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Use uppercase subscripts
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mhostetter committed May 17, 2024
1 parent d4b41f7 commit 94a0b04
Showing 1 changed file with 9 additions and 9 deletions.
18 changes: 9 additions & 9 deletions src/sdr/_estimation/_time.py
Original file line number Diff line number Diff line change
Expand Up @@ -25,14 +25,14 @@ def tdoa_crlb(
Calculates the Cramér-Rao lower bound (CRLB) on the time difference of arrival (TDOA) estimation.
Arguments:
snr1: The signal-to-noise ratio (SNR) of the first signal $\gamma_1 = S_1 / (N_0 B_n)$ in dB.
snr2: The signal-to-noise ratio (SNR) of the second signal $\gamma_2 = S_2 / (N_0 B_n)$ in dB.
snr1: The signal-to-noise ratio (SNR) of the first signal $\gamma_1 = S_1 / (N_0 B_N)$ in dB.
snr2: The signal-to-noise ratio (SNR) of the second signal $\gamma_2 = S_2 / (N_0 B_N)$ in dB.
time: The integration time $T$ in seconds.
bandwidth: The signal bandwidth $B_s$ in Hz.
rms_bandwidth: The root-mean-square (RMS) bandwidth $B_{s,\text{rms}}$ in Hz. If `None`, the RMS bandwidth
is calculated assuming a rectangular spectrum, $B_{s,\text{rms}} = B_s/\sqrt{12}$.
noise_bandwidth: 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.
bandwidth: The signal bandwidth $B_S$ in Hz.
rms_bandwidth: The root-mean-square (RMS) bandwidth $B_{S,\text{rms}}$ in Hz. If `None`, the RMS bandwidth
is calculated assuming a rectangular spectrum, $B_{S,\text{rms}} = B_S/\sqrt{12}$.
noise_bandwidth: 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 time difference of arrival (TDOA) estimation standard deviation
Expand All @@ -42,10 +42,10 @@ def tdoa_crlb(
The Cramér-Rao lower bound (CRLB) on the time difference of arrival (TDOA) estimation standard deviation
$\sigma_{\text{TDOA}}$ is given by
$$\sigma_{\text{TDOA}} = \frac{1}{2 \pi B_{s,\text{rms}}} \frac{1}{\sqrt{B_n T \gamma}} .$$
$$\sigma_{\text{TDOA}} = \frac{1}{2 \pi B_{S,\text{rms}}} \frac{1}{\sqrt{B_N T \gamma}} .$$
The effective signal-to-noise ratio (SNR) $\gamma$ is improved by the coherent integration gain, which is the
time-bandwidth product $B_n T$. The product $B_n T \gamma$ is the output SNR of the matched filter
time-bandwidth product $B_N T$. The product $B_N T \gamma$ is the output SNR of the matched filter
or correlator.
The time measurement accuracy is inversely proportional to the bandwidth of the signal and the square root of
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