Bandwidth and Resolution

The ability of an instrument to differentiate between adjacent absorption or emission peaks is the resolving power of the instrument Usually, this is defined as 

A UV/VIS spectrometer is a classic, but not the indispensable tool of forensic chemistry it once was. UV absorption is all but relegated to use as part of a detector system in instruments, whereas visible interactions are used as part of color analysis and characterization. Color is an important descriptor of presiunptive 

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From this equation, it apparent that the bandwidth should be as small as possible to minimize the spacing and avoid missing important data. The typical number of lines of resolution is 100 to 12,800, but it is important to make sure that the selected bandwidth and resolution include all... 

However, the calculation requires different algorithms for each specific case. Hence, an explicit mathematical solution providing basing retention data (elution volumes or times, bandwidths, and resolution) is much more frequently used in analytical gradient chromatography, even at a cost of some simplifications [9-11]. 

Once the retention volume of a solute is calculated for a particular gradient profile, corresponding bandwidth and resolution can be determined by introducing the appropriate instantaneous retention factor A, at the elution of the peak maximum calculated from the gradient function tpf = /(Tr) and from the retention equation A, = f chromatographic mode and gradient function used [851 ... 

At this point, it seems useful to consider the bandwidth, resolution, and sensitivity limits of Fourier transform spectroscopy in some depth. The commonly quoted expressions for bandwidth and resolution are, respectively. 

In practice, however, bandwidth and resolution are usually not given by Equations 56. One is almost always limited to a finite number of channels, n, in which data can be stored and averaged. The usable bandwidth F is limited by the sampling theorem, which requires that a periodic signal be sampled at least twice per cycle, leading to Equation 54, where At is the time resolution between points. The total time for which data may be accumulated is then n-At, leading directly to Equation 53 for the attainable frequency resolution. 

Tissue Vibration. Noncontact methods are preferred for measuring the surface motion of tissues. Laser vibrometers are commercially available with sufficient bandwidth and resolution for most studies. A direct mass load on the skin, together with the skin s elasticity, forms a mechanical low-pass filter (see Simple Lumped Models in Sec. 10.3.1). If a device is to be mounted directly on the skin, it must be of low weight (e.g., <3 g) and possess a comparatively large attachment area (e.g., >S cm ), in order for vibration to be recorded without attenuation of the motion at 80 Hz. An upper frequency limit of 200 Hz is theoretically achievable (—3dB) with a transducer and skin mount weighing 3 g and an attachment area of 1.8 cm. ... 

The combination of bandwidth and lines of resolution selected for each machine-train must effect separation of the unique frequency components that represent a machine s operating dynamics. Resolution can be improved by reducing Fmax> increasing the lines of resolution, or a combination of both. 

Theoretically, 8/lp in the resonant wavelength shift scheme is independent of resonance shape or resonant bandwidth, and should be determined merely by instrument resolution, typically less than 10 pm. However, in reality, noise can perturb resonance spectra such that accurate determination of resonant wavelength shift becomes difficult for a broad resonance curve. To enhance accuracy in detecting wavelength shift, narrower resonance is required. This is equivalent to obtaining higher-g resonance behavior. To take into account noise-included detectability of 8/lp, 8/lp can be simply described as a fraction (p) of the full width at half maximum (FWHM) bandwidth of resonance, A7.. WnM. In this fashion, optical detection limit becomes pA/.. WnMAS or p/-vl(QS). In practice, p can be chosen as a reasonable value of 0.1. In the intensity variation scheme, 87 is determined by noise from environment and photodetectors. It can reach as low as several nanowatts with care. 

Bandwidth is the width of the wavelength band that is allowed to exit a monochromator. The narrowness of this band is called the resolution. High resolution corresponds to a very narrow bandwidth and vice versa. 

Resolution, on the other hand, is a more technical term. It refers to the distance between adjacent bands relative to their bandwidths and acknowledges the fact that proteins are distributed in Gaussian profiles with overlapping distributions. The numerical expression for resolution is obtained by dividing the distance between the centers of adjacent bands by some measure of their average bandwidths. It expresses the distance between band centers in units of bandwidth and gives a measure of the overlap between two adjacent bands. For preparative applications, when maximal purity is desired, two proteins to be isolated should be separated by at least a bandwidth. In many applications it is sufficient to be able to simply discern that two bands are distinct. In this case bands can be less than a bandwidth apart. 

In packed columns, all three terms contribute to band broadening. For open tubular columns, the multiple path term, A, is 0, so bandwidth decreases and resolution increases. In capillary electrophoresis (Chapter 26), both A and C go to 0, thereby reducing plate height to submicron values and providing extraordinary separation powers. 

One advantage of NMR spectroscopy is that it can be used on almost all polymeric materials through either solid-state or solution NMR methods. However, the techniques and resolution of the two methods are radically different. For example, the proton NMR spectrum of the water is sharp and narrow with a bandwidth of 1 Hz, while the proton NMR spectrum of ice is extremely broad with a bandwidth of 20 KHz. The differences... 

An important component in any instrument is a set of one or more mechanical slits (or holes) that limit the spatial width of radiation (i) The "source slit" limits the width of the radiation source seen by the sample (ii) the "detector slit" limits the width of the radiation emitted by the sample and sent to the detector. On the one hand, the narrower the slit (or hole), the better the resolution, but on the other hand, the narrower the slit or hole, the weaker the signal seen by the detector. The slits also control the effective bandwidth of the instrument Very narrow slits yield the smallest bandwidth and the best energy or wavelength resolution, but this comes at a price of a weaker signal. 

Figure 3.3-17 Relation between the optimal spectral band width Ai>o = t /Rp and the line width AC i/2, Rq is the resolving power of the spectrometer. For a the sensitivity is low since the spectral band width is larger than the line width, also the background contributes to the noise, for b the resolution of the spectrometer is sufficient to yield a high sensitivity, c Relation between bandwidth and resolving power.

Sensor measurements are recorded in situ by a battery-powered, eight-channel, miniature digital tape recorder (MDTR) (Oxford Medical Systems Ltd.) with range, 0-1.023 V resolution, 0.1% scan period, 15 s (alternatively 5 s) duration (C120 cassette), 25 h (alternatively 8 h) and data capacity, 6000 scans. Direct (bandwidth 100 Hz) or pulse-width modulated analog signals (range, 120 mV bandwidth, 0-8 Hz and resolution, 1-2 %) can be recorded on the additional three tracks of the same cassette. 

The abihty to distinguish closely spaced peaks in spectroscopy has received much attention in the classical literature, and many of the same principles apply to Raman spectroscopy. Raman does require fairly high frequency precision and resolution, since one is observing relatively small frequency shifts from a particular laser frequency (see Chapter 10 for more detail). In the context of spectrometer evaluation, it should be noted that most analytical Raman apph-cations involve liquids and solids in which Raman bandwidths are significantly greater than those in the gas phase. The narrowest linewidths encountered in most liquid and solid samples are in the range of 3 to 10 cm". ... 

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