BeginnerTopic 9 of 101Signals, Control & Comms
Linearity and time invariance
LTI systems obey superposition and respond the same regardless of when input is applied.
In depth
A linear system obeys superposition: H{ax₁+bx₂}=aH{x₁}+bH{x₂}. A time-invariant system gives the same response for delayed input (shift in, shift out). LTI systems are completely described by impulse response or transfer function, enabling Fourier and Laplace analysis.
Key points
- Test with sums and shifts
- Nonlinear examples: y=x²
- Time-varying: y=t·x
ExampleAn ideal capacitor and resistor network is LTI.
What is Linearity and time invariance?
LTI systems obey superposition and respond the same regardless of when input is applied.
Why is it important to understand linearity and time invariance?
Linear time-invariant systems can be analysed completely with simple tools, which is why LTI models are used almost everywhere in engineering.
What does linearity mean?
Superposition holds, so the response to a sum of inputs is the sum of the individual responses, and scaling an input scales the output.
What does time invariance mean?
A time shift in the input produces the same time shift in the output, so the system does not change with time.
How is Linearity and time invariance used in practice?
Typical applications include predicting the output of a filter using superposition of its inputs, assuming a circuit responds the same way on Monday as on Friday and designing a controller with transfer function methods based on LTI assumptions.
What should I learn before Linearity and time invariance?
Study "Systems" first (topic 8 of 101 in Signals, Control & Comms). It covers the groundwork this topic builds on.
What are the key points of Linearity and time invariance?
Test with sums and shifts. Nonlinear examples: y=x². Time-varying: y=t·x.
Can you give an example of Linearity and time invariance?
An ideal capacitor and resistor network is LTI.
What comes after Linearity and time invariance?
The next topic is "Causality and stability", which builds directly on this one.
How difficult is Linearity and time invariance?
It is classed as Beginner in Signals, Control & Comms (topic 9 of 101). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.