Convolution difference filter

A more useful filter for resolution enhancement is the convolution difference filter, given by... 

Fig. 16. (A) The first increment of a NOESY spectrum of a 3 mM sample of a 26 amino acid peptide from the hepatitis B envelope protein recorded in 90% H2O at 273 K on a 500 MHz spectrometer. (B) Same as (A) but after applying the shifted time-domain convolution difference filter. (From Sodano and Delepierre with permission.)...

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. 

Except for its lower protein concentration, glomerular filtrate at the top of the nephron is chemically identical to the plasma. The chemical composition of the urine is however quantitatively very different to that of plasma, the difference is due to the actions of the tubules. Cells of the proximal convoluted tubule (PCT) are responsible for bulk transfer and reclamation of most of the filtered water, sodium, amino acids and glucose (for example) whereas the distal convoluted tubule (DCT) and the collecting duct are concerned more with fine tuning the composition to suit the needs of the body. 

The site of action of the thiazide and thiazide-like diuretics differs slightly from one species to another. In humans, however, it appears safe to conclude that all of these diuretics block the reabsorption of Na (and, thereby, the reabsorption of Cr) in the distal convoluted tubules by inhibiting the luminal membrane-bound Na /CI cotransport system (Fig. 18-5). Thus, all diuretics in this class arc responsible for the urinary loss of about 5 to 8% of the filtered load of Na. Although they differ in their potencies (i.c., the amount of drug needed to produce a given diuretic response), they are equally efficacious (i.c., they can all exert a similar maximal diuretic response).-"-As a result of their action at site 3, the thiazide and thia-zidc-like diuretics secondarily alter the renal excretion rate of important ions other than Na and Cl. Inhibition of Na ... 

In the simplest (and most localized) member of the Daubechies family, the four coefficients [Cq, Cj, C2, C3] represent the low-pass filter H that is applied to the odd rows of the transformation matrix. The even rows perform a different convolution by the coefficients [C3, -C2, Cj, -Cq] that represent the high-pass filter G. H acts as a coarse filter (or approximation filter) emphasizing the slowly changing (low-frequency) features, and G is the detail filter that extracts the rapidly changing (high-frequency) part of the data vector. The combination of the two filters H and G is referred to as a filter bank. 

Without apodisation, the basis functions of the Fourier set correspond to sharp pulses in the frequency domain. With apodisation, these pulses become convoluted with the transform of the apodisation function, e.g. a Gaussian. The convolution of a pulse with some shape moves this shape to the position of the pulse. As a consequence, the frequency domain is not cut up in disjoint frequencies, but in a series of overlapping Gaussians. Note that this is no more than a different view of the filtering effect of the transform of the apodisation function. 

For an impulse response that differs from zero on, let us say, four points, several aspects of the pyramid algorithm are less obvious. We need to be able to drop half the points and still represent the signal using the output of the LP and HP filters. In other words, we need to step the linear convolution of signal and impulse response by two points. The Haar wavelet basis is also special in the sense that, as the impulse responses are only two points wide, we do not lose points at the extremes when performing a linear convolution. For wider impulse responses, something has to be done about those extremes, e.g. a circular convolution, which puts an additional constraint on the shapes of those impulse responses. 

The Forman phase correction algorithm, presented in Chap. 2, is shown in Fig. 3.6. Initially, the raw interferogram is cropped around the zero path difference (ZPD) to get a symmetric interferogram called subset. This subset is multiplied by a triangular apodization function and Fourier transformed. With the complex phase obtained from the FFT a convolution Kernel is obtained, which is used to filter the original interferogram and correct the phase. Finally the result of the last operation is Fourier transformed to get the phase corrected spectrum. This process is repeated until the convolution Kernel approximates to a Dirac delta function. 

If we take the DFT of a signal and then take the inverse DFT of that, we of course get back to where we started. The cepstrum calculation differs in two important respects. First we are only using the magnitude of the spectrum - in effect we are throwing away the phase information. The inverse DFT of a magnitude spectrum is already very different fi om the inverse DFT of a normal (complex) spectrum. The log operation scales the harmonics which emphasises the periodicity of the harmonics. It also ensures that the cepstrum is the sum of the source and filter components and not their convolution. 

As it has been stressed at the beginning of this survey, it is important for an experimenter to be able to calculate theoretical responses. Once the experimental response has been properly manipulated > e,g. the noise has been filtered off, the background has been subtracted, convolution or deconvolution operations have been possibly performed the problem is how to perform the comparison with theoretical responses in order to 0 establish the nature of the mechanism ii) determine the relevant thermodynamic and Idnetic parameters. The first step, i.e. the identification of the type of the operative mechanism, can be simply based on the comparison of the trends of some significant quantities of experimental and computed responses. As an example, in the case of cyclic voltammetric experiments, the forward peak potential, the ratio between backward and forward peak currents, the difference... 

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