Generate orthogonal variable spreading factor (OVSF) code from set of orthogonal codes

Spreading Codes

The OVSF Code Generator block generates an OVSF code from a set of orthogonal codes. OVSF codes were first introduced for 3G communication systems. OVSF codes are primarily used to preserve orthogonality between different channels in a communication system.

OVSF codes are defined as the rows of an N-by-N matrix, C_{N},
which is defined recursively as follows. First, define C_{1} =
[1]. Next, assume that C_{N} is defined and let
C_{N}(k) denote the kth row of C_{N}.
Define C_{2N} by

$${C}_{2N}=\left[\begin{array}{cc}{C}_{N}(0)& {C}_{N}(0)\\ {C}_{N}(0)& -{C}_{N}(0)\\ {C}_{N}(1)& {C}_{N}(1)\\ {C}_{N}(1)& -{C}_{N}(1)\\ \mathrm{...}& \mathrm{...}\\ {C}_{N}(N-1)& {C}_{N}(N-1)\\ {C}_{N}(N-1)& -{C}_{N}(N-1)\end{array}\right]$$

Note that *C*_{N }is
only defined for *N* a power of 2. It follows by
induction that the rows of *C*_{N }are
orthogonal.

The OVSF codes can also be defined recursively by a tree structure, as shown in the following figure.

If [C] is a code length 2^{r} at depth
r in the tree, where the root has depth 0, the two branches leading
out of C are labeled by the sequences [C C] and [C -C], which have
length 2^{r+1}. The codes at depth r in the
tree are the rows of the matrix C_{N}, where N
= 2^{r}.

Note that two OVSF codes are orthogonal if and only if neither
code lies on the path from the other code to the root. Since codes
assigned to different users in the same cell must be orthogonal, this
restricts the number of available codes for a given cell. For example,
if the code C_{41} in the tree is assigned to
a user, the codes C_{10}, C_{20},
C_{82}, C_{83}, and so on,
cannot be assigned to any other user in the same cell.

You specify the code the OVSF Code Generator block outputs by
two parameters in the block's dialog: the **Spreading factor**,
which is the length of the code, and the **Code index**,
which must be an integer in the range [0, 1, ... , N - 1], where N
is the spreading factor. If the code appears at depth r in the preceding
tree, the **Spreading factor** is 2^{r}.
The **Code index** specifies how far down the column
of the tree at depth r the code appears, counting from 0 to *N* -
1. For C_{N, k} in the preceding diagram, *N* is
the **Spreading factor** and *k* is
the **Code index**.

You can recover the code from the **Spreading factor **and
the **Code index** as follows. Convert the **Code
index** to the corresponding binary number, and then add
0s to the left, if necessary, so that the resulting binary sequence *x*_{1} *x*_{2} ... *x*_{r} has
length *r*, where *r* is the
logarithm base 2 of the **Spreading factor**. This
sequence describes the path from the root to the code. The path takes
the upper branch from the code at depth *i* if
x_{i} = 0, and the lower branch if x_{i} =
1.

To reconstruct the code, recursively define a sequence of codes
C_{i} for as follows. Let C_{0} be
the root [1]. Assuming that C_{i} has been defined,
for *i* < *r*, define C_{i+1} by

$${C}_{i+1}=\{\begin{array}{ll}{C}_{i}{C}_{i}\hfill & \text{if}{x}_{i}=0\hfill \\ {C}_{i}(-{C}_{i})\hfill & \text{if}{x}_{i}=1\hfill \end{array}$$

The code C_{N} has the specified **Spreading
factor** and **Code index**.

For example, to find the code with **Spreading factor** `16`

and **Code
index** `6`

, do the following:

Convert 6 to the binary number 110.

Add one 0 to the left to obtain 0110, which has length 4 = log

_{2}16.Construct the sequences C

_{i}according to the following table.i x _{i}C _{i}0 C _{0}= [1]1 0 C _{1}= C_{0}C_{0}= [1] [1]2 1 C _{2}= C_{1}-C_{1}= [1 1] [-1 -1]3 1 C _{3}= C_{2}-C_{2}= [1 1 -1 -1] [-1 -1 1 1]4 0 C _{4}= C_{3}C_{3}= [1 1 -1 -1 -1 -1 1 1] [1 1 -1 -1 -1 -1 1 1]

The code C_{4 }has **Spreading factor** `16`

and **Code
index** `6`

.

**Spreading factor**Positive integer that is a power of 2, specifying the length of the code.

**Code index**Integer in the range [0, 1, ... , N - 1] specifying the code, where N is the

**Spreading factor**.**Sample time**A positive real scalar specifying the sample time of the output signal.

**Frame-based outputs**Determines whether the output is frame-based or sample-based.

**Samples per frame**The number of samples in a frame-based output signal. This field is active only if you select

**Frame-based outputs**.**Output data type**The output type of the block can be specified as an

`int8`

or`double`

. By default, the block sets this to`double`

.

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