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## The zeros of prediction filters

Because prediction filters are finite-impulse-response filters, they can be characterized by the zeros of their z-transform. From equation (2), we know that the zeros of the prediction filter are .Therefore, we can express this filter as follows:

 (10)
If we scan the amplitude spectrum of this filter over the s plane, we can find L notches at

 (11)
that locate all the zeros of the z-transform of this filter. Similarly, we can express the prediction filter as follows:

 (12)
where denotes the phases of the Mth order complex roots of the unity. Now, if we scan the amplitude spectrum of over the s plane, we can find notches at

 (13)
M times as many notches as that of .Comparing equation (13) with equation (11), it is apparent that these two equations become identical when is equal to zero. Thus, L out of zeros of are the zeros of .Our goal is to identify these L zeros when is known. If the component of data at frequency is not spatially aliased, then has L zeros between two vertical lines and , which are L zeros of . However, if the component of data at frequency is spatially aliased, the task of identifying the zeros becomes complicated and requires sophisticated algorithms.

Next: Dealiasing prediction filters with Up: DEALIASING THE PREDICTION FILTERS Previous: DEALIASING THE PREDICTION FILTERS
Stanford Exploration Project
11/18/1997