Md. Asif Uddin
    I.11.X03

    Finite boundaries break a shift identity

    counterexample▲▲△

    Cross-correlate (1,2,3)(1,2,3) with (1,1,1)(1,1,1) using zero padding one on both sides and stride one. Compare correlating its cropped right shift (0,1,2)(0,1,2) with taking the cropped right shift of its original output.

    Hint

    A cropped right shift discards the last entry and inserts zero on the left.

    Solution

    The original output is (3,6,5)(3,6,5). Correlating the shifted input gives (1,3,3)(1,3,3). Shifting the original output gives (0,3,6)(0,3,6). The two disagree at the boundary positions.

    This does not contradict I.11.T1: the finite crop removes values that an unrestricted-grid translation would retain. A boundary convention is part of the operator, not an implementation detail outside its mathematics. Circular shifts with compatible circular padding give a different case.

    Draws on