Use Z-transform tables and properties of the Z-transform to determine the Z-transform along with the respective ROC of the following signals (sequences) a) $x_1[n] = 2^n \sin(\pi n / 6)u[-n]$, b) $x_2[n] = 4^n \cos(\pi n / 4)u[n-2]$, c) $x_3[n] = 0.5x_1[n] - 0.8x_2[n]$.
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First, we need to determine the Z-transform of a basic sequence. Let's say we have a sequence x[n] = a^n, where a is a constant. The Z-transform of x[n] = a^n is given by X(z) = 1 / (1 - az^(-1)), where |z| > |a|. Show more…
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Given the z-transform pair: x(n) ↔ X(z) = z^(-1) / (1 + 0.8z^(-1)) with ROC: |z| > 0.8, use the z-transform properties to determine the z-transform of the following sequences: a. y[n] = x[n + 2], b. y[n] = (1/4)^n * x[n]
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Determine the z-transform of (a) $3^{k}(b)(-3)^{k}$ (a) From 6 in Table $79.1, Z\left\{a^{k}\right\}=\frac{z}{z-a}$ If $\mathrm{a}=3$, then $\quad Z\left\{3^{k}\right\}=\frac{z}{z-3}$ (b) From 6 in Table 79.1, $Z\left\{a^{k}\right\}=\frac{z}{z-a}$ If $\mathrm{a}=-3$, then $\boldsymbol{Z}\left\{(-3)^{k}\right\}=\frac{z}{z--3}=\frac{z}{z+3}$
Determine the z-transform of $2 e^{-3 k}$ From 9 in Table 79.1, $Z\left\{e^{-a k}\right\}=\frac{z}{z-e^{-a}}$ Hence, $Z\left\{2 e^{-3 k}\right\}=2 Z\left\{e^{-3 k}\right\}=\frac{2 z}{z-e^{-3}}$
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