The frequency domain

The previous sections have shown how arbitrary periodic signals can be composed ofhar-monics with appropriate amplitudes and phases. It is often very useful to study the amplitude and phases directly 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. 

Hgure 10.6 Fovirier synthesis of a square wave for various numbers of harmonics, (a) The square wave and sine waves of the first harmonic, (b) Multiphcation of square wave and sine wave of the first harmonic, (c) The square wave and sine wave of the second harmonic. 

Rgure 10.7 The two ways of showing the full spectrum for a square wave (a) real spectrum, 

The difference between the two square waves is in their relative position in time - one is slightly later than the other. This agrees with what we expect from human hearing if we play a note on a piano and then play the same note at some other time, tiie notes sound exactly the same, even thongh their position with respect to some reference time t = 0 

---

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... 

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 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... 

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... 

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. 

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-... 

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 3.9(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. 

A computer digitizes the time domain spectmm f(t) and carries out the Fourier transformation to give a digitized F(v). Then digital-to-analogue conversion gives the frequency domain spectmm F(v) in the analogue form in which we require it. 

There is one important point, however, that we have neglected so far. Real spectra in the frequency domain do not look like those in Figures 3.7(b)-3.9(b) the lines in the spectra are not stick-like and infinitely sharp but have width and shape. 

For radiofrequency and microwave radiation there are detectors which can respond sufficiently quickly to the low frequencies (<100 GHz) involved and record the time domain specttum directly. For infrared, visible and ultraviolet radiation the frequencies involved are so high (>600 GHz) that this is no longer possible. Instead, an interferometer is used and the specttum is recorded in the length domain rather than the frequency domain. Because the technique has been used mostly in the far-, mid- and near-infrared regions of the spectmm the instmment used is usually called a Fourier transform infrared (FTIR) spectrometer although it can be modified to operate in the visible and ultraviolet regions. 

Figure 9.45(b) shows fhe resulf of Fourier transformation (see Section 3.3.3.2) of the signal in Figure 9.45(a) from the time to the frequency domain. This transformation shows clearly that two vibrations, with frequencies of about 3.3 THz (= 3.3 x lo ... 

Pulsed ft mode esr instmments have appeared beginning in the mid-1980s. These collect digitized time-domain spectra which may be processed into the frequency domain as are nmr data. Pulse durations are much shorter than in nmr with typical 90° times of 8—20 ns. 

Finding the values of G allows the determination of the frequency-domain spectrum. The power-spectrum function, which may be closely approximated by a constant times the square of G f), is used to determine the amount of power in each frequency spectrum component. The function that results is a positive real quantity and has units of volts squared. From the power spectra, broadband noise may be attenuated so that primary spectral components may be identified. This attenuation is done by a digital process of ensemble averaging, which is a point-by-point average of a squared-spectra set. 

Bickel, H.J., and Rothschild, R.S., Real-Time Signal Processing in the Frequency Domain, Federal Scientific Monograph 3, March 1973. 

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. 

---