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Is there any relationship between the FFT and linear predictive methods? Can an FFT result or the input to an FFT be modified to do non-circular prediction/extrapolation from the FFT results.

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One thing that comes to mind is that a linear predictor can be implemented using the LMS algorithm, and the LMS algorithm can be implemented in block fashion using FFTs (see Chapter 7 of Haykin's Adaptive Filter Theory). I wouldn't say that implies any kind of relationship, but FFTs can be used to accelerate the implementation of some linear predictors.

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Well, the linear prediction problem can be thought as a filtering operation. Actually many linear predictors are based on FIR filters. So Technically you can implement the FIR filter with an FFT (And in a more efficient way depending on the size of the data).

As a Reference, try "The Theory of Linear Prediction - Vaidyanathan"

http://www.amazon.com/Theory-Prediction-Synthesis-Lectures-Processing/dp/1598295756

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