Complex Numbers for Audio Signal Processing

참고 : https://www.youtube.com/watch?v=fMqL5vckiU0&list=PL-wATfeyAMNrtbkCNsLcpoAyBBRJZVlnf


1. Why use complex number in audio?

To express both (1) frequency & (2) phase!

  • complex numbers : \(c = a+ib\)
    • where \(a,b \in \mathbb{R}\).
    • \(a\) : REAL part
    • \(ib\) : IMAGINARY part


Plotting complex numbers in …

(1) Cartesian coordinates

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(2) Polar coordinates

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  • express using \(c\) and \(\gamma\)


2. Polar Coordinates


\(\cos (\gamma)=\frac{a}{ \mid c \mid }\).

\(\rightarrow\) \(a= \mid c \mid \cdot \cos (\gamma)\)


\(\sin (\gamma)=\frac{b}{ \mid c \mid }\).

\(\rightarrow\) \(b= \mid c \mid \cdot \sin (\gamma)\)


\[c = a+ib\]

\(\rightarrow\) \(c= \mid c \mid \cdot(\cos (\gamma)+i \sin (\gamma))\).


\(\text{tan}(\gamma) = \frac{\sin (\gamma)}{\cos (\gamma)}=\frac{b}{a}\).

\(\rightarrow\) \(\gamma=\arctan \left(\frac{b}{a}\right)\).


3. Eular Formula

(1) Euler formula

\(e^{i \gamma}=\cos (\gamma)+i \sin (\gamma)\).


(2) Euler identity

\(e^{i\pi} +1=0\).

\(\rightarrow\) \(e^{i\pi}=-1\).


(3) Polar coordinates 2.0

\(c= \mid c \mid \cdot(\cos (\gamma)+i \sin (\gamma))\).

\(e^{i \gamma}=\cos (\gamma)+i \sin (\gamma)\).

\(\rightarrow\) \(c= \mid c \mid e^{i\gamma}\).

  • can express complex number \(c\) with
    • (1) \(\mid c \mid\) : magnitude
    • (2) \(\gamma\) : direction


Interpretation

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\(\rightarrow\) why not use MAGNITUDE & PHASE as polar coordinates?


4. Fourier Transform using Complex Number

(1) Magnitude & Phase

Magnitude

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Phase

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  • meaning of \(-\) : rotate “clock-wise”


(2) Continuous audio signal

\(g(t) \quad g: \mathbb{R} \rightarrow \mathbb{R}\).

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(3) COMPLEX Fourier Transform

\(\hat{g}(f)=c_f\).

  • \(\hat{g}: \mathbb{R} \rightarrow \mathbb{C}\).


Mapping into a complex space!

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Mathematical Expression

\(\hat{g}(f)=\int g(t) \cdot e^{-i 2 \pi f t} d t\).


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(4) Complex Fourier Transform coefficients

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(5) Magnitude & Phase

\(c_f=\frac{d_f}{\sqrt{2}} \cdot e^{-i 2 \pi \varphi_f}\).


a) Magnitude

ABSOLUTE value of \(\hat{g}(f)\)

= REAL part of \(c_f\)

= \(d_f=\sqrt{2} \cdot \mid \hat{g}(f) \mid\).


b) Phase

IMAGINARY value of \(\hat{g}(f)\)

= \(\varphi_f=-\frac{\gamma_f}{2 \pi}\).


5. Fourier & Inverse Fourier Transform

\(\begin{gathered} \hat{g}(f)=\int g(t) \cdot e^{-i 2 \pi f t} d t \\ g(t)=\int c_f \cdot e^{i 2 \pi f t} d f \end{gathered}\).

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