True or False conceptual check: Is the Z-transform a non-linear operation?

Introduction / Context:
The Z-transform is a cornerstone tool in discrete-time signal processing and control. A key property is linearity, which allows superposition in the Z-domain mirroring superposition in the time domain.

Given Data / Assumptions:

• Standard bilateral or unilateral Z-transform definitions.
• Region of convergence (ROC) exists such that the sums converge.
• Zero initial conditions for simplicity when needed.

Concept / Approach:

Linearity means Z{a x[n] + b y[n]} = a Z{x[n]} + b Z{y[n]} provided a common ROC exists or is appropriately handled. This property underpins transfer function algebra and block-diagram manipulations.

Step-by-Step Solution:

Verification / Alternative check:

All standard transform pairs and properties (convolution theorem, time shifting) rely on linearity. Nonlinearity would invalidate these core identities.

Why Other Options Are Wrong:

• True: contradicts the linearity shown above.
• Conditional statements tying linearity to signal length, causality, or ROC alone are incorrect; linearity is inherent to the transform definition.

Common Pitfalls:

• Confusing linearity of the transform with linearity of systems analyzed using it.
• Misinterpreting ROC issues as linearity issues.

Final Answer:

False