sdr.toa_crlb

sdr.toa_crlb(snr: ArrayLike, time: ArrayLike, bandwidth: ArrayLike, rms_bandwidth: ArrayLike | None = None, noise_bandwidth: ArrayLike | None = None) → NDArray[float64]

Calculates the Cramér-Rao lower bound (CRLB) on time of arrival (TOA) estimation.

Parameters:¶

snr: ArrayLike¶: The signal-to-noise ratio (SNR) of the signal $γ = S / (N_{0} B_{n})$ in dB.
time: ArrayLike¶: The integration time $T$ in seconds.
bandwidth: ArrayLike¶: The signal bandwidth $B_{s}$ in Hz.
rms_bandwidth: ArrayLike | None = None¶: The root-mean-square (RMS) bandwidth $B_{s, rms}$ in Hz. If None, the RMS bandwidth is calculated assuming a rectangular spectrum, $B_{s, rms} = B_{s} / \sqrt{12}$ .
noise_bandwidth: ArrayLike | None = None¶: The noise bandwidth $B_{n}$ in Hz. If None, the noise bandwidth is assumed to be the signal bandwidth $B_{s}$ .

Returns:¶

The Cramér-Rao lower bound (CRLB) on the time of arrival (TOA) estimation error standard deviation $σ_{toa}$ in seconds.

See also

sdr.rms_bandwidth

Notes

The Cramér-Rao lower bound (CRLB) on the time of arrival (TOA) estimation error standard deviation $σ_{toa}$ is given by

σ_{toa} = \frac{1}{π \sqrt{8} B_{s, rms}} \frac{1}{\sqrt{B_{n} T γ}}

B_{s, rms} = \sqrt{\frac{\int_{- \infty}^{\infty} (f - μ_{f})^{2} \cdot S (f - μ_{f}) d f}{\int_{- \infty}^{\infty} S (f - μ_{f}) d f}}

where $γ$ is the signal-to-noise ratio (SNR), $S (f)$ is the power spectral density (PSD) of the signal, and $μ_{f}$ is the centroid of the PSD.

Note

The constant terms from Stein’s original equations were rearranged. The factor of 2 was removed from $γ$ and the factor of $2 π$ was removed from $B_{s, rms}$ and incorporated into the CRLB equation.

The signal-to-noise ratio (SNR) $γ$ is improved by the coherent integration gain, which is the time-bandwidth product $B_{n} T$ . The product $B_{n} T γ$ is the output SNR of the matched filter or correlator, which is equivalent to $E / N_{0}$ .

B_{n} T γ = B_{n} T \frac{S}{N_{0} B_{n}} = \frac{S T}{N_{0}} = \frac{E}{N_{0}}

Warning

According to Stein, the CRLB equation only holds for output SNRs greater than 10 dB. This ensures there is sufficient SNR to correctly identify the time/frequency peak without high $P_{f a}$ . Given the rearrangement of scaling factors, CRLB values with output SNRs less than 7 dB are set to NaN.

The time measurement precision is inversely proportional to the bandwidth of the signal and the square root of the output SNR.

Examples

In [1]: snr = 10

In [2]: time = np.logspace(-6, 0, 101)

In [3]: plt.figure(); \
   ...: plt.loglog(time, sdr.toa_crlb(snr, time, 1e5), label="100 kHz"); \
   ...: plt.loglog(time, sdr.toa_crlb(snr, time, 1e6), label="1 MHz"); \
   ...: plt.loglog(time, sdr.toa_crlb(snr, time, 1e7), label="10 MHz"); \
   ...: plt.loglog(time, sdr.toa_crlb(snr, time, 1e8), label="100 MHz"); \
   ...: plt.legend(title="Bandwidth"); \
   ...: plt.xlim(1e-6, 1e0); \
   ...: plt.ylim(1e-12, 1e-6); \
   ...: plt.xlabel("Integration time (s), $T$"); \
   ...: plt.ylabel(r"CRLB on TOA (s), $\sigma_{\text{toa}}$"); \
   ...: plt.title(f"Cramér-Rao lower bound (CRLB) on TOA estimation error\nstandard deviation with {snr}-dB SNR");
   ...: 