Linearity Test

The linearity test determines whether a system obeys superposition, which is the foundation of all Fourier, Laplace, and Z-transform analysis. A system that fails the linearity test cannot be analyzed with standard LTI tools, so checking linearity is the first step before applying any transform-based method.

Core Concept

A system T is linear if it satisfies two conditions: additivity and homogeneity. Together these are called the superposition principle. Additivity means the response to a sum of inputs equals the sum of individual responses. Homogeneity means scaling the input by any constant scales the output by the same constant.

The standard test procedure is: compute the output y1(t) for input x1(t) and y2(t) for input x2(t) separately. Then compute the output for the combined input a*x1(t) + b*x2(t) directly. If the result equals a*y1(t) + b*y2(t) for all constants a, b and all inputs x1, x2, the system is linear.

Systems with initial conditions stored from before t=0 are not linear in the strict sense. The response to zero input is not zero. This is why transfer function analysis assumes zero initial conditions: it restricts attention to the zero-state behavior, where the system does behave linearly.

Key Formula

Formally, system T is linear if and only if: T{a*x1(t) + b*x2(t)} = a*T{x1(t)} + b*T{x2(t)} for all scalars a, b and all valid inputs x1, x2. To test, compute the left side by applying the system rule to the combined input. Compute the right side by scaling the individual outputs. If they are identical as functions of t, the system is linear. One counterexample (one set of inputs where equality fails) is sufficient to prove nonlinearity.

Test linearity of y(t) = t * x(t)

Step 1: Find output for x1(t)
  y1(t) = t * x1(t)

Step 2: Find output for x2(t)
  y2(t) = t * x2(t)

Step 3: Compute a*y1 + b*y2
  a*y1 + b*y2 = a*t*x1(t) + b*t*x2(t)
             = t * [a*x1(t) + b*x2(t)]

Step 4: Apply system to combined input a*x1 + b*x2
  T{a*x1 + b*x2} = t * [a*x1(t) + b*x2(t)]

Step 5: Compare
  T{a*x1 + b*x2} = a*y1 + b*y2   (they match)

Conclusion: y(t) = t*x(t) is LINEAR.

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Test linearity of y(t) = x^2(t)

Step 1: y1 = x1^2,  y2 = x2^2
Step 2: a*y1 + b*y2 = a*x1^2 + b*x2^2
Step 3: T{a*x1 + b*x2} = (a*x1 + b*x2)^2
                       = a^2*x1^2 + 2ab*x1*x2 + b^2*x2^2
Step 4: These are NOT equal in general.

Conclusion: y(t) = x^2(t) is NONLINEAR.

Exam Tip: Systems with a non-zero additive constant term like y(t) = 2*x(t) + 3 are nonlinear because T{0} = 3, not 0. A necessary condition for linearity is that zero input must produce zero output. Test this first: if y(t) is nonzero when x(t)=0, stop there and declare the system nonlinear without further work.

Properties Summary

• Additivity: T{x1+x2} = T{x1}+T{x2} for all input pairs
• Homogeneity: T{a*x} = a*T{x} for all scalars a
• Superposition: combines both: T{a*x1+b*x2} = a*y1+b*y2
• Necessary condition: zero input must give zero output; T{0}=0
• Nonlinear examples: squaring, absolute value, multiplication of two signals, constant offset
• Linear examples: differentiation, integration, multiplication by a fixed function of t (not x)
• Non-zero initial conditions make a system non-linear in full generality; zero initial conditions restore linearity

Quick Revision

• Linearity requires both additivity and homogeneity to hold
• Standard test: compare T{a*x1+b*x2} with a*T{x1}+b*T{x2}
• First quick check: does zero input give zero output? If not, nonlinear.
• Multiplying by t is linear; squaring or taking absolute value is not
• A constant additive offset in the output always destroys linearity
• Transfer function analysis requires linearity and time-invariance both
• One counterexample is enough to disprove linearity
• Exam trap: concluding a system with y(t) = 3*x(t) + 5 is linear because the 3*x(t) part obeys superposition; the constant 5 violates T{0}=0 and makes the whole system nonlinear