Help with Z Transform Calculations

[DrunkEngineer]

Homework Statement

I. Find the z transform and ROC of each of the ff sequence
1.x(n) = 2\delta{n} + 3(\frac{1}{2})^{n}u[n] - (1/4)^{n}u(n)


II. Use the Z transform to perform the convolution of the following sequence.
x[n] = 3^{n}u(-n)
h[n] = (0.5)^{n}u(n)


part III. Find the causal signal x(n) fo the following z transforms
X(Z) = \frac{ 1-z^{-1} }{(1+z^{-1})(1+\frac{1}{2}z^{-1})}
pls go to number 3 because latex cannot be edited

Homework Equations

properties and table of z transform are found here
http://en.wikipedia.org/wiki/Z_transform

The Attempt at a Solution

I. Find the z transform and ROC of each of the ff sequence
1.x(n) = 2\delta{n} + 3(\frac{1}{2})^{n}u[n] - (1/4)^{n}u(n)
X(z) = 2(1) + 3\frac{1}{ 1-\frac{1}{2}z^{-1} } - \frac{1}{1-\frac{1}{4}z^{-1} }
ROC: All z, 1/2 <z and 1/4 < z
the total ROC is z > 1/2

II. Use the Z transform to perform the convolution of the following sequence
x[n] = 3^{n}u(-n)
since x(-n)'s z transform is X(z^{-1})

X_1(Z) = \frac{1}{1-3z}


h[n] = (0.5)^{n}u(n)
X_2(Z) = \frac{1}{1-\frac{1}{2}z^-1}
x[n]*h[n] is equivalent to X_1(Z)X_2(Z)

X_1(Z)X_2(Z) =\frac{1}{(1-3z)(1-\frac{1}{2}z^-1)}

using wolfram alpha to solve partial fractions
\frac{2z}{(1-3z)(2z-1)}

\frac{2}{3z-1} - \frac{2}{2z-1}

then simplify
\frac{ 2z^{-1}\frac{1}{3} }{ 1-\frac{1}{3}z^{-1} } - \frac{ 2z^{-1}\frac{1}{2} }{1-\frac{1}{2}z^{-1} }
the region of convergence is 1/3 < z and 1/2 < z
then the total ROC is 1/2

the inverse z transform is : using the time shifting property z^-1X(z) = u(n-1)
\frac{2}{3}(\frac{1}{3})^{n}u(n-1) -(\frac{1}{2})^{n}u(n-1)

part III. Find the causal signal x(n) fo the following z transforms
X(Z) = \frac{ 1-z^{-1} }{(1+z^{-1})(1+\frac{1}{2}z^{-1})}
using wolfram alpha
\frac{2(z-1)z}{z+1}{2z+1}
\frac{3}{2z+1}-\frac{4}{z+1}+1

Making it into a z transform
\frac{\frac{1}{2}3z^{-1}}{1-(-\frac{1}{2})z^{-1}} -\frac{4z^{-1}}{(1-(-z^{-1})} +1
the region of convergence is
-1/2 < z, -1 < z, and the entire plane of z
the total region of convergence is the entire plane of z

the inverse z transform is
x(n) = \frac{3}{2}(-\frac{1}{2})^{n}u(n-1) - 4(-1)^{n}u(n-1)

can you check if this is correct?
i mean the ROC etc.

 

[DrunkEngineer]

Anybody kno z transform?
bump for 1st page