The Frequency Domain Spectrum

In general, we live in the time domain, but perceive many aspects of objects and processes in the frequency domain. Recall that in Section 2.4.2 we discussed the 3 0 Hz transition, which refers to the fact that events happening much faster than 30 times per second are perceived as spectral events in frequency, rather than gestural events in time. 

Now we will see that sine waves are extremely important for yet one more reason. We ll learn that just as light can be passed through a prism to break it into the individual light frequencies from which it is composed, sound can be separated into individual simple frequencies (sine waves). The set of individual amplitudes and phases of the sines that make up a sound are called a frequency spectrum. The mathematical technique used to turn a time-domain waveform into a frequency-domain spectrum is called a transform. The math used to turn a frequency-domain spectrum back into a time-domain waveform is called an inverse transform. Using the frequency spectrum to inspect aspects of a sound is called spectral analysis. 

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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... 

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. 

The evolution period tl is systematically incremented in a 2D-experiment and the signals are recorded in the form of a time domain data matrix S(tl,t2). Typically, this matrix in our experiments has the dimensions of 512 points in tl and 1024 in t2. The frequency domain spectrum F(o l, o 2) is derived from this data by successive Fourier transformation with respect to t2 and tl. 

The frequency-domain spectrum is computed by Fourier transformation of the FIDs. Real and imaginary components v(co) and ifi ct>) of the NMR spectrum are obtained as a result. Magnitude-mode or powermode spectra P o)) can be computed from the real and imaginary parts of the spectrum through application of the following equation ... 

Stonehouse and Keeler developed an intriguing method for the accurate determination of scalar couplings even in multiplets with partially convoluted peaks (one- or two-dimensional). They recognized that the time domain signal is completely resolved and that convolution of the frequency domain spectrum is a consequence of the Fourier transform of the signal decay. The method requires that the multiplet be centred about zero frequency and this was achieved by the following method ... 

In the conventional NMR experiment, a radio-frequency field is applied continuously to a sample in a magnetic field. The radio-frequency power must be kept low to avoid saturation. An NMR spectrum is obtained by sweeping the rf field through the range of Larmor frequencies of the observed nucleus. The nuclear induction current (Section 1.8.1) is amplified and recorded as a function of frequency. This method, which yields the frequency domain spectrum f(ai), is known as the steady-state absorption or continuous wave (CW) NMR spectroscopy [1-3]. 

In Figure 3.16, the effect of T2 line broadening is illustrated by adding a small amount of iron to the sample. It is quite apparent what this addition does to the linewidth of peaks in the frequency domain spectrum, what is less appreciated is what this addition does to the time domain data (FID). As mentioned above, T2 determines the amount of time that M remains in the xy plane therefore, if T2 relaxation is short (broad signal), the FID decays to zero much faster than if T2 is long (sharp signal). 

FIGURE 5.6 Fourier transformation of a series of FIDs like the ones in Figure 5.5 (C) to give the frequency-domain spectrum as both a peak and as contours. The contour plot also shows a projection parallel to F2. 

Figure 3.9(a) shows a time domain spectrum corresponding to the frequency domain spectrum 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... 

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

An algorithm developed by Cooley and Tukey simplified this extremely time-consuming calculation, bringing it within the capability of modern microcomputers. Today, the transformation takes only a few seconds, after which the frequency-domain spectrum Fj can be plotted. (The frequency-domain spectrum corresponding to Figure 3.16 will be discussed in Chapter 5.)... 

Pulse sequence in FT-NMR. The FID data in (a) are averaged and Fourier transformed to yield the frequency-domain spectrum shown in (b). 

Figure 9.40. (a) Single decaying line wave and a plot of the tingle frequency it ttprescnts. (b) Two decaying sine waves and a plot of the two frequencies they (present, (c) Many co-addcd sine waves in a free induction decay of the Cnmr spectrum of 3-ethylpyridine and the frequency domain spectrum they represent... 

In this type of experiment, the echo and antiecho are linearly combined with the same amplitude to yield an amplitude-modulated signal in Pure absorption lineshapes may then be obtained in the frequency domain spectrum after a two-dimensional Fourier transform is performed. The disadvantage of this method is that it is not possible to discriminate the sign of the MQ coher-... 

First, either a Lorentzian or Gaussian filter is applied to the FID to reduce the amount of noise. The choice of lineshape will depend on the shape of the frequency domain spectrum, the lineshape is related to how the fluorine spins interact with their environment. The filter linewidth is generally similar to or slightly less than the T2 value (T2 can be estimated from the spectral linewidth). After application of the time domain filter, a fast Fourier transform (FFT) is performed. The resultant frequency domain spectrum will then need to undergo phase adjustment to obtain a pure absorption spectrum. The amount of receiver dead time (time lost between the end of the excitation pulse and the first useful detection time point) will determine the presence and extent of baseline artifact present as well as how difficult phase adjustment will be to accomplish. 

The time and frequency domains are related by a simple function, one being the inverse of the other (Fig. 2.17). The complicating factor is that a genuine FID is usually composed of potentially hundreds of components of differing frequencies and amplitude, in addition to noise and other possible artefacts, and in such cases the extraction of frequencies by direct inspection is impossible. By far the most widely used method to produce the frequency domain spectrum is the mathematical procedure of Fourier transformation, which has the general form ... 

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