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Digital Signal Processing: Discrete-time LTI systems and difference equations · Linearity and Time Invariance

How to tell whether a discrete-time system is linear and time-invariant, and why that pair of properties lets one impulse response describe the whole system.

Digital Signal Processing · 24 questions

Try 3 questions

1. Is the system y[n] = 2x[n] + 3 linear?

  1. Yes, because it has no squared terms
  2. Yes, because its graph is a straight line
  3. No, a zero input gives a nonzero output
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Answer: No, a zero input gives a nonzero output

A linear system must map zero input to zero output. The constant 3 makes it affine, so doubling the input does not double the output.

2. The superposition principle that defines a linear system combines which two properties?

  1. Additivity and homogeneity
  2. Time invariance and memorylessness
  3. Causality and stability
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Answer: Additivity and homogeneity

Additivity means the response to a sum is the sum of responses; homogeneity means scaling the input scales the output by the same factor.

3. Which correctly describes the squaring system y[n] = x[n]²?

  1. Linear but not time-invariant
  2. Time-invariant but not linear
  3. Linear and time-invariant
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Answer: Time-invariant but not linear

Doubling the input quadruples the output, so it is not linear. But the rule is the same at every n, so delaying the input simply delays the output.

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