Video summary

Time-Invariant and Time-Variant Systems

Main summary

Key takeaways

Educational

Main ideas / concepts

  • The lecture transitions from previously discussed system types (static/dynamic, causal/non-causal) to a new classification: time-invariant vs. time-variant systems.
  • This property is later used in the context of LTI (Linear Time-Invariant) systems.
  • A system is tested for time invariance by checking whether shifting (delaying) the input by (T_0) results in the same shift in the output.

Notation setup

  • Input to the system: (x(t)) (referred to in subtitles as (X_t))
  • Output of the system: (y(t)) (referred to in subtitles as (Y_t))
  • Consider a delay of (T_0).

Time invariance test (methodology / steps)

Given a system mapping (x(t) \to y(t)), test time invariance as follows:

Step 1: Delay the output

  1. Feed the original input (x(t)) into the system to obtain output (y(t)).
  2. Form the “delayed-output scenario” by delaying the output by (T_0), giving the expected form:
    • [ y(t - T_0) ]

Step 2: Delay the input

  1. Delay the input by (T_0):

    • [ x(t - T_0) ]

  2. Feed the delayed input into the system and denote the new output by:

    • [ y’(t) ] (subtitles use forms like (Y’_t)).

Decision

  • If the outputs match:

    • [ y’(t) = y(t - T_0) ] → Time-invariant system

  • If the outputs do not match:

    • [ y’(t) \ne y(t - T_0) ] → Time-variant system

Definition stated in the lecture

  • Time-invariant system definition (as given):
    • “A system in which any delay provided an input must be reflected in the output.”

Example 1: Time-variant due to time scaling

  • System output:
    • [ y(t) = x^2(t) ]

Apply the test:

  • Step 1 (delay output):

    • [ y(t - T_0) = (x(t - T_0))^2 = x^2(t - T_0) ]

    • The subtitles indicate a mismatch due to how the time shift interacts with the operation (the key takeaway is the classification result).

  • Step 2 (delay input):

    • Feed the delayed input (x(t - T_0)) into the system.
    • Since the system effectively involves time scaling / involving time inside the signal argument, the subtitles warn not to incorrectly redistribute the time-scaling/shifting.

  • Comparison result:

    • The two outputs are not the same, so:
    • → The system is time-variant

Conceptual takeaway: Systems that effectively scale time (i.e., modify the independent-time variable inside the signal argument) tend to be time-variant.

Example 2: Time-invariant due to amplitude shifting

  • System output:
    • [ y(t) = 2 + x(t) ]

Apply the test:

  • Step 1 (delay output):

    • [ y(t - T_0) = 2 + x(t - T_0) ]

  • Step 2 (delay input):

    • Feed (x(t - T_0)) into the system:
    • [ y’(t) = 2 + x(t - T_0) ]

  • Comparison result:

    • The outputs match exactly:
    • [ y’(t) = y(t - T_0) ] → The system is time-invariant

Conceptual takeaway: Systems that do amplitude shifting (e.g., adding a constant like (+2)) are time-invariant.

Final conclusion / rule of thumb

  • If the system performs time scaling → time-variant
  • If the system performs amplitude shifting (adding a constant) → time-invariant

Speakers / sources featured

  • Unidentified speaker (lecture presenter) — no name given in the subtitles.

Original video