Frequency-domain representation

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.

Once a smooth signal has been constructed, how is the trend represented Most of the available techniques do not provide a framework for the representation (and thus, interpretation) of trends, because their representations (in the frequency or time domains) do not include primitives that capture the salient features of a trend, such as continuity, discontinuity, linearity, extremity, singularity, and locality. In other words, most of the approaches used to represent process signals are in fact data compaction techniques, rather than trend representation approaches. Furthermore, whether an approach employs a frequency or a time-domain representation, it must make several major decisions before the data are compacted. For frequency-domain representations, assumptions about the... 

FFT Fast Fourier Transform algorithms 33 are used to obtain the frequency domain representation of the sample and reference waveforms sampled. 

Figure 4.4 Frequency-domain representation of the dynamically controlled decoherence rate in various limits (Section 4.4). (a) Golden-Rule limit, (b) Anti-Zeno effect (AZE) limit (c) Quantum Zeno effect (QZE) limit. Here, F,( ) and G(w) are the modulation and bath spectra, respectively and F are the interval of change and width of G( ), respectively and is the interruption rate.

Apart from some special drift processes that we will treat separately, the noise in the measurements is expected to be the result of random processes much faster than the changes in the useful signal itself. Fourier transform spectral methods exploit this difference in frequency for separating the two components by considering a frequency-domain representation of the signal instead of its original time domain representation. 

The frequency domain representation F of a function f depending on time t is defined by the Fourier transform... 

In an ordinary Fourier transform NMR experiment the time-domain signal (the FID) is converted into a frequency-domain representation (the spectrum) thus a function of time, S(t2), is converted into a function of frequency, S(f2). The very simple basic idea of 2D NMR is to treat the period preceding the recording of the FID (known as the evolution period ) as the second time variable. During this period, tu the nuclear spins are manipulated in various ways. In the 2D experiment a series of S(t2) FID s are recorded, each for a different t u and the result is considered a function of both time variables, S(tu t2). A twofold application of the Fourier transformation (see Fig. 82) then yields a 2D spectrum, S(fi,f2), which has two frequency... 

The magnitude values of the frequency domain representation are converted to a 1/3-critical band energy representation. This is done by adding the magnitude values within a threshold calculation partition. 

The frequency domain representation of the data is calculated via an FFT after applying a Hann window with a window length of 1024 samples. The calculation is done with a shift length equal to the block structure of the coding system. As described below, the shift length is 576 samples for Layer 3 of the ISO/MPEG-Audio system. 

The separate calculation of the frequency domain representation is necessary because the filter bank output values (polyphase filter bank used in Layer 1/2 or hybrid filter bank used in Layer 3) can not easily be used to get a magnitude/phase representation of the input sequence as needed for the estimation of tonality. 

The divergence of all the global characteristic times for anomalous diffusion—as defined in their conventional sense (which is a natural consequence of the underlying Levy distribution), rendering them useless as a measure of the relaxation behavior—signifies the importance of characteristic times for such processes in terms of the frequency-domain representation of... 

In general, it is useful to regard 5 as a complex number s = a jw in these analyses (41, 42). Then one can calculate real-axis and imaginary-axis frequency domain representations of a function. For example, the real-axis transform of E t) is... 

The ear is relatively insensitive to phase, and so the magnitude spectrum is normally sufficient to characterise the frequency domain representation of a signal. 

While an essential part of speech analysis, the windowing process unfortunately introduces some unwanted effects in the frequency domain. These are easily demonstrated by calculating the DFT of the sinusoid. The true frequency domain representation for this, calculated in continuous time with a Fourier transform, is a single delta peak at the frequency of the sinusoid. Figures 12.2 show the DFT and log DFT for the three window types. The DFT of the rectangular window... 

To understand the basic concepts of modulation, we first review time- and frequency-domain representations of signals. The information present in m(t) can be completely specified by a complex function of speech amplitude vs. frequency, M f), obtained using the Fourier transform. Since M f) contains aU of the information in m t), it is possible to go back and forth between m t) and M f) using the forward and inverse Fourier transforms. 

Figure 20.42 shows the frequency-domain representation of an ideal and a nonideal filter and the time-domain representation of the input and output signals. Figures 20.42(a), 20.42(b), and 20.42(c) show, respectively, the frequency response of the ideal filter, the input signal waveform, and the output signal waveform, which is a time delayed version of the input waveform. Both the filter response and the input signal are bandlimited to B Hz. 

For the case of USB modulation in which the frequency error is zero but the phase error is nonzero, the demodulated output is obtained by setting A/ equal to zero in Eq. (20.26). This gives y(t), the demodulated output signal and its frequency domain representation Y(/), respectively, as... 

But the frequency domain representation of the Hilbert transform of the message signal, mh f), can be expressed as... 

Frequency domain representation The portrayal of a signal in the frequency domain representing a signal by displaying its sine wave components the signal spectrum. 

Spectrum The frequency domain representation of a signal wherein it is represented by displaying its... 

Among the alternative simplified procedures [e.g., see examples discussed by Neagu and Neagu (2000)], the one most frequently used for passing from the time-domain experimental data into a frequency-domain representation of the dielectric losses (e") is that originally proposed by Hamon (1952). This can be expressed as... 

In the eighteenth century, Fourier proposed that complex vibrations could be analysed as a set of parallel sinusoidal frequencies, separated by a fixed integer ratio for example, lx, 2x, 3x, etc., where x is the lowest frequency of the set. Helmholtz then developed Fourier s analysis for the realm of musical acoustics. He proposed that our auditory system identifies musical timbres by decomposing their spectrum into sinusoidal components, called harmonic series. In other words, Helmholtz proposed that the ear naturally performs Fourier s analysis to distinguish between different sounds. The differences are perceived because the loudness of the individual components of the harmonic series differs from timbre to timbre. The representation of such analysis is called frequency-domain representation (Figure 5.1). The main limitation of this representation is that it is timeless. That is, it represents the spectral components as infinite, steady sinusoids. 

Figure 5.1 Frequency-domain representation is timeiess because it represents the spectrum of a sound as infinite, steady sinusoids...

Figure 5.2 Frequency-domain representation snapshots can show how the spectrum of a sound evoives with time...

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