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FINITE DIFFERENCE S INTRODUCTION Suppose x(0), x(1),...x(n) are sample input signals and y(0), y(1), ....y(n) output signals as given in Fig. Then we ca n form a table of values x x o x 1 x 2 x 3 . . . x n-1 x n y y o y 1 y 2 y 3 . . . x n-1 y n If we subtract from each value of y (except y 0 ) the proceeding value of y we get y 1 - y 0 , y 2 - y 1 , .....y n - y n-1 . These are called as first order forward differences of y and are denoted by ∆y. i. e ∆y 0 = y 1 - y 0 , ∆y 1 = y 2 - y 1 , .... ∆y n-1 = y n - y n-1 . These are called first differences. From these we can form second order differences ∆ 2 y 0 = ∆y 1 - ∆y 0 , ∆ 2 y 1 = ∆y 2 - ∆y 1 , ... ∆ 2 y n-1 = ∆y 2 - ∆y 1 . If x 0 , x 1 , x 2 , ,, x n are equally spaced t hen the input sig nals can be written as x 0 + h, x 0 + 2h, ..... x n + h , so tha x 1 - x 0 = h = x 2 - x 1 etc. We now define ∆f(x)= f(x + h) - f(x). Backward Differences: The differences of the form ∇ y 1 = y 1 - y 0 , ∇ y 2 = y 2 - y 1 , .... ∇ y n = y n - y n-1 .

Lecture Notes Finite Differences

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