What Is Fourier Transform

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Accordingly, the magnitude of energy density produces the spectrum of function, which is commonly indicated as color plots. To investigate the signal around time t of interest, window function has chosen that is peaked around t. Therefore, the modified signal S_t (τ) is short and its Fourier transform is called short-time Fourier transform (Satish 1998). The principle of STFT is to divide the initial waveform signal into small segments with short-time window, apply Fourier transform to each segment. However, due to the signal segmentation, this method has limitation in time–frequency resolution. It can only be applied to analyze the segmented stationary signal or the approximate stationary signal, but for the non-stationary signal, when…show more content…
A continuous wavelet transform is expressed as: Where x(t)is the waveform signal, a is the scale parameter, b is the time parameter and ψ(t)is the mother wavelet, which is a zero mean oscillatory function centred around zero with a finite energy,∫_(-∞)^(+∞)▒〖ψ(t)dt=0〗, and “*” denotes complex conjugate. By dilating via the scale parameter a and translations via the time parameter b, wavelet transform of a waveform signal decomposes the signal to a number of oscillatory functions with different frequencies and time (Jardine 2006). Similar as Fourier transform decomposes a signal into a series of complex sinusoids, the wavelet transform decomposes a signal into a family of wavelets. The difference is that sinusoids are smooth, symmetric, and regular but wavelets can be ether symmetric or asymmetric, sharp or smooth, regular or irregular. The dilated and translated versions of a prototype function are contained in the family of wavelets. The prototype function is defined as a mother wavelet. The scale parameter a and time parameter b of wavelets formulate how the mother wavelet dilates and translates along the time or space axis. Different types of wavelets can be chosen for different forms of signals to best match the features of the signal. The flexibility of selecting wavelets and the characteristic of wavelets make wavelet transform a beneficial tool for obtaining reliable results and
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