----- > [!proposition] Proposition. ([[converting from complex exponentials to sinusoids]]) > Let $z \in \cc$ and $t \in \rr$. We have $ze^{i \omega t} + \overline{z} e^{- i\omega t} = A\cos (\omega t + \theta),$ where $A= 2 | z |$ and $\theta =$ [[argument of a complex number]] $z$. > [!proof]- Proof. ([[converting from complex exponentials to sinusoids]]) > Let $z=a+bj$. Using [[Euler's formula]] we have > $\begin{align} ze^{i \omega t} + \overline{z}e^{-i \omega t} = & (a + bi)e^{i \omega t} + (a-bi)e^{-i\omega t} \\= & ae^{i \omega t} + bi e ^{i \omega t} + ae^{-i \omega t} - bie^{-i \omega t}\\ = & a\cos \omega t + ai \sin \omega t + bi\cos \omega t + bi^{2} \sin \omega t \\+ & a\cos (-\omega t) + a i\sin(- \omega t) -bi\cos(-\omega t)-bi^{2} \sin (-\omega t) . \end{align}$ Using parity of $\sin$ and $\cos$ and that $i^{2}=-1$, the above simplifies into $2a\cos \omega t - 2b \sin \omega t. (*)$ Using the [[cosine sum formula]] this implies that there is $A \in \cc$, $\theta \in [0,2\pi)$ s.t. $(*)=A \cos (\omega t + \theta)$ where $2a=A\cos \theta$ and $2b=A\sin \theta$. Dividing we get $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{b}{a},$ hence $\theta = \tan ^{-1} (\frac{b}{a})=\arg z$. We also see that $A^{2}\cos ^{2}\theta + A^{2}\sin ^{2}\theta = A^{2} = (2a)^{2} + (2b)^{2},$ therefore $A=\sqrt{ 4a^{2} + 4b^{2} }=2\sqrt{ a^{2} + b^{2} }=2|z |$. ----- #### > [!basicexample]- > 1. Consider $s=4e^{jt} + 4e^{-jt}$. We have $z=4=4+0j$. Then $z=|z |,$ hence $A=2\cdot {4} = 8$. Clearly $\arg z=0$. Thus $s=8\cos ( t)$. > 2. Consider $s=-je^{2jt}+je^{-2jt}$. We have $z=0+1j$ and $\overline{z}=0-1j$. Hence $A=2|z |=2$ and $\arg z = \arg i = \frac{\pi}{2}$: $s=2\cos\left( 2t + \frac{\pi}{2} \right)$. > 3. Consider $s = (4+4j)e^{6jt} + (4-4j)e^{-6jt}$. We have $|z | = \sqrt{ 32 }$, and $\arg z=\frac{\pi}{4}$. So, $s=2\sqrt{32 }\cos\left( 6t + \frac{\pi}{4} \right)$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```