Difference between revisions of "Aufgaben:Exercise 3.8: OVSF Codes"

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The spreading codes for UMTS should
 
The spreading codes for UMTS should
 
*be orthogonal, in order to avoid mutual influence of the participants,
 
*be orthogonal, in order to avoid mutual influence of the participants,
*at the same time also allow a flexible realization of different spreading factors  $J$ .
+
*at the same time also allow a flexible realization of different spreading factors  $J$.
  
  
An example are the  "Orthogonal Variable Spreading Factor Codes"  $\rm (OVSF)$, which provide the spreading codes of lengths from  $J = 4$  to  $J = 512$ .
+
An example are the  "Orthogonal Variable Spreading Factor Codes"  $\rm (OVSF)$, which provide the spreading codes of lengths from  $J = 4$  to  $J = 512$.
  
As shown in the graphic, these can be created with the help of a code tree.  In doing so, each branching from a code   $\mathcal{C}$ results in two new codes   $(+\mathcal{C}\  +\mathcal{C})$  and  $(+\mathcal{C} \ –\mathcal{C})$.
+
As shown in the graphic, these can be created with the help of a code tree.  In doing so, each branching from a code  $\mathcal{C}$ results in two new codes  $(+\mathcal{C}\  +\mathcal{C})$  and  $(+\mathcal{C} \ –\mathcal{C})$.
  
 
The diagram illustrates the principle given here using the following example  $J = 4$.  If you number the spreading sequences from  $0$  to  $J -1$, the spreading sequences result
 
The diagram illustrates the principle given here using the following example  $J = 4$.  If you number the spreading sequences from  $0$  to  $J -1$, the spreading sequences result
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It should be noted that no predecessor or successor of a code may be used by other participants.  
 
It should be noted that no predecessor or successor of a code may be used by other participants.  
*In the example, four spreading codes with spreading factor  $J = 4$  could be used, or  
+
*In the example, four spreading codes with spreading factor $J = 4$  could be used, or  
*the three codes highlighted in yellow – once with  $J = 2$  and twice with   $J = 4$.
+
*the three codes highlighted in yellow – once with $J = 2$  and twice with $J = 4$.
  
  
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''Notes:''
 
''Notes:''
*This task belongs to the chapter  [[Modulation_Methods/Spreizfolgen_für_CDMA|Spreizfolgen für CDMA]].
+
*This task belongs to the chapter  [[Modulation_Methods/Spreizfolgen_für_CDMA|Spreading sequences for CDMA]].
*Particular reference is made to the page  [[Modulation_Methods/Spreizfolgen_für_CDMA#Codes_mit_variablem_Spreizfaktor_.28OVSF.E2.80.93Code.29|Codes mit variablem Spreizfaktor (OVSF–Code)]].
+
*Particular reference is made to the page  [[Modulation_Methods/Spreading_Sequences_for_CDMA#Codes_with_variable_spreading_factor_.28OVSF_codes.29|Codes with variable spreading factor]].
 
   
 
   
  
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<quiz display=simple>
 
<quiz display=simple>
  
{Construct the tree diagram for&nbsp; $J = 8$. What are the resulting OVSF codes?
+
{Construct the tree diagram for&nbsp; $J = 8$.&nbsp; What are the resulting OVSF codes?
 
|type="[]"}
 
|type="[]"}
 
+ $\langle c_\nu^{(1)}\rangle = +\hspace{-0.05cm}1 \ +\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1$,
 
+ $\langle c_\nu^{(1)}\rangle = +\hspace{-0.05cm}1 \ +\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1$,
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$K \ = \ $ { 5 }
 
$K \ = \ $ { 5 }
  
{The tree structure applies to&nbsp; $J = 32$. &nbsp;Is the following assignment feasible: &nbsp; <br>twice &nbsp; $J = 4$, once&nbsp; $J = 8$, once&nbsp; $J = 164$&nbsp; and eight times&nbsp; $J = 32$?
+
{The tree structure applies to&nbsp; $J = 32$. &nbsp;Is the following assignment feasible: &nbsp; Twice &nbsp; $J = 4$, once&nbsp; $J = 8$, once&nbsp; $J = 164$&nbsp; and eight times&nbsp; $J = 32$?
 
|type="()"}
 
|type="()"}
 
+ Yes.
 
+ Yes.
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[[File:P_ID2263__Bei_A_4_6a.png|right|frame|OVSF tree structure for $J = 8$]]
 
[[File:P_ID2263__Bei_A_4_6a.png|right|frame|OVSF tree structure for $J = 8$]]
'''(1)'''&nbsp; The following graphic shows the OVSF tree structure for $J = $8 users.  
+
'''(1)'''&nbsp; The following graphic shows the OVSF tree structure for&nbsp; $J = $8&nbsp; users.  
  
 
*From this it can be seen that the <u>solutions 1, 3 and 4</u> apply, but not the second.
 
*From this it can be seen that the <u>solutions 1, 3 and 4</u> apply, but not the second.
  
  
'''(2)'''&nbsp; If each user is assigned a spreading code with the spreading degree $J = 8$, $K_{\rm max} \ \underline{= 8}$ users can be supplied.
 
  
 +
'''(2)'''&nbsp; If each user is assigned a spreading code with the spreading degree&nbsp; $J = 8$,&nbsp; $K_{\rm max} \ \underline{= 8}$&nbsp; users can be supplied.
 +
 +
 +
 +
'''(3)'''&nbsp; If three users are supplied with&nbsp; $J = 4$, only two users can be served by a spreading sequence with&nbsp; $J = 8$&nbsp; (see example yellow background in the graphic)  $\  \Rightarrow \ \ \underline{K = 5}$.
  
'''(3)'''&nbsp; If three users are supplied with $J = 4$, only two users can be served by a spreading sequence with $J = 8$ (see example yellow background in the graphic)  $\  \Rightarrow \ \ \underline{K = 5}$.
 
  
  
 
'''(4)'''&nbsp;  We denote
 
'''(4)'''&nbsp;  We denote
*$K_{4} = 2$ as the number of spreading sequences with $J = 4$,
+
*$K_{4} = 2$&nbsp; as the number of spreading sequences with&nbsp; $J = 4$,
*$K_{8} = 1$ as the number of spreading sequences with $J = 8$,
+
*$K_{8} = 1$&nbsp; as the number of spreading sequences with&nbsp; $J = 8$,
*$K_{16} = 2$ as the number of spreading sequences with $J = 16$,
+
*$K_{16} = 2$&nbsp; as the number of spreading sequences with&nbsp; $J = 16$,
*$K_{32} = 8$ as the number of spreading sequences with $J = 32$,
+
*$K_{32} = 8$&nbsp; as the number of spreading sequences with&nbsp; $J = 32$,
  
  
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:$$K_4 \cdot \frac{32}{4} + K_8 \cdot \frac{32}{8} +K_{16} \cdot \frac{32}{16} +K_{32} \cdot \frac{32}{32} \le 32\hspace{0.3cm}
 
:$$K_4 \cdot \frac{32}{4} + K_8 \cdot \frac{32}{8} +K_{16} \cdot \frac{32}{16} +K_{32} \cdot \frac{32}{32} \le 32\hspace{0.3cm}
 
\Rightarrow \hspace{0.3cm} K_4 \cdot8 + K_8 \cdot 4 +K_{16} \cdot 2 +K_{32} \cdot1 \le 32 \hspace{0.05cm}.$$
 
\Rightarrow \hspace{0.3cm} K_4 \cdot8 + K_8 \cdot 4 +K_{16} \cdot 2 +K_{32} \cdot1 \le 32 \hspace{0.05cm}.$$
*Because $2 \cdot 8 + 1 \cdot 4 + 2 \cdot 2 + 8 = 32$ the desired assignment is just allowed  &nbsp; &rArr; &nbsp;  <u>The answer is YES</u>.  
+
*Because&nbsp; $2 \cdot 8 + 1 \cdot 4 + 2 \cdot 2 + 8 = 32$&nbsp; the desired assignment is just allowed  &nbsp; &rArr; &nbsp;  <u>The answer is YES</u>.  
*For example, providing the $J = 4$ twice blocks the upper half of the tree, after providing a $J = 8$ spreading code, $3$ of the $8$ branches remain to be occupied at the $J = 8$ level, and so on and so forth.
+
*For example, providing the&nbsp; $J = 4$&nbsp; twice blocks the upper half of the tree, after providing a&nbsp; $J = 8$&nbsp; spreading code,&nbsp; $3$&nbsp; of the&nbsp; $8$&nbsp; branches remain to be occupied at the&nbsp; $J = 8$&nbsp; level, and so on and so forth.
  
 
{{ML-Fuß}}
 
{{ML-Fuß}}
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[[Category:Exercises for Mobile Communications|^3.4 Characteristics of UMTS^]]
+
[[Category:Mobile Communications: Exercises|^3.4 Characteristics of UMTS^]]

Latest revision as of 13:37, 23 March 2021

Tree diagram to construct
an OVSF–Code

The spreading codes for UMTS should

  • be orthogonal, in order to avoid mutual influence of the participants,
  • at the same time also allow a flexible realization of different spreading factors  $J$.


An example are the  "Orthogonal Variable Spreading Factor Codes"  $\rm (OVSF)$, which provide the spreading codes of lengths from  $J = 4$  to  $J = 512$.

As shown in the graphic, these can be created with the help of a code tree.  In doing so, each branching from a code  $\mathcal{C}$ results in two new codes  $(+\mathcal{C}\ +\mathcal{C})$  and  $(+\mathcal{C} \ –\mathcal{C})$.

The diagram illustrates the principle given here using the following example  $J = 4$.  If you number the spreading sequences from  $0$  to  $J -1$, the spreading sequences result

$$\langle c_\nu^{(0)}\rangle = \ {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},$$
$$ \langle c_\nu^{(1)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
$$\langle c_\nu^{(2)}\rangle = \ {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
$$ \langle c_\nu^{(3)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm}.$$

According to this nomenclature, there are the spreading sequences  $\langle c_\nu^{(0)}\rangle, \text{...} ,\langle c_\nu^{(7)}\rangle$  for the spreading factor  $J = 8$. 

It should be noted that no predecessor or successor of a code may be used by other participants.

  • In the example, four spreading codes with spreading factor $J = 4$  could be used, or
  • the three codes highlighted in yellow – once with $J = 2$  and twice with $J = 4$.




Notes:


Questionnaire

1

Construct the tree diagram for  $J = 8$.  What are the resulting OVSF codes?

$\langle c_\nu^{(1)}\rangle = +\hspace{-0.05cm}1 \ +\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1$,
$\langle c_\nu^{(3)}\rangle = +\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1$,
$\langle c_\nu^{(5)}\rangle = +\hspace{-0.05cm}1 \ -\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1$,
$\langle c_\nu^{(7)}\rangle = +\hspace{-0.05cm}1 \ -\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ +\hspace{-0.08cm}1 \ -\hspace{-0.08cm}1$.

2

How many UMTS users can be served with  $J = 8$  at maximum?

$K_{\rm max} \ = \ $

3

How many users can be supplied with  $J = 8$  if three of them should use a spreading code with  $J = 4$ ?

$K \ = \ $

4

The tree structure applies to  $J = 32$.  Is the following assignment feasible:   Twice   $J = 4$, once  $J = 8$, once  $J = 164$  and eight times  $J = 32$?

Yes.
No.


Solution

OVSF tree structure for $J = 8$

(1)  The following graphic shows the OVSF tree structure for  $J = $8  users.

  • From this it can be seen that the solutions 1, 3 and 4 apply, but not the second.


(2)  If each user is assigned a spreading code with the spreading degree  $J = 8$,  $K_{\rm max} \ \underline{= 8}$  users can be supplied.


(3)  If three users are supplied with  $J = 4$, only two users can be served by a spreading sequence with  $J = 8$  (see example yellow background in the graphic) $\ \Rightarrow \ \ \underline{K = 5}$.


(4)  We denote

  • $K_{4} = 2$  as the number of spreading sequences with  $J = 4$,
  • $K_{8} = 1$  as the number of spreading sequences with  $J = 8$,
  • $K_{16} = 2$  as the number of spreading sequences with  $J = 16$,
  • $K_{32} = 8$  as the number of spreading sequences with  $J = 32$,


Then the following condition must be fulfilled:

$$K_4 \cdot \frac{32}{4} + K_8 \cdot \frac{32}{8} +K_{16} \cdot \frac{32}{16} +K_{32} \cdot \frac{32}{32} \le 32\hspace{0.3cm} \Rightarrow \hspace{0.3cm} K_4 \cdot8 + K_8 \cdot 4 +K_{16} \cdot 2 +K_{32} \cdot1 \le 32 \hspace{0.05cm}.$$
  • Because  $2 \cdot 8 + 1 \cdot 4 + 2 \cdot 2 + 8 = 32$  the desired assignment is just allowed   ⇒   The answer is YES.
  • For example, providing the  $J = 4$  twice blocks the upper half of the tree, after providing a  $J = 8$  spreading code,  $3$  of the  $8$  branches remain to be occupied at the  $J = 8$  level, and so on and so forth.