MIT 12 864 - 4. Discrete Observations (4 pages)

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4. Discrete Observations



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4. Discrete Observations

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Pages:
4
School:
Massachusetts Institute of Technology
Course:
12 864 - Inference from Data and Models
Inference from Data and Models Documents

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4 Discrete Observations 16 4 0 1 Sampling The above results show that a band limited function can be reconstructed perfectly from an in nite set of perfect samples Similarly the Fourier transform of a time limited function can be reconstructed perfectly from an in nite number of perfect samples the Fourier Series frequencies In observational practice functions must be both band limited one cannot store an in nite number of Fourier coe cients and time limited one cannot store an in nite number of samples Before exploring what this all means let us vary the problem slightly Suppose we have w with Fourier transform v and we sample w at uniform intervals p w without paying attention initially as to whether it is band limited or not What is the relationship between the Fourier transform of the sampled function and that of w That is the above development does not tell us how to compute a Fourier transform from a set of samples One could use 3 2 interpolating before computing the Fourier integral As it turns out this is unnecessary We need some way to connect the sampled function with the underlying continuous values The function proves to be the ideal representation Eq 2 13 produces a single sample at time wp The quantity LLL w w 4 X w q w 4 1 q 4 vanishes except at w t w for any integer t The value associated with LLL w at that time is found by integrating it in an in nitesimal interval t w w t w and one nds immediately that LLL t w t w Note that all measurements are integrals over some time interval no matter how Continued on next page 4 D ISC R E T E O B SE RVAT IO N S 17 short perhaps nanoseconds Because the function is in nitesimally broad in time the briefest of measurement integrals is adequate to assign a value 1 Let us Fourier analyze LLL w and evaluate it in two separate ways I Direct sum Z 4 LLL v 4 w p w h 2 lvw gw p 4 4 X 4 X w p w h 2 lvp w 4 2 p 4 II By convolution v F LLL v What is F P4 p 4 4 X w p w 4 3 w p w We have by direct integration 4 4 X X w p w h



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