[1.5 EVEN AND](#page-7-0) ODD FUNCTIONS

← Back to LINEAR SYSTEMS AND SIGNALS Overview

1.5 EVEN AND ODD FUNCTIONS

A function xe(t) is said to be an even function of t if it is symmetrical about the vertical axis. A function xo(t) is said to be an odd function of t if it is antisymmetrical about the vertical axis. Mathematically expressed, these symmetry conditions require

x_e(t) = x_e(-t)

and $x_o(t) = -x_o(-t)$ (1.15)

An even function has the same value at the instants t and −t for all values of t. On the other hand, the value of an odd function at the instant t is the negative of its value at the instant −t. An example even signal and an example odd signal are shown in Figs. 1.23a and 1.23b, respectively.

1.5-1 Some Properties of Even and Odd Functions

Even and odd functions have the following properties:

even function ×odd function = odd function odd function ×odd function = even function even function ×even function = even function

The proofs are trivial and follow directly from the definition of odd and even functions [Eq. (1.15)].

AREA

Because of the symmetries of even and odd functions about the vertical axis, it follows from Eq. (1.15) [or Fig. 1.23] that

\int_{-a}^{a} x_e(t) dt = 2 \int_{0}^{a} x_e(t) dt \quad \text{and} \quad \int_{-a}^{a} x_o(t) dt = 0 \quad (1.16)

These results are valid under the assumption that there is no impulse (or its derivatives) at the origin. The proof of these statements is obvious from the plots of even and odd functions. Formal proofs, left as an exercise for the reader, can be accomplished by using the definitions in Eq. (1.15).

Because of their properties, study of odd and even functions proves useful in many applications, as will become evident in later chapters.

1.5-2 Even and Odd Components of a Signal

Every signal x(t) can be expressed as a sum of even and odd components because

x(t) = \underbrace{\frac{1}{2}[x(t) + x(-t)]}_{\text{even}} + \underbrace{\frac{1}{2}[x(t) - x(-t)]}_{\text{odd}}

(1.17)

From the definitions in Eq. (1.15), we can clearly see that the first component on the right-hand side is an even function, while the second component is odd. This is apparent from the fact that replacing t by −t in the first component yields the same function. The same maneuver in the second component yields the negative of that component.

EXAMPLE 1.8 Finding the Even and Odd Components of a Signal

Find and sketch the even and odd components of x(t) = e−atu(t).

Based on Eq. (1.17), we can express x(t) as a sum of the even component xe(t) and the odd component xo(t) as

x(t) = x_e(t) + x_o(t)

where

x_e(t) = \frac{1}{2} [e^{-at}u(t) + e^{at}u(-t)] \quad \text{and} \quad x_o(t) = \frac{1}{2} [e^{-at}u(t) - e^{at}u(-t)]

The function e−atu(t) and its even and odd components are illustrated in Fig. 1.24.

EXAMPLE 1.9 Finding the Even and Odd Components of a Complex Signal

Find the even and odd components of ejt.

From Eq. (1.17),

ejt ⁼ xe(t)+xo(t)

where

xe(t) ⁼ ¹ ² [ejt ⁺e−jt] = cos ^t and xo(t) ⁼ ¹ ² [ejt −e−jt] = jsin t

A MODIFICATION FOR COMPLEX SIGNALS

While a complex signal can be decomposed into even and odd components, it is more common to decompose complex signals using conjugate symmetries. A complex signal x(t) is said to be conjugate-symmetric if x(t) = x∗(−t). A conjugate-symmetric signal is even in the real part and odd in the imaginary part. Thus, a real conjugate-symmetric signal is an even signal. A signal is conjugate-antisymmetric if x(t) = −x∗(−t). A conjugate-antisymmetric signal is odd in the real part and even in the imaginary part. A real conjugate-antisymmetric signal is an odd signal. Any signal x(t) can be decomposed into a conjugate-symmetric portion xcs(t) plus a conjugate-antisymmetric portion xca(t). That is,

x(t) = x_{cs}(t) + x_{ca}(t)

where

x_{cs}(t) = \frac{x(t) + x^*(-t)}{2}

and $x_{ca}(t) = \frac{x(t) - x^*(-t)}{2}$

The proof is similar to the one for decomposing a signal into even and odd components. As we shall see in later chapters, conjugate symmetries commonly occur in real-world signals and their transforms.