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= [[:Category:Problem solving|Practice Problem]] on Z-transform computation  =
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[[Category:problem solving]]
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'''[[Digital_signal_processing_practice_problems_list|Practice Question on "Digital Signal Processing"]]'''
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Topic: Computing a z-transform
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----
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==Question==
  
 
Compute the compute the z-transform (including the ROC) of the following DT signal:  
 
Compute the compute the z-transform (including the ROC) of the following DT signal:  
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<br>  
 
<br>  
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:<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  ===
 
=== Answer 3  ===
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</span>  
 
</span>  
  
<br> [[2013 Fall ECE 438 Boutin|Back to ECE438 Fall 2013 Prof. Boutin]]  
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:<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>
  
 
=== Answer 7  ===
 
=== Answer 7  ===
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       = 9z^{3} + 4z^{2} + z
 
       = 9z^{3} + 4z^{2} + z
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:<span style="color:red"> TA's comment: ROC is everywhere except z=infinity.</span>
  
 
=== Answer 8  ===
 
=== Answer 8  ===
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<math>X(z) = \sum_{n=-\infty}^{+\infty} x[n] z^{-n}</math>  
 
<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
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<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
 
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</span>  
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:<span style="color:red"> TA's comment: Show your derivation</span>
  
 
=== Answer 9  ===
 
=== Answer 9  ===
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<br>  
 
<br>  
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:<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  ===
 
=== Answer 10  ===
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<br>  
 
<br>  
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:<span style="color:red"> TA's comment: ROC is everywhere except z=infinity.</span>
  
 
=== Answer 11  ===
 
=== Answer 11  ===
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[[Category:ECE301]] [[Category:ECE438]] [[Category:ECE438Fall2013Boutin]] [[Category:Problem_solving]] [[Category:Z-transform]]
 
[[Category:ECE301]] [[Category:ECE438]] [[Category:ECE438Fall2013Boutin]] [[Category:Problem_solving]] [[Category:Z-transform]]
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:<span style="color:red"> TA's comment: Simple and straightforward.</span>
  
 
=== Answer 12  ===
 
=== Answer 12  ===
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<math>X(z) = \sum_{n=-3}^{0} n^2 z^{-n}</math>&nbsp;
 
<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>
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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
 
X(z) = 9z<sup>3</sup> + 4z<sup>2</sup> + z
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:<span style="color:red"> TA's comment: In the third step, it's better write it as a summation. </span>

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] = n2(u[n + 2] − u[n − 1]).

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

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

= 9z3 + 4z2 + z

= z3(9 + 4z − 1 + z − 2)

= X(z) = (9 + 4z − 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] = n2(u[n + 3] − u[n − 1])

x[n] = n2(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) = 9z3 + 4z2 + 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) = n2(δ(n + 3) + δ(n + 2) + δ(n + 1) + δ(n) + δ(n − 1))zn

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]z3 + x[ − 2]z2 + x[ − 1]z1

      = 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) = (9z + 3 + 4z + 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) = 9z + 3 + 4z +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) = 9z + 3 + 4z + 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] = n2   (-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.

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