Fourier time-frequency

The time resolution of the instrument determines the wavenumber-dependent sensitivity of the Fourier-transformed, frequency-domain spectrum. A typical response of our spectrometer is 23 fs, and a Gaussian function having a half width... 

In single-scale filtering, basis functions are of a fixed resolution and all basis functions have the same localization in the time-frequency domain. For example, frequency domain filtering relies on basis functions localized in frequency but global in time, as shown in Fig. 7b. Other popular filters, such as those based on a windowed Fourier transform, mean filtering, and exponential smoothing, are localized in both time and frequency, but their resolution is fixed, as shown in Fig. 7c. Single-scale filters are linear because the measured data or basis function coefficients are transformed as their linear sum over a time horizon. A finite time horizon results infinite impulse response (FIR) and an infinite time horizon creates infinite impulse response (HR) filters. A linear filter can be represented as... 

Fourier pairs not only exist in time-/frequency domain but also in any other domain combined by a quantity q and the belonging dimension-inverted quantity l/q. 

Portnoff, 1980] Portnoff, M. R. (1980). Time-Frequency Representation of Digital Signals and Systems Based on Short-Time Fourier Analysis. IEEE Trans. Acoustics, Speech, Signal Processing, ASSP-28 55-69. 

Fourier analysis. Among several different kinds of time-frequency distributions, we employ the Wigner distribution function [12], defined as... 

Fourier Transform Frequency Analysis of the Time Domain MR Signal... 

In this paper, we utilized the method of short-time Fourier transform (spectrogram). The time-frequency ( -a>) distributions P(, at) of the time-dependent signal s(t) is given as follows ... 

Similar to a one-dimensional experiment where is varied, in a two-dimensional experiment, a series of FIDs from experiments carried out at varying tj are processed. However, in the two-dimension experiment, the amplitudes of the signals at particular frequencies are read and Fourier-transformed a second time. The resulting signals are then a function of the two times tx and t2 and after Fourier transformation, frequencies, v and v2. The plot is usually given as a contour. 

Each metabolite has one or more resonant frequencies. Besides concentrations, the task is also to reconstruct these frequencies, known as chemical shifts. The FFT cannot retrieve them because it is limited to the preassigned Fourier grid frequencies as a function of the total acquisition time. Overall, the FFT by itself cannot reconstruct chemical shifts or the actual heights of the components, since it can only provide the total shape spectra. Thus, as mentioned earlier, in practice, such spectra are typically postprocessed via fitting, with the attendant drawbacks. 

These correlations constitute a fundamental difference of any time-frequency (or scale) resolved analysis to time independent Fourier analysis, where, neighboring frequencies are asymptotically uncorrelated. 

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).

When localisation is an issue, the intuitive solution still making use of the Fourier transform would be to cut up the signal and to transform the pieces. This approach is called the short-time Fourier transform, it adds a dimension to the Fourier transform, namely time, as it allows following frequencies over time. Where the Fourier transform is a frequency analysis, the short-time Fourier transform is a time-frequency analysis. Instead of describing the signal in either the time or the frequency domain, we describe it in both, a joint time-frequency domain. When we do this, we are faced with a fundamental limitation we cannot localise in the time domain and the frequency domain at the same time. 

This type of time-frequency analysis gives us a very rich view of the data, but it is poor when we are after an efficient representation. The results of the Fourier transforms of two neighbouring, not disjoint but severely overlap-... 

A wavelet basis allows a time-frequency analysis similar to that of a short-time Fourier basis. It is different in that its time-localisation is better hence its frequency localisation is worse, for high frequencies than for low fre-... 

Time-domain techniques record the intensity of the signal as a function of time, frequency-domain techniques record the phase and the amplitude of the signal as a function of frequency. Time domain and frequency domain are connected via the Fourier transform. Therefore, the time domain and the frequency domain are generally equivalent. However, this does not imply an equivalence between time-domain and frequency-domain recording techniques or the instruments used for each. An exhaustive comparison of the techniques is difficult and needs to include a number of different electronic design principles and applications. 

Many of the applications of Fourier analysis involve the time-frequency domain. A time-dependent signal f(t) can be expressed as... 

Fourier integrals in the time-frequency domain have the form... 

The Bayesian spectral density approach approximates the spectral density matrix estimators as Wishart distributed random matrices. This is the consequence of the special structure of the covariance matrix of the real and imaginary parts of the discrete Fourier transforms in Equation (3.53) [295]. Another approximation is made on the independency of the spectral density matrix estimators at different frequencies. These two approximations were verified to be accurate at the frequencies around the peaks of the spectmm. The spectral density estimators in the frequency range with small spectral values will become dependent since aliasing and leakage effects have a greater impact on their values. Therefore, the likelihood function is constructed to include the spectral density estimators in a limited bandwidth only. In particular, the loss of information due to the exclusion of some of the frequencies affects the estimation of the prediction-error variance but not the parameters that govern the time-frequency structure of the response, e.g., the modal frequencies or stiffness of a structure. 

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 commonly used algorithms for time-frequency representations of spectral estimates include the short-time Fourier transform (STFT), the Wigner-Ville transform, wavelet transforms, etc. A good summary of the mathematical basis of each of these techniques can be found in Ref. 11. 

Both methods for time domain (Fourier) and frequency domain (direct) filters are equivalent and are related by the convolution theorem, and both have similar aims, to improve the quality of spectroscopic or chromatographic or time series data. Two functions, /"and g, are said to be convoluted to give h, if... 

The equality of transformations of the mixed time—frequency data and the completely Fourier transformed data is a consequence of Parseval s theorem (5.4) and has been previously discussed [10,11]. It can also be understood by taking into account that the spectral reconstruction is achieved by relating two direct dimensions via an indirect dimension, which in turn is discarded. Whether two frequency domains are correlated via a common time domain or a common frequency domain is therefore equivalent. It is also equivalent to Noda s model of relating two IR wavenumber dimensions via a common perturbation stemming from either time or sample space. From the matrix representation, it can be seen that Noda s synchronous matrix O, Eqs. (5.6) and (5.17) corresponds to the covariance map according to Eq. (5.16), if the data matrices yielding O are composed of... 

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