Time-frequency domain

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. 

J. R. Alcala, I. Scott, J. Parker, B. Atwater, C. Yu, R. Fischer, and K. Bellingrath, Real time frequency-domain fiber optic sensor for intra-arterial blood oxygen measurements, in SPIE 1885, Proceedings of SPIE Biomedieal Optics (J. R. Lakowicz, ed.), Los Angeles, pp. 306-316 (1993). 

An example of the decomposition of a sinusoidal tone burst in the time-frequency domain is given in Fig. 1.4. It should be realised that these time constants xonly give an exponential approximation, at the distance of half a window length, of the time-domain masking functions. 

A filter bank is used to decompose the input signal into subsampled spectral components (time/frequency domain). Together with the corresponding filter bank in the decoder it forms an analysis/synthesis system. 

The optimal values of the parameters a/reg and a time depend on the sampling of the time-frequency domain. For the values used in our implementation, A z = 0.2 Bark and A t = 20 ms (total window length is about 40 ms), the optimal values of the parameters in the model were found to be a freq = 0.8, a time = 0.6 and y = 0.04. The dependence of the correlation on the time-domain masking parameter OQj me turned out to be small. 

Sondhi and Schroeter, 1987] Sondhi, M. M. and Schroeter, J. (1987). A hybrid time-frequency domain articulatory speech synthesizer. IEEE Trans. Acoustics, Speech, Signal Processing, ASSP-35(7) 955-967. 

We calculate the laser propagation along the propagation axis using a onedimensional (ID) model and display the results using the time-frequency analysis. A distribution function in the time-frequency domain [11] is a powerful tool to analyze the chirp structure that cannot be revealed by a standard... 

FIGURE 4.37 Joint time-frequency domains of the AE signal for slurry 1 (a) and slurry 2 (b). The AE signal was filtered by a Debouche 05 wavelet filter. Only middle bands were selected and processed by the marching pursuit joint time-frequency domain algorithm (from Ref 27). 

Viscoelastic function in the whole time frequency domain Thus the relaxation modulus may be calculated from a very limited number of... 

The experimental setup for the broadband CARS is rather simple because only two pulses are needed for three-color CARS emission, as shown in Fig. 5.4a a broadband first pulse impulsively promotes molecules to vibrationally excited states through a two-photon Raman process, and a delayed narrowband second pulse induces anti-Stokes Raman emission from coherent superpositions to the ground state [29]. By changing the delay time for the second pulse, therefore, one can expect to probe dynamical behaviors of multiple RS-active modes. Such a two-dimensional observation in the time-frequency domains should be effective for detailed analysis of nanomaterials. 

In what follows we focus on long time, frequency-domain Raman scattering, which is easier to analyze. To simplify notation we denote the initial state, 11, vi, ki) by zzz) and the final state 1, v, k) by out. We also assume that a single zero-photon excited state Is ) = 2, V2,0) is close to resonance with the incident radiation, that is, in(vi,[Pg.652]

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. 

Fig. 6 Different tilings of the time-frequency domain for a signal that is 32 points long.

Fig. 12 Time-frequency domain tiling for the pyramid algorithm, (a) First application of a pair of filters to 16-point signal (h) the filters applied to the low-frequency part of (a) (c) and (d) further cut up of the lower frequencies analogous to (h).

Fig. 15 Some selections of the grid of Fig. 14(b) representing wavelet packet bases (left), and the corresponding tilings of the time-frequency domain (right).

Being able to zoom in is a nice feature, but what if one does not know what to zoom in on, which is the most likely situation in chemical applications. We do not usually know what tiling of the time-frequency domain is most suited for, let us say, our NIR spectrum. Fortunately, one does not need to know this in advance. Techniques exist that select the best basis for a particular situation from the wealth of bases offered by wavelets and wavelet packets. 

Transform the noisy signal into the time-frequency domain by decomposing the signal on a set of orthonormal wavelet basis functions. 

Let us consider the Discrete Wavelet Transform (DWT), applied to the set of m signals (e.g. spectra) of length n each, presented in the form of matrix X. If all signals are decomposed by DWT with the same filter and to the same decomposition level, they can be presented as m vectors of length n each in the time-frequency domain, forming matrix Z (see Fig. 2). The information content of... 

Now we would like to compress the data in the time-frequency domain, without loss of an essential information about the data variability. This can be done, based on the variance of the wavelet coefficients (matrix Z). For each column of matrix Z, a variance can be calculated and in this way vector V (1 X n) is obtained, which contains n elements, each of them describing the variance of one column of matrix Z (see Fig. 3). The jth element of vector v is defined as ... 

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... 

Extraction of monitoring features can be conducted in time domain or frequency and time-frequency domain. The aim of feature extraction is to derive characteristic values which describe the relevant information about the process conditions. 

E. Song, Y.-J. Shin, RE. Stone, etal, Detection and Location of Multiple Wiring Faults via Time-Frequency-Domain Reflectometry, IEEE Transactions on Electromagnetic Compatibility, vol. 51, no. 1, pp. 131-138, 2009. 

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