sdr.threshold_factor

sdr.threshold_factor(p_fa: ArrayLike, detector: 'coherent' | 'linear' | 'square-law' = 'square-law', complex: bool = True, n_c: int = 1, n_nc: int | None = None, db: bool = False) → NDArray[float64]

Computes the theoretical detection threshold factor \(\alpha\).

Parameters:¶

p_fa: ArrayLike¶

The desired probability of false alarm \(P_{fa}\) in \((0, 1)\).

detector: 'coherent' | 'linear' | 'square-law' = 'square-law'¶

The detector type.

"coherent": A coherent detector,
\[T(x) = \mathrm{Re}\left\{\sum_{i=0}^{N_c-1} x[n-i]\right\} .\]
"linear": A linear detector,
\[T(x) = \sum_{j=0}^{N_{nc}-1}\left|\sum_{i=0}^{N_c-1} x[n-i-jN_c]\right| .\]
"square-law": A square-law detector,
\[T(x) = \sum_{j=0}^{N_{nc}-1}\left|\sum_{i=0}^{N_c-1} x[n-i-jN_c]\right|^2 .\]

complex: bool = True¶

Indicates whether the input signal is real or complex. This affects how the SNR is converted to noise variance.

n_c: int = 1¶

The number of samples to coherently integrate \(N_c\).

n_nc: int | None = None¶

The number of samples to non-coherently integrate \(N_{nc}\). Non-coherent integration is only allowable for linear and square-law detectors.

db: bool = False¶

Indicates whether to return the detection threshold \(\alpha\) in dB.

Returns:¶

The detection threshold factor \(\alpha\).

Notes

The detection threshold factor \(\alpha\) is defined as the ratio of the detection threshold \(\gamma\) to the mean of the detector output under the null hypothesis. This is true for linear and square-law detectors.

\[\alpha = \frac{\gamma}{\frac{1}{N} \sum_{i=1}^{N}\{T(x[i]) \mid \mathcal{H}_0\}}\]

For coherent detectors, the detection threshold factor \(\alpha\) is defined as the ratio of the detection threshold \(\gamma\) to the mean of the square of the detector output under the null hypothesis. This is required because the mean of the coherent detector output is zero.

\[\alpha = \frac{\gamma}{\frac{1}{N} \sum_{i=1}^{N}\{T(x[i])^2 \mid \mathcal{H}_0\}}\]

Examples

In [1]: p_fa = np.logspace(-16, -1, 101)  # Probability of false alarm

In [2]: plt.figure(); \
   ...: plt.semilogx(p_fa, sdr.threshold_factor(p_fa, detector="coherent"), label="Coherent"); \
   ...: plt.semilogx(p_fa, sdr.threshold_factor(p_fa, detector="linear"), label="Linear"); \
   ...: plt.semilogx(p_fa, sdr.threshold_factor(p_fa, detector="square-law"), label="Square-Law"); \
   ...: plt.xlabel("Probability of false alarm, $P_{fa}$"); \
   ...: plt.ylabel(r"Detection threshold factor, $\alpha$"); \
   ...: plt.legend(title="Detector"); \
   ...: plt.title("Detection threshold factor across false alarm rate");
   ...: 

../../_images/sdr_threshold_factor_1.png