Dirac comb

Dirac comb

Signal Analysis

Published in Russell L. Herman, An Introduction to Fourier Analysis, 2016

Another approach to the proof of the Sampling Theorem is to use the Dirac comb function. As shown in Figure 7.54, the Fourier transform y(w) is multiplied by the gate function, Gn(w) to produce the bandlimited function yn(w). One then computes the inverse Fourier transform of this function to obtain yn(t). Sampling of this function, using the Dirac comb function, gives (yn * combn)(t). The Fourier transform of this function, F[yn * combn](w), is then computed. This results in a periodic function as shown in the last plot of Figure 7.54. One then gets back the transform of the bandlimited function by multiplying by the gate function.

Identification of Impulse Responses in Heat Transfer: Parameterization, Doses, Partial Time Moments

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Journal Information

Published in Heat Transfer Engineering, 2023

Denis Maillet, Benjamin Rémy

Exactly the same parameterization can be implemented for the input and also for the output since a series of discrete observations or of one of these functions can be considered as the result of a sampling process, that is a projection of the corresponding continuous function over a one-sided Dirac comb defined as: where is the Dirac distribution. So, if the sampling of each of the three functions involved in the convolution model is fine enough, its output coincides with its parameterized version at each time of the discrete observation grid, with:

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Signal Analysis

Identification of Impulse Responses in Heat Transfer: Parameterization, Doses, Partial Time Moments