• Support PF! Buy your school textbooks, materials and every day products Here!

Help with z transform

  • #1

Homework Statement


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


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


part III. Find the causal signal x(n) fo the following z transforms
[tex]X(Z) = \frac{ 1-z^{-1} }{(1+z^{-1})(1+\frac{1}{2}z^{-1})}[/tex]
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.[tex] x(n) = 2\delta{n} + 3(\frac{1}{2})^{n}u[n] - (1/4)^{n}u(n)[/tex]
[tex]X(z) = 2(1) + 3\frac{1}{ 1-\frac{1}{2}z^{-1} } - \frac{1}{1-\frac{1}{4}z^{-1} }[/tex]
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
[tex]x[n] = 3^{n}u(-n)[/tex]
since [tex]x(-n)[/tex]'s z transform is [tex]X(z^{-1})[/tex]

[tex]X_1(Z) = \frac{1}{1-3z}[/tex]


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

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

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

[tex]\frac{2}{3z-1} - \frac{2}{2z-1}[/tex]

then simplify
[tex]\frac{ 2z^{-1}\frac{1}{3} }{ 1-\frac{1}{3}z^{-1} } - \frac{ 2z^{-1}\frac{1}{2} }{1-\frac{1}{2}z^{-1} }[/tex]
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)
[tex]\frac{2}{3}(\frac{1}{3})^{n}u(n-1) -(\frac{1}{2})^{n}u(n-1)[/tex]

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

Making it into a z transform
[tex]\frac{\frac{1}{2}3z^{-1}}{1-(-\frac{1}{2})z^{-1}} -\frac{4z^{-1}}{(1-(-z^{-1})} +1[/tex]
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
[tex]x(n) = \frac{3}{2}(-\frac{1}{2})^{n}u(n-1) - 4(-1)^{n}u(n-1)[/tex]

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

Answers and Replies

  • #2
Anybody kno z transform?
bump for 1st page
 

Related Threads on Help with z transform

  • Last Post
Replies
2
Views
456
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
2
Views
741
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
1
Views
7K
  • Last Post
Replies
1
Views
6K
Replies
3
Views
2K
  • Last Post
2
Replies
28
Views
3K
  • Last Post
Replies
4
Views
538
Top