that are used as the basis functions extend infinitely in time.

*1.2 Mathematical Formulation (Continuous Fourier Transform)*

For a continuous signal f(t)*, the Fourier Transform *F(ω) is defined as:



Where:

f(t) is the signal in the time domain.
F(ω) is the signal in the frequency domain.
ω is the angular frequency (ω = 2πf, where f is the frequency in Hertz).
j is the imaginary unit (√-1).
e^(-jωt) is a complex exponential, which can be expressed as cos ...

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