# Help with z transform

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.

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