Short-time Fourier transform windowing

The windowed Fourier transform [2,3] (also called the short time Fourier transform) was introduced so that the frequency information about a signal could be localised with respect to time. Instead of analysing the function f(t) as a whole, the windowed Fourier transform performs a Fourier transform on pieces of the function. The pieces are obtained by using a windowing function G(t) which slides across the function. The windowed Fourier transform of f(t) is defined as... 

Looking at Figure 9.36, it is obvious that the signal could be represented by only one or two frequency components if the transformation was performed within a moving time window. This kind of time-frequency analysis is available in the so-called short-time Fourier transform (STFT). The problem with STFT is that the resolution in time and... 

Some of the methods used in TED are short time Fourier transform (STFT), wavelet transform (WT) and wavelet packet transform (WPT). In general, the difference between these methods is the partitioning of the time-scale axis. In short-time Fourier transform (STFT), the EMG signal is mapped into frequency components that present within an interval of time (window). A suitable window size must be determined prior to this as small window will give good time resolution but poor frequency resolution and vice versa. The partitioning ratio of the STFT is fixed once specified, each cell has an identical aspect ratio. To overcome the resolution problem in STFT, WT was developed. 

Together, the sixteen elements of the columns a" and "d" form an alternative representation of the signal. We could say that we have just performed a basis transformation. The basis functions are presented in Table 2. The first element of column a is the inproduct of the signal and the basis function given in the first column of Table 2. The inproduct means that we calculate the product of the first element of the signal and the first element of the basis function, the product of the second element of the signal and the second element of the basis function, etc. Then we sum the products. As all but the first two products are zero, it is easy to see that the inproduct boils down to the sum we calculated earlier. The basis of Table 2 can be regarded a short-time Fourier basis for a window width of 2 points. 

Another non - parametric approach is deconvolution by discrete Fourier transformation with built - in windowing. The samples obtained in pharmacokinetic applications are, however, usually short with non - equidistant sample time points. Therefore, a variety of parametric deconvolution methods have been proposed (refs. 20, 21, 26, 28). In these methods an input of known form depending on unknown parameters is assumed, and the model response predicted by the convolution integral (5.66) is fitted to the data. 

Whenever coherences in the upper manifold are particularly short lived, doorway channel A will dominate the evolution of the polarization at least at later times. In that case, the fringes seen in the 4WM signal as the time between the doorway and window stages is altered (with a delay line) reflect those Raman Ifequencies in the ground state that can be spectrally embraced by the femtosecond pulse. Then the Fourier transform of the fringes leads to the conventional spontaneous RRS of the ground state. Indeed, in the absence of electronic resonance, channel B reverts to a purely nonresonant doorway event (Dj. ) and only channel A reveals Raman resonances—those in the electronic ground state [62]. 

The essential point is the complementary nature of the descriptions in the time and frequency domains, a complementarity most familiar to us in the form of the time energy uncertainty principle. For our purpose we want a somewhat more detailed statement, a statement whose physical content can be loosely stated as the overall shape of the spectrum is determined by very short time dynamics, higher resolution corresponds to longer time evolution. A fully resolved spectrum is equivalent to a complete knowledge of the dynamics. We now proceed to make this into a technical statement by an appeal to the convolution theorem for the Fourier transform (51). A preliminary requirement for this development is the definition of the operation of smoothing. To erase details in a function (in our case, the spectrum) we convolute it with a localized window function. A convolution operation is defined by... 

Figure 5 The unit window function in frequency, W rn) and its Fourier transform in the time domain, W, (/). The widths of the transform pair are inverse to one another, and this mathematical result is true in general (51). (The physical implication of this theorem is, of course, the time-frequency uncertainty principle. In the present context, this theorem implies that short time dynamics determines the broad features of the spectrum, and vice versa). Convoluting the spectrum with the unit window function is the simplest form of coarse graining cf. Eq. (10).

The computed survival probability shown in Figure 9b may be compared with the absolute square of the Fourier transform of the experimental overtone spectrum. Compared with the experimental results, the survival probability shown in Figure 9b shows more rapid falloff at short times, t < 5000 au. This is expected, since the experimental spectra are available over a limited (-200 cm ) frequency window. After t = 20,000 au (0.5 ps), the computed survival shows a sequence of low-amplitude recurrence oscillations this amplitude agrees qualitatively with the experimental results. 

The STFT thus results in a spectrum that depends on the time instant to which the window is shifted. The choice of Gaussian functions for the short-duration window gives excellent localization properties despite the fact that the functions are not limited in time. Alternatively, STFT can also be viewed as filtering the signal at all times using a bandpass filter centered around a given frequency/whose impulse response is the Fourier transform of the short-duration window modulated to that frequency. However, the duration and bandwidth of the window remain the same for all frequencies. 

After windowing, Fourier analysis is performed on each frame, resulting in short-time Discrete Fourier Transform (DFT). Then derived values are then grouped together in critical bands and weighted by a triangular filter bank called mel-spaced filter banks. 

There are other variants of Gabor functions some use different envelope functions, or vary the envelope width at different positions. They also have various other names, such as short-time or windowed Fourier transforms. Most technical details concerning these functions have been omitted here, but can be found in a basic text. ... 

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