Fourier Transform and Discrete Function Continuation

DISCRETE FOURIER TRANSFORM AND DISCRETE FUNCTION CONTINUATION 

As with the continuous Fourier transform, we could treat the equations of the discrete Fourier transform (DFT) completely independently, derive all the required theorems for them, and work entirely within this closed system. However, because the data from which the discrete samples are taken are usually continuous, some discussion of sampling error is warranted. Further, the DFT is inherently periodic, and the limitations and possible error associated with a periodic function should be discussed. 

In these equations, N is the number of sample points, and n and k are integers. 

To illustrate more appropriately the relationship between the (continuous) Fourier transform and the DFT, the alternative form given in Chapter 1 will be employed. Accordingly, we define this transform as 

Then to recover the original function f(x) we must define the inverse transform as 

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IV. Discrete Fourier Transform and Discrete Function Continuation... 

IV. DISCRETE FOURIER TRANSFORM AND DISCRETE FUNCTION CONTINUATION... 

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