Exponential filters double

Substituting (17-13) into (17-11) and rearranging gives the following expression for the double exponential filter ... 

The advantage of the double exponential filter over the exponential filter of Eq. 17-9 is that it provides better filtering of high-frequency noise, especially if 7 = a. On the other hand, it is sometimes difficult to tune 7 and a properly for a given application or data set. It is also hard to tune a controller in series with a double exponential filter. 

Although the double exponential filter is beneficial in some cases, the single exponential filter is more widely used in process control applications. 

Platinum and palladium porphyrins in silicon rubber resins are typical oxygen sensors and carriers, respectively. An analysis of the characteristics of these types of polymer films to sense oxygen is given in Ref. 34. For the sake of simplicity the luminescence decay of most phosphorescence sensors may be fitted to a double exponential function. The first component gives the excited state lifetime of the sensor phosphorescence while the second component, with a zero lifetime, yields the excitation backscatter seen by the detector. The excitation backscatter is usually about three orders of magnitude more intense in small optical fibers (100 than the sensor luminescence. The use of interference filters reduce the excitation substantially but does not eliminate it. The sine and cosine Fourier transforms of/(f) yield the following results ... 

Deconvolution of the observed decay pulse was necessary because of the finite width (approx. 1.9ns FWHM) of the excitation pulse. This was carried out using a Fourier transform method (4). Despite the non-trlvlal computational requirements and the need to filter out the high frequency components of the Fourier division before taking the Inverse transform (5) It was felt that this method was the most direct way of generating the true Impulse response function of the system. The resulting fluorescence decay curves were fit to a single (or double) exponential. The phase plane plot method of Demas (6) and others (2), was also used for deconvolution but with less success. 

At the start of a pulse saturation experiment the sample is saturated, which is equivalent, in terms of parameter (Ok, to setting the value of the transfer function equal to zero, that is, implying zero resultant magnetization. This is analogous to establishing a short circuit between the terminals of the second capacitor in the double RC-filter model. After the saturating pulse the amplitude of the microwave field. Hi, does not saturate the material, and one then observes an exponential increase in the value of the magnetization. 

Another useful digital filter is the double exponential or second-order filter, which offers some advantages for dealing with signal drift the second-order filter is equivalent to two first-order filters in series where the second filter input is the output signal yp k) from the exponential filter in Eq. 17-9. The second filter (with output yp k) and filter constant 7) can be expressed as... 

Fig. 4. Two-dimensional (2D) spectra of cyclo(Pro-Gly), 10 mM in 70/30 volume/volume DMSO/H2O mixture at CLio/27r = 500 MHz and T = 263 K. (A) TCX SY, t = 55 ms. (B) NOESY, Tm = 300 ms. (C) ROESY, = 300 ms, B, = 5 kHz. (D) T-ROESY, Tin = 300 ms, Bi = 10 kHz. Contours are plotted in the exponential mode with the increment of 1.41. Thus, a peak doubles its intensity every two contours. All spectra are recorded with 1024 data points, 8 scans per ti increment, 512 fi increments repetition time was 1.3 s and 90 = 8 ps 512x512 time domain data set was zero filled up to 1024 x 1024 data points, filtered by Lorentz to Gauss transformation in u>2 domain (GB = 0.03 LB = -3) and 80° skewed sin" in u), yielding a 2D Fourier transformation 1024 x 1024 data points real spectrum. (Continued on subsequent pages)...

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