Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

B.6 EVEN AND ODD FUNCTIONS

Definition: Even and Odd Even function gE(x) and odd function gO(x) are defined by having the following properties about x = 0:

(B.29)

Any function g(x) can be decomposed as follows:

(B.30)

where

(B.31)

An example of this decomposition is shown in Figure B.6. The only type of function that is even and odd, satisfying both equations in (B.31), is a constant function.

Figure B.6 Example even and odd parts of a piecewise linear function. (a) g(x). (b) g(−x). (c) gE(x). (d) gO(x).

Several properties of even and odd functions are provided in Table B.1 without proof. It is easy to see that for finite a:

Table B.1 Properties of Even and Odd Functions

Function	Property
g1E(x)g2E(x)	Even
g1O(x)g2O(x)	Even
gE(t)gO(x)	Odd
g1E(x)+g2E(x)	Even
g1O(x)+g2O(x)	Odd
gE(t)+gO(x)	Neither
dgE(x)/dx	Odd
dgO(x)/dx	Even

(B.32)

which is useful when calculating moments of a distribution. When an integral of the following form is obtained

(B.33) ...