static sdr.ReplicaCorrelator.p_fa(threshold: ArrayLike, energy: ArrayLike, sigma2: ArrayLike, complex: bool = True) → NDArray[float_]

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

Parameters:
threshold: ArrayLike

The threshold \(\gamma'\).

energy: ArrayLike

The received energy \(\mathcal{E} = \sum_{i=0}^{N-1} \left| s[n] \right|^2\).

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:

\[P_{FA} = Q\left( \frac{\gamma'}{\sqrt{\sigma^2 \mathcal{E}}} \right)\]

For complex signals:

\[P_{FA} = Q\left( \frac{\gamma'}{\sqrt{\sigma^2 \mathcal{E} / 2}} \right)\]

References

  • Steven Kay, Fundamentals of Statistical Signal Processing: Detection Theory, Equations 4.12 and 13.6.