Discrete-Time Systems and Classification

What Is a Discrete-Time System?

A discrete-time system transforms an input sequence $x[n]$ into an output sequence $y[n]$:

\[y[n] = \mathcal{T}\{x[n]\}\]

The system is an operator or mapping from input space to output space. Examples:

• $y[n] = 2x[n]$ (scaling)
• $y[n] = x[n-1]$ (delay)
• $y[n] = \frac{1}{M}\sum_{k=0}^{M-1} x[n-k]$ (moving average)

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System Classification: The Five Critical Properties

Every system can be classified by these five independent properties. LTI systems (Linear + Time-Invariant) are the “sweet spot”—powerful theory, realizable, and ubiquitous.

Property	Test	Pass =
Static (Memoryless) vs Dynamic	Output depends only on current input?	Static
Time-Invariant vs Time-Variant	Delayed input → delayed output?	TI
Linear vs Nonlinear	Superposition holds?	Linear
Causal vs Non-Causal	Output depends only on present/past?	Causal
BIBO Stable vs Unstable	Bounded input → bounded output?	Stable

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1. Static vs Dynamic (Memory)

Static (Memoryless): $y[n] = f(x[n])$ — output at $n$ depends only on input at $n$.

Dynamic (With Memory): $y[n]$ depends on past/future inputs $x[n-k]$.

System	Classification	Reason
$y[n] = x^2[n]$	Static	Only $x[n]$
$y[n] = x[n] + 2x[n-1] + x[n-2]$	Dynamic	Uses $x[n-1], x[n-2]$
$y[n] = e^{x[n]}$	Static	Only $x[n]$

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