A Polarimetric Approach for Constraining the Dynamic Foreground Spectrum for Cosmological Global 21 cm Measurements

Consider a coordinate system fixed to the sky and a pair of crossed-dipole antennas with their boresight aligned along the z-axis at $\theta =0^\circ$ and the horizon is at $\theta =90^\circ$ on the xy-plane for the antenna's coordinate system. We further assume that the sky coordinate is not observable below the horizon at $\theta \leqslant 90^\circ$, although the ionosphere can extend the visible sky below the horizon due to refraction in the low frequencies (e.g., Vedantham et al. 2013). From a given direction $(\theta ,\phi )$ in the sky at a given instant, the broadband incoming unpolarized radio signal can be represented as quasi-monochromatic electric field with polar and azimuthal components ${{\boldsymbol{E}}}_{\mathrm{in}}(\theta ,\phi ,\nu )=\{{E}_{\theta }(\theta ,\phi ,\nu )\hat{\theta }+{E}_{\phi }(\theta ,\phi ,\nu )\hat{{\boldsymbol{\phi }}}\}$. The incoming E-field is projected and received by the crossed dipoles, which have an X and Y oriented E-fields that are orthogonal to each other. The observed field, ${{\boldsymbol{E}}}_{\mathrm{out}}(\theta ,\phi ,\nu )$ in Cartesian coordinates for such a system, is a product of the incoming field with a Jones matrix (Trippe 2014),

$$\begin{eqnarray}{{\boldsymbol{E}}}_{\mathrm{out}}(\theta ,\phi ,\nu ) & = & {E}_{X}(\theta ,\phi ,\nu )\hat{{\boldsymbol{x}}}+{E}_{Y}(\theta ,\phi ,\nu )\hat{{\boldsymbol{y}}}\\ & = & [{{\boldsymbol{J}}}_{\mathrm{ant}}^{X}(\theta ,\phi ,\nu )+{{\boldsymbol{J}}}_{\mathrm{ant}}^{Y}(\theta ,\phi ,\nu )]{{\boldsymbol{E}}}_{\mathrm{in}}(\theta ,\phi ,\nu )\\ & = & {{\boldsymbol{J}}}_{\mathrm{ant}}(\theta ,\phi ,\nu ){{\boldsymbol{E}}}_{\mathrm{in}}(\theta ,\phi ,\nu ),\end{eqnarray} \tag{ 3 }$$

where the superscripts on the Jones matrices indicate the X and Y dipoles. In a realistic system, the Jones matrix also describes other instrumental systematics as well as external influences on the electric field such as ionospheric effects (Kraus 1986; Heiles 2002; Trippe 2014). Here, we only consider the projection of the incoming E-field onto the dipole antennas, which is described by ${{\boldsymbol{J}}}_{\mathrm{ant}}$.

The antenna Jones matrix ${{\boldsymbol{J}}}_{\mathrm{ant}}$ is the composite far-field patterns of both crossed dipoles at each frequency, which is represented as

$$\begin{eqnarray}{{\boldsymbol{J}}}_{\mathrm{ant}}(\theta ,\phi ,\nu ) & = & {\left[\begin{array}{cc}{J}_{\theta }^{X} & {J}_{\phi }^{X}\\ {J}_{\theta }^{Y} & {J}_{\phi }^{Y}\end{array}\right]}_{(\theta ,\phi ,\nu )}\\ & = & {\left[\begin{array}{cc}| {E}_{\theta }^{X}| {e}^{j{{\rm{\Phi }}}_{\theta }^{X}} & | {E}_{\phi }^{X}| {e}^{j{{\rm{\Phi }}}_{\phi }^{X}}\\ | {E}_{\theta }^{Y}| {e}^{j{{\rm{\Phi }}}_{\theta }^{Y}} & | {E}_{\phi }^{Y}| {e}^{j{{\rm{\Phi }}}_{\phi }^{Y}}\end{array}\right]}_{(\theta ,\phi ,\nu )},\end{eqnarray} \tag{ 4 }$$

where the complex far-field components for the X and Y oriented dipoles are given in terms of their magnitude $| E|$ and phase Φ for each direction $(\theta ,\phi )$.

Components of the Stokes vector at a given direction can be calculated as linear combinations of the observed E-field's autocorrelation and cross-correlation,

$$\begin{eqnarray}{I}_{\mathrm{out}}^{\nu }(\theta ,\phi ) & = & \langle {E}_{X}{E}_{X}^{* }\rangle +\langle {E}_{Y}{E}_{Y}^{* }\rangle ,\\ {Q}_{\mathrm{out}}^{\nu }(\theta ,\phi ) & = & \langle {E}_{X}{E}_{X}^{* }\rangle -\langle {E}_{Y}{E}_{Y}^{* }\rangle ,\\ {U}_{\mathrm{out}}^{\nu }(\theta ,\phi ) & = & \langle {E}_{X}{E}_{Y}^{* }\rangle +\langle {E}_{X}^{* }{E}_{Y}\rangle ,\\ {V}_{\mathrm{out}}^{\nu }(\theta ,\phi ) & = & j(\langle {E}_{X}{E}_{Y}^{* }\rangle -\langle {E}_{X}^{* }{E}_{Y}\rangle ),\end{eqnarray} \tag{ 5 }$$

where the bracket represents a time average of the product between the observed E-field components and their complex conjugates, with $j=\sqrt{-1}$. The spatial and frequency dependence on the E-field are suppressed for the ease of reading.

Assuming the E-field from all directions add incoherently, the net composite field measured at the crossed dipoles are sums of squares instead of the square of sums of individual electric field. Hence, the Stokes vector representing the projection-induced polarization of the foreground at a given instant is a vector summation of the individual Stokes vectors from each direction on the Poincaré sphere,

$$\begin{eqnarray}{I}_{\mathrm{net}}^{\nu } & = & \displaystyle \frac{1}{{{\rm{\Omega }}}_{{XY}}}\displaystyle \sum _{\phi =0}^{2\pi }\ \displaystyle \sum _{\theta =0}^{\pi /2}{I}_{\mathrm{out}}^{\nu }(\theta ,\phi ){\rm{\Delta }}{\rm{\Omega }},\\ {Q}_{\mathrm{net}}^{\nu } & = & \displaystyle \frac{1}{{{\rm{\Omega }}}_{{XY}}}\displaystyle \sum _{\phi =0}^{2\pi }\ \displaystyle \sum _{\theta =0}^{\pi /2}{Q}_{\mathrm{out}}^{\nu }(\theta ,\phi ){\rm{\Delta }}{\rm{\Omega }},\\ {U}_{\mathrm{net}}^{\nu } & = & \displaystyle \frac{1}{{{\rm{\Omega }}}_{{XY}}}\displaystyle \sum _{\phi =0}^{2\pi }\ \displaystyle \sum _{\theta =0}^{\pi /2}{U}_{\mathrm{out}}^{\nu }(\theta ,\phi ){\rm{\Delta }}{\rm{\Omega }},\\ {V}_{\mathrm{net}}^{\nu } & = & \displaystyle \frac{1}{{{\rm{\Omega }}}_{{XY}}}\displaystyle \sum _{\phi =0}^{2\pi }\ \displaystyle \sum _{\theta =0}^{\pi /2}{V}_{\mathrm{out}}^{\nu }(\theta ,\phi ){\rm{\Delta }}{\rm{\Omega }},\end{eqnarray} \tag{ 6 }$$

where ${{\rm{\Omega }}}_{{XY}}={\sum }_{\phi =0}^{2\pi }{\sum }_{\theta =0}^{\pi /2}F(\theta ,\phi ,\nu ){\rm{\Delta }}{\rm{\Omega }}$ is the beam normalization factor of the averaged antenna radiation pattern $F(\theta ,\phi ,\nu )=({F}^{X}+{F}^{Y})/2$, where ${F}^{X,Y}(\theta ,\phi ,\nu )\,=| {E}_{\theta }^{X,Y}(\theta ,\phi ,\nu ){| }^{2}+| {E}_{\phi }^{X,Y}(\theta ,\phi ,\nu ){| }^{2}$ are the radiation patterns of antenna X and Y, with the differential solid angle element ${\rm{\Delta }}{\rm{\Omega }}=\sin \theta {\rm{\Delta }}\theta {\rm{\Delta }}\phi$.

Since the sky revolves about the celestial pole periodically every sidereal day, this dynamic characteristic is also carried over to the resulting projection-induced polarization. To illustrate this, we simplify the projection by aligning the crossed dipoles' boresight with the NCP at $\theta =0^\circ$. This is analogous to placing the antenna at the Geographic North Pole (GNP, at latitude ${\phi }_{\oplus }=90^\circ$), such that the same sky revolves about the celestial pole continuously at an angular frequency ${\omega }_{\mathrm{sky}}$, where ${\omega }_{\mathrm{sky}}/2\pi \ll \nu$.

Because the coordinate system we adopted is fixed to the sky, the $(\theta ,\phi )$ coincide with the declination decl. and right ascension R.A. in the equatorial system, respectively. As the sky revolves relative to the fixed antennas on the ground, the net induced polarization reaches its maximal or minimal values when it is parallel or orthogonal to one of the dipoles. For the same dipole, the net induced polarization is detected twice per revolution, or twice diurnal. This gives rise to the cyclic Stokes parameters ${Q}_{\mathrm{net}}^{\nu }(t)$ and ${U}_{\mathrm{net}}^{\nu }(t)$ with an angular frequency equaling twice the sky revolving rate, i.e., $2{\omega }_{\mathrm{sky}}$.

In such a coordinate system, the observed E-field is produced as if having the antenna rotated relative to the fixed sky. By reciprocity, we can model the antenna response coupled to the incoming E-field from direction $(\theta ,\phi )$ as one emitted by an infinitesimal horizontal crossed dipoles placed at the origin of the coordinate system. For the dipole along x-axis, the complex far field measured at distance r has components

$$\begin{eqnarray}{E}_{\theta }^{X}(\theta ,\phi ,\nu ) & \simeq & -j\displaystyle \frac{\omega \mu {I}_{0}{{le}}^{-j{\boldsymbol{k}}\cdot {\boldsymbol{r}}}}{4\pi r}\cos \theta \cos \phi ,\\ {E}_{\phi }^{X}(\theta ,\phi ,\nu ) & \simeq & +j\displaystyle \frac{\omega \mu {I}_{0}{{le}}^{-j{\boldsymbol{k}}\cdot {\boldsymbol{r}}}}{4\pi r}\sin \phi ,\end{eqnarray} \tag{ 7 }$$

where $\omega =2\pi \nu$, $| {\boldsymbol{k}}| =2\pi /\lambda$ is the wave number of the observed wavelength, μ is the permeability, and I₀ is the current excited across the infinitesimal dipole with length l by the source (Balanis 2016). Similarly, for the y-oriented dipole, components of the corresponding far field are

$$\begin{eqnarray}{E}_{\theta }^{Y}(\theta ,\phi ,\nu ) & \simeq & -j\displaystyle \frac{\omega \mu {I}_{0}{{le}}^{-j{\boldsymbol{k}}\cdot {\boldsymbol{r}}}}{4\pi r}\cos \phi ,\\ {E}_{\phi }^{Y}(\theta ,\phi ,\nu ) & \simeq & -j\displaystyle \frac{\omega \mu {I}_{0}{{le}}^{-j{\boldsymbol{k}}\cdot {\boldsymbol{r}}}}{4\pi r}\cos \theta \sin \phi .\end{eqnarray} \tag{ 8 }$$

As the sky revolves about the boresight of the antenna over some time t, a sky emission from direction $({\theta }_{0},{\phi }_{0})$, that was originally coupled to the antenna response at $({\theta }_{0},{\phi }_{0})$, is now received to the antenna response $(\theta ^{\prime} ,\phi ^{\prime} )=({\theta }_{0},{\phi }_{0}+{\omega }_{\mathrm{sky}}t)$. By substituting the fields from Equation (8) into Equations (3)–(5), the resulting Stokes parameters corresponding to $(\theta ,\phi )$ is a function of time and are written as

$$\begin{eqnarray}{I}_{\mathrm{out}}^{\nu }(\theta ,\phi ,t) & = & {g}_{0}^{2}({E}_{\theta }^{2}{\cos }^{2}\theta +{E}_{\phi }^{2}),\\ {Q}_{\mathrm{out}}^{\nu }(\theta ,\phi ,t) & = & {g}_{0}^{2}({E}_{\theta }^{2}{\cos }^{2}\theta -{E}_{\phi }^{2})\cos (2{\omega }_{\mathrm{sky}}t)\\ & & -2{g}_{0}^{2}{E}_{\theta }{E}_{\phi }\cos \theta \sin (2{\omega }_{\mathrm{sky}}t),\\ {U}_{\mathrm{out}}^{\nu }(\theta ,\phi ,t) & = & {g}_{0}^{2}({E}_{\theta }^{2}{\cos }^{2}\theta -{E}_{\phi }^{2})\sin (2{\omega }_{\mathrm{sky}}t)\\ & & +2{g}_{0}^{2}{E}_{\theta }{E}_{\phi }\cos \theta \cos (2{\omega }_{\mathrm{sky}}t),\\ {V}_{\mathrm{out}}^{\nu }(\theta ,\phi ,t) & = & 0,\end{eqnarray} \tag{ 9 }$$

where ${g}_{0}=\omega \mu {I}_{0}l/4\pi r$, noting that the incoming E-field components are time independent. The net induced polarization, which is estimated by the vector sum of all Stokes vectors from all direction above the horizon as Equation (6), subsequently consists of this twice-diurnal component as a function of time. This is the dynamic feature that we exploit to measure the foreground separately from the static background.

It is trivial that the net polarization can become relatively small, if not zero, when the overall foreground anisotropy decreases. For example, if the incoming E-field components are identical for all directions, the vector sum of the Stokes vectors reduces to zero. This is similar to the symmetry argument for the isotropic background signal. As a result, the magnitude of twice diurnal component can be altered. However, this is very unlikely in a real observation except for some extreme cases, such as horizon obstruction at low latitude discussed in Section 5.6. A more comprehensive analysis is elaborated through simulations with a realistic foreground map in Section 4 and discussions in Section 5.

For a continuous signal g(t), its PSD can be estimated by the Fourier decomposition in the frequency domain. For discrete signal data with length of N and sampling interval ${\rm{\Delta }}t$, one of the optimized PSD estimates (Heinzel et al. 2002) is

$$\begin{eqnarray}&&{S}_{g}^{\nu }(f)=\displaystyle \frac{{({\rm{\Delta }}t)}^{2}}{{s}_{1}^{2}}{\left|\displaystyle \sum _{t=1}^{N}w(t)g(t){e}^{-j2\pi {ft}}\right|}^{2},\end{eqnarray} \tag{ 10 }$$

where the window function w(t) prevents spectral leakage between frequency channels in the PSD and the normalization factor ${s}_{1}={\sum }_{t=0}^{N}w(t)$. In this paper, we adopt the commonly used Blackman-Harris window function for the PSD calculations. At each observed frequency, a Stokes PSD can be estimated by replacing g(t) with each of the Stokes parameters. Here we explicitly denote the dynamic frequency as f to distinguish it from the observed frequency ν. In the context of the sky rotation, $f={\omega }_{\mathrm{sky}}/2\pi$.

A harmonic of order n will appear in the PSD if g(t) consists of a frequency component at f, where $n=1/({fN}{\rm{\Delta }}t)$. Hence, the available power associated to the components of twice the angular frequency in Stokes Q_net(t) and U_net(t) are estimated as ${S}_{Q}^{\nu }(n=2)$ and ${S}_{U}^{\nu }(n=2)$. Meanwhile, in case of the constant total intensity from ${I}_{\mathrm{net}}^{\nu }(t)$, the power is located at the zeroth harmonic ${S}_{I}^{\nu }(n=0)$.

Since the ν-dependence of the Stokes parameters is not affected by the Fourier transformation, the spectral dependence of the sky measurement is carried over to the Stokes PSD. The measured ${S}_{Q}^{\nu }(n=2)$ and ${S}_{U}^{\nu }(n=2)$ originated from the dynamic component of the foreground polarization are model-independent, high-fidelity, and scalable replicas of the foreground spectrum, which do not contain any contribution from the isotropic 21 cm background. The static background combined with the foreground is measured at the zero-frequency component of the total intensity spectrum ${S}_{I}^{\nu }(n=0)$.

The sky-averaged antenna temperature ${T}_{\mathrm{ant}}(\nu )$ for a given antenna radiation pattern $F(\theta ,\phi ,\nu )$ is defined as

$$\begin{eqnarray}&&{T}_{\mathrm{ant}}(\nu )=\displaystyle \frac{{\int }_{0}^{2\pi }{\int }_{0}^{\pi /2}{T}_{\mathrm{sky}}(\nu ,\theta ,\phi )F(\theta ,\phi ,\nu )\sin \theta d\theta d\phi }{{\int }_{0}^{2\pi }{\int }_{0}^{\pi /2}F(\theta ,\phi ,\nu )\sin \theta d\theta d\phi },\end{eqnarray} \tag{ 11 }$$

where ${T}_{\mathrm{sky}}(\nu ,\theta ,\phi )$ is the brightness temperature distribution of the total sky (Kraus 1986; Wilson et al. 2009). In an ideal case, the sky temperature is simply a linear combination of the foreground emission ${T}_{{fg}}(\nu )$ and the background 21 cm signal $\delta {T}_{b,21\mathrm{cm}}(\nu )$. The total-power spectrum at zero-frequency ${S}_{I}^{\nu }(n=0)$ is equivalent to ${T}_{\mathrm{ant}}(\nu )$ and can be parameterized as

$$\begin{eqnarray}&&{S}_{I,0}^{\nu }\equiv {S}_{I}^{\nu }(n=0)={A}_{1}{k}_{{\rm{B}}}[{T}_{{fg}}(\nu )+\delta {T}_{b,21\mathrm{cm}}(\nu )]+{A}_{0}.\end{eqnarray} \tag{ 12 }$$

Similarly, the second-harmonic Stokes spectra are

$$\begin{eqnarray}{S}_{Q,2}^{\nu }\equiv {S}_{Q}^{\nu }(n=2) & = & {B}_{1}{k}_{{\rm{B}}}{T}_{{fg}}(\nu )+{B}_{0},\\ {S}_{U,2}^{\nu }\equiv {S}_{U}^{\nu }(n=2) & = & {C}_{1}{k}_{{\rm{B}}}{T}_{{fg}}(\nu )+{C}_{0},\end{eqnarray} \tag{ 13 }$$

where the As, Bs, and Cs are the scaling factors and offsets. In practice, these scaling coefficients are corrupted by instrumental effects and become frequency-dependent. By assuming optimal instrument calibration, the coefficients can be treated as constants for the purpose of this paper.

Since all of the Stokes spectra are measured simultaneously in the same observation, by solving for the scaling factors, the foreground spectrum can be subtracted to extract the background signal, without the need of high-order polynomial fitting using, for instance, Equation (2). Although either equation in Equation (13) can be used to solve for T_fg because both Stokes Q_net(t) and U_net(t) measure the same linear polarization but with a phase difference of 90°, using both measurements at the same time can improve the precision of the n = 2 harmonic since we know both Stokes parameters share the same angular frequency component and a unique relative phase difference of 90° apart.

For simplicity and without loss of generality, by substituting the Q-spectrum into Equation (12) and solving for the background signal we have

$$\begin{eqnarray}&&\delta {T}_{b,21\mathrm{cm}}(\nu )=\displaystyle \frac{1}{{k}_{{\rm{B}}}}\left(\displaystyle \frac{{S}_{I,0}^{\nu }-{A}_{0}}{{A}_{1}}-\displaystyle \frac{{S}_{Q,2}^{\nu }-{B}_{0}}{{B}_{1}}\right),\end{eqnarray} \tag{ 14 }$$

though utilizing both Stokes measurement, along with Stokes V measurement, can be useful in practice. We can obtain a relation between A₁ and B₁ by taking the derivative on both sides of Equation (14) as

$$\begin{eqnarray}&&\displaystyle \frac{{A}_{1}}{{B}_{1}}=\left[\displaystyle \frac{{{dS}}_{I,0}^{\nu }}{d\nu }-{A}_{1}{k}_{{\rm{B}}}\displaystyle \frac{d\delta {T}_{b,21\mathrm{cm}}(\nu )}{d\nu }\right]{\left(\displaystyle \frac{{{dS}}_{Q,2}^{\nu }}{d\nu }\right)}^{-1}.\end{eqnarray} \tag{ 15 }$$

If the global 21 cm spectrum consists of an absorption feature at ${\nu }_{\min }$ as predicted by the models such as ones shown in Figure 1, there exists a global minimum at ${\nu }_{\min }$ that can be determined by its first and second derivatives as shown in Figure 3. Hence, we can solve for ${A}_{1}/{B}_{1}$ based entirely on the derivatives of the measured Stokes spectra since the derivative of $\delta {T}_{b,21\mathrm{cm}}(\nu )$ is zero if we evaluate Equation (15) at ${\nu }_{\min }$

$$\begin{eqnarray}&&\displaystyle \frac{{A}_{1}}{{B}_{1}}={\displaystyle \frac{{{dS}}_{I,0}^{\nu }}{d\nu }{\left(\displaystyle \frac{{{dS}}_{Q,2}^{\nu }}{d\nu }\right)}^{-1}|}_{\nu ={\nu }_{\min }}.\end{eqnarray} \tag{ 16 }$$

In principle, the first derivatives of the emission features at the higher end of the passband in Figure 1 also provide a zero crossing to constrain ${A}_{1}/{B}_{1}$. However, the emission feature is more likely to be outside the band since it is predicted to be broader and less prominent than the absorption one.

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By comparing (15) and (16), it is apparent that the sources of error in solving for ${A}_{1}/{B}_{1}$ are the accuracy in estimating the derivatives of the measured Stokes spectra as well as determining ${\nu }_{\min }.$ In general, there exists truncation error in numerical differentiation in the order of the frequency step size ${ \mathcal O }({\rm{\Delta }}\nu )$ (Hamming 2012). Hence, keeping the spectral resolution low can improve the accuracy of calculating the derivatives. However, determining the frequency of the absorption feature in the global 21 cm signal is not straightforward since the background signal is unknown. Possible ways helping to constrain ${\nu }_{\min }$ and the sensitivity of this scheme are further discussed in Sections 5.2 and 5.3. It is worth noting that the foreground removal discussed here only applies when the an absorption feature exists in the observing band as predicted by theory ($\sim 40\lt \nu \lt 200$ MHz).

In this section, we illustrate how the projection-induced polarization from a realistic sky map can give rise to the second-harmonic spectrum and how a model global 21 cm spectrum can be recovered using the procedure presented in the previous section.

For simplicity, the broad far-field beam patterns of the crossed dipoles are approximated by circular Gaussian beams. Referencing the typical antenna beam sizes from other global 21 cm experiments like EDGES (Mozdzen et al. 2016), we adopt a full width at half maximum of 60° for the Gaussian beams.

The Gaussian beams are centered at the NCP of a set of foreground maps, which are extrapolated from the Haslam full-sky survey at 408 MHz, as shown in Figure 4 (Haslam et al. 1982), to 40–120 MHz using a power-law parameterization with a mean spectral index $\beta =2.47$ as suggested by previous observations (Bowman et al. 2008),

$$\begin{eqnarray}&&{T}_{{fg}}(\theta ,\phi ,\nu )={T}_{\mathrm{Haslam}}(\theta ,\phi ){\left(\displaystyle \frac{\nu }{408\mathrm{MHz}}\right)}^{-\beta }.\end{eqnarray} \tag{ 17 }$$

For each of the ARES global 21 cm models, at each frequency, an isotropic 21 cm background map of brightness temperature $\delta {T}_{b,21\mathrm{cm}}(\nu )$ for all directions is linearly combined with the foreground map ${T}_{{fg}}(\nu ,\theta ,\phi )$. Since the spatial resolution is not critical in the global 21 cm measurement, it is sufficient to set the spatial resolution of both the beam patterns and the sky map to $1^\circ \times 1^\circ$ in our simulations. We also assume the Gaussian beams to be frequency independent across the passband, which is not necessarily true for the real antenna beams.

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Although intrinsic linear polarization is known to exist in the Galactic synchrotron foreground, we assume the foreground itself to be unpolarized (or randomly polarized) in the simulations since the sky-averaged intrinsic polarization is expected to be less dominant than net projection-induced polarization. For completeness, both the chromatic beam effects and intrinsic polarization are elaborated in Section 5.

For an unpolarized foreground, half of the averaged total power is equally received by each of the antenna polarization. According to the formalism presented in Section 3.1, we estimate the incoming E-field from the foreground as ${{\boldsymbol{E}}}_{\mathrm{in}}(\theta ,\phi ,\nu )=\{{E}_{0}(\theta ,\phi ,\nu )\hat{{\boldsymbol{\theta }}}+{E}_{0}(\theta ,\phi ,\nu )\hat{{\boldsymbol{\phi }}}\}$ where ${E}_{0}(\theta ,\phi ,\nu )=\sqrt{{k}_{{\rm{B}}}{T}_{{fg}}(\theta ,\phi ,\nu )/2}$ with the Boltzmann constant k_B.

After the observed E-field is calculated as in Equation (3), the net Stokes parameters are computed as the foreground revolves about the NCP at a rate of one cycle per 24 sidereal hours. The resulting Stokes parameters are Fourier transformed to compute the PSD. Zeroth- and second-harmonic spectra, ${S}_{I,0}^{\nu }$ and ${S}_{Q,2}^{\nu }$, are constructed by assembling the magnitude of the FFT output at n = 0 and n = 2 for all frequencies. Figure 5 summarizes the basic steps to produce the Stokes spectra.

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For a circular beam, the same sky region is observed continuously over time as the foreground revolves about the NCP, hence the ${I}_{\mathrm{net}}^{\nu }(t)$ is constant at each frequency as shown in Figure 6. This translates to a zero-frequency component in its periodogram as shown in Figure 7. On the contrary, the linear net polarization represented by the Stokes ${Q}_{\mathrm{net}}^{\nu }(t)$ is a sinusoidal function with angular frequency that equals twice the sky's revolution rate, or two cycles per 24 sidereal hours, as illustrated in Figure 8. Subsequently, its periodogram consists of only a harmonic at n = 2 in Figure 9. This second harmonic only arises from the anisotropy in the foreground map. Hence, by constructing a second-harmonic spectrum ${S}_{Q,2}^{\nu }$ in a manner similar to ${S}_{I,0}^{\nu }$, we obtain a replica of the foreground spectrum, free from any isotropic background signal. This is confirmed by calculating the spectral index β recovered from the ${S}_{Q,2}^{\nu }$, which is found to be identical to the input β of 2.47 in Figure 10.

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In this ideal scenario, where instrumental systematics and other contaminations can be removed, the total-power spectrum is identical to ${S}_{I,0}^{\nu }$, so A₁ is unity with zero offset A₀. Since ${S}_{Q,2}^{\nu }$ is a scalable replica of the the foreground spectrum, only the scaling coefficient B₁ is needed and no offset value B₀ is expected. After scaling the ${S}_{Q,2}^{\nu }$ up to the ${S}_{I,0}^{\nu }$ with B₁ as in Equation (16), the foreground spectrum is subtracted to reveal the underlying global 21 cm model with Equation (14), as shown in Figures 10 and 11.

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In this simulation, we assume that the Stokes measurements have been integrated long enough to achieve a minimal measurement precision and the only remaining error in the extracted 21 cm model is the uncertainty ${\sigma }_{{AB}}$ from determining the scaling factor ${A}_{1}/{B}_{1}$, or $1/{B}_{1}$ in this case. Details on error propagation are elaborated on in following section.

As mentioned in Section 3.4, the dominant sources of error originate from estimating the derivatives of the Stokes spectra as well as determining the ${\nu }_{\min }$ at which the first derivative of absorption feature in the background signal vanishes. Because both errors are functions of the frequency resolution, the confidence levels on the extracted Model 1 and Model 2 as shown above are attributed to the resolution chosen for this simulation, i.e., ${\rm{\Delta }}\nu =1\,\mathrm{MHz}$.

Assuming the absorption feature exists at ${\nu }_{\min }$ within the instrument's passband, but the estimated ${\nu }_{\min }^{\prime }={\nu }_{\min }\pm {\rm{\Delta }}\nu$ with error ${\rm{\Delta }}\nu$ is used for Equation (15), then a non-zero $d\delta {T}_{b,21\mathrm{cm}}/d\nu$ is unknowingly assumed to be zero when computing ${A}_{1}/{B}_{1}$ with Equation (16). In fact, when evaluating Equation (15) at ${\nu }_{\min }^{\prime }\ne {\nu }_{\min },$ we obtain a non-zero value on the right-hand side (rhs) in the following equation:

$$\begin{eqnarray}&&\displaystyle \frac{{A}_{1}}{{B}_{1}}-\displaystyle \frac{{{dS}}_{I,0}^{\nu }}{d\nu }{\left(\displaystyle \frac{{{dS}}_{Q,2}^{\nu }}{d\nu }\right)}^{-1}={A}_{1}{k}_{{\rm{B}}}\displaystyle \frac{d\delta {T}_{b,21\mathrm{cm}}(\nu )}{d\nu }{\left(\displaystyle \frac{{{dS}}_{Q,2}^{\nu }}{d\nu }\right)}^{-1}.\end{eqnarray} \tag{ 18 }$$

Since the frequency resolution of the simulations is set to 1 MHz, the first derivative of $\delta {T}_{b,21\mathrm{cm}}$ does not equal to zero exactly at the estimated ${\nu }_{\min }^{\prime }$. So we estimate the ${\sigma }_{{AB}}$ to be the unknown difference between the values of ${A}_{1}/{B}_{1}$ at the correct ${\nu }_{\min ,1}$ and at the ${\nu }_{\min }^{\prime }$. This error is essentially the term on the rhs but we are not using the derivative of our input Model 1 directly.

By applying a basic error propagation to Equation (14), the uncertainty of the extracted 21 cm signal ${\sigma }_{21\mathrm{cm}}$ can be estimated as

$$\begin{eqnarray}{\sigma }_{21\mathrm{cm}}(\nu ) & = & \displaystyle \frac{1}{{k}_{{\rm{B}}}{A}_{1}}\left[{\sigma }_{I,0}^{2}(\nu )+{\left(\displaystyle \frac{{A}_{1}}{{B}_{1}}\right)}^{2}{\sigma }_{Q,2}^{2}(\nu )\right.\\ & & +{{({S}_{Q,2}^{\nu }-{B}_{0})}^{2}{\sigma }_{{AB}}^{2}]}^{1/2},\end{eqnarray} \tag{ 19 }$$

where ${\sigma }_{I,0}$ and ${\sigma }_{Q,2}$ are the measurement uncertainties of the Stokes spectra ${S}_{I,0}^{\nu }$ and ${S}_{Q,2}^{\nu }$, defined by the radiometer equation in Equation (21). In the simulations above, we assume the measurement uncertainty of the Stokes spectra have been reduced to a minimal level after long-term integration, so the overall uncertainty of the extracted signal which defines the confidence levels reduces to

$$\begin{eqnarray}&&{\sigma }_{21\mathrm{cm}}(\nu )=({S}_{Q,2}^{\nu }-{B}_{0}){\sigma }_{{AB}}={S}_{Q,2}^{\nu }{\sigma }_{{AB}}.\end{eqnarray} \tag{ 20 }$$

Except for the derivatives of the Stokes spectra on the left-hand side of Equation (18), the scaling factor and the background signal's first derivative are unknowns. As an observer, we are simply searching for a frequency of the absorption feature in the background so that the second terms on the rhs vanishes even without knowing where it is. We here propose a blind search to determine an optimal ${\nu }_{\min }$ during the foreground subtraction. After an initial foreground subtraction by choosing a random frequency as ${\nu }_{\min }$ in the passband, the process of scaling and subtracting ${S}_{Q,2}^{\nu }$ from ${S}_{I,0}^{\nu }$ can be refined iteratively (ideally from one end of the passband to the other) until potential spectral structures may appear in the residual spectrum, as illustrated in Figure 13.

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Without the background signal and its absorption feature as a priori information, the iterative scaling process only converges when it locates a zero crossing in the global 21 cm spectrum's derivative. By assuming that each of the chosen ${\nu }_{\min }$ can potentially be the desired value, we can simply calculate ${A}_{1}/{B}_{1}$ by ignoring the extra term on the rhs of Equation (18). Figure 14 shows how ${A}_{1}/{B}_{1}$ varies as a function of ${\nu }_{\min }$. By the formulation of Equation (18), it is not surprising that the curve is proportional to the negative value of Model 1's first derivative in Figure 3. More importantly, since local and global minima can be distinguished by the magnitude of the second derivative, we can determine the desired ${\nu }_{\min ,1}$ simply by taking the first derivative of ${A}_{1}/{B}_{1}$ respect to ${\nu }_{\min }$, as shown in Figure 14, without any knowledge of the background signal.

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It is worth noting that the approach presented here only applies to an idealized instrument with optimized calibration, which is free from any spurious contamination and unwanted distortion. Presence of realistic instrumental systematics and other measurement uncertainties can complicate this foreground subtraction procedure. More sophisticated estimation algorithms, which utilize theoretical constraints from the 21 cm physics and instrument models in terms of the Bayesian statistics, such as the Monte Carlo Markov Chain (MCMC, Harker et al. 2012; Mirocha et al. 2015) and SVD, are needed in parallel. This will be investigated in future work.

In the full sky simulations above, the Stokes spectra are assumed to have been integrated long enough to achieve an optimal measurement precision such that only ${\sigma }_{{AB}}$ dominates the overall error in extracting the 21 cm models as shown in Figures 10 and 11. By considering only the contribution of thermal noise after other systematics have been corrected and treating ${\sigma }_{{AB}}$ as a second-order effect, we estimate the required integration time to achieve a ${\sigma }_{I,0}/{k}_{{\rm{B}}}$ of at least 10 part per million (ppm) of the expected sky noise temperature with the radiometer equation.

The generic radiometer equation in Equation (21) suggests the measuring uncertainty (or minimal detectable temperature change) at each frequency channel ${\rm{\Delta }}{T}_{\min }(\nu )$ is dictated by the systematic noise temperature ${T}_{\mathrm{sys}}(\nu )$ and decreases as a function of the overall integration time ${\tau }_{\mathrm{int}}$ and the available bandwidth ${\rm{\Delta }}\nu$ in the absence of any receiver gain variations,

$$\begin{eqnarray}&&{\rm{\Delta }}{T}_{\min }(\nu )=\displaystyle \frac{{T}_{\mathrm{sys}}(\nu )}{\sqrt{{\tau }_{\mathrm{int}}{\rm{\Delta }}\nu }}.\end{eqnarray} \tag{ 21 }$$

From the simulation, at 40 MHz, the total sky temperature is about 9400 K. By assuming a receiver noise temperature ${T}_{{rx}}=300$ K, the total system noise temperature ${T}_{\mathrm{sys}}\sim 9700$ K, the radiometer equation gives rudimentary estimates of integration time required for a 10 ppm uncertainty over different bandwidth values. As shown in Table 1, a reasonably short integration time is needed for ${\rm{\Delta }}\nu \gt 100\,\mathrm{kHz}$ to achieve the required sensitivity under the given assumptions.

Table 1.  Integration Time Estimates for Different Spectral Resolution

${\rm{\Delta }}\nu$ $(\mathrm{Hz})$	${\tau }_{\mathrm{int}}\ (\mathrm{Hr})$
$1.0\times {10}^{4}$	$2.46\times {10}^{4}$
$5.0\times {10}^{4}$	$4.92\times {10}^{3}$
$1.0\times {10}^{5}$	$2.46\times {10}^{3}$
$5.0\times {10}^{5}$	$4.92\times {10}^{2}$
$1.0\times {10}^{6}$	$2.46\times {10}^{2}$

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In global 21 cm experiments, it is desirable to have a broadband antenna with relatively low variations in the beam patterns as a function of frequency. Since the intrinsic physical size and surrounding environment of a resonant antenna structure determine the available bandwidth and its efficiency, it is unavoidable to have an antenna radiation pattern $F(\theta ,\phi ,\nu )$ varies as a function of frequencies. The beam patterns are related to the antenna directive gain, which determines the antenna's effective FOV. Hence, the effective collecting area is altered by the frequency-dependent antenna gain, which perturbs the total observable power. This chromatic dependence of the antenna beams corrupts the spectrally smooth foreground by introducing unwanted spectral structures into the sky measurements (Vedantham et al. 2013; Bernardi et al. 2015; Mozdzen et al. 2016).

The projection-induced polarization relies on the overall foreground anisotropy to a produce a net Stokes vector, but the level of anisotropy depends on the beams' effective FOV. Hence the variations in the antenna gain with frequency will perturb the measured net Stokes vector. The spectral shape of the constructed Stokes spectra, like ${S}_{I,0}^{\nu }$ and ${S}_{Q,2}^{\nu }$, are also therefore distorted. Depending on the magnitude of these gain variations, the resulting unwanted spectral structures can be as large as, if not greater than, the predicted global 21 cm signal.

In addition to the perturbed FOV due to the antenna gain variations, geometry of the beam patterns can also affect the spectral shape of the Stokes spectra. In practice, it is difficult to design an antenna with a circular beam across the entire passband. Due to the intrinsic characteristics of a dipole resonator, a certain level of beam elongation (or ellipticity) is often present in the beam patterns.

If the crossed dipoles have circular beams, they cover the same sky region the entire time so that the cross-correlation between the X and Y oriented antenna are identical. However, as illustrated in Figure 15, when the dipoles have two orthogonal elongated beams, only the overlapping region of the two beams covers the same sky at the entire time. This overlapping region is maximized when the two beams are aligned, in the case of circular ones. The sky portion outside of this overlapping region is only sampled whenever it is aligned with the dipoles. This contributes the fourth-order harmonic component (n = 4) to the Stokes parameters I and I_p. As the beam elongates differently from frequency to frequency, this introduces spectral variations to the Stokes measurements and thus to the foreground spectrum.

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These chromatic distortions can compromise the detection sensitivity unless the antenna beams are perfectly known. One of the probable ways to mitigate this effect is to reference the measurements to numerical antenna beam models. Careful incorporation of the antenna structure design and environment allows sophisticated electromagnetic simulation software, such as the Computer Simulation Technology (CST) or the High Frequency Structural Simulator (HFSS), to produce high-fidelity antenna beam models. By bootstrapping these beam models to the measurements, the spectral-dependent beam distortions can be mitigated, if not removed. Rudimentary analysis suggests that as long as the antenna beam pattern can be constrained with accuracy down to a few tens of ppm, the chromatic effects on the sky measurements can potentially be corrected to a desired level. In principle, the fourth-order harmonic is tracking the beam elongation and potentially can be used as feedback information to the beam modeling and calibration. However, the full potential of such beam correction using numerical beam model will be explored in future work.

Our simulations assume a pair of crossed-dipole antennas located at the GNP (at latitude ${\phi }_{\oplus }=90^\circ$) to have a full FOV the northern sky centered at the NCP. For practical and logistical reasons, such an instrument can only be deployed at locations of latitude between $0^\circ \lt {\phi }_{\oplus }\lt 90^\circ$, where the Equator is at ${\phi }_{\oplus }=0^\circ$. The antenna will be pointing toward the NCP at a tilt angle δ, which is a function of the observer's latitude.

From simple image theory formulation, it is shown that the far-field antenna beams for a horizontal dipole above a finite ground plane at height h are smooth pattern resembling the Gaussian beams (Balanis 2016). However, as the antenna and its ground plane are being tilted to point at the celestial pole, instead of a single image, multiple images of the dipole are reflected across the ground screen as well as the actual ground as illustrated in Figure 16. The different images from different propagation distances result in phase delays such that interferometric fringes are introduced into the beams. These ground effects can be mitigated by situating the antenna system on a slope instead of on the flat ground as in Figure 17. The slope angle needs to be consistent with the tilt angle δ. Given the proposed configuration, any potential ground effects on the antenna beam will be present in the measured Stokes parameters as constant with time. It is worth noting that such beam distortion due to the ground can very likely impose an n = 1 harmonic, if not at higher orders ($n\gt 4$), to the Stokes measurement instead of at n = 2 due to the relative geometry of the antenna and the sky.

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As the latitude decreases toward the Equator, the FOV is partially obstructed by the horizon, thus the visible sky can be separated into two parts. Only the inner region of the sky about the boresight, as shown within the dashed circle in Figure 18, is observable the entire time, whereas the sky region outside the dashed circle rises and sets once per revolution.

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As noted in previous sections, the magnitude of the second harmonic in projection-induced polarization depends on the overall anisotropy of the sky region that can be observed the entire time continuously. Hence, with the presence of the horizon, the sky region that contributes to the second harmonic in ${S}_{Q,2}^{\nu }$ and ${S}_{U,2}^{\nu }$ is this inner sky region within the dashed circle. As a result, smaller radii of the inner sky region at lower latitudes imply lower accuracy of the second harmonics relative to overall system noise. Additionally, as the outer sky region rises and sets, it contributes additional terms to the projection-induced polarization with an angular frequency of at least once per revolution to produce an n = 1 harmonic. Meanwhile, the sharp cutoff of the horizon will also introduce high-frequency components to the periodic waveforms in the Stokes parameters, which can be identified at $n\gt 4$ in the PSD. Similar to other effects mentioned above, although these additional harmonics can take over the power from the second harmonic and lower its sensitivity, their effects are attenuated because the antenna gain decreases at larger angles away from the boresight for the outer sky region.

By incorporating the observer's latitude, effects of the horizon obstruction on the Stokes measurement can be characterized and corrected if needed. A balance is also needed between site selection, RFI prevention, SNR of the projection-induced Stokes parameters, as well as minimal ionospheric disturbance. For example, one can select a radio-quiet zone with a latitude range that can accommodate a minimal horizon obstruction, yet is not close to the Earth's magnetic poles, at which the strong interaction between the ionosphere and cosmic rays is well known (Newell et al. 2001). An alternative to avoid the Earth's horizon effect is to adapt this technique into a space-based instrument, such as what is being proposed for DARE.

By design, the accuracy of measuring the second-harmonic component in the projection-induced polarization relies on the antenna pointing. If the antenna boresight is not aligned with the celestial poles, the foreground region centered at the celestial pole is not concentric with the antenna beams, as illustrated in Figure 19. As the sky revolves around, the foreground appears to "wobble" about the off-centered pointing.

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Similar to the horizon obstruction, the effective sky region that produces the n = 2 component (within the dot-dashed circle) is reduced, which decreases the power of the second-harmonic spectrum thus limiting the accuracy of the Stokes measurements. As the off-centered sky revolves, amplitudes and angular frequencies of the Stokes parameters are modulated such that additional harmonics are introduced in their PDS, for example an additional n = 4 component can be introduced to the measured intensity I and I_p. These modulations depend on the frequency-dependent beam sizes, hence they will also cause unwanted spectral structures to appear in the measured sky spectrum because the modulations correlate to the frequency-dependent beam sizes. Being said, these effects can easily be mitigated by improving the pointing accuracy. Based on our simulations, we find that the error is acceptable for a pointing error of 1° or less relative to the celestial pole based.

Synchrotron emission Galactic foreground is well known to have a linearly polarized component. As an example, Figure 20 illustrates the spatial distribution of the intrinsic foreground polarized brightness temperature observed from the DRAO 1.4 GHz sky survey (Wolleben et al. 2006).

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As the polarized incoming signal propagates through the magnetized interstellar medium (ISM), the polarization angle of the signal rotates, known as Faraday rotation. In Gaussian cgs units, the change in propagation phase angle is given as (Wilson et al. 2009)

$$\begin{eqnarray}&&{\rm{\Delta }}\psi ={\lambda }^{2}\mathrm{RM}=\displaystyle \frac{{e}^{2}}{2\pi {m}_{e}^{2}{c}^{2}}\displaystyle \frac{1}{{\nu }^{2}}{\int }_{0}^{L}{B}_{\parallel }(l){n}_{e}(l){dl},\end{eqnarray} \tag{ 22 }$$

where the rotation measure $\mathrm{RM}$ is defined to be the integration of the electron number density n_e(l) with the ISM magnetic field component that is parallel to the direction of propagation ${B}_{\parallel }(l)$ at each distance increment ${dl}$ along the path of the medium with thickness L. The other physical constants are the charge of an electron e, electron mass m_e, and the speed of light in a vacuum c.

Due to irregular distribution of electron density in the ISM as well as the Galactic magnetic field, the frequency-dependent Faraday rotation of the incoming polarized foreground emission will be different across the band. Hence, unwanted spectral structures can also be introduced to the measured spectrum. In fact, the intrinsic foreground polarization has been identified as the main cause of the polarization leakage, which complicates the instrument calibration in the interferometric EoR experiments (Jelić et al. 2014, 2015; Asad et al. 2015).

A single polarization dipole system in conventional global experiment may be insensitive to linearly polarized portion of the foreground emission if the antenna is not aligned with the orientation of the foreground polarization. On the other hand, a full-Stokes measurement as proposed in this study can measure the intrinsically polarized portion of the foreground and improves the accuracy of the foreground spectrum measurement. Since the intrinsic polarization follows the apparent foreground rotation relative to the antenna on the ground, its contribution to the net projection-induced polarization can be represented as a vector summation between the projection Stokes vector ${{\boldsymbol{S}}}_{\mathrm{net}}$ and the intrinsic Stokes vector ${{\boldsymbol{S}}}_{\mathrm{intr}}$. As a result, the total measured polarization ${{\boldsymbol{S}}}_{\mathrm{tot}}$ also contains a component of twice the sky revolution rate. A total-power spectrum and a second-harmonic spectrum can be reconstructed as before. Assuming only the polarizations are linear, Figure 21 illustrates that the vector summation between Stokes vectors can be represented by a 2D version of the Poincaré sphere with the Q and U axes.

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The foreground continuum emission has been observed to have a direction-dependent spectral index β depending on the pointing, i.e., $\beta =\beta (\theta ,\phi )$. The spectral index distribution can be estimated by extrapolating from a power law between the Haslam all-sky survey at 408 MHz and a 45 MHz map (Guzmán et al. 2011), as shown in Figure 22. This may potentially complicate the foreground measurement using the second-harmonic Stokes spectrum. By writing Equation (23) with a non-constant β as

$$\begin{eqnarray}&&{T}_{{fg}}(\theta ,\phi ,\nu )={T}_{\mathrm{Haslam}}(\theta ,\phi ){\left(\displaystyle \frac{\nu }{408\mathrm{MHz}}\right)}^{-\beta (\theta ,\phi )},\end{eqnarray} \tag{ 23 }$$

it is apparent that the chromatic antenna beam pattern can couple to the sky differently at different frequencies. As a result, the net projection-induced polarization can vary and contribute some unwanted spectral feature to the Stokes spectra, especially the second-harmonic one, if the antenna beam is not corrected.

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However, since the induced-polarization approach measures both the total-power spectrum and the second-harmonic spectrum simultaneously from the same sky region, as the simulations suggested, the measured mean spectral index of ${S}_{Q,2}^{\nu }$ is still identical to the foreground spectral index in ${S}_{I,0}^{\nu }$ for any given mean value of β averaged over the sky. In fact, one of the biggest challenges for the global signal experiment is to ensure any such spectral structures detected in the residual are indeed from the background signal. This degeneracy can be broken by carrying out multiple observation at different sky regions. For example, on the ground, if the residual spectral features measured at the NCP and SCP are consistent, it is very likely they originated from the same source, possibly from the background itself.

One of the major sources of contamination is RFI. Traditionally, radio-quiet sites, such as South Africa's Karoo desert and Murchison Radio-astronomy Observatory (MRO) in Western Australia, are chosen for minimal contamination. Recently, DARE proposed to carry out observations on the lunar farside when the spacecraft orbits around the Moon (Burns et al. 2012). Nonetheless, regardless how quiet a site is, there is always low-level RFI that is either reflected or scattered from satellites, space debris, and the ionosphere (Offringa et al. 2013).

A common practice is to utilize RFI excision algorithms, such as one based on kurtosis of the incoming signal's statistics (Antoni 2004; De Roo 2009; Nita & Gary 2010) to remove the corrupted signal channels. This approach is more effective at detecting strong instead of low-level RFI, especially ones that are equivalent to the system's noise level. Hence, unknown spectral structures can also be introduced to the measured spectrum. However, since most of the incoming RFI are direction-dependent, they do not necessarily introduce any dynamic variations to the measured linear Stokes parameters, unless they are affected by the ionosphere during propagation. More importantly, the RFI does not contain an n = 2 component, hence it does not have a major effect on the second-harmonic spectrum used to constrain the foreground spectrum.

The ionosphere has been well known for causing refraction, absorption, and self-emission to radio propagation. It was not until recently that careful studies have started investigating the effects of ionospheric distortion to the ground-based 21 cm measurements (Vedantham et al. 2013; Rogers et al. 2015; Sokolowski et al. 2015b; Datta et al. 2016). The ionosphere consists of plasma layers with complex structures whose characteristics are position and time dependent. The total electron content, which represents the total electron column density for a given direction, varies throughout the day. These variations are pseudo-periodic from one day to another, but are not repeatable and are difficult to quantify. Current total-power global 21 cm experiments (Rogers et al. 2015; Sokolowski et al. 2015b) have only provided a limited understanding on how these ionospheric effects can impact the accuracy and precision of the global 21 cm signal. Attempts to utilize Global Positioning System ionospheric measurements are considered insufficient at calibrating the ionospheric effect (Datta et al. 2016). However, by measuring the dynamic ionospheric variations, our polarimetric approach may provide some constraints on their influence on the overall sky measurement as a function of time and observed frequency. A more thorough study in this area will be conducted in future work.

We have shown that the harmonic decomposition of the dynamic Stokes parameters provides additional information on different types of measurement and systematic uncertainties. In most cases, the experiment can be designed and configured to modulate the foreground measurement in attempt to distinguish these perturbations from the signal of interest. Based on our simulations and analysis, Table 2 provides a rudimentary list of harmonic components that can potentially be used to identify different signal sources and bound the uncertainty in constraining the foreground spectrum. Harmonics at higher orders ($n\gt 4$) are indicated as "H" for reference.

Table 2.  Summary of Harmonic Analysis for the Projection-induced Stokes Spectra

Signal Sources	Harmonic Order n					Notes
Stokes Spectrum:	${S}_{I}^{\nu }$	${S}_{{I}_{p}}^{\nu }$	${S}_{Q}^{\nu }$	${S}_{U}^{\nu }$	${S}_{V}^{\nu }$
21 cm background	0	...	...	...	...	for circular Gaussian beams
Projection-induced polarization	...	...	2	2	...	for circular Gaussian beams
Beam gain distortion and elongation	0, 4	4	H	H	H	need correction with beam models
Horizon obstruction	0, 1	1	1	1	1	latitude and sky dependent
Boresight pointing error	4	4	Ha	H	H	⋯
RFI	0	H	H	H	H	location and time dependent
Ionospheric distortion	0	H	H	H	H	time and latitude dependent, pseudo-periodic

Note.

^aH—possible high-order harmonics, i.e., $n\gt 4$.

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In analogy to the CMB anisotropy studies (i.e., Tegmark & Efstathiou 1996; Ichiki 2014), the foreground spatial anisotropy and hence its induced polarization can also be decomposed into spherical harmonic ${Y}_{l}^{m}(\theta ,\phi ,\nu )$ with different modes $l=0,1,2,\ldots$ and $m=0,\ldots ,l$. We speculate that such decomposition is complimentary to the harmonic analysis. In fact, by combining the two, it may be possible to characterize different contribution and systematics in the measurement by mapping out different modes and harmonics, similar to Table 2. This is outside the scope of the current study.