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'''1.7''' ROAC= all complex numbers z with |z|<0.5.  
 
'''1.7''' ROAC= all complex numbers z with |z|<0.5.  
  
'''1.8''' ROAC= all finite complex numbers z with 0<|z|<0.5.  
+
'''1.8''' ROAC= all complex numbers z with 0<|z|<0.5.  
  
 
'''1.9''' ROAC= all complex numbers z with |z|<3.  
 
'''1.9''' ROAC= all complex numbers z with |z|<3.  
  
'''1.10''' ROAC= all finite complex numbers z with 0<|z|<3.  
+
'''1.10''' ROAC= all complex numbers z with 0<|z|<3.  
  
'''1.11''' ROAC= all finite complex numbers z with 2<|z|<3.
+
'''1.11''' ROAC= all complex numbers z with 2<|z|<3.  
 
+
'''1.12''' ROAC= all finite complex numbers z with 0.5<|z|<2.  
+
  
 +
'''1.12''' ROAC= all complex numbers z with 0.5<|z|<2.
  
 
==Question 2==
 
==Question 2==
 
 
Compute the z-transform of the signal
 
Compute the z-transform of the signal
  
<math>x[n]= \left( \frac{1}{2} \right)^n u[-n] </math>
+
<math>x[n]= 6^n u[n-1] \ </math>
 
+
==Questions 2==
+
  
 +
==Questions 3==
 
Compute the z-transform of the signal
 
Compute the z-transform of the signal
  
<math>x[n]= 5^n u[n-3] \ </math>
+
<math>x[n]= \left( \frac{1}{5} \right)^n u[-n] </math>
 
+
==Questions 3==
+
  
 +
==Questions 4==
 
Compute the z-transform of the signal
 
Compute the z-transform of the signal
  
<math>x[n]= 5^{-|n|} \ </math>
+
<math>x[n]= 3^{-|n+1|} \ </math>
 
+
== Question 4  ==
+
  
 +
== Question 5  ==
 
Compute the z-transform of the signal
 
Compute the z-transform of the signal
  
<math>x[n]= 2^{n}u[n]+ 3^{n}u[-n+1]  \ </math>
+
<math>x[n]= 2^{n}u[n]- 3^{n}u[-n+1]  \ </math>
 
+
== Question 4 ==
+
  
 +
== Question 6 ==
 
Compute the inverse z-transform of  
 
Compute the inverse z-transform of  
  
<math>X(z)=\frac{1}{1+z}, \text{ ROC } |z|<1 </math>
+
<math>X(z)=\frac{7}{1+z}, \text{ ROC } |z|<1 </math>
  
  
== Question 5 ==
+
== Question 7 ==
 
+
 
Compute the inverse z-transform of  
 
Compute the inverse z-transform of  
  
<math>X(z)=\frac{1}{1+2 z}, \text{ ROC } |z|> \frac{1}{2} </math>
+
<math>X(z)=\frac{1}{1-3 z}, \text{ ROC } |z|> \frac{1}{3} </math>
 
+
== Question 6 ==
+
  
 +
== Question 8 ==
 
Compute the inverse z-transform of  
 
Compute the inverse z-transform of  
  
<math>X(z)=\frac{1}{1+2 z}, \text{ ROC } |z|< \frac{1}{2} </math>
+
<math>X(z)=\frac{1}{1+z^2}, \text{ ROC } |z|< 1</math>
  
== Question 7 ==
+
== Question 9 ==
  
 
Compute the inverse z-transform of  
 
Compute the inverse z-transform of  
Line 93: Line 85:
  
  
== Question 8 ==
+
== Question 10 ==
  
 
Compute the inverse z-transform of  
 
Compute the inverse z-transform of  
Line 99: Line 91:
 
<math>X(z)=\frac{1}{(1+ z)(3-z)}, \text{ ROC }  |z|>3</math>
 
<math>X(z)=\frac{1}{(1+ z)(3-z)}, \text{ ROC }  |z|>3</math>
  
== Question 9 ==
+
== Question 11 ==
  
 
Compute the inverse z-transform of  
 
Compute the inverse z-transform of  

Revision as of 11:59, 2 November 2016


Homework 8, ECE438, Fall 2016, Prof. Boutin

Hard copy due in class, Wednesday November 9, 2016.


Question 1

Below we describe the ROAC of the transfer function of an LTI system. For each ROAC, determine whether each of these applies. (Just list the letters of the properties that apply next to the system.)

a) the system is causal
b) the system is BIBO stable,
c) the system has a well defined and finite frequency response function
d) the system is FIR
e) The system is IIR
f) the unit impulse response of the system is right-sided
g) the unit impulse response of the system is left-sided

1.1 ROAC= all finite complex numbers, but not infinity.

1.2 ROAC= all complex numbers, including infinity.

1.3 ROAC= all complex numbers z with |z|>0.5, including infinity.

1.4 ROAC= all finite complex numbers z with |z|>0.5, but not infinity.

1.5 ROAC= all complex numbers z with |z|>3, including infinity.

1.6 ROAC= all finite complex numbers z with |z|>3, but not infinity.

1.7 ROAC= all complex numbers z with |z|<0.5.

1.8 ROAC= all complex numbers z with 0<|z|<0.5.

1.9 ROAC= all complex numbers z with |z|<3.

1.10 ROAC= all complex numbers z with 0<|z|<3.

1.11 ROAC= all complex numbers z with 2<|z|<3.

1.12 ROAC= all complex numbers z with 0.5<|z|<2.

Question 2

Compute the z-transform of the signal

$ x[n]= 6^n u[n-1] \ $

Questions 3

Compute the z-transform of the signal

$ x[n]= \left( \frac{1}{5} \right)^n u[-n] $

Questions 4

Compute the z-transform of the signal

$ x[n]= 3^{-|n+1|} \ $

Question 5

Compute the z-transform of the signal

$ x[n]= 2^{n}u[n]- 3^{n}u[-n+1] \ $

Question 6

Compute the inverse z-transform of

$ X(z)=\frac{7}{1+z}, \text{ ROC } |z|<1 $


Question 7

Compute the inverse z-transform of

$ X(z)=\frac{1}{1-3 z}, \text{ ROC } |z|> \frac{1}{3} $

Question 8

Compute the inverse z-transform of

$ X(z)=\frac{1}{1+z^2}, \text{ ROC } |z|< 1 $

Question 9

Compute the inverse z-transform of

$ X(z)=\frac{1}{(1+ z)(3-z)}, \text{ ROC } |z|<1 $


Question 10

Compute the inverse z-transform of

$ X(z)=\frac{1}{(1+ z)(3-z)}, \text{ ROC } |z|>3 $

Question 11

Compute the inverse z-transform of

$ X(z)=\frac{1}{(1+ z)(3-z)}, \text{ ROC } 1< |z|<3 $



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