Q13.
Find the z-transform of each of the following sequences:
(a) $x(n) = 3\delta(n) + \delta(n - 2) + \delta(n + 2)$
(b) $x(n) = u(n) - u(n - 10)$
(a) Because this sequence is finite in length, the z-transform is a polynomial,
$X(z) = 3 + z^{-2} + z^2$
and the region of convergence is $0 < |z| < \infty$. Note that because $x(n)$ has nonzero values for $n < 0$, the
region of convergence does not include $|z| = \infty$, and because $x(n)$ has nonzero values for $n > 0$, the region of
convergence does not include the point $z = 0$.
(b) For this sequence,
$X(z) = \sum_{n=0}^9 z^{-n} = \frac{1 - z^{-10}}{1 - z^{-1}}$
which converges for all $|z| > 0$. Note that the roots of the numerator are solutions to the equation
$z^{10} = 1$
These roots are
$z = e^{j2\pi k/10}$ $k = 0, 1, \dots, 9$
which are 10 equally spaced points around the unit circle. Thus, the pole at $z = 1$ in the denominator of $X(z)$
is canceled by the zero at $z = 1$ in the numerator, and the z-transform may also be expressed in the form
$X(z) = \prod_{k=1}^9 (1 - e^{j2\pi k/10}z^{-1})$
