Frequency domain filter

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... 

In an attempt to provide some meaningful comparisons between the two approaches, the authors have reformulated the time-domain filter described in reference (1) to accept frequency-domain data. In the same way as the time-domain filter accepts data samples at each time Increment so the frequency-domain filter accepts real and imaginary components of the frequency... 

A full derivation of the frequency domain filter is given by Mottershead and Stanway (7). Essentially the unknown parameters are defined as state variables in much the same way as in the time-domain formulation, i.e. x = C, x. = C, X = C and x = C. However unlMe the titile domain ormulatlon, he physical variables (i.e. displacements and velocities) are not included in the state vector, which obviously reduces its dimension, i.e. 

Taking note of the implications of equations (7) and (8), the frequency-domain filter can be written down directly from the time-domain version (equations (4) and (5)) simply by substituting the frequency interval n in place of the time interval T. 

The frequency-domain filter is supplied sequentially with the real and Imaginary parts of the observations in (ai) and the filter equations are then solved at each frequency step using any suitable numerical technique. Note that in order to supply response spectra in this form the observed input and output time-series... 

The results presented in the paper extend previous work in that the feasibility of employing a frequency domain filter has been demonstrated. However further work (preferably using common data) is still required to provide a direct comparison between the time and frequency domain approaches. 

MOTTERSHEAD, J. E. and STANWAY, R. Identification of structural vibration parameters using a frequency domain filter, to appear In J. Sound and Vibration. 

Frequency domain filtering is carried out by using the polynomial filters where each point x, is replaced by x ... 

In the analysis of vibration data there is often the need to transform the data from the time domain to the frequency domain or, in other words, to obtain a spectrum analysis of the vibration. The original and inexpensive system to obtain this analysis is the tuneable swept-filter analyzer. Because of inherent limitations of this system, this process, despite the use of automated sweep, is time-consuming when analyzing low frequencies. When the spectra data needs to be digitized for computer inputing, there are further limitations in capability of tuneable filter-analysis systems. 

The first method uses high-frequency vibration components that result from oscillating rotor bars. Typically, these frequencies are well above the normal maximum frequency used to establish the broadband signature. If this is the case, a high-pass filter such as high-frequency domain can be used to monitor the condition of the rotor bars. 

Filtering and smoothing are related and are in fact complementary. Filtering is more complicated because it involves a forward and a backward Fourier transform. However, in the frequency domain the noise and signal frequencies are distinguished, allowing the design of a filter that is tailor-made for these frequency characteristics. 

The shaip cutoff at the limits -i and t, as illustrated by the boxcar function, often occurs in the frequency domain. In this case the boxcar acts as a low-pass filter in applications in electronics. All frequencies below [l] areunaltgEed, in this ideal case all higher ones are suppressed. 

Processing of time domain data may cause artefacts in the frequency domain. One example for these distortions are truncations at the beginning or at the end of the FID which could lead to severe baseline artefacts which can be reduced by an appropriate filter. Undesired resonances leading to broad lines in the final spectra can be more easily eliminated in time domain by truncating the first few data points. Furthermore, the model functions in time domain are mathematically simpler to handle than the frequency domain analogues, which leads to a reduction of computation time. The advantage of the frequency domain analysis is that the quantification process can be directly interpreted visually. 

The actual limit of the summation is the extent of the weighting filter. Zero padding is used to ensure that the discretized matrices have sizes which are a power of two so that the computation can be done in the frequency domain using fast fourier transform (FFT) techniques. The effective discretized density, pin, Wj), is then given by... 

The simplest way to sharpen up such data in the frequency domain is to use a Wiener filter (Press et al. 1986 Kino 1987). In the time domain, each... 

R. W. Field Prof. Rabitz, I like the idea of sending out a scout to map a local region of the potential-energy surface. But I get the impression that the inversion scheme you are proposing would make no use of what is known from frequency-domain spectroscopy or even from nonstandard dynamical models based on multiresonance effective Hamiltonian models. Your inversion scheme may be mathematically rigorous, unbiased, and carefully filtered against a too detailed model of the local potential, but I think it is naive to think that a play-and-leam scheme could assemble a sufficient quantity of information to usefully control the dynamics of even a small polyatomic molecule. 

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