2장 Sinusoids(3)_ 복소수/ 극좌표/ 극좌표계와 직교좌표계/ 오일러 공식/ 복소 지수/ 오일러 공식의 실수 부분 Re()/ 복소 진폭/ 복소 지수 신호의 실수 부분과 허수 부분

COMPLEX NUMBERS

• To solve $z^{2}=-1$
  • $z=j$
  • Math and Physics use $z=i$
• Complex number : $z=x+jy$

데카르트 좌표계

$i$대신 $j$ 사용
y가 imaginary number, x가 real number
x가 $cos\theta$ , y가 $sin\theta$ 이므로 $cos\theta + jsin\theta = e^{j\theta}$
$z = re^{j\theta}= rcos\theta + jrsin\theta$

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PLOT COMPLEX NUMBERS

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COMPLEX ADDITION = VECTOR Addition

복소수 합 = 벡터 합

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POLAR FORM

• Vector Form
  • Length = 1
  • Angle = $\theta$
• Common Values
  • $j$ has angle of $0.5\pi$
  • -1 has angle of $\pi$
  • $-j$ has angle of $1.5\pi$
  • also, angle of $-j$ could be $-0.5\pi=1.5\pi-2\pi$
  • because the PHASE is AMBIGUOUS

j는 노란색 선, -j는 파란색 선 // 벡터 크기가 1 // 반대로 되면 부호가 달라지므로 위상이 애매모호함.

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POLAR ↔ RECTANGULAR

• Relate $(x,\; y)$ to $(r, \;\theta)$

$r^{2}=x^{2}+y^{2}$
$\theta = tan^{-1}(\frac{y}{x})$

$x = rcos\theta $

$y = rsin\theta $

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Euler's FORMULA

• Complex Exponential
  • Real part is cosine.
  • Imaginary part is sine.
  • Magnitude is one.

복소 지수 : 실수 부분은 cos, 허수 부분은 sine, 크기는 1

$e^{j\theta}=cos(\theta)+jsin(\theta)$

$re^{j\theta}=rcos(\theta)+jrsin(\theta)$

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COMPLEX EXPONENTIAL

$e^{j\omega t}=cos(\omega t)+jsin(\omega t)$

• Interpret this as a Rotating Vector
  • $\theta=\omega t $
  • Angle changes vs. time
  • ex : $\omega=20\pi\;rad/s $
  • Rotates $0.2\pi$ in 0.01 secs

$e^{j\theta}=cos(\theta)+jsin(\theta)$

1초에 $20\pi$니까 0.01초에는 $0.2\pi$임.
$\omega=\frac{\theta}{t} $임.

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cos = REAL PART

Real Part of Euler's (오일러 공식에서 실수부를 다음과 같이 표현함)

$cos(\omega t)=\Re \left\{e^{j\omega t} \right\}$

General Sinusoid

$x(t)=Acos(\omega t+\varphi )$

So, 

$Acos(\omega t+\varphi )=\Re \left\{Ae^{j(\omega t+\varphi )} \right\}=\Re \left\{Ae^{j\varphi }e^{j\omega t} \right\}$

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REAL PART EXAMPLE

 $Acos(\omega t+\varphi )=\Re \left\{Ae^{j\varphi }e^{j\omega t} \right\}$

 Evaluate :  

$x(t) = \Re \left\{-3je^{j\omega t} \right\}$

 Answer :

$x(t) = \Re \left\{(-3j)e^{j\omega t} \right\}=\Re \left\{3e^{-j0.5\pi}e^{j\omega t} \right\}=3cos(\omega t-0.5\pi)$

-3j는? : imaginary line의 아래로 3만큼 내려간 하나의 벡터 (-3인 벡터)
-3은 $1.5\pi=-0.5\pi$ 따라서 $3e^{-j0.5\pi}$

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COMPLEX AMPLITUDE

General Sinusoid

$x(t) = Acos(\omega t+\varphi )=\Re \left\{Ae^{j\varphi}e^{j\omega t} \right\}$

Complex AMPLITUDE = X                                                        ↕

$z(t) = Xe^{j\omega t}\;\;\;\;\;\;\;\;\;\;\;X=Ae^{j\varphi }$

Then, any Sinusoid = REAL PART of $Xe^{j\omega t}$

$x(t) = \Re \left\{Xe^{j\omega t} \right\}=\Re \left\{{Ae^{j\varphi }e^{j\omega t}} \right\}$

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Real and imaginary parts of the complex exponential signal

$z(t) = 20e^{j(2\pi(40)t-0.4\pi)}=20e^{j(80\pi t-0.4\pi)}$

        $=20cos(80\pi t-0.4\pi)+j20cos(80\pi t$$-0.9\pi$$)$

$j20sin(80\pi t-0.4\pi)$을 $cos$으로 바꾸어서 $j20cos(80\pi t-0.9\pi) $가 된 것.