Sources:

1. B. P. Lathi & Roger Green. (2021). Chapter 1: Signals and Systems. Signal Processing and Linear Systems (2nd ed., pp. 64-136). Oxford University Press.

Even and odd functions

A function $x_e(t)$ is said to be an even function of $t$ if it is symmetrical about the vertical axis.

A function $x_o(t)$ is said to be an odd function of $t$ if it is antisymmetrical about the vertical axis.

Mathematically expressed, these symmetry conditions require $$\begin{array}{}\text{(1)}& x_e(t)=x_e(-t)\text{ and }{x}_o(t)=-x_o(-t)\end{array}$$

where e stands for even, o stands for odd.

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.

Even and odd components of a signal

Every signal $x(t)$ can be expressed as a sum of even and odd components because $$\begin{array}{}\text{(2)}& x(t)=\underset{\text{even }}{\underbrace{\frac{1}{2}[x(t)+x(-t)]}}+\underset{\text{odd }}{\underbrace{\frac{1}{2}[x(t)-x(-t)]}}\end{array}$$

From the definitions in $\text{(1)}$ , we can clearly see that the first component on the right-hand side is an even function, while the second component is odd.

Finding the even and odd components of a signal

Given a signal $x(t)=e^{-at}u(t)$, based on $\text{(2)}$, we can expressas a sum of the even component $x_e(t)$ and the odd component $x_o(t)$ as $$x(t)=x_e(t)+x_o(t)$$ where $$x_e(t)=\frac{1}{2}[e^{-at}u(t)+e^{at}u(-t)]\text{ and }{x}_o(t)=\frac{1}{2}[e^{-at}u(t)-e^{at}u(-t)]$$

The function $e^{-at}u(t)$ and its even and odd components are illustrated in Fig. 1.24.

Figure 1.24

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 $x(t)=x^{\ast }(-t)$.
  • 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 $x(t)=-x^{\ast }(-t)$.
  • 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 $x_{cs}(t)$ plus a conjugate-antisymmetric portion $x_{ca}(t)$. That is, $$x(t)=x_{cs}(t)+x_{ca}(t)$$ where $$x_{cs}(t)=\frac{x(t)+x^{\ast }(-t)}{2}\text{ and }{x}_{ca}(t)=\frac{x(t)-x^{\ast }(-t)}{2}$$

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.