True signal function

Therefore, signal functions y(z) always represent a convolution of the true signal function y z)trUe and the characteristic function of the analytical instrument h(z) which characterizes all the insufficiencies of the measuring system ... 

A signal function y(z), which is mostly treated as a time-dependent function, /(f), can be regarded as consisting of the original true signal function yt rue (S ) which is superposed by a noise function n(t) see Fig. 2.15i... 

Component of the signal function in the time domain that is considered being true (influenced by noise)... 

FIGURE 10.4 Discrete convolution of two functions (a) Gaussian broadening function (b) true signal (dotted line) and broadened result (solid line) of convolution with the Gaussian function. 

FIGURE 10.12 Time-domain smoothing of the noisy data in Figure 10.1 with the impulse response function of Figure 10.7, processed from left to right in this spectrum. The true signal is shown as a dotted line. Note the significant filter lag in this example. 

As a consequence, the results typically reflect what was put into the trial function. Consider calculating the momentum distribution. Suppose the trial function has a Fermi surface. Then the momentum distribution will exhibits a discontinuity ai kf signaling the presence of a Fermi surface. This does not imply that the true wave function has a sharp Fermi surface. 

Here, y k) represents the estimate of the true signal. Further, / is the filter constant, or, in other words, the filtering bandwidth. By a judicious choice of / , one can remove high-frequency noise components from the signal and retain the relevant signal characteristics. Figure 6.10 shows the frequency response of a first-order filter and how the bandwidth changes as a function of / . 

The kinetic determination of any concentration as a function of time yields k, as in Eqs. (3-61) and (3-63). This is true no matter whether one follows [A] or [P],. The latter point, although correct, can sometimes seem illogical. Suppose one measures the buildup of P2 (say), by monitoring an infrared peak, an ultraviolet band, or an NMR signal. Assume that neither A nor any product other than P2 contributes to the signal. Surely then, will it not be k2 that is obtained from the kinetic analysis The answer is no. Consider the result from Eq. (3-63), which gives the concentration of the one product in terms of its absorbance (per unit optical path) and molar absorptivity (62) ... 

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