Hey!

The sequence of: \( x[n] = 1, \frac{1}{2}^1,\frac{1}{2}^2,\frac{1}{2}^3,... \)

was asked to be expressed using the z-transform:

\(X(z) = 1 +\frac{1}{2}^1z^-^1 +\frac{1}{2}^2z^-^2+\frac{1}{2}^3z^-^3+... \)

What I can't do is express the final result into the one on the sheet which is:
\( \frac{z}{z-0.5}\)

The notes say to use the linear difference equation:

\(\frac{x[n]+x[n-1]}{2} \)

I found x[n-1] just by subtracting one from each value for x[n] above so when I added the numerator I got 1 and the denominator as 2...clearly it is wrong.

Thank you in advance for helping!

 

This is the sum of a geometric series.

\(X(z)=1+\frac{1}{2z}+\frac{1}{4z^2}+\frac{1}{8z^3} + ....\)

Since the series is infinite we may write ...

\(\frac{1}{2z}X(z)=X(z)-1\)

\(\frac{1}{2z}X(z)-X(z)=-1\)

\((\frac{1}{2z}-1)X(z)=-1\)

\((\frac{1-2z}{2z})X(z)=-1\)

\(X(z)=\frac{-2z}{1-2z}\)

\(X(z)=\frac{2z}{(2z-1)}\)

\(X(z)=\frac{z}{(z-0.5)}\)