Domain, frequency

The previous sections have shown how arbitrary periodie signals can be composed of harmonics with appropriate amplitudes and phases. It is often very useful to study the amplitude and phases direetly as a function of frequency. Such representations are referred to as spectra and are described as being in the frequency domain. By contrast, waveforms as a function of time are said to be in the time domain. 

Because the harmonic coefficients are complex, we can plot the spectrum in either Cartesian or polar form. The Cartesian form has a real spectrum and imaginary spectrum as shown 

Now consider the square wave that is identical to the one we have been analysing, except that it is shifted 1/4 period to the left. If we subject this to Fourier analysis, we get the spectra 

This is an important fact for speech analysis and synthesis. For analysis, we generally only need to study the magnitude spectra and can ignore phase information. For s mthesis, our signal 

2 Phase information is however used to localise sounds, that is, we use the phase difference between the signal arriving at each ear to estimate where the source of the signal is 

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In the work presented here, a slightly different two-parameter transient model has been used. Instead of specifying a center frequency b and the bandwidth parameter a of the amplitude function A(t) = 6 , a simple band pass signal with lower and upper cut off frequencies and fup was employed. This implicitly defined a center frequency / and amplitude function A t). An example of a transient prototype both in the time and frequency domain is found in Figure 1. 

Figure 1 Example of signal prototype in the time and frequency domains.

UTDefect is basically working in the frequency domain so pulse problems are solved by superposing a number of frequencies. In principal any frequency spectrum can be used (an experimental one, for instance). As the standard spectrum the following simple one is used... 

Standard procedures that are used for testing of construction materials are based on square pulse actions or their various combinations. For example, small cyclic loads are used for forecast of durability and failure of materials. It is possible to apply analytical description of various types of loads as IN actions in time and frequency domains and use them as analytical deterministic models. Noise N(t) action as a rule is represented by stochastic model. 

An idea of investigation of AE response of the material to different types of loads and actions seems to be useful for building up a dynamic model of the material. In this ease AE is representing OUT data, and it is possible to take various AE parameters for this purpose. It is possible to consider a single AE pulse in time or frequency domain or AE pulses sequence as... 

Figure 8 mother wavelet y/(t) (left) and wavelet built out of the mother wavelet by time shift b, and dilatation a. Both functions are represented in the time domain and the frequency domain. 

However, it is easily shown that if the mother wavelet is located in the frequency domain "around"/o (fig 8), then the wavelet a.b(t) is located around f(/a. That is to say, by the mean of the formal identification f = fata it is possible to interpret a time-scale representation as a time-frequency representation [4]. 

The results of both experiments showed that the analysis in the frequency domain provides new technological possibilities of testing characteristics of austenitic steels. Using known phase-frequency characteristics of structural noises it is possible to construct algorithms for separation of useful signal from the defect, even through amplitude values of noise and signal are close in value. 

In space-frequency domain, the back-scattering transfer function is given by ... 

In the remainder of this paper, we exhibit the solution of the deconvolution problem in the frequency domain, but it is possible to establish an analogy with tlie temporal solution exposed by G. Demoment [5,6]. 

If X, y and h are functions with Fourier transforms X, Y and H (real problem), we can write equation (9) in the frequency domain ... 

While the data are collected in the time domain by scaiming a delay line, they are most easily interpreted in the frequency domain. It is straightforward to coimect the time and frequency domains tln-ough a Fourier transform... 

Figure Bl.2.7. Time domain and frequency domain representations of several interferograms. (a) Single frequency, (b) two frequencies, one of which is 1.2 times greater than the other, (c) same as (b), except the high frequency component has only half the amplitude and (d) Gaussian distribution of frequencies.

In turn, an expression for is obtained, which, in the frequency domain, consists of a numerator containing a product of (.s + 1) transition moment matrix elements and a denominator of. s complex energy... 

Johnson A E and Myers ABA 1996 A comparison of time- and frequency-domain resonance Raman spectroscopy in triiodide J. Cham. Phys. 104 2497-507... 

This is the description of NMR chemical exchange in the time domain. Note that this equation and equation (B2.4.11)) are Fourier transfomis of each other. The time-domain and frequency-domain pictures are always related in this way. 

Binsch [6] provided the standard way of calculating these lineshapes in the frequency domain, and implemented it in the program DNMR3 [7], Fonnally, it is the same as the matrix description given in section (B2.4.2.3). The calculation of the matrices L, R and K is more complex for a coupled spin system, but that should not interfere witii the understanding of how the method works. This work will be discussed later, but first the time-domain approach will be developed. 

The Bloch equation approach (equation (B2.4.6)) calculates the spectrum directly, as the portion of the spectrum that is linear in a observing field. Binsch generalized this for a frilly coupled system, using an exact density-matrix approach in Liouville space. His expression for the spectrum is given by equation (B2.4.42). Note that this is fomially the Fourier transfomi of equation (B2.4.32). so the time domain and frequency domain are coimected as usual. 

Fast Fourier Transformation is widely used in many fields of science, among them chemoractrics. The Fast Fourier Transformation (FFT) algorithm transforms the data from the "wavelength" domain into the "frequency" domain. The method is almost compulsorily used in spectral analysis, e, g., when near-infrared spectroscopy data arc employed as independent variables. Next, the spectral model is built between the responses and the Fourier coefficients of the transformation, which substitute the original Y-matrix. 

In order to analyze the vibrations of a single molecule, many molecular dynamics steps must be performed. The data are then Fourier-transformed into the frequency domain to yield a vibrational spectrum. A given peak can be selected and transformed back to the time domain. This results in computing the vibra-... 

Other types of mass spectrometer may use point, array, or both types of collector. The time-of-flight (TOF) instrument uses a special multichannel plate collector an ion trap can record ion arrivals either sequentially in time or all at once a Fourier-transform ion cyclotron resonance (FTICR) instrument can record ion arrivals in either time or frequency domains which are interconvertible (by the Fourier-transform technique). 

The process of going from the time domain spectrum f t) to the frequency domain spectrum F v) is known as Fourier transformation. In this case the frequency of the line, say too MFtz, in Figure 3.7(b) is simply the value of v which appears in the equation... 

Figure 3.7 (a) The time domain spectrum and (b) the corresponding frequency domain spectrum... 

Figure 3.9(a) shows a time domain specttum corresponding to the frequency domain specttum in Figure 3.9(b) in which there are two lines, at 25 and 100 MHz, with the latter having half the intensity of the former, so that... 

Conceptually, the problem of going from the time domain spectra in Figures 3.7(a)-3.9(a) to the frequency domain spectra in Figures 3.7(b)-3.9(b) is straightforward, at least in these cases because we knew the result before we started. Nevertheless, we can still visualize the breaking down of any time domain spectrum, however complex and irregular in appearance, into its component waves, each with its characteristic frequency and amplitude. Although we can visualize it, the process of Fourier transformation which actually carries it out is a mathematically complex operation. The mathematical principles will be discussed only briefly here. 

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