static sdr.EnergyDetector.p_fa(threshold: ArrayLike, N_nc: ArrayLike, sigma2: ArrayLike, complex: bool = True) → NDArray[float64]

Computes the probability of false alarm \(P_{fa}\).

Parameters:
threshold: ArrayLike

The threshold \(\gamma'\).

N_nc: ArrayLike

The number of samples \(N_{nc}\) to non-coherently integrate.

sigma2: ArrayLike

The noise variance \(\sigma^2\).

complex: bool = True

Indicates whether the signal is complex.

Returns:

The probability of false alarm \(P_{fa}\).

Notes

For real signals:

\[P_{fa} = Q_{\chi_N^2}\left( \frac{\gamma'}{\sigma^2} \right)\]

For complex signals:

\[P_{fa} = Q_{\chi_{2N}^2}\left( \frac{\gamma'}{\sigma^2 / 2} \right)\]

References

  • Steven Kay, Fundamentals of Statistical Signal Processing: Detection Theory, Equation 5.2.