#### Table I

Comparison of FS and SA

Dominant noise term | Detector backgroundD ≫ M + BΔω | Background lightBΔω ≫ M + D | Narrowband backgroundM ≫ BΔω + D |
---|

Γ_{θf}/Γ_{θs} |
$$\pi {\phantom{\rule{0.2em}{0ex}}\left(\frac{{a}_{f}}{{a}_{s}}\right)}^{2}\phantom{\rule{0.2em}{0ex}}\frac{1}{{\tau}_{0}\phantom{\rule{0.1em}{0ex}}\mathrm{\Delta}\omega}\phantom{\rule{0.1em}{0ex}}\simeq \phantom{\rule{0.1em}{0ex}}1$$ |
$$\pi {\phantom{\rule{0.2em}{0ex}}\left(\frac{{a}_{f}}{{a}_{s}}\right)}^{2}\phantom{\rule{0.2em}{0ex}}\frac{{a}_{s}}{{\tau}_{0}\phantom{\rule{0.1em}{0ex}}\mathrm{\Delta}\omega}\phantom{\rule{0.2em}{0ex}}\frac{\mathrm{\Delta}\omega}{\mathrm{\Delta}{\omega}_{p}}\phantom{\rule{0.2em}{0ex}}<\phantom{\rule{0.2em}{0ex}}1$$ |
$$\pi {\phantom{\rule{0.2em}{0ex}}\left(\frac{{a}_{f}}{{a}_{s}}\right)}^{2}\phantom{\rule{0.2em}{0ex}}\frac{b}{{\tau}_{0}\mathrm{\Delta}\omega}\phantom{\rule{0.2em}{0ex}}<\phantom{\rule{0.2em}{0ex}}1$$ |

ρ_{θ,Rf}/ρ_{θ,Rs} | ∽1 | ∽1 | ∽1 |

ρ_{θ,Mf}/ρ_{θ,Ms} | ≫1 | ≫1 | ≫1 |

Note: Γ

_{θ} is the Fisher number for

θ. Subscript

f indicates FS; subscript s indicates SA.

ρ is a correlation coefficient. A large Fisher number infers a good estimate, a large

ρ indicates that lack of knowledge about a secondary parameter (

R or

M) has a large (negative) effect on the estimation of

θ. It is assumed that SA is done in parallel, if not Γ

_{θ,s} must be divided by the number

N of spectral values; the ratio Γ

_{θf}/Γ

_{θs} is then enhanced in favor of FS.

#### Table II

Performance of Different Filters

| Γ_{θ} | ρ | T_{a} |
---|

Optimum rectangular filter (no implementation given) |
$$0.12{\phantom{\rule{0.2em}{0ex}}\left(\frac{R}{\theta}\right)}^{2}\phantom{\rule{0.2em}{0ex}}\frac{{T}_{a}}{D\phantom{\rule{0.1em}{0ex}}+\phantom{\rule{0.1em}{0ex}}B\mathrm{\Delta}\omega}$$ | | |

FS derived filter, a two-arm interferometer 1 |
$$0.14{\phantom{\rule{0.2em}{0ex}}\left(\frac{R}{\theta}\right)}^{2}\phantom{\rule{0.2em}{0ex}}\frac{{T}_{a}}{D\phantom{\rule{0.1em}{0ex}}+\phantom{\rule{0.1em}{0ex}}B\mathrm{\Delta}\omega}$$ | >0.2 | T |

Gaussian filters |
$$0.06{\phantom{\rule{0.2em}{0ex}}\left(\frac{R}{\theta}\right)}^{2}\phantom{\rule{0.2em}{0ex}}\frac{{T}_{a}}{D\phantom{\rule{0.1em}{0ex}}+\phantom{\rule{0.1em}{0ex}}B{\phantom{\rule{0.2em}{0ex}}(2\pi \theta )}^{1/2}}$$ | >0.2 | T/2 |

FS derived filter, a two-arm interferometer 2 |
$$0.12{\phantom{\rule{0.2em}{0ex}}\left(\frac{R}{\theta}\right)}^{2}\phantom{\rule{0.2em}{0ex}}\frac{{T}_{a}}{0.69\phantom{\rule{0.1em}{0ex}}R\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}2B\mathrm{\Delta}\omega \phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}D}$$ | >0.27 | T/2 |

Note: Δ

ω can be considered as a prefilter bandwidth that necessarily is larger than √

θ. In general there is an optimum Δ

ω. It has been assumed here that Δ

ω ≫ √

θ.

T_{a} is the averaging time and

T the observation, or measuring time. For the three first filters it is assumed that

R ≪ 2

BΔ

ω +

D.