Chapter 2 - Signal Properties¶
Symmetry¶
Worked Example 1:¶
Find the even and odd components of
\[
f(t) = \cos{\left(2\pi t - \tfrac{\pi}{4}\right)}
\]
Show answer
The even and odd components are given by:
\[
\begin{cases}
f_{e}(t) &= \frac{\sqrt{2}}{2}\cos{\left(2\pi t\right)} \\
f_{o}(t) &= \frac{\sqrt{2}}{2}\sin{\left(2\pi t\right)} \\
\end{cases}
\]
where
\[
f(t) = f_{e}(t) + f_{o}(t)
\]
Method A:¶
Show solution
We can use a trig identity to solve this.
\[
f(t) = \cos{\left(2\pi t - \tfrac{\pi}{4}\right)}
\]
\[
\cos{(a-b)} = \cos{(a)}\cos{(b)} + \sin{(a)}\sin{(b)}
\]
So for this signal, we have:
\[
\begin{align*}
f(t) &= \cos{\left(2\pi t - \frac{\pi}{4}\right)} \\
&= \cos{\left(2\pi t\right)}\cos{\left(\tfrac{\pi}{4}\right)} + \sin{\left(2\pi t\right)}\sin{\left(\tfrac{\pi}{4}\right)} \\
&=\frac{\sqrt{2}}{2}\cos{\left(2\pi t\right)} + \frac{\sqrt{2}}{2}\sin{\left(2\pi t\right)}
\end{align*}
\]
So clearly our even and odd components are:
\[
\begin{cases}
f_{e}(t) &= \frac{\sqrt{2}}{2}\cos{\left(2\pi t\right)} \\
f_{o}(t) &= \frac{\sqrt{2}}{2}\sin{\left(2\pi t\right)} \\
\end{cases}
\]
Method B:¶
Show solution
The other method for this is to use the formlas directly
\[
f(t) = \cos{\left(2\pi t - \tfrac{\pi}{4}\right)}
\]
For the even component:
\[
\begin{align*}
f_{e}(t) &= \tfrac{1}{2}\left(f(t) + f(-t)\right) \\
&= \tfrac{1}{2}\left(\cos{\left(2\pi t - \tfrac{\pi}{4}\right)} + \cos{\left(-2\pi t - \tfrac{\pi}{4}\right)}\right) \\
&= \tfrac{1}{2}\left(\cos{\left(2\pi t - \tfrac{\pi}{4}\right)} + \cos{\left(2\pi t +\tfrac{\pi}{4}\right)}\right) \\
\end{align*}
\]
Now using the trig identity:
\[
\cos{(a)}\cos{(b)} = \tfrac{1}{2}\left(\cos{(a - b)} + \cos{(a + b)}\right)
\]
We get:
\[
f_{e}(t) = \cos{\left(2\pi t\right)}\cos{\left(\tfrac{\pi}{4}\right)} = \tfrac{\sqrt{2}}{2}\cos{\left(2\pi t\right)}
\]
For the odd component:
\[
\begin{align*}
f_{o}(t) &= \tfrac{1}{2}\left(f(t) - f(-t)\right) \\
&= \tfrac{1}{2}\left(\cos{\left(2\pi t - \tfrac{\pi}{4}\right)} - \cos{\left(-2\pi t - \tfrac{\pi}{4}\right)}\right) \\
&= \tfrac{1}{2}\left(\cos{\left(2\pi t - \tfrac{\pi}{4}\right)} - \cos{\left(2\pi t +\tfrac{\pi}{4}\right)}\right) \\
\end{align*}
\]
Now using the trig identity:
\[
\sin{(a)}\sin{(b)} = \tfrac{1}{2}\left(\cos{(a - b)} - \cos{(a + b)}\right)
\]
We get:
\[
f_{o}(t) = \sin{\left(2\pi t\right)}\sin{\left(\tfrac{\pi}{4}\right)} = \tfrac{\sqrt{2}}{2}\sin{\left(2\pi t\right)}
\]