Aufgaben:Exercise 5.4Z: OVSF Codes

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Construction of an OVSF code

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

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

	Code word 1:   $ \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{0.15cm}{-\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},$
	Code word 3:   $ \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{0.15cm}{+\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}$ ,
	Code word 5:   $ \langle c_\nu^{(5)}\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{0.15cm}{+\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}$,
	Code word 7:   $ \langle c_\nu^{(7)}\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{0.15cm}{-\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}$.

2 What is the maximum number of UMTS subscribers  $(K_{\rm max})$  that can be served with  $J = 8$ ?

$K_{\rm max} \ = \ $

3 How many subscribers  $(K)$  can be served if three of these subscribers are to use a spreading code with  $J = 4$ ?

$K \ = \ $

4 Assume a tree structure for  $J = 32$.   Is the following assignment feasible:
Twice  $J = 4$,  once  $J = 8$,  twice  $J = 16$  and eight times  $J = 32$?

	Yes.
	No.

Solution

Solution

OVSF tree structure for  $J = 8$

(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.