Exercise 5.4Z: OVSF Codes
The spreading codes for UMTS should
- all be orthogonal to each other in order to avoid mutual interference between subscribers,
- additionally allow a flexible realization of different spreading factors $J$.
One example of this is the so-called "Orthogonal Variable Spreading Factor" code, which provide spreading codes with lengths from $J = 4$ to $J = 512$.
These can be created using a code tree, as shown in the diagram. In this process, two new codes $(+C \ +C)$ and $(+C \ -C)$ are created from a code $C$ at each branching.
The diagram illustrates the principle given here with the example $J = 4$:
- If the spreading sequences are numbered from $0$ to $J -1$ the spreading sequences are as follows
- $$\langle c_\nu^{(0)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm},\hspace{0.3cm} \langle c_\nu^{(1)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},$$
- $$\langle c_\nu^{(2)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.05cm},\hspace{0.3cm} \langle c_\nu^{(3)}\rangle = {+\hspace{-0.05cm}1}\hspace{0.15cm} {-\hspace{-0.05cm}1} \hspace{0.15cm} {-\hspace{-0.05cm}1}\hspace{0.15cm} {+\hspace{-0.05cm}1} \hspace{0.05cm}.$$
- According to this nomenclature, for the spreading factor $J = 8$ there are the spreading sequences $\langle c_\nu^{(0)}\rangle $, ... , $\langle c_\nu^{(7)}\rangle $.
- Note that no predecessor and successor of a code may be used for another participant.
- So, 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:
- The exercise belongs to the chapter Spreading Sequences for CDMA.
- Reference is made in particular to the section Codes with variable spreading factor (OVSF codes) in the theory part.
- We would also like to draw your attention to the (German language) interactive SWF module OVSF.
Questions
Solution
(1) The diagram shows the OVSF tree structure for $J = 8$ users. From this it can be seen that solutions 1, 3 and 4 apply, but not the second one.
(2) If each user is assigned a spreading code with $J = 8$, $K_{\rm max}\hspace{0.15cm}\underline{ = 8}$ subscribers can be served.
(3) When three subscribers are served by $J = 4$
⇒ only two subscribers can still be served by a spreading sequence with $J = 8$ (see exemplary yellow background in the diagram) ⇒ $K\hspace{0.15cm}\underline{ = 5}$.
(4) We denote by
- $K_4 = 2$ the number of spreading sequences with $J = 4$,
- $K_8 = 1$ the number of spreading sequences with $J = 8$,
- $K_{16} = 2$ the number of spreading sequences with $J = 16$,
- $K_{32} = 8$ the number of spreading sequences with $J = 32$.
Then the following condition must be satisfied:
- $$K_4 \cdot \frac{32}{4} + K_8 \cdot \frac{32}{8} +K_{16} \cdot \frac{32}{16} +K_{32} \cdot \frac{32}{32} \le 32 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} K_4 \cdot8 + K_8 \cdot 4 +K_{16} \cdot 2 +K_{32} \cdot1 \le 32 \hspace{0.05cm}.$$
- Because of $2 · 8 + 1 · 4 + 2 · 2 + 8 = 32$, the desired occupancy is just allowed ⇒ answer YES.
- For example, supplying the spreading factor $J = 4$ twice blocks the upper half of the tree.
- After providing one spreading $J = 8$, three of the eight branches remain to be occupied on the $J = 8$ level, and so on.
