Transformation of the independent variable

Shifting and folding (reflecting about the vertical axis)
Given the sequence x(n), here n is the independent variable, we have the 2 transformation operations as stated below:

• Shifting in time by k units, where k is an integer, is represented by x(n-k). The sequence x(n-k) represents sequence x(n) shifted by k samples, to right if the k is positive, or to the left if the k is negative in nature. Parenthesize n and then replace it by (n-k) for the k units of delay, or by (n+k) for the k units of time advancement.
• Folding (a.k.a. time reversal or reflecting about the vertical axis) can be denoted by x(-n). The signal x(-n) corresponds to reflecting x(n) about the time origin n = 0. Now reverse the sign of n (replace n with -n).

As in the case of continuous-time signals the operations of shifting and folding are not commutative. In other words, the result of first shifting and then folding is not the same as that of first folding and then shifting.

The 3rd operation is scaling, which is discussed later in this unit.

Example Delta function- Given the delta function δ(n), first it reflects and then shifts by 2 units. The other possibility is 1st to shift by 2 units and then reflect.

The result of "reflect then shift" is shown below left. The word Reflect means to change the sign of n; then shifting can be done by replacing (n) by (n-2). The result is: (1) δ(n) → δ(-n) and (2) δ(-n) = δ(-(n))→ δ(-(n-2)) = δ(-n+2).

By continuing the example, the second possibility, the result of "shift then reflect" can be shown above right. Shift means replacing (n) by (n-2); then reflects by changing the sign of n. The result is δ(- n-2). 

The result is: (1) δ(n) = δ((n))→ δ((n-2)) and (2) δ((n-2)) = δ(n-2) → δ(-n-2).

Note that the 2 end results are not the same. This serves to show that the 2 operations of shifting and reflecting are not commutative in nature:

Fold(Shift(δ(n))) ≠ Shift(Fold(δ(n)))

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