Question

Determine whether the systems given by following equations are Linear/Non linear.

Shift Variant / Shift Invariant

y(t) = 2x(t-3) + x(2-t)

y(t) = x²(t-t₀) + 2

Answer 1

Step-by-Step Explanation of Visuals

System 1: y1(t)=2x(t−3)+x(2−t)y_1(t) = 2x(t-3) + x(2-t)

1. Input Signal:

   • The blue curve represents the original input x(t)x(t), a rectangular pulse active between t=0t=0 and t=5t=5.

2. Shifted Signal (x(t−3)x(t-3)):

   • The orange curve is x(t−3)x(t-3), where the input is shifted to the right by 3 units.

3. Scaled Shifted Signal (2x(t−3)2x(t-3)):

   • The purple curve is the scaled version of x(t−3)x(t-3). Multiplication by 2 doubles the amplitude.

4. Flipped Signal (x(2−t)x(2-t)):

   • The green curve is x(2−t)x(2-t), the input flipped around t=2t=2. Notice how the rectangular pulse is mirrored.

5. Final Output (y1(t)y_1(t)):

   • The red curve is the sum of 2x(t−3)2x(t-3) and x(2−t)x(2-t). This shows the combined contributions of the shifted, scaled, and flipped components.

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System 2: y2(t)=x2(t−t0)+2y_2(t) = x^2(t-t_0) + 2

1. Shifted Signal (x(t−t0)x(t-t_0)):

   • The orange curve is x(t−t0)x(t-t_0), where the input is shifted to the right by t0=2t_0 = 2.

2. Squared Shifted Signal (x2(t−t0)x^2(t-t_0)):

   • The purple curve represents squaring x(t−t0)x(t-t_0). Squaring amplifies the signal's positive values and eliminates any negative values.

3. Final Output (y2(t)y_2(t)):

   • The red curve adds 2 to x2(t−t0)x^2(t-t_0), producing the final output. This vertical shift raises the entire signal by 2 units.

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Key Observations for New Learners

1. System 1:

   • Combines linear transformations (scaling, shifting, flipping).
   • Linear but Shift-Variant because of the dependency on t=2t=2.

2. System 2:

   • Includes nonlinear operations (squaring) and constant addition.
   • Nonlinear but Shift-Invariant because it handles input shifts uniformly.

These plots should help visualize the system behaviors step-by-step. Let me know if you'd like further clarifications or additional analysis!