This post introduce the the concept of convolution, Linear System and the Linear Filters.

Convolution

Linear System

What is a system?

A system is defined as a method that modifies a signal.

How to define a Linear system?

Linear systems satisfy some conducive properties of linearity.

Properties

Homogeneity（齐次性）：If the input to a linear system is scaled, the output would also be scaled by the same factor.

Additivity（可加性）：The independent responses (output) of multiple different input signals are added when the inputs are added. This implies that each signal is passed through the system independently without interacting with others.

Shift Invariance（平移不变性）：Finally, the output of a shifted input is also shifted.

When that data passes through multiple systems, the properties of linearity assures the following.

Commutative（可交换性）：If two linear systems are applied to a signal in a cascaded manner (i.e. in series), the order of their application does not matter. Given two linear systems and ,

Superposition（叠加性）：If each input generates multiple outputs in a linear system, the addition of the inputs will generate an additions of the outputs.

Response of a Linear System

What is an impulse?

An impulse, , is a discrete signal with only one non-zero sample.

is a special kind of impulse whose non-zero sample is at and has a value . Therefore, has a normalized spike at 0. Considering each sample to be of unit width and height proportional to its value, the area covered by a delta is therefore 1. is considered the simplest signal.