Convolution

• categories: Functional analysis, Definition

Definition

Continuous Convolution

The convolution of two functions $f(t)$ and $g(t)$, denoted as $(f\ast g)(t)$, is defined as:

$$(f\ast g)(t)={\int }_{-\infty }^{\infty }f(\tau )g(t-\tau )d\tau .$$

Discrete Convolution

For discrete sequences $f[n]$ and $g[n]$, the convolution is defined as:

$$(f\ast g)[n]=\sum _{k=-\infty }^{\infty }f[k]g[n-k].$$

Key Intuition

1. Sliding and Summing:

   • In continuous convolution, the function $g(t-\tau )$ represents a flipped and shifted version of $g(t)$ that is “slid” over $f(t)$. At each position, the product of overlapping values is summed.
   • In discrete convolution, the same idea applies, with summation replacing integration.

2. Output Interpretation:
   The output of the convolution $(f\ast g)(t)$ represents how the shape of $g(t)$ “modulates” the shape of $f(t)$.

Properties

1. Commutativity:

   $$f\ast g=g\ast f.$$

2. Associativity:

   $$(f\ast g)\ast h=f\ast (g\ast h).$$

3. Distributivity:

   $$f\ast (g+h)=(f\ast g)+(f\ast h).$$

4. Scaling:

   $$a(f\ast g)=(af)\ast g=f\ast (ag),a\in \mathbb{R}.$$

5. Shift Invariance:
   For $g(t-t_0)$, convolution shifts the output by $t_0$:

   $$(f\ast g(t-t_0))(t)=(f\ast g)(t-t_0).$$

Applications

1. Signal Processing:
   Describes the output of a linear time-invariant (LTI) system when an input signal $f(t)$ is passed through a system with impulse response $g(t)$.

2. Image Processing:
   Used for filtering operations, such as blurring and edge detection.

3. Data Science:
   Core operation in Convolutional Neural Networks (CNNs).

4. Physics:
   Models systems where the output depends on the interaction of two functions (e.g., Green’s function in differential equations).