Avoiding Deconvolution

We examined this problem, working under the restrictions of the same laser in source and target systems and of higher resolution in the source instrument compared to the target instrument. The former restriction assures that the information contents of the two spectra are the same. The latter restrietion allows the calculation to be restricted to convolution, which is stable, and avoiding deconvolution, which is not [10]. 

In fact a sensor measures a flow and proceeds an integration of across a surface, which operates as a spacial lowpass filter. To avoid a critical deconvolution, the error due to this integration must be kept negligible. 

The Wiener filter therefore avoids noise amplification and provides the best solution according to some quality criterion. We will see that these features are common to all other methods which correctly solve the deconvolution inverse problem. The result of applying Wiener inverse-filter to the simulated image is shown in Fig. 3b. 

The effect of instrumental broadening can be eliminated by deconvolution (see p. 38) of the instrumental profile from the measured spectrum. If deconvolution shall be avoided one can make assumptions on the type19 of both the instrumental profile and of the remnant line profile. In this case the deconvolution can be carried out analytically, and the result is an algebraic relation between the integral breadths of instrumental and ideal peak profile. From such a relation a linearizing plot can be found (e.g., measured peak breadths vs. peak position ) in which the instrumental breadth effect can be eliminated (Sect. 8.2.5.8). 

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. 

With time, the CW-EPR spectral pattern slowly changes the proportion of the components varies and a third component appears. The CW-EPR spectrum taken 1 h after mixing can be deconvoluted into the earlier-mentioned triplet and quintet, and into a septuplet at giSO,3 = 1.9793, rtfiSO,3 1.0 x 10" 4 cm 1 (=3.0 MHz)], in 6 3 1 ratio. Other minor species also appear, but they were not included in the simulations to avoid over-parameterization.87 Based on the H hyperfine coupling, the three components are suggested to correspond to [CrOti/.v-O1,02-29)21 (complex II,... 

These properties carry back to the discrete formulation. We shall use both discrete and continuous formulations in this volume, changing back and forth as needs require. The continuous regime allows us to avoid consideration of sampling effects when such consideration is not of immediate concern. Deconvolution algorithms, on the other hand, are numerically implemented on sampled data, and we find the discrete representation indispensable in such cases. 

With convolution and deconvolution, one must be careful to avoid end-point error with this type of function. Convolution with the function beyond the end point of the data will extend inside the interval containing the data about half the length of the impulse response function, so the error will extend about half the length of the impulse response function also (assuming the impulse response function is approximately symmetrical). To minimize this error, the function extending beyond the end points should... 

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