sdr.gold_code

sdr.gold_code(length: int, index: int = 0, poly1: PolyLike | None = None, poly2: PolyLike | None = None, verify: bool = True, output: 'binary' = 'binary') → ndarray[Any, dtype[int64]]

sdr.gold_code(length: int, index: int = 0, poly1: PolyLike | None = None, poly2: PolyLike | None = None, verify: bool = True, output: 'field' = 'binary') → FieldArray

sdr.gold_code(length: int, index: int = 0, poly1: PolyLike | None = None, poly2: PolyLike | None = None, verify: bool = True, output: 'bipolar' = 'binary') → ndarray[Any, dtype[float64]]

Generates the Gold code/sequence of length $n = 2^m - 1$.

Parameters:¶

length: int¶

The length $n = 2^m - 1$ of the Gold code/sequence.

index: int = 0¶

The index $i$ in $[-2, n)$ of the Gold code.

poly1: PolyLike | None = None¶

The primitive polynomial of degree $m$ over $\mathrm{GF}(2)$ for the first $m$-sequence. If None, a preferred pair is found using sdr.preferred_pairs().

poly2: PolyLike | None = None¶

The primitive polynomial of degree $m$ over $\mathrm{GF}(2)$ for the second $m$-sequence. If None, a preferred pair is found using sdr.preferred_pairs().

verify: bool = True¶

Indicates whether to verify that the provided polynomials are a preferred pair using sdr.is_preferred_pair().

output: 'binary' = 'binary'¶

output: 'field' = 'binary'

output: 'bipolar' = 'binary'

The output format of the Gold code/sequence.

"binary" (default): The Gold code with binary values of 0 and 1.
"field": The Gold code as a Galois field array over $\mathrm{GF}(2)$.
"bipolar": The Gold sequence with bipolar values of 1 and -1.

Returns:¶

The Gold code/sequence of length $n = 2^m - 1$ and index $i$.

Notes

Gold codes are generated by combining two preferred pair $m$-sequences, $u$ and $v$, using the formula

\[\begin{split} c = \begin{cases} u & \text{if $i = -2$} \\ v & \text{if $i = -1$} \\ u \oplus T^i v & \text{otherwise} , \end{cases} \end{split}\]

where $i$ is the code index, $\oplus$ is addition in $\mathrm{GF}(2)$ (or the XOR operation), and $T^i$ is a left shift by $i$ positions.

Gold codes are PN sequence with good auto-correlation and cross-correlation properties. The Gold code set contains $2^m + 1$ sequences of length $2^m - 1$. The correlation sides are guaranteed to be less than or equal to $t(m)$.

\[\begin{split} t(m) = \begin{cases} 2^{(m+1)/2} + 1 & \text{if $m$ is odd} \\ 2^{(m+2)/2} + 1 & \text{if $m$ is even} \end{cases} \end{split}\]

There are no preferred pairs with degree $m$ divisible by 4. Therefore, there are no Gold codes with degree $m$ divisible by 4.

References

John Proakis, Digital Communications, Chapter 12.2-5: Generation of PN Sequences.

Examples

Create a Gold code and sequence of length 7.

In [1]: sdr.gold_code(7, 1)
Out[1]: array([1, 0, 1, 0, 1, 1, 0])

In [2]: sdr.gold_code(7, 1, output="bipolar")
Out[2]: array([-1.,  1., -1.,  1., -1., -1.,  1.])

In [3]: sdr.gold_code(7, 1, output="field")
Out[3]: GF([1, 0, 1, 0, 1, 1, 0], order=2)

Create several Gold codes of length 63.

In [4]: x1 = sdr.gold_code(63, 0, output="bipolar"); \
   ...: x2 = sdr.gold_code(63, 1, output="bipolar"); \
   ...: x3 = sdr.gold_code(63, 2, output="bipolar");
   ...: 

In [5]: plt.figure(); \
   ...: sdr.plot.time_domain(x1 + 3); \
   ...: sdr.plot.time_domain(x2 + 0); \
   ...: sdr.plot.time_domain(x3 - 3)
   ...: 

Examine the auto-correlation of the Gold sequences.

In [6]: plt.figure(); \
   ...: sdr.plot.correlation(x1, x1, mode="circular"); \
   ...: sdr.plot.correlation(x2, x2, mode="circular"); \
   ...: sdr.plot.correlation(x3, x3, mode="circular"); \
   ...: plt.ylim(0, 63);
   ...: 

Examine the cross-correlation of the Gold sequences.

In [7]: plt.figure(); \
   ...: sdr.plot.correlation(x1, x2, mode="circular"); \
   ...: sdr.plot.correlation(x1, x3, mode="circular"); \
   ...: sdr.plot.correlation(x2, x3, mode="circular"); \
   ...: plt.ylim(0, 63);
   ...: 