Linear and Non-Linear Systems

Main ideas / concepts conveyed

• The lecture is about classifying systems based on the property of linearity.
• It builds on earlier categories (static/dynamic, causal/non-causal, time-invariant/time-variant) and introduces the next property: linearity.
• Systems are divided into:
  • Linear systems
  • Nonlinear systems

Linearity is determined by whether the system follows the principle of superposition.

The principle of superposition is described as a necessary and sufficient condition for linearity.

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Methodology: How to test whether a system is linear

1) Principle of superposition (definition)

A system is linear if it satisfies both:

• Law of additivity (part of superposition)
• Law of homogeneity (part of superposition)

If the system violates either law, it violates superposition and is nonlinear.

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2) Test the Law of Additivity

Let the system be the same throughout.

1. Apply input (X_1(t)) to the system

   • Output is (Y_1(t))

2. Apply input (X_2(t)) to the same system

   • Output is (Y_2(t))

3. Compute output by adding the outputs

   • (Y_1(t) + Y_2(t))

4. Compute output by adding the inputs

   • Form a new input: (X_1(t) + X_2(t))
   • Feed it to the system
   • Output becomes (Y’(t)) (the response to the summed input)

5. Compare

   • If (Y’(t) = Y_1(t) + Y_2(t)) → additivity holds
   • If (Y’(t) \ne Y_1(t) + Y_2(t)) → additivity fails → system is nonlinear

The speaker notes: if additivity fails, you can stop and conclude nonlinearity (no need to test homogeneity).

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3) Test the Law of Homogeneity

Again, keep the system unchanged.

1. Apply input (X(t))

   • Output is (Y(t))

2. Compute output by scaling the output

   • Multiply the output by constant (K)
   • Result: (K\,Y(t))

3. Compute output by scaling the input

   • Multiply the input by (K)
   • New input: (KX(t))
   • Feed it to the system
   • Output becomes (Y’‘(t))

4. Compare

   • If (Y’‘(t) = K\,Y(t)) → homogeneity holds
   • If (Y’‘(t) \ne K\,Y(t)) → homogeneity fails → system is nonlinear

Optional order mentioned: you may test homogeneity first, and if it fails, you can conclude nonlinearity without testing the other.

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4) Conclusion rule

• If the system satisfies both additivity and homogeneity ⇒ it satisfies superposition ⇒ the system is linear
• If either fails ⇒ the system is nonlinear

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Example worked in the lecture

Given system

• (Y(t) = X\sin(t))

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Step A: Check additivity

Let inputs be (X_1) and (X_2).

1. Output for (X_1):

   • (Y_1(t) = X_1\sin(t))

2. Output for (X_2):

   • (Y_2(t) = X_2\sin(t))

3. Add outputs:

   • (Y_1(t) + Y_2(t) = X_1\sin(t) + X_2\sin(t))

4. Add inputs and pass through system:

   • Input: (X_1 + X_2)
   • Output: ((X_1 + X_2)\sin(t) = X_1\sin(t) + X_2\sin(t))

5. Comparison:

   • Results match ⇒ additivity holds

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Step B: Check homogeneity

1. Output scaling:

   • (Y(t) = X\sin(t))
   • (K\,Y(t) = KX\sin(t))

2. Input scaling:

   • Input becomes (KX)
   • Output becomes (KX\sin(t))

3. Comparison:

   • Results match ⇒ homogeneity holds

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Final conclusion for the example

• Since both additivity and homogeneity hold, the system satisfies superposition ⇒ the system is linear.

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Speakers / sources featured

• Unidentified single speaker/lecturer (no name provided in the subtitles).