Discussion
Find the Z-transform of the given finite duration, right-hand sequence.
\(x(n) = \{1, 2, 4, 0, 7\}\)
Answer: A
The Z-transform of a sequence is \(X(z) = \sum x(n) z^{-n}\). A right hand finite duration sequence written as \(\{1, 2, 4, 0, 7\}\) starts at \(n = 0\) with the first value, so the samples are at \(n = 0, 1, 2, 3, 4\).
\[ \begin{aligned} X(z) \\ &= 1 + 2z^{-1} + 4z^{-2} + 0 \cdot z^{-3} + 7z^{-4} \\ &= 1 + \frac{2}{z} + \frac{4}{z^2} + \frac{7}{z^4} \end{aligned} \]
The zero sample at \(n = 3\) drops out, so the last term has \(z^4\), not \(z^3\), which excludes the third and fourth options. The options with minus signs change the signs of the samples, which are all positive. The answer is option A.
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