PX275 - H7b - examples of convolutions

convolution with a $δ$ function

\begin{aligned} f * g (x^{'}) & = \int_{- \infty}^{\infty} f (y^{'}) g (x^{'} - y^{'}) d y^{'} \\ f * δ (x - y) & = \int_{- \infty}^{\infty} f (z) δ (x - y - z) d z \\ = \int_{- \infty}^{\infty} f (z) δ (z - (x - y)) d z \\ = f (x - y) \end{aligned}

this shows that convolutions with $δ (x - y)$ shifts the function by $x = y$

\begin{aligned} f * δ (x - y) & = F^{- 1} (\tilde{f} (k) \tilde{g} (k)) \\ = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{i k x} \tilde{f} (k) (δ (x - y)) d k \\ = \frac{1}{2 π} \int_{- \infty}^{\infty} e^{i k (x - y)} \tilde{f} (k) d k \\ = f (x - y) \end{aligned}

convolve with a set of $δ$ functions

\begin{matrix} g (x) = \sum_{i = 1}^{N} δ (x - y_{i}) \\ f * g (x) = \sum_{i = 1}^{N} f (x - y_{i}) \end{matrix}

the convolution is a sum of the function shifted to each $y_{i}$

convolution of two gaussians

\begin{matrix} f (x) = \sqrt{\frac{1}{2 π σ_{1}^{2}}} \exp (- x^{2} / 2 σ_{1}^{2}) \\ g (x) = \sqrt{\frac{1}{2 π σ_{2}^{2}}} \exp (- x^{2} / 2 σ_{2}^{2}) \\ \tilde{f} (k) = \exp (- k^{2} σ_{1}^{2} / 2) \\ f * g (x) = \int_{- \infty}^{\infty} e^{i k x} \exp (- x^{2} / 2 σ_{1}^{2}) \exp (- x^{2} / 2 σ_{2}^{2}) \\ = \int_{- \infty}^{\infty} e^{i k x} \underset{gaussian with varaince added}{\underset{⏟}{\exp (\frac{1}{2} (σ_{1}^{2} + σ_{2}^{2}) k^{2})}} d k \\ = \frac{1}{\sqrt{2 π (σ_{1}^{2} + σ_{2}^{2})}} \exp (- \frac{x^{2}}{2 (σ_{1}^{2} + σ_{2}^{2})}) \end{matrix}

therefore, gaussian convolutions broadens/smears the function, with the result having combined variance
this is called gaussian blurring