Time-and frequency-domain signals

Multiple-pulse FT-NMR gives biochemists unparaiieied control over the information content and display of spectra, and to take full advantage of the technique, we need to understand how radiofrequency pulses work to excite a spin system and how the signal is monitored and interpreted. 

It is sometimes useful to compare the quantum mechanical and classical pictures of magnetic nuclei pictured as tiny bar magnets. A bar magnet in an externally applied magnetic field undergoes the motion called precession as it twists around the direction of the field (Fig. 13.18). The rate of precession is proportional to the strength of the applied field and is in fact equal to y l2n) Bo, which in this context is called the Larmor precession frequency, Vl. 

As time passes, the individual spins move out of step (partly because they are precessing at slightly different rates, as we explain later), so the magnetization vector shrinks exponentially with a time constant Tj and induces an ever weaker signal in the detector coil. The form of the signal that we can expect is therefore the osdllating-decaying firee-induction decay (FID) shown in Fig. 13.23. 

24 (a) A free-induction decay signal of a sample of an AX species and (b) its analysis into its frequency components. 

Now consider a two-spin system. We can think of the magnetization vector of an AX spin system with /= 0 as consisting of two parts, one formed by the A spins and the other by the X spins. When the 90° pulse is applied, both magnetization vectors are tipped into the perpendicular plane. However, because the A and X nuclei precess at different frequencies, they induce two signals in the detector coils, and the overall FID curve may resemble that in Fig. 13.24a. The composite FID curve is the analog of the struck bell emitting a rich tone composed of all the frequencies at which it can vibrate. 

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The inlerconversion of time- and frequency-domain signals is complex and mathematically tedious when more than a few lines are involved the operation is only practical with a computer. Today fast Fourier transform algorithms allow calculation of frequency-domain spectra from time-domain spectra in seconds or less. 

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.

Apparently, the time-domain and frequency-domain signals are interlinked with one another, and the shape of the time-domain decaying exponential will determine the shape of the peaks obtained in the frequency domain after Fourier transformation. A decaying exponential will produce a Lorentzian line at zero frequency after Fourier transformation, while an exponentially decaying cosinusoid will yield a Lorentzian line that is offset from zero by an amount equal to the frequency of oscillation of the cosinusoid (Fig. 1.23). 

Before discussing the Fourier transform, we will first look in some more detail at the time and frequency domain. As we will see later on, a FT consists of the decomposition of a signal in a series of sines and cosines. We consider first a signal which varies with time according to a sum of two sine functions (Fig. 40.3). Each sine function is characterized by its amplitude A and its period T, which corresponds to the time required to run through one cycle (2ti radials) of the sine function. In this example the frequencies are 1 and 3 Hz. The frequency of a sine function can be expressed in two ways the radial frequency to (radians per second), which is... 

As said before, there are two main applications of Fourier transforms the enhancement of signals and the restoration of the deterministic part of a signal. Signal enhancement is an operation for the reduction of the noise leading to an improved signal-to-noise ratio. By signal restoration deformations of the signal introduced by imperfections in the measurement device are corrected. These two operations can be executed in both domains, the time and frequency domain. 

Fig. 3.5. Connection between characteristic signal functions in time and frequency domain...

The Fourier transform (FT) relates the function of time to one of frequency—that is, the time and frequency domains. The output of the NMR spectrometer is a sinusoidal wave that decays with time, varies as a function of time and is therefore in the time domain. Its initial intensity is proportional to Mz and therefore to the number of nuclei giving the signal. Its frequency is a measure of the chemical shift and its rate of decay is related to T2. Fourier transformation of the FID gives a function whose intensity varies as a function of frequency and is therefore in the frequency domain. 

This comparison between time and frequency domain measurements is performed at submegahertz frequencies in order to avoid the issue of deconvolution of time domain signals. At megahertz frequencies time domain measurements encounter an additional limitation, these signals must be deconvoluted to isolate the sensor response from the instrument response. The need for deconvolutions adds extra software and computation time, which limits the versatility of time domain techniques for real-time applications. No deconvolutions are necessary in the frequency domain as shown below. 

The laser used to generate the pump and probe pulses must have appropriate characteristics in both the time and the frequency domains as well as suitable pulse power and repetition rates. The time and frequency domains are related through the Fourier transform relationship that hmits the shortness of the laser pulse time duration and the spectral resolution in reciprocal centimeters. The limitation has its basis in the Heisenberg uncertainty principle. The shorter pulse that has better time resolution has a broader band of wavelengths associated with it, and therefore a poorer spectral resolution. For a 1-ps, sech -shaped pulse, the minimum spectral width is 10.5 cm. The pulse width cannot be <10 ps for a spectral resolution of 1 cm . An optimal choice of time duration and spectral bandwidth are 3.2 ps and 3.5 cm. The pump pulse typically is in the UV region. The probe pulse may also be in the UV region if the signal/noise enhancements of resonance Raman... 

While in the frequency domain all the spectroscopic information regarding vibrational frequencies and relaxation processes is obtained from the positions and widths of the Raman resonances, in the time domain this information is obtained from coherent oscillations and the decay of the time-dependent CARS signal, respectively. In principle, time- and frequency-domain experiments are related to each other by Fourier transform and carry the same information. However, in contrast to the driven motion of molecular vibrations in frequency-multiplexed CARS detection, time-resolved CARS allows recording the Raman free induction decay (RFID) with the decay time T2, i.e., the free evolution of the molecular system is observed. While the non-resonant contribution dephases instantaneously, the resonant contribution of RFID decays within hundreds of femtoseconds in the condensed phase. Time-resolved CARS with femtosecond excitation, therefore, allows the separation of nonresonant and vibrationally resonant signals [151]. 

Continuous functions and signals in the time domain are denoted by lower case letters with the argument in parentheses, e.g. x(t). Sampling at constant intervals A t produces a discrete approximation x[n] to the continuous signal, defined at times f = n A t, n = 0,1,2. Square brackets are used for the arguments of discrete functions. The Fourier transform establishes the connection between the time and frequency domains [76] ... 

The matter of sampling and limited representation of frequencies requires a second look at the representation of data in the time and frequency domains, as well as the transformation between those domains. Specifically, we need to consider the Fourier transformation of bandwidth-limited, finite sequences of data so that the S/N enhancement and signal distortion of physically significant data can be explored. We begin with an evaluation of the effect of sampling, and the sampling theorem, on the range of frequencies at our disposal for some set of time-domain data. 

Fig. 11.21. The Fourier transform the link between the time and frequency domain, (a) A simple sinusoidal signal (b) The first two components of a square...

Figure 4.1 Time and frequency domain data in signal processing in the noiseless case using the fast Fourier transform (FFT) and fast Pad6 transform (FPT). Top panel (i) the input FID (to avoid clutter, only the real part of the time signal is shown). Middle panel (ii) absorption total shape spectrum (FFT). Bottom panel (iii) absorption component (lower curves FPT) and total (upper curve FPT) shape spectra. Panels (ii) and (iii) are generated using both the real and imaginary parts of the FID.

The data processing software includes the options of signal correction, correction of electrode polarization and dc-conductivity, and different fitting procedures both in time and frequency domain [86]. 

Measurements and information. Signal processing allows both time- and frequency-domain information to be obtained depending on the type of specimen studied. In addition, the use of frequency-tunable laser to generate density variations within the sample results in thermal expansion by absorption of light. Compared to spontaneous Brillouin scattering, the SNR in this forced Brillouin spectrometry is substantially enhanced by the generation of coherent phonons within the sample. 

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