Video Summary - Causal and Non-Causal Systems

Main ideas and concepts

Motivation/structure of the lecture
- After discussing static vs. dynamic systems, the lecture moves to causal vs. non-causal systems.
- Causal and non-causal systems are covered together because they are closely related.
- The approach to determining causality is to examine the relationship between output and input across present, past, and future.
How to test whether a system is causal
- Check whether the output at time (t) depends on future values of the input.
- Identify whether the required inputs come from present/past only, or whether future input is involved.

Definition
- A system is causal if its output is independent of future values of the input.
Equivalent description
- The output at time (t) depends only on:
  - present input and/or
  - past input
- It must not depend on future input.
Examples
- Example 1: ( y(t) = x(t) )
  - Depends on present input only → causal.
- Example 2: ( y(t) = x(t) + x(t-1) )
  - Depends on present input (x(t)) and past input (x(t-1)) → causal.
Practical/physical realizability point
- All physically realizable (real-life) systems are causal.
Delay-based intuition using two systems
- System 1 (delay = 0):
  - Input (x(t)) given at (t=0) → output (y_1(0)) is determined by (x(0)) (present value).
  - Output depends on present input → causal behavior.
- System 2 (delay = 1 second):
  - Input (x(0)) at (t=0) produces output (y_2(0)) that depends on (x(-1)) (a past value), because of the delay.
  - Output depends on past input → still causal.
- In both cases, the output does not rely on future input.

Definition
- A system is non-causal if the output at any instant depends on future values of the input.
Examples
- Example 1: ( y(t) = x(t) + 2 )
  - The lecture treats (x(t+2)) as the “future value of input” (even though the written line appears as (x(t)+2)).
  - Intended concept: output depends on future input → non-causal.
- Example 2: ( y(t) = x(t) + x(t-1) + x(t+1) )
  - Depends on present (x(t)), past (x(t-1)), and future (x(t+1)).
  - Because it depends on future input, it is → non-causal.

Definition
- Anti-causal systems are the opposite of causal systems:
  - The output depends only on future values of the input.
  - i.e., no dependence on past/present input.
Example
- The lecture uses an example consistent with output depending only on future input (aligned with the style of reasoning like (y(t)=x(t+2))).
Logical relationship stated
- Anti-causal ⇒ non-causal (always).
- Non-causal ⇏ anti-causal (not every non-causal system is anti-causal).