Solution Notes -- Example Problem Set 4

1.

The new Fourier Transform will be:

2.

The new Fourier Transform will be: $F(\mu)e^{- i\alpha \mu}$.

3.

The new Fourier Transform will be: $\vert\alpha \vert F(\alpha\mu)$.

4.

The Fourier Transform $H(\mu)$ of the convolution result h(x) from convolving f(x) with another function g(x), having respective Fourier Transforms $F(\mu)$ and $G(\mu)$, is the product of the two Transforms: $H(\mu) = F(\mu)G(\mu)$.

5.

The Laplacian operator $\nabla^{2}$ is defined as:

$$\nabla^{2} \equiv \left(\frac{d^{2}}{dx^{2}} + \frac{d^{2}}{dy^{2}} \right)$$

and the 2D Fourier Transform of $\nabla^{2}f(x,y)$ is:

$$\nabla^{2}f(x,y)\equiv \left(\frac{d^{2}}{dx^{2}} + \frac{d^{... ...stackrel{2DFT}{\Longrightarrow} -(\mu^{2}+\nu^{2})F(\mu,\nu).$$