Linear Time-Invariant System

A Linear Time-Invariant (LTI) System is a Linear Transformation that has Time Invariance

Linear Transformation

$$\begin{array}{rl}T(x_1[n]+x_2[n])& =T(x_1[n])+T(x_2[n])\\ T(\lambda x_1[n])& =\lambda T(x_1[n])\end{array}$$

Time Invariance

$$T(x[n-n_0])=y[n-n_0]$$

Categorical Structure

In Category Theory, an LTI system is an Endomorphism on the Space of Discrete-Time Signals: it is a Morphism whose Domain and Codomain are the same signal space.

Time-invariance is the condition that $T$ commutes with the Delay Operator $\mathcal{R}$:

$T\circ \mathcal{R}=\mathcal{R}\circ T$

This is expressed as a Commutative Diagram: applying a delay before or after $T$ yields the same result.

Discrete-Time LTI System

Discrete-Time Linear Time-Invariant System

An Linear Time-Invariant System that is Discrete-Time

Representations

There are many ways to represent an DT LTI system. Each representation has its own advantages.

Verbal Description

Verbal description is good for communicating intuitively and accessibly, but potentially imprecise.

Next year, your account will contain $p$ times your balance from this year plus the money that you added this year.

Linear Difference Equation

Using a Difference Equation is mathematically precise and compact, but lacks the potential for conceptual expressiveness and intuition that language can provide. A difference equation is declarative. Algebraic equivalences that are trivially spottable in algebraic form, can be extremely unintuitive to spot in either of the other forms.

$$y[n+1]=x[n]+p\cdot y[n]$$

Operator Polynomial Equation

Using Operator Notation is as mathematically precise as the Linear Difference Equation, while being even more compact. The algebraic form admits standard polynomial operations: factoring, root finding, partial fraction decomposition, etc. without requiring sample-by-sample reasoning. This form is extremely useful, as it enables the powerful tools of algebra to intuitively be applied to systems. Its strength is abstraction. Note that the order of the polynomial corresponds to the order of the system.

$$(1-p\mathcal{R})Y=\mathcal{R}X$$

Block Diagram

Using a Block Diagram is good for conveying the step-by-step procedure of a system, like a circuit schematic, but is potentially unintuitive in expressively conveying ultimate effect of the system and requires tracing to understand.

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Continuous-Time LTI System

Continuous-Time Linear Time-Invariant System

An Linear Time-Invariant System that is Continuous-Time

Representations

Verbal Description

Your account will grow in proportion to your balance plus the rate at which you deposit.

Linear Differential Equation

$$y^{\prime }(t)=x(t)+p\cdot y(t)$$

See Linear Differential Equation

Operator Polynomial Equation

$$(1-p\mathcal{A})Y=\mathcal{A}X$$

Block Diagram

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