sdr.kasami_code(length: int, index: int | tuple[int, int] = 0, poly: PolyLike | None = None, output: 'binary' = 'binary') → ndarray[Any, dtype[int64]]
sdr.kasami_code(length: int, index: int | tuple[int, int] = 0, poly: PolyLike | None = None, output: 'field' = 'binary') → FieldArray
sdr.kasami_code(length: int, index: int | tuple[int, int] = 0, poly: PolyLike | None = None, output: 'bipolar' = 'binary') → ndarray[Any, dtype[float64]]

Returns the Kasami code/sequence of length \(N\).

Parameters:
length: int

The length \(N = 2^n - 1\) of the Kasami code/sequence. The degree \(n\) must be even.

index: int | tuple[int, int] = 0

The index of the Kasami code.

  • int: The index \(m\) in \([-1, 2^{n/2} - 1)\) from the Kasami code small set. There are \(2^{n/2}\) codes in the small set.

  • tuple[int, int]: The index \((k, m)\) from the Kasami code large set, with \(k \in [-2, 2^n - 1)\) and \(m \in [-1, 2^{n/2} - 1)\). There are \((2^n + 1) \cdot 2^{n/2}\) codes in the large set.

poly: PolyLike | None = None

The primitive polynomial of degree \(n\) over \(\mathrm{GF}(2)\). The default is None, which uses the default primitive polynomial of degree \(n\), i.e. galois.primitive_poly(2, n).

output: 'binary' = 'binary'
output: 'field' = 'binary'
output: 'bipolar' = 'binary'

The output format of the Kasami code/sequence.

  • "binary" (default): The Kasami code with binary values of 0 and 1.

  • "field": The Kasami code as a Galois field array over \(\mathrm{GF}(2)\).

  • "bipolar": The Kasami sequence with bipolar values of 1 and -1.

Returns:

The Kasami code/sequence of length \(N\).

References

Examples

Create a Kasami code and sequence of length 15.

In [1]: sdr.kasami_code(15, 1)
Out[1]: array([1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0])

In [2]: sdr.kasami_code(15, 1, output="bipolar")
Out[2]: 
array([-1., -1., -1.,  1.,  1., -1.,  1.,  1.,  1.,  1.,  1., -1., -1.,
        1.,  1.])

In [3]: sdr.kasami_code(15, 1, output="field")
Out[3]: GF([1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0], order=2)

Create several Kasami codes of length 63 from the small set.

In [4]: seq1 = sdr.kasami_code(63, 0, output="bipolar"); \
   ...: seq2 = sdr.kasami_code(63, 1, output="bipolar"); \
   ...: seq3 = sdr.kasami_code(63, 2, output="bipolar");
   ...: 

In [5]: plt.figure(); \
   ...: sdr.plot.time_domain(seq1 + 3, label="Index 0"); \
   ...: sdr.plot.time_domain(seq2 + 0, label="Index 1"); \
   ...: sdr.plot.time_domain(seq3 - 3, label="Index 2")
   ...:
../../_images/sdr_kasami_code_1.png

Examine the autocorrelation of the Kasami sequences.

In [6]: lag = np.arange(-seq1.size + 1, seq1.size); \
   ...: acorr12 = np.correlate(seq1, seq1, mode="full"); \
   ...: acorr13 = np.correlate(seq2, seq2, mode="full"); \
   ...: acorr23 = np.correlate(seq3, seq3, mode="full");
   ...: 

In [7]: plt.figure(); \
   ...: sdr.plot.time_domain(lag, np.abs(acorr12), label="0"); \
   ...: sdr.plot.time_domain(lag, np.abs(acorr13), label="1"); \
   ...: sdr.plot.time_domain(lag, np.abs(acorr23), label="2"); \
   ...: plt.xlabel("Lag"); \
   ...: plt.title("Autocorrelation of length-63 Kasami sequences");
   ...:
../../_images/sdr_kasami_code_2.png

Examine the cross correlation of the Kasami sequences.

In [8]: lag = np.arange(-seq1.size + 1, seq1.size); \
   ...: xcorr12 = np.correlate(seq1, seq2, mode="full"); \
   ...: xcorr13 = np.correlate(seq1, seq3, mode="full"); \
   ...: xcorr23 = np.correlate(seq2, seq3, mode="full");
   ...: 

In [9]: plt.figure(); \
   ...: sdr.plot.time_domain(lag, np.abs(xcorr12), label="0 and 1"); \
   ...: sdr.plot.time_domain(lag, np.abs(xcorr13), label="0 and 2"); \
   ...: sdr.plot.time_domain(lag, np.abs(xcorr23), label="1 and 2"); \
   ...: plt.xlabel("Lag"); \
   ...: plt.title("Cross correlation of length-63 Kasami sequences");
   ...:
../../_images/sdr_kasami_code_3.png