Design Article
A Larger Subset of Pseudo-Orthogonal Spreading Codes for WCDMA
Joo da Silva Pereira and Henrique Jos Almeida da Silva
12/12/2001 12:00 AM EST
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A Larger Subset of Pseudo-Orthogonal
Spreading Codes for WCDMA
João da Silva Pereira, Instituto
Politécnico de Leiria, Portugal |
are used mainly in modern communication systems
with Spread Spectrum Code Division Multiple Access (SS-CDMA). These
applications often require large sets of codes with highly peaked
auto-correlation and minimum cross-correlation. The principal
features of these sequences are the ratio between the peak
auto-correlation value and the peak cross-correlation value, as
well as the average value of this ratio. In a noisy communication
channel with multiple users, these parameters are fundamental for
the correct bit information detection.
Among a large range of different families of binary and
non-binary PN-sequences
, the more classical one is the set of
Gold-sequences. The purpose of this paper is to analyze only the
set of Gold-sequences that derive from m-sequences.
The objective is to be able to construct more
PN-sequences using the same circuit generator of Gold-sequences
with minimum changes. However, the problem of designing families of
sequences having low values of the periodic and aperiodic
cross-correlation functions is a difficult one.
The new design is based in the same Gold-sequences generated in reverse mode. If all Gold-sequences are generated in reverse mode, than these sequences have the same peaks of periodic and aperiodic cross-correlation, and the periodic and aperiodic auto-correlations are still the same as for the original set of Gold sequences. Assembling these two sets, and calculating the new peak of periodic and aperiodic cross-correlation, it is possible to observe a small increase of these values. However, this increase seems to be not significant when the Gold-sequences are long.
More new sequences can be generated by mixing two sequences of the Gold set with a logical exclusive-or operation, when at least one of them is in reverse mode. The result is to increase four times the quantity of PN-sequences, keeping an acceptable level of cross-correlation.
by:
This is equivalent to:
The y and x are binary m-sequences of Nc bits constructed by means of two polynomial generators of degree n.
The y sequence is combined by a logical exclusive-or operation with each row of the X matrix, which is a set of all cyclic rotations of sequence x.
The notation yr is used to identify the y sequence in reverse mode, and the set of rows of the matrix Xr is also the set of x sequences in reverse mode.
The notation of the new set is:
Analyzing individually the two sets G(y,x) and Gr (y,x) we can verify that they have identical auto-correlation and cross-correlation properties.
Now, mixing the two sets we obtain a new set:
The set P(y,x) has (4Nc+4) different sequences and the Gold set has only (Nc+2). Then P(y,x) has almost four times more sequences than the Gold set.
An advantage of this set is that you can generate it with few changes in the classical Gold circuit generator:
- Adding a shift register at the end circuit generator synchronized with clock frequency fc
- Adding two clock frequencies, fc and fc(Nc-1), for each m-sequence generator for x and y sequences.
Figure 1 shows a P(y,x) generator comprising two polynomial generators with octal representations 45 and 67.
Figure 1: This P(y,x) generator comprises two polynomial generators with different octal representations
The purpose of the second clock frequency fc(Nc-1) in Figure 1 is to generate sequences in reverse mode which are "filtered" by a shift register at the end. However, for long sequences, this second frequency fc(Nc-1) would be too high for the operation of the shift registers. A solution is to implement a new circuit, where the m-sequences G1(x) and G2(x) are maintained in a fast read-only memory (ROM) that may be read in ascending or descending order before the last logical "exclusive-or" operation.
Figure 2: Periodic auto-correlation for the sequences generated by the circuit of Figure 1.
The maximum value of the absolute peak periodic and aperiodic
cross-correlation is equal to 25; there exists a difference of
19.4% relative to the peak auto-correlation value (Nc=31). This may
be acceptable in some applications, but it is not high enough for
use in wireless communication systems. For this reason, the next
results were obtained with longer segments of some subsets of
Gold-sequences, which may be considered as an alternative for use
in UMTS.
The results of periodic and aperiodic cross-correlation are presented in Tables 1 and 2, respectively. Only the first 512 sequences of the Gold set have been considered in this simulation. The results of the Gold subset are in the shaded cells (column 2) of Tables 1 and 2. These tables do not show the cross-correlation values of the total P(y,x) set, because the m-sequences x, y, xr, and yr are the only four distinct sequences with good correlation properties.
{y 
X} with |
{y
X} |
{y
Xr} |
{yr
X} |
{yr
Xr} |
| Maximum Peak | 1202 | 1250 | 1294 | 1426 |
| Average of Max. Peak | 821.05 | 816.67 | 816.83 | 817.32 |
| Average of Min. Peak | -816.05 | -817.30 | -816.82 | -817.18 |
| Minimum Peak | -1226 | -1244 | -1328 | -1292 |
| {yr
X} with |
{y
Xr} |
{yr
X} |
{yr
Xr} |
| Maximum Peak | 1304 | 1204 | 1258 |
| Average of Max. Peak | 817.27 | 817.87 | 816.87 |
| Average of Min. Peak | -816.79 | -812.73 | -817.39 |
| Minimum Peak | -1194 | -1184 | -1230 |
| {y
Xr} with |
{y
Xr} |
{yr
Xr} |
| Maximum Peak | 1246 | 1286 |
| Average of Max. Peak | 817.06 | 817.35 |
| Average of Min. Peak | -812.80 | -817.63 |
| Minimum Peak | -1196 | -1210 |
| {yr
Xr} with |
{yr
Xr} |
| Maximum Peak | 1322 |
| Average of Max. Peak | 821.53 |
| Average of Min. Peak | -815.82 |
| Minimum Peak | -1172 |
Table 1: Periodic cross-correlation of subset P(y,x)
| {y
X} with |
{y
X} |
{y
Xr} |
{yr
X} |
{yr
Xr} |
| Maximum Peak | 1252 | 1248 | 1212 | 1212 |
| Average of Max. Peak | 818.92 | 816.99 | 816.59 | 817.63 |
| Average of Min. Peak | -819.29 | -817.37 | -817.45 | -817.20 |
| Minimum Peak | -1204 | -1250 | -1220 | -1250 |
| {yr
X} with |
{y
Xr} |
{yr
X} |
{yr
Xr} |
| Maximum Peak | 1320 | 1184 | 1270 |
| Average of Max. Peak | 817.20 | 815.13 | 816.83 |
| Average of Min. Peak | -817.57 | -815.09 | -817.26 |
| Minimum Peak | -1222 | -1282 | -1260 |
| {y
Xr} with |
{y
Xr} |
{yr
Xr} |
| Maximum Peak | 1236 | 1194 |
| Average of Max. Peak | 815.65 | 817.05 |
| Average of Min. Peak | -815.44 | -817.50 |
| Minimum Peak | -1210 | -1220 |
| {yr
Xr} with |
{yr
Xr} |
| Maximum Peak | 1262 |
| Average of Max. Peak | 819.14 |
| Average of Min. Peak | -819.12 |
| Minimum Peak | -1244 |
Table 2: Aperiodic cross-correlation of subset P(y,x)
The maximum absolute peak (of periodic and aperiodic cross-correlation) for the Gold subset of 512 different sequences is 1252. Therefore, the difference relative to the auto-correlation value (38,400) is 96.74%, for this subset. The maximum peak (of periodic and aperiodic cross-correlation) for the subset P(y,x) of 2048 different sequences is 1426; the difference relative to the auto-correlation (38,400) is 96.29%. The difference between the Gold subset and the P(y,x) subset, less than 0.5%, is not significant.
X} and {y
Xr} are slightly better subsets, with a lower absolute
average peak cross-correlation, as compared with the Gold subset {y
X}.
These results indicate that, by using long sequences of a Gold set, it is possible to increase almost four times the quantity of PN-sequences maintaining a low cross-correlation between all sequences. You can thus use the set P(y,x) in many systems where the maximum quantity of available PN-sequences is primordial for better resource usage.
When you use the complete set P(y,x), it is possible to reduce by four the required chip rate if the degree of the primitive polynomial generators of the Gold set is reduced by two, maintaining the same available quantity of PN-sequences. However, high quality WCDMA systems require the use of high degree polynomial generators.
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