sdr.non_coherent_gain(n_nc: ArrayLike, snr: ArrayLike, p_fa: ArrayLike = 1e-06, snr_ref: 'input' | 'output' = 'input', extrapolate: bool = True) → NDArray[float64]

Computes the SNR improvement by non-coherently integrating \(N_{NC}\) samples.

Parameters:
n_nc: ArrayLike

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

snr: ArrayLike

The reference SNR in dB.

p_fa: ArrayLike = 1e-06

The desired probability of false alarm \(P_{FA}\). This is used to compute the necessary thresholds before and after integration. The non-coherent gain is slightly affected by the \(P_{FA}\).

snr_ref: 'input' | 'output' = 'input'

The SNR reference.

  • "input": The SNR is referenced at the input of the non-coherent integrator.

  • "output": The SNR is referenced at the output of the non-coherent integrator.

extrapolate: bool = True

Indicates whether to extrapolate \(G_{NC}\) using smaller values of \(N_{NC}\). This is only done when the non-coherent gain cannot be explicitly solved for due to lack of floating-point precision. If False, the function will return np.nan for any \(N_{NC}\) that cannot be solved for.

Returns:

The non-coherent gain \(G_{NC}\) in dB.

Notes

\[y[m] = \sum_{n=0}^{N_{NC}-1} \left| x[m-n] \right|^2\]
\[\text{SNR}_{y,\text{dB}} = \text{SNR}_{x,\text{dB}} + G_{NC}\]

Examples

See the Non-coherent integration example.

Compute the non-coherent gain for various integration lengths at 10-dB SNR.

In [1]: sdr.non_coherent_gain(1, 10)
Out[1]: 0.0

In [2]: sdr.non_coherent_gain(2, 10)
Out[2]: 2.499445060713011

In [3]: sdr.non_coherent_gain(10, 10)
Out[3]: 8.666092814306324

In [4]: sdr.non_coherent_gain(20, 10)
Out[4]: 11.457842213037324

Plot the non-coherent gain parameterized by input SNR.

In [5]: plt.figure(); \
   ...: n = np.logspace(0, 3, 51); \
   ...: plt.semilogx(n, sdr.coherent_gain(n), color="k");
   ...: 

In [6]: for snr in np.arange(-30, 40, 10):
   ...:     plt.semilogx(n, sdr.non_coherent_gain(n, snr, snr_ref="input"), label=f"{snr} dB")
   ...: 

In [7]: plt.legend(title="Input SNR", loc="upper left"); \
   ...: plt.xlabel("Number of samples, $N_{NC}$"); \
   ...: plt.ylabel("Non-coherent gain, $G_{NC}$"); \
   ...: plt.title("Non-coherent gain for various input SNRs");
   ...:
../../_images/sdr_non_coherent_gain_1.png

Plot the non-coherent gain parameterized by output SNR.

In [8]: plt.figure(); \
   ...: n = np.logspace(0, 3, 51); \
   ...: plt.semilogx(n, sdr.coherent_gain(n), color="k");
   ...: 

In [9]: for snr in np.arange(-30, 40, 10):
   ...:     plt.semilogx(n, sdr.non_coherent_gain(n, snr, snr_ref="output"), label=f"{snr} dB")
   ...: 

In [10]: plt.legend(title="Output SNR", loc="upper left"); \
   ....: plt.xlabel("Number of samples, $N_{NC}$"); \
   ....: plt.ylabel("Non-coherent gain, $G_{NC}$"); \
   ....: plt.title("Non-coherent gain for various output SNRs");
   ....:
../../_images/sdr_non_coherent_gain_2.png

Examine the non-coherent gain across input SNR and false alarm rate for non-coherently integrating 10 samples. Notice that the non-coherent gain is affected by both. The coherent integration gain, however, is a constant 10 dB across both.

In [11]: plt.figure(); \
   ....: snr = np.linspace(-40, 12, 101); \
   ....: n_nc = 10;
   ....: 

In [12]: for p_fa in [1e-14, 1e-12, 1e-10, 1e-8, 1e-6, 1e-4, 1e-2]:
   ....:     g_nc = sdr.non_coherent_gain(n_nc, snr, p_fa)
   ....:     plt.plot(snr, g_nc, label=f"{p_fa:1.0e}")
   ....: 

In [13]: plt.legend(title="$P_{FA}$"); \
   ....: plt.ylim(0, 10); \
   ....: plt.xlabel("Input signal-to-noise ratio (dB)"); \
   ....: plt.ylabel("Non-coherent gain (dB)"); \
   ....: plt.title("Non-coherent gain for $N_{NC} = 10$");
   ....:
../../_images/sdr_non_coherent_gain_3.png