Aufgaben:Exercise 3.8: OVSF Codes: Difference between revisions
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{{quiz-Header|Buchseite= | {{quiz-Header|Buchseite=Mobile_Communications/Characteristics_of_UMTS | ||
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[[File:EN_Mob_A_3_9.png|right|frame|Tree diagram to construct <br> | [[File:EN_Mob_A_3_9.png|right|frame|Tree diagram to construct <br>an OVSF–Code]] | ||
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 | 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 $(+\mathcal{C}\ +\mathcal{C})$ | 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 | 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 | ||
:$$\langle c_\nu^{(0)}\rangle = \ {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},$$ | :$$\langle c_\nu^{(0)}\rangle = \ {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},$$ | ||
:$$ \langle c_\nu^{(1)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$ | :$$ \langle c_\nu^{(1)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$ | ||
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:$$ \langle c_\nu^{(3)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm}.$$ | :$$ \langle c_\nu^{(3)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm}.$$ | ||
According to this nomenclature, | 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. | 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 | *In the example, four spreading codes with spreading factor $J = 4$ could be used, or | ||
*the three codes highlighted in yellow | *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 [[ | *This task belongs to the chapter [[Modulation_Methods/Spreizfolgen_für_CDMA|Spreading sequences for CDMA]]. | ||
*Particular reference is made to the page [[ | *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]]. | ||
=== | ===Questionnaire=== | ||
<quiz display=simple> | <quiz display=simple> | ||
{Construct the tree diagram for $J = 8$. What are the resulting OVSF codes? | {Construct the tree diagram for $J = 8$. 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 $J = 32$. Is the following assignment feasible: | {The tree structure applies to $J = 32$. Is the following assignment feasible: Twice $J = 4$, once $J = 8$, once $J = 164$ and eight times $J = 32$? | ||
|type="()"} | |type="()"} | ||
+ Yes. | + Yes. | ||
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</quiz> | </quiz> | ||
=== | ===Solution=== | ||
{{ML-Kopf}} | {{ML-Kopf}} | ||
[[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)''' The following graphic shows the OVSF tree structure for $J = $8 users. | '''(1)''' The following graphic shows the OVSF tree structure for $J = $8 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)''' 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 | '''(4)''' We denote | ||
*$K_{4} = 2$ as the number of spreading sequences with $J = 4$, | *$K_{4} = 2$ as the number of spreading sequences with $J = 4$, | ||
*$K_{8} = 1$ as the number of spreading sequences with $J = 8$, | *$K_{8} = 1$ as the number of spreading sequences with $J = 8$, | ||
*$K_{16} = 2$ as the number of spreading sequences with $J = 16$, | *$K_{16} = 2$ as the number of spreading sequences with $J = 16$, | ||
*$K_{32} = 8$ as the number of spreading sequences with $J = 32$, | *$K_{32} = 8$ as the number of spreading sequences with $J = 32$, | ||
Then the following condition must be fulfilled: | Then the following condition must be fulfilled: | ||
:$$K_4 \cdot \frac{32}{4} + K_8 \cdot \frac{32}{8} +K_{16} \cdot \frac{32}{16} +K_{32} \cdot \frac{32}{32} \le 32\hspace{0.3cm} | :$$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 ⇒ <u>The answer is YES</u>. | ||
*Because $2 \cdot 8 + 1 \cdot 4 + 2 \cdot 2 + 8 = 32$ the desired assignment is just allowed ⇒ <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 $J = 4$ 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. | |||
{{ML-Fuß}} | {{ML-Fuß}} | ||
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[[Category: | [[Category:Mobile Communications: Exercises|^3.4 Characteristics of UMTS^]] | ||
[[de:Exercises:Exercise_3.8:_OVSF_Codes]] | |||
Latest revision as of 15:48, 16 March 2026
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:
- This task belongs to the chapter Spreading sequences for CDMA.
- Particular reference is made to the page Codes with variable spreading factor.
Questionnaire
Solution
(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.
