Signal processing convolution

Keywords Digital signal processing Convolution Application-based teaching... 

By means of numerical convolution one can obtain Xg t) directly from sampled values of G t) and Xj(t) at regular intervals of time t. Similarly, numerical deconvolution yields Xj(t) from sampled values of G(t) and Xg(t). The numerical method of convolution and deconvolution has been worked out in detail by Rescigno and Segre [1]. These procedures are discussed more generally in Chapter 40 on signal processing in the context of the Fourier transform. 

One of the most useful relations in signal processing is the convolution property. The convolution of two discrete (sampled) functions x(t) and y(t) is defined as... 

The goat of the signal processing is the parameter determination of the state of mixedness (i.e. Bodenstein number, mean residence time, etc.). The identiflcation of the model parameters is done by using an off-line identification scheme. The identification can take place either in the time or in the frequency domain. Since the convolution in the time domain becomes a multiplication in the frequency domain, it is advantageous to execute the identification in the frequency domain to save computational costs. Additionally, the presentation of the results in the frequency domain offers new insights. 

The correct (ideal in a provable digital signal processing sense) way to perform interpolation is convolution with the sine function, defined as ... 

In the case of very thin samples in transmission measurements, Fabry-Perot interference fringes can arise as a result of sample-support interactions which can manifest as an oscillatory signal that convolutes the data and further complicates the data interpretation process. Though proper modeling of these feamres can produce the tme absorbance while also revealing additional information about the sample, a workaround is to utilize a diffuse reflectance or absorption configuration as detailed below. 

R.G. Keys, Cubic convolution interpolation for digital image processing. IEEE Trans. Acoust. Speech Signal Process ASSP-29(6), 1153-1160 (1981)... 

This shows that the convolution of two functions in the space domain is equivalent to multiplication in the frequency domain. It is this simplifying property that demonstrates the advantage of conducting signal processing in the frequency domain rather than the space domain. 

For physicists, it models the behavior of a signal processing device with an input/f) and an output git), related by the transfer function K(t) featuring the device. Note that the convolution product is interesting for physicists because it helps distinguish the transfer function from the input signal. 

Convolution is another important concept in signal processing. For a linear system, let h t) be the impulse response (the output of the system to the input of delta function) of the system, then for any input x(t), the output y t) is obtained by convolution. 

We discuss convolution and correlation in one dimension, but they can be done in more than one dimension. The process is generally the same - not much more difficult than the one-dimensional case. Most signal processing programs that use Fourier processes allow the use of at least two dimensions. 

Like correlation, the mathematical definition of convolution is a combination of two other functions (or sampled data sets). Unlike correlation, g and h are normally very different. As used in signal processing, either gorh represents an input, and the other represents some process, and the convolution of g and h represents the output after the input is processed . The process h could represent the effect of an electrical circuit, and the input g could be the electrical signal as function of time the convolution would be the electrical signal at the output. Or g could be the irradiance from a target, and the process h could be the responsivity of a detector - both as functions of X and y. 

For real-world signals with finite records, the convolution processes leading to decomposition and reconstruction of a signal, require data in regions beyond the signal s endpoints. Assuming a mirror image of the... 

By way of illustration the spectrometry example is worked out. Two functions are involved in the process, the signal/(X,) and the convolution function h(k). Both functions should be measured in the same domain and should be digitized with the same interval and at the same r-values (in spectrometry X-values). Let us furthermore assume that the spectrum/(A,) and convolution function h(k) have a simple triangular shape but with a different half-height width. 

The partial convolution of two peaks poses special problems and with high signal to noise, it is easy to see how the first derivative would help this process. Differentiation accentuates the noise, so this will not be a universally successful procedure. 

For a correct analysis of photoionization processes studied by electron spectrometry, convolution procedures are essential because of the combined influence of several distinct energy distribution functions which enter the response signal of the electron spectrometer. In the following such a convolution procedure will be formulated for the general case of photon-induced two-electron emission needed for electron-electron coincidence measurements. As a special application, the convolution results for the non-coincident observation of photoelectrons or Auger electrons, and for photoelectrons in coincidence with subsequent Auger electrons are worked out. Finally, the convolutions of two Gaussian and of two Lorentzian functions are treated. 

Various models are used in the literature to account for the kinetics of the excitons involved in optical processes. In the simplest cases, the signal evolution n(t) can be reproduced by considering either a single exponential or multiexponential time dependences. This model is well suited for solutions or solids in which monomolecular mechanisms happen alone. Since in most transient experiments the temporal response is a convolution of a Gaussian-shaped pulse and of the intrinsic kinetics, the rate of change with time of the excited-state population decaying exponentially is given by... 

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