sdr.fdoa_crlb

sdr.fdoa_crlb(snr1: ArrayLike, snr2: ArrayLike, time: ArrayLike, bandwidth: ArrayLike, rms_integration_time: ArrayLike | None = None, noise_bandwidth: ArrayLike | None = None) → NDArray[float64]

Calculates the Cramér-Rao lower bound (CRLB) on frequency difference of arrival (FDOA) estimation.

Parameters:¶

snr1: ArrayLike¶: The signal-to-noise ratio (SNR) of the first signal $γ_{1} = S_{1} / (N_{0} B_{n})$ in dB.
snr2: ArrayLike¶: The signal-to-noise ratio (SNR) of the second signal $γ_{2} = S_{2} / (N_{0} B_{n})$ in dB.
time: ArrayLike¶: The integration time $T$ in seconds.
bandwidth: ArrayLike¶: The signal bandwidth $B_{s}$ in Hz.
rms_integration_time: ArrayLike | None = None¶: The root-mean-square (RMS) integration time $T_{rms}$ in Hz. If None, the RMS integration time is calculated assuming a rectangular power envelope, $T_{rms} = T / \sqrt{12}$ .
noise_bandwidth: ArrayLike | None = None¶: The noise bandwidth $B_{n}$ in Hz. If None, the noise bandwidth is assumed to be the signal bandwidth $B_{s}$ . The noise bandwidth must be the same for both signals.

Returns:¶

The Cramér-Rao lower bound (CRLB) on the frequency difference of arrival (FDOA) estimation error standard deviation $σ_{fdoa}$ in Hz.

See also

sdr.rms_integration_time

Notes¶

The Cramér-Rao lower bound (CRLB) on the frequency difference of arrival (FDOA) estimation error standard deviation $σ_{fdoa}$ is given by

σ_{fdoa} = \frac{1}{π \sqrt{8} T_{rms}} \frac{1}{\sqrt{B_{n} T γ}}

\frac{1}{γ} = \frac{1}{γ_{1}} + \frac{1}{γ_{2}} + \frac{1}{γ_{1} γ_{2}}

T_{rms} = \sqrt{\frac{\int_{- \infty}^{\infty} (t - μ_{t})^{2} \cdot {| x (t - μ_{t}) |}^{2} d t}{\int_{- \infty}^{\infty} {| x (t - μ_{t}) |}^{2} d t}}

where $γ$ is the effective signal-to-noise ratio (SNR), ${| x (t) |}^{2}$ is the power envelope of the signal, and $μ_{t}$ is the centroid of the power envelope.

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 $T_{rms}$ and incorporated into the CRLB equation.

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

Examples¶

In [1]: snr = 10

In [2]: bandwidth = np.logspace(5, 8, 101)

In [3]: plt.figure(); \
   ...: plt.loglog(bandwidth, sdr.fdoa_crlb(snr, snr, 1e-6, bandwidth), label="1 μs"); \
   ...: plt.loglog(bandwidth, sdr.fdoa_crlb(snr, snr, 1e-5, bandwidth), label="10 μs"); \
   ...: plt.loglog(bandwidth, sdr.fdoa_crlb(snr, snr, 1e-4, bandwidth), label="100 μs"); \
   ...: plt.loglog(bandwidth, sdr.fdoa_crlb(snr, snr, 1e-3, bandwidth), label="1 ms"); \
   ...: plt.legend(title="Integration time"); \
   ...: plt.xlim(1e5, 1e8); \
   ...: plt.ylim(1e0, 1e6); \
   ...: plt.xlabel("Bandwidth (Hz), $B$"); \
   ...: plt.ylabel(r"CRLB on FDOA (Hz), $\sigma_{\text{fdoa}}$"); \
   ...: plt.title(f"Cramér-Rao lower bound (CRLB) on FDOA estimation error\nstandard deviation with {snr}-dB SNR");
   ...: 