Find the *z-*transforms of

(a) **x**=[1 0 0 1]

(b) **x**=[1 1 1 1 ...]

(c) **x**=[1 2 3 4 ...]

(d) **x**=[1 1/2 1/3 1/4 ...]

*Solution.
*

(a) For a sequence of numbers x[n], the z-transform is written as:

(b) The signal **x** can be written as:

Thus

for |

(c) When we define unit step function

Then the signal **x** is equivalent to:

Thus we can write

From the

for |

(d) The signal **x** is equivalent to:

From the definition of the z-transform,

This expression can be simplified by using the series

Then X(z) can be expressed as