Sine-squared function

The raw band pass of an AOTF has a sine squared function lineshape with sidebands, which if ignored, may amount tol0% of the pass optical energy in off-centre wavelengths. This apodization issue is normally addressed by careful control of the transducer coupling to the crystal. 

Fig. 19 Deconvolving s(x) = sinc(jc) and s(x) = sinc2(x) convolved data. Trace (a) is the original spectrum o(x), trace (b) the result of convolving with an eight-point sine function, trace (c) the result of unconstrained deconvolution using the same sine function for just 10 iterations. Trace (d) is the result of 100 iterations using zero clipping. Trace (e) is the original spectrum convolved with an 8 /T-point sine-squared function, trace (f) the result deconvolving trace (e) with the same sine-squared function for 100 iterations using the Jansson-type relaxation function r k)[oik X)(x)].

Fig. 15 Apodization, or the reduction of artifacts in the spectral line by the multiplication of the interferogram by a window function that tapers to zero at the end point of the interferogram. (a) Cosine interferogram of Fig. 13(a) multiplied by the triangular window function of Fig. 14(d). (b) Resulting spectral line, the sine-squared function.

Figure 3. TOCSY and NOESY contour plots are shown for Ppep-4 for the amide/aromatic region in Hj0(90%)/Dj0(10%) (left) and for the aH region in DjO (right). Ppep-4 peptide concentration was 20 mg/mL in 20 mM potassium phosphate, pH 6.3, 40°C. The NOESY mixing time was 0.2 s, and the TOCSY spin iock time was 40 ms. The data were zero-fiiied to 1024 in t1. The raw data were multiplied by a 30° shifted sine-squared function in t1 and t2 prior to Fourier transformation.

In the case of narrow unwanted signals a notch filter introduced by Marion et al. [80] and Cross [81] can be more efficient. These filters are based on convolution difference and usually employ a Gaussian function or sine-square function as a notch filter. 

It is obvious from eqs. (7a) and (7b) that the diffraction efficiencies of s- and p- components oscillate in the form of a sine square function asynchronously of which the primitive periods are Ts=an and Tp=7t/a, respectively. Therefore, when 0,2 has a large diffraction angle near 90°, the parameter a has a relative small value. This condition results a smaller value of Ts and a larger value of Tp, and the peak values of s- and p- diffraction efficiencies leave far away each other. The smaller value of Ts means a smaller required phase modulation. Therefore, in the condition of a small phase modulation value ttid, we can obtain a desired result of /s=100% and Tjp 0 and complete the purpose of polarization beam splitting effectively. Shown in Fig. 5 is the relation of diffraction efficiencies v.s. x considering 0,2=85°. It is obviously that when the value of x equals 0.46, corresponding to an effective index modulation Ni=0.15, we can obtain j/s=100% and j/p=1.89%. 

Figure 5 Filter function F x) (solid curve) as a function of kx. The dashed curve is a sine-squared function, (a) Q2 2.00, S2 = =1.50, (b) Q2 2 > (c)...

Now consider the case where the energy is only somewhat larger than the potential in Region II (E > Vn). The sine-squared function in Equation 5-35 varies from a minimum of zero (complete transmission), to a maximum of one (a non-zero reflection probability). Complete transmission of the particle will occur when the width of Region n, a, times the constant ku is equal to some positive integer, n, multiple of n. 

The sine-bell functions are attractive because, having only one adjustable parameter, they are simple to use. Moreover, they go to zero at the end of the time domain, which is important when zero-filling to avoid artifacts. Generally, the sine-bell squared and the pseudoecho window functions are the most suitable for eliminating dispersive tails in COSY spectra. 

With COSY and 2D y-resolved spectra, it is normally necessary to apply weighting functions in both dimensions. Multiplication with a sine-bell squared function is recommended. 

In optics and spectroscopy, resolution is often limited by diffraction. To a good approximation, the spread function may appear as a single-slit diffraction pattern (Section II). If equal-intensity objects (spectral lines) are placed close to one another so that the first zero of one sine-squared diffraction pattern is superimposed on the peak of the adjacent pattern, they are said to be separated by the Rayleigh distance (Strong, 1958). This separation gives rise to a 19% dip between the peaks of the superimposed patterns. 

Brief reflection on the sampling theorem (Chapter 1, Section IV.C) with the aid of the Fourier transform directory (Chapter 1, Fig. 2) leads to the conclusion that the Rayleigh distance is precisely two times the Nyquist interval. We may therefore easily specify the sample density required to recover all the information in a spectrum obtained from a band-limiting instrument with a sine-squared spread function evenly spaced samples must be selected so that four data points would cover the interval between the first zeros on either side of the spread function s central maximum. In practice, it is often advantageous to place samples somewhat closer together. 

What meaning do these two-point resolution criteria have in describing the deconvolution process, that is, resolution before and after deconvolution Although width criteria may be applied to derive suitable before-after ratios, the Rayleigh criterion raises an interesting question. Because the diffraction pattern is an inherent property of the observing instrument, would it not be best to reserve this criterion to describe optical performance The effective spread function after deconvolution is not sine squared anyway. 

We recommend that the Rayleigh criterion and Fourier frequency cutoff Q be used as fundamental specifications of optical performance, but that other criteria such as full width at half maximum be used in describing widths before and after deconvolution and in specifying spread functions having other than sine-squared shape. 

Typically, t(co) is small for co large. A spectrometer suppresses high frequencies. If the data i(x) have appreciable noise content at those frequencies, it is certain that the restored object will show the noise in a more-pronounced way. It is clearly not possible to restore frequencies beyond the band limit Q by this method when such a limit exists. (Optical spectrometers having sine or sine-squared response-function components do indeed band-limit the data.) Furthermore, where the frequencies are strongly suppressed, the signal-to-noise ratio is poor, and T(cu) will amplify mainly the noise, thus producing a noisy and unusable object estimate. 

A review of deconvolution methods applied to ESCA (Carley and Joyner, 1979) shows that Van Cittert s method has played a big role. Because the Lorentzian nature of the broadening does not completely obliterate the high Fourier frequencies as does the sine-squared spreading encountered in optical spectroscopy (its transform is the band-limiting rect function), useful restorations are indeed possible through use of such linear methods. Rendina and Larson (1975), for example, have used a multiple filter approach. Additional detail is given in Section IV.E of Chapter 3. 

Fig. 26 Fourier transform spectrum of v2 of ammonia. Trace (a) is a section of the infrared absorption spectrum of ammonia recorded on a Digilab Fourier transform spectrometer at a nominal resolution of 0.125 cm-1. In this section of the spectrum near 848 cm-1 the sidelobes of the sine response function partially cancel, but the spectrum exhibits negative absorption and some sidelobes. Trace (b) is the same section of the ammonia spectrum using triangular apodiza-tion to produce a sine-squared transfer function. Trace (c) is the deconvolution of the sine-squared data using a Jansson-type weight constraint.

So you can just set sbl = nilswl and sbsl = —sbl for a 90°-shifted sine-bell, and sbl = nil(2 x swl) and sbsl = 0 for an unshifted sine-bell. Bruker uses the parameter wdw (in both F and To) to set the window function (SINE = sine-bell, QSINE = sine-squared, etc.) and ssb for the sine-bell shift. For example, if ssb = 2, the sine function is shifted 90° (180°/ssb) and we get a simple cosine-bell window. For an unshifted sine-bell, use ssb = 0. 

Here Sq is the magnitude of the field and co is the frequency. The pulse envelope function,/(r), is typically chosen to represent either a sine-squared or a trapezoidal pulse. If we are interested in calculating quantities relevant to a specific intensity, Iq = ceI/8k, we choose a pulse envelope function that rises over several optical cycles to its maximum value (1.0) and is then held constant for 20-30 cycles. The pulse rise must involve at least a few cycles or the calculated results can be contaminated by unphysical transients. 

Figure 12. Transfer efficiencies P2 to 12) (upper row) and P3 to 13) (lower row) as functions of the detunings Ap and As (in units of fimax) at the end of the pulses for the intuitive (left column) and counterintuitive (right column) sequences of delayed sine-squared pulses eith the same peak amplitude flmax and a large temporal area fimaxx = 500 (x is the pulse length and the delay is x/2). The efficient pupulation transfer are bounded by Ap = 0 and As = 0 (thick full lines) and the branches of hyperbolas (dashed lines). The areas bounded by the full lines are labeled by the cases...

The shape of all of these weighting functions are altered subtly by squaring them to give the sine bell squared functions these are also shown in Fig. 4.13. The weighting function is then... 

It is apparent from Check it 3.3.2.1 that the 7i/2-shifted Sine-Bell squared window function is the most appropriate apodization procedure for the 2D IR phase sensitive COSY spectrum, see Fig. 3.16. The reason that the Sine-Bell squared function is the best choice is because the last data points are zero and this type of window function ensures that there is no discontinuity in the FID. However the position of the function also has an important effect on the intensity of the data points in the first third of FID and this is why several values of SSB should be tried prior to making a final selection. 

Fourier transformation of a FID, which has not decayed to zero intensity causes a distortion ("wiggles") at the base of peaks in the spectrum. By applying a suitable window function WDW the FID will decay smoothly to zero. A variety of window functions options are available, none, exponential EM, gaussian GM, sine SINE, squared sine QSINE and trapezoidal TRAP function. The best type of window function depends on the appearance of the FID and the resulting spectrum. Consequently where possible it is best to fit the window function interactively. 

A simple way to do this is to multiply by a symmetrical shaping function, such as the sine-bell function (Marco and Wuethrich, 1976), which is zero in the beginning, rises to a mciximum, and then falls to zero again, resembling a broad inverted cone (Fig. 1.36g). One problem with this function is that we cannot control the point at which it is centered, and its use can lead to severe distortions in line shape. A modification of the function, the phase-shifted sine bell (Wagner et al, 1978) (Fig. 1.36h), allows us to adjust the position of the maximum. This leads to a lower reduction in the signal-to-noise ratio and improved line shapes in comparison to the sine-bell function. The sine-bell squared and the corresponding phase-shifted sine-bell squared functions have also been employed (see Section 3.2.2. also). 

Shifted squared sine bell function. An apodization function with the amplitude of a squared sinusoidal pattern starting at a maximum and dropping to zero. The first quarter of a squared cosine waveform. 

By monitoring T using a photomultiplier, R can be determined. Note that the sign of R is unknown since T varies with the sine squared of R hence, one must determine the sign by another method. Also T is a multivalued function of so that the order of the retardation must be established via another route. 

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