In continuous-time signal classification, the Dirac delta δ(t) is best described as which type of signal?
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AAn energy signal
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BA power signal
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CNeither an energy nor a power signal
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DBoth an energy and a power signal
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EA periodic signal with finite energy
Answer
Correct Answer: Neither an energy nor a power signal
Explanation
Introduction / Context:The Dirac delta δ(t) is a generalized function (distribution) frequently used to model impulsive events and ideal sampling. Classifying it within the conventional energy/power framework highlights important distinctions between ordinary signals and distributions used in system theory.
Given Data / Assumptions:
- δ(t) is the unit impulse with ∫{−∞}^{∞} δ(t) dt = 1.
- Energy is E = ∫ |x(t)|^2 dt; average power is P = lim{T→∞} (1/(2T)) ∫_{−T}^{T} |x(t)|^2 dt.
Concept / Approach:For ordinary signals, finite energy implies zero average power, and periodic power signals have infinite energy but finite average power. However, δ(t) is not a classical function; |δ(t)|^2 is not defined in the usual sense, so E and P are not finite, nor meaningfully defined. Thus δ(t) is neither an energy signal nor a power signal in the traditional sense used for LTI analysis.
Step-by-Step Solution:
Attempting to compute energy E = ∫ |δ(t)|^2 dt is not meaningful (distribution squared is undefined).Average power P also cannot be coherently defined for δ(t) using standard formulas.Therefore, δ(t) does not fit into energy or power categories.Verification / Alternative check:
In practical engineering, δ(t) is used as an ideal input to probe LTI systems via their impulse response h(t); classification is not necessary for its utility.Why Other Options Are Wrong:
Energy signal or power signal: Both require finite, well-defined integrals of |x(t)|^2, which δ(t) lacks.Both energy and power / periodic with finite energy: Contradict definitions and properties of δ(t).Common Pitfalls:
Assuming δ(t) has finite energy because its area is 1; area is not energy, and squaring δ(t) is not defined.Final Answer:
Neither an energy nor a power signal