Continuous Space Fourier Transform of 2D Signals
| Continuous Space Fourier Transform (2D Fourier Transform) | ||
|---|---|---|
| 1D Continuous Space Fourier Transform(CSFT) definitions and its inverse transform | ||
| Continous Space Fourier Transform | $ \mathcal{X}(f)=\int_{-\infty}^{\infty}x(t)e^{-j2\pi ft} dt $ | |
| Inverse Continous Space Fourier Transform | $ x(t)=\int_{-\infty}^{\infty}\mathcal{X}(2\pi f)e^{j2\pi ft} df $ | |
| 2D Continuous Space Fourier Transform(CSFT) definitions and its inverse transform | ||
|---|---|---|
| Forward transform | $ \mathcal{F}(u,v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux+vy)} dxdy } $ | |
| Inverse transform | $ f(x,y) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{F(u,v)e^{j2\pi(ux+vy)} dudv } $ | |
| 2D Continuous Space Fourier Transform(CSFT) Properties | ||||
|---|---|---|---|---|
| $ \displaystyle x(t) $ | $ \longrightarrow $ | $ \mathcal{X}(f) $ | ||
| Linearity | $ \displaystyle af_1(x,y)+bf_2(x,y) $ | $ \displaystyle aF_1(u,v)+bF_2(u,v) $ | ||
| Scaling | $ f(\frac{x}{a},\frac{y}{b}) $ | $ \displaystyle|ab|F(au,bv) $ | ||
| Shifting | $ \displaystyle f(x-x_o,y-y_o) $ | $ \displaystyle F(u,v)e^{-j2\pi(ux_o+vy_o)} $ | ||
| Modulation | $ \displaystyle f(x,y)e^{j2\pi(xu_o+yv_o)} $ | $ \displaystyle F(u-u_o,v-v_o) $ | ||
| Reciprocity | $ \displaystyle F(x,y) $ | $ \displaystyle f(-u,-v) $ | ||
| Other Properties | ||
|---|---|---|
| Parseval’s relation | $ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{|f(x,y)|^2dxdy }=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{|F(u,v)|^2dudv } $ | |
| Initial Value | $ F(0,0)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)dxdy } $ | |
| Symmetry Properties for Continuous Space Fourier Transform | |
|---|---|
| If f(x,y) is real, the magnitued of F(u,v) is an even function; the angle of F(u,v) is an odd function. | |
| $ \displaystyle F(u,v)=A(u,v)e^{j\theta(u,v)} $ | |
| $ \displaystyle F(u,v)=F^{*}(-u,-v) $ | |
| $ \displaystyle A(u,v)=A(-u,-v) $ | |
| $ \displaystyle \theta(u,v)=-\theta(-u,-v) $ | |
| $ f(x,y)=2 \int_0 ^{\infty}\int_{-\infty}^{\infty}{ A(u,v)cos[2 \pi(ux+vy)+ \theta(u,v)] dudv} $ | |
| Separability | ||||
|---|---|---|---|---|
| $ \displaystyle g(x) $ | $ \longrightarrow $ | $ \displaystyle G(u) $ | ||
| $ \displaystyle h(x) $ | $ \longrightarrow $ | $ \displaystyle H(v) $ | ||
| $ \displaystyle f(x,y) $ | $ \longrightarrow $ | $ \displaystyle F(u,v) $ | ||
| $ \displaystyle f(x,y)=g(x)h(y) $ | $ \longrightarrow $ | $ \displaystyle F(u,v)=G(u)H(v) $ | ||
| *For example, | ||||
| $ \displaystyle rect(x,y)=rect(x)rect(y) $ | $ \longrightarrow $ | $ \displaystyle sinc(u)sinc(v)=sinc(u,v) $ | ||
- Notes: If we are trying to draw rect(x,y) from a top view, it will just look like a square. In the 3D plot, keep the top view as a base, making the height as 1. The plot is a cube. Similar as sinc(u,v).
| $ \displaystyle circ(x,y) $ | $ \longrightarrow $ | $ \displaystyle jinc(u,v) $ |
- Notes: if we are trying to draw circ(x,y) from a top view, it will look like a circle with a radius of ½. In the 3D plot, we keep the top view as a base, making the height as 1. The plot is a cylinder.
Before we go to the important transform pairs, the separability is a very important property of 2D signals. It enables us to transform 2D signals to our familiar 1D signals. given,
Other important transform pairs:
- $ \displaystyle \delta (x,y)---CSFT---1 $
- $ \displaystyle 1---CSFT--- \delta (u,v) $
- $ \displaystyle rect(x)---CSFT---sinc(u) \delta (v) $
- $ \displaystyle \delta (x)--CSFT---\delta (v) $
- $ e^{j2\pi(u_ox+v_oy)}---CSFT--- \delta (u-u_o,v-v_o) $
- $ cos[2\pi(u_ox+v_oy)]---CSFT---\frac{1}{2} [\delta (u-u_o,v-v_o)+\delta (u+u_o,v+v_o)] $
In 2D, we can also change the coordianting system from rectangular to polar form,
- 'x = rc'o's(θ)----------------u = ρc'o's(φ)
- 'y = rs'i'n(θ)----------------v = ρs'i'n(φ)
- Forward transform-$ F(\rho,\phi)=\int_0 ^{2\pi}\int_0 ^{\infty}{f(r,\theta)e^{-j2\pi\rho r cos(\phi -\theta)}r dr d \theta } $
- Inverse transform-$ f(r,\theta)=\int_0 ^{2\pi}\int_0 ^{\infty}{F(\rho,\phi)e^{j2\pi\rho r cos(\phi -\theta)}\rho d \rho d \phi } $
Some properties related to the polar representations
- Rotation $ \displaystyle f(r,\theta +\alpha )-------CSFT-------F(\rho ,\phi + \alpha) $
- Circular Symmetry $ \displaystyle f(r,\theta)=f_o(r)---------->F(\rho , \phi)=F_o(\rho) $
Convolution Theorem
- $ \displaystyle f_1(x,y)**f_2(x,y)---CSFT---F_1(u,v)F_2(u,v) $
Product Theorem
- $ \displaystyle f_1(x,y)f_2(x,y)---CSFT---F_1(u,v)**F_2(u,v) $
