Discussion
Determine whether the given signal is a power or energy signal? What is its power or energy level?
\(X(n) = (1/2)^n u(n)\)
Answer: C
The signal \(x(n) = (1/2)^n u(n)\) decays to zero, so its total energy is finite and its average power is zero. It is an energy signal.
\[ \begin{aligned} E \\ &= \sum_{n=0}^{\infty} \left|\left(\tfrac12\right)^n\right|^2 \\ &= \sum_{n=0}^{\infty} \left(\tfrac14\right)^n \\ &= \frac{1}{1 - 1/4} = \frac{4}{3} \end{aligned} \]
The sum is a geometric series with ratio 1/4. The options calling the signal a power signal are wrong, since a power signal has finite nonzero power and infinite energy. The value 3/4 is the reciprocal of the energy. The answer is energy signal, 4/3, option C.
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