Difference between revisions of "Practice z transform computation 1 ECE438F13" - Rhea

Latest revision as of 11:52, 26 November 2013

Practice Question on "Digital Signal Processing"

Topic: Computing a z-transform

Question

Compute the compute the z-transform (including the ROC) of the following DT signal:

$x[n]= n^2 \left( u[n+3]- u[n-1] \right)$

(Write enough intermediate steps to fully justify your answer.)

Share your answers below

You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!

No need to write your name: we can find out who wrote what by checking the history of the page.

Answer 1

Andrei Henrique Patriota Campos $x [n] = n 2 (u [n + 2] - u [n - 1])$ .

$X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}$

$= \sum_{n=-3}^{0} n^2 z^{-n}$

$= 9 z 3 + 4 z 2 + z$

$= z 3 (9 + 4 z - 1 + z - 2)$

$= X (z) = (9 + 4 z - 1 + z - 2) / (z - 3)$ , for all z in complex plane.

TA's comment: z can not be

\infty

for the z transform to converge

Answer 2

$x [n] = n 2 (u [n + 3] - u [n - 1])$

$x [n] = n 2 (δ(n + 3) + δ(n + 2) + δ(n + 1) + δ(n))$

$X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}$

$X(z) = \sum_{n=-\infty}^{+\infty} n^2(\delta(n+3)+\delta(n+2)+\delta(n+1)+\delta(n)) z^{-n}$

$X (z) = 9 z 3 + 4 z 2 + z + 1$ for all z in complex plane

TA's comment: When n=0,x[n]=0. So the constant term is 0. ROC is everywhere except z=infinity

Answer 3

Write it here.

Answer 4

Write it here.

Answer 5

Tony Mlinarich

$X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}$

$X (z) = n 2 (δ(n + 3) + δ(n + 2) + δ(n + 1) + δ(n) + δ(n - 1)) z - n$

$X(z) = 9z3 + 4z2 + z + 1/z<\span>$

TA's comment: u[n+3]-u[n-1] is non-zero only when n=-3,-2,-1,0. So x[n]= n²(δ(n + 3) + δ(n + 2) + δ(n + 1) + δ(n)). ROC is everywhere except z=infinity

Answer 7

Yixiang Liu

$X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}$

$X(z) = \sum_{n=-\infty}^{+\infty} n^{2}[{u[n+3]-u[n-1]}]z^{-n}$

This expression equals to zero except n = -3, -2, -1

so $X (z) = x [ - 3] z 3 + x [ - 2] z 2 + x [ - 1] z 1$

      = 9z^{3} + 4z^{2} + z

TA's comment: ROC is everywhere except z=infinity.

Answer 8

Xi Wang

$X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}$

$= X (z) = (9 z + 3 + 4 z + 2 + z). The range of the value of z is from negative infinity to positive infinity$

TA's comment: Show your derivation

Answer 9

$X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}$

$X(z) = \sum_{n=-3}^{+1} x[n] z^{-n}$

$= X (z) = 9 z + 3 + 4 z +2 + z + 1$ for all z in complex plane

TA's comment: In your second step, the summation should be from -3 to 0 . There should be no constant termsince x[0]=0. ROC is everywhere except z=infinity

Answer 10

Cary Wood

$X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}$

$X(z) = \sum_{n=-3}^{0} x[n] z^{-n}$

$= X (z) = 9 z + 3 + 4 z + 2 + z, for all z in complex plane$

TA's comment: ROC is everywhere except z=infinity.

Answer 11

Shiyu Wang

x[n] = n²(u[n + 3] − u[n − 1])

x[n] = n^{2 (-3=< n < 1)}

$X(z) = \sum_{n=-3}^{0} n^2 z^{-n}$

x(z)=9z³+4z²+z, for all z in complex plane except z=infinity

TA's comment: Simple and straightforward.

Answer 12

Matt Miller

x[n] = n²(u[n+3]-u[n-1])

x[n] = n²u[n+3] - n²u[n-1]

x[n] = n²|⁰_-3

$X(z) = \sum_{n=-3}^{0} n^2 z^{-n}$

X(z) = (-3)²z³ + (-2)²z² + (-1)²z¹ + (0)²z⁰

X(z) = 9z³ + 4z² + z

lim z->inf X(1/2) = 0, lim z->0 X(1/2) = inf --> valid for all Z in complex plane.

TA's comment: In the third step, it's better write it as a summation.

@@ Line 1: / Line 1: @@
-= [[:Category:Problem solving|Practice Problem]] on Z-transform computation  =
+[[Category:problem solving]]
+<center><font size= 4>
+'''[[Digital_signal_processing_practice_problems_list|Practice Question on "Digital Signal Processing"]]'''
+</font size>
+Topic: Computing a z-transform
+</center>
+----
+==Question==
 Compute the compute the z-transform (including the ROC) of the following DT signal:
@@ Line 9: / Line 19: @@
 ----
-== Share your answers below ==
+== Share your answers below  ==
 You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
 '''No need to write your name: we can find out who wrote what by checking the history of the page.'''
-----
-=== Answer 1 ===
+----
+=== Answer 1  ===
+Andrei Henrique Patriota Campos
 <span class="texhtml">''x''[''n''] = ''n''<sup>2</sup>(''u''[''n'' + 2] − ''u''[''n'' − 1])</span>.
@@ Line 31: / Line 41: @@
 <span class="texhtml"> = ''X''(''z'') = (9 + 4''z''<sup> − 1</sup> + ''z''<sup> − 2</sup>) / (''z''<sup> − 3</sup>)</span>, for all z in complex plane.
-=== Answer 2 ===
+:<span style="color:red"> TA's comment: z can not be <math>\infty</math> for the z transform to converge </span>
-Muhammad Syafeeq Safaruddin
+=== Answer 2  ===
 <span class="texhtml">''x''[''n''] = ''n''<sup>2</sup>(''u''[''n'' + 3] − ''u''[''n'' − 1])</span>
@@ Line 47: / Line 57: @@
 <br>
-=== Answer 3 ===
+:<span style="color:red"> TA's comment: When n=0,x[n]=0. So the constant term is 0. ROC is everywhere except z=infinity</span>
+=== Answer 3  ===
 Write it here.
-=== Answer 4 ===
+=== Answer 4  ===
 Write it here.
-=== Answer 5 ===
+=== Answer 5  ===
 Tony Mlinarich
 <math>X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}</math>
-<math>X(z) = n^2(\delta (n+3)+\delta (n+2)+\delta (n+1)+\delta (n)+\delta (n-1)) z^{-n}</math>
+<span class="texhtml">''X''(''z'') = ''n''<sup>2</sup>(δ(''n'' + 3) + δ(''n'' + 2) + δ(''n'' + 1) + δ(''n'') + δ(''n'' − 1))''z''<sup> − ''n''</sup></span>
-<span class="texhtml">''X''(''z'') = 9''z''<sup>3</sup> + 4''z''<sup>2</sup> + ''z'' + ''1/z''<\span>
+<span class="texhtml">''X''(''z'') = 9''z''<sup>3</sup> + 4''z''<sup>2</sup> + ''z'' + ''1/z''&lt;\span&gt;
+</span>
+:<span style="color:red"> TA's comment: u[n+3]-u[n-1] is non-zero only when n=-3,-2,-1,0. So x[n]= ''n''<sup>2</sup>(δ(''n'' + 3) + δ(''n'' + 2) + δ(''n'' + 1) + δ(''n'')). ROC is everywhere except z=infinity</span>
-[[2013 Fall ECE 438 Boutin|Back to ECE438 Fall 2013 Prof. Boutin]]
+=== Answer 7  ===
+Yixiang Liu
+<math>X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}</math>
+<math>X(z) = \sum_{n=-\infty}^{+\infty} n^{2}[{u[n+3]-u[n-1]}]z^{-n}</math>
+This expression equals to zero except n = -3, -2, -1
+so <span class="texhtml">''X''(''z'') = ''x''[ − 3]''z''<sup>3</sup> + ''x''[ − 2]''z''<sup>2</sup> + ''x''[ − 1]''z''<sup>1</sup></span>
+       = 9z^{3} + 4z^{2} + z
+:<span style="color:red"> TA's comment: ROC is everywhere except z=infinity.</span>
+=== Answer 8  ===
+Xi Wang
+<math>X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}</math>
+<span class="texhtml"> = ''X''(''z'') = (9''z''<sup> + 3</sup> + 4''z''<sup> + 2</sup> + ''z''). The range of the value of z is from negative infinity to positive infinity
+</span>
+:<span style="color:red"> TA's comment: Show your derivation</span>
+=== Answer 9  ===
+<math>X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}</math>
+<math>X(z) = \sum_{n=-3}^{+1} x[n] z^{-n}</math>
+<span class="texhtml"> = ''X''(''z'') = 9''z''<sup> + 3</sup> + 4''z''<sup> +2</sup> + ''z'' + 1</span> for all z in complex plane
+<br>
+:<span style="color:red"> TA's comment: In your second step, the summation should be from -3 to 0 . There should be no constant termsince x[0]=0. ROC is everywhere except z=infinity </span>
+=== Answer 10  ===
+Cary Wood
+<math>X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}</math>
+<math>X(z) = \sum_{n=-3}^{0} x[n] z^{-n}</math>
+<span class="texhtml"> = ''X''(''z'') = 9''z''<sup> + 3</sup> + 4''z''<sup> + 2</sup> + z, for all z in complex plane</span>
+<br>
+:<span style="color:red"> TA's comment: ROC is everywhere except z=infinity.</span>
+=== Answer 11  ===
+Shiyu Wang
+x[n] = n<sup>2</sup>(u[n + 3] − u[n − 1])
+x[n] = n<sup>2 &nbsp; (-3=&lt; n &lt; 1)</sup>
+<math>X(z) = \sum_{n=-3}^{0} n^2 z^{-n}</math>&nbsp; <br>
+x(z)=9z<sup>3</sup>+4z<sup>2</sup>+z, for all z in complex plane except z=infinity
 [[Category:ECE301]] [[Category:ECE438]] [[Category:ECE438Fall2013Boutin]] [[Category:Problem_solving]] [[Category:Z-transform]]
-=== Answer 7 ===
+:<span style="color:red"> TA's comment: Simple and straightforward.</span>
+=== Answer 12  ===
+Matt Miller
+x[n] = n<sup>2</sup>(u[n+3]-u[n-1])
+x[n] = n<sup>2</sup>u[n+3] - n<sup>2</sup>u[n-1]
+x[n] = n<sup>2</sup>|<sup>0</sup><sub>-3</sub>
+<math>X(z) = \sum_{n=-3}^{0} n^2 z^{-n}</math>&nbsp;
+X(z) = (-3)<sup>2</sup>z<sup>3</sup> + (-2)<sup>2</sup>z<sup>2</sup> + (-1)<sup>2</sup>z<sup>1</sup> + (0)<sup>2</sup>z<sup>0</sup>
+X(z) = 9z<sup>3</sup> + 4z<sup>2</sup> + z
-Yixiang Liu
+lim z->inf X(1/2) = 0, lim z->0 X(1/2) = inf --> valid for all Z in complex plane.
-<math>X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}</math>
+ <br>
-<math>X(z) = \sum_{n=-\infty}^{+\infty} n^{2}{u[n+3]-u[n-1]}z^{-n}</math>
+:<span style="color:red"> TA's comment: In the third step, it's better write it as a summation. </span>

Difference between revisions of "Practice z transform computation 1 ECE438F13" - Rhea

Latest revision as of 11:52, 26 November 2013

Contents

Question

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Answer 1

Answer 2

Answer 3

Answer 4

Answer 5

Answer 7

Answer 8

Answer 9

Answer 10

Answer 11

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