[신호처리] Lec 04 - Fourier Transform

Precaution

본 게시글은 서울대학교 이종호 B 교수님의 SNU FastMRI Challange, 2021 Signal Processing을 바탕으로 제작되었습니다.

Fourier Transform

• Transformation to frequency domain

\[x(t) \stackrel{\mathcal{F}}{\rightarrow}X(f)\]

• 2D Fourier Transform

\[a(x,y) \rightarrow A(k_x, k_y)\]

Mathematical Definition (1D)

• Fourier Transform

\[X(f) = \int_{-\infty}^\infty x(t) e^{-j\cdot 2\pi f t}dt = \mathcal{F}(x(t))\]

• Inverse Fourier Transform

\[x(t) = \int_{-\infty}^\infty X(f) e^{+ j\cdot 2\pi f t}dt = \mathcal{F}^{-1}(X(f))\]

Mathematical Definition (2D)

• Fourier Transform

\[A(k_x, k_y) = \int_{-\infty}^\infty\int_{-\infty}^\infty a(x,y) e^{-j\cdot 2\pi (k_x x + k_y y)}dxdy\]

• Inverse Fourier Transform

\[a(x,y) = \int_{-\infty}^\infty\int_{-\infty}^\infty A(k_x,k_y) e^{+j\cdot 2\pi (k_x x + k_y y)}dk_xdk_y\]

• Useful Formulas i.e. time delay / image shift transformed expotential : Linear phase

\[\begin{aligned} \delta(t) \ &\leftrightarrow \ 1 \\ \delta(at) \ &\leftrightarrow \ \frac{1}{|a|} \\ \delta(t-t_0) \ &\leftrightarrow \ e^{-j 2\pi f t_0} \\ \delta(x-x_0, y) \ &\leftrightarrow \ e^{-j 2\pi k_x x_0} \end{aligned}\]

• Quarter-pixel shift : 1/4 픽셀만큼 shift하는 것은 복잡함. 이 때 영상을 F.T. 한 다음 Linear phase를 곱하고 다시 IFT 하면 쉽게 shift할 수 있음.

\[\begin{aligned} 1 \ &\leftrightarrow \ \delta(f) \\ \text{rect}(t) \ &\leftrightarrow \ \text{sinc}(f) \\ \land (t) = \text{rect}(t) * \text{rect}(t) \ &\leftrightarrow \ \text{sinc}^2 (f) \\ \cos{(2\pi f_0 t)}=\frac{e^{j 2\pi f_0 t}+e^{-j 2\pi f_0 t}}{2} \ &\leftrightarrow \ \frac{\delta(f-f_0) + \delta (f+f_0)}{2} \\ e^{-at}u(t) \ &\leftrightarrow \ \frac{1}{a+j\cdot 2\pi f} \end{aligned}\]

• $u(t)$ : unit step function, a: real positive number F.T. of exponential function is so-called Lorentzian

rect and sinc function

Lorentzian function

F.T. Related to Shah Function

• Shah (or III) function and its F.T. i.e. Sampling function.

\[\begin{aligned} III(t) = \sum_{k=-\infty}^\infty \delta(t-k) &\leftrightarrow \ III(f) \\ f(at) &\leftrightarrow \ \frac{1}{|a|} \mathcal{F}(\frac{f}{a}) \\ \frac{1}{T} III(\frac{t}{T}) &\leftrightarrow \ III(Tf) \end{aligned}\]

Properties of F.T.

• 가장 큰 이점 : convolution 계산을 단순 곱연산으로 바꾸어 준다.

\[\begin{aligned} y(t) &= x(t) * h(t) \\ Y(f) &= X(f) \cdot H(f)\\ y(t) &= \mathcal{F}^{-1}\big( \mathcal{F}(x(t)) \cdot \mathcal{F}(h(t))\big) \end{aligned}\]

• Modulation property

\[y(t) = s(t) \cdot p(t) \leftrightarrow \ Y(f) = S(f) * P(f)\]