CALIBRATION OF THE EDGES HIGH-BAND RECEIVER TO OBSERVE THE GLOBAL 21 cm SIGNATURE FROM THE EPOCH OF REIONIZATION

Figure 2 presents a conceptual block diagram of the EDGES instrument. The receiver that operates in the field corresponds to the actual unit calibrated in the lab and it is part of an identical setup, except for the antenna and ground plane which only exist in the field.

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Physically, the receiver consists of a metal enclosure that houses the first-stage low-noise amplifier (LNA), noise references for relative calibration, additional stages of amplification, filtering, and conditioning, and electronics for remote measurement of the antenna reflection coefficient. The receiver operates at 25°C at all times in order to keep the noise and reflection characteristics of the LNA stable during calibration and sky observations. Temperature stability to better than $0.1^\circ {\rm{C}}$ is achieved using a thermal controller that reads out a thermistor mounted inside the receiver. Based on the reading, the controller acts on a hot/cold plate attached to the bottom of the receiver to compensate for temperature drifts.

From the output of the receiver the RF signal is taken along a coaxial cable into an equipment rack where it is amplified and filtered by back-end electronics. The signal is then sent to a PC-based 400 MS/s analog-to-digital converter (ADC) that samples with 14 bit resolution. Finally, blocks of 65536 time samples are Fourier transformed to the frequency domain to obtain spectra with 32768 points and 6.1 kHz resolution using a four-term Blackman–Harris window function.

The LNA input switches continuously between the antenna and an internal calibration position using switch SW2 in Figure 2. When in the calibration position, the LNA receives sequentially two noise levels: (1) load, and (2) load + noise source. In the load state, the noise comes from the output of a 30 dB attenuator, and in the load + noise source state the active noise source connected to the input of the attenuator is turned on for additional noise.

The noise from the antenna only represents a few percent of the total power at the receiver output. Most of the output power is due to out-of-band noise injected below 30 MHz as part of the conditioning stage. This injected noise remains constant during the input switching and provides a stable signal level to the back-end electronics and digitizer, improving linearity and dynamic range in the measurements.

Each of the three input levels is measured for 13 s, which corresponds to an antenna duty cycle of 33.3%. A total of 40960 spectra are accumulated for each position. An implicit assumption of this switching scheme is that the amplification chain remains stable within each 39 s cycle.

The circuitry for measurement of antenna reflections, shown in Figure 2, functions as a remote calibration unit for a vector network analyzer (VNA). It is centered around a four-position mechanical RF switch where three of the ports are connected to open, short, and matched calibration standards, and the fourth port serves as a pass-through to the antenna.

The efficiency of the instrument is highest below 195 MHz. Therefore, the calibration described in the rest of this paper was conducted in the range 90–190 MHz.

Here we summarize the calibration strategy used by EDGES, which is based on the method introduced in Rogers & Bowman (2012).

For each three-position cycle of the receiver, we use the power spectral density (PSD) from the antenna (${P}_{{\rm{ant}}}$), load (${P}_{{\rm{L}}}$), and load + noise source (${P}_{{\rm{L+NS}}}$), to compute an initial uncalibrated antenna temperature,

$$\begin{eqnarray}&&{T}_{{\rm{ant}}}^{* }={T}_{{\rm{NS}}}\displaystyle \frac{({P}_{{\rm{ant}}}-{P}_{{\rm{L}}})}{({P}_{{\rm{L+NS}}}-{P}_{{\rm{L}}})}+{T}_{{\rm{L}}},\end{eqnarray} \tag{ 1 }$$

where ${T}_{{\rm{L}}}$ and ${T}_{{\rm{NS}}}$ represent realistic assumptions for the noise temperatures of the load and noise source, respectively. This computation serves to calibrate out the time-dependent system gain, which includes the complex bandpass of the filters, amplifiers, cables, and ADC. All the parameters in Equation (1), and in what follows, are frequency-dependent. We do not explicitly show this dependence for simplicity of notation.

To derive the expression for calibration of ${T}_{{\rm{ant}}}^{* }$ it is necessary to write the PSDs in Equation (1) in terms of the specific instrument response contributions. The PSD for the antenna is given by:

$$\begin{eqnarray}{P}_{{\rm{ant}}} & = & g[{T}_{{\rm{ant}}}(1-| {{\rm{\Gamma }}}_{{\rm{ant}}}{| }^{2}| F{| }^{2}\\ & & +{T}_{{\rm{unc}}}| {{\rm{\Gamma }}}_{{\rm{ant}}}{| }^{2}| F{| }^{2}\\ & & +{T}_{\cos }| {{\rm{\Gamma }}}_{{\rm{ant}}}| | F| \cos \alpha \\ & & +{T}_{\sin }| {{\rm{\Gamma }}}_{{\rm{ant}}}| | F| \sin \alpha \\ & & +{T}_{0}],\end{eqnarray} \tag{ 2 }$$

with

$$\begin{eqnarray}&&F=\displaystyle \frac{\sqrt{1-| {{\rm{\Gamma }}}_{{\rm{rec}}}{| }^{2}}}{1-{{\rm{\Gamma }}}_{{\rm{ant}}}{{\rm{\Gamma }}}_{{\rm{rec}}}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\alpha =\arg ({{\rm{\Gamma }}}_{{\rm{ant}}}F).\end{eqnarray} \tag{ 4 }$$

Here, ${T}_{{\rm{ant}}}$ corresponds to the calibrated antenna temperature. The quantities g and T₀ represent the system gain referenced to the receiver input, and the receiver noise offset, respectively. ${{\rm{\Gamma }}}_{{\rm{ant}}}$ is the reflection coefficient of the antenna and ${{\rm{\Gamma }}}_{{\rm{rec}}}$ is the reflection coefficient of the receiver, both referenced to a 50 Ω system impedance. The temperatures ${T}_{{\rm{unc}}}$, T_cos, and T_sin are called noise wave parameters following the formalism introduced by Meys (1978). They are associated with the noise emitted by the LNA input toward the antenna. This noise is reflected back due to imperfect impedance match and re-enters the receiver with phase α. The ${T}_{{\rm{unc}}}$ temperature represents the portion of input noise that is uncorrelated with the noise at the LNA output, while T_cos and T_sin are components of the correlated portion.

The PSDs for the internal load and load + noise source follow the same form as Equation (2). However, at this stage we assume that the reflection coefficients of the load and noise source are zero. In reality, they are not zero but very low ($\lt -40$ dB). Thus, these PSDs are modeled as:

$$\begin{eqnarray}&&{P}_{{\rm{L}}}={g}^{* }[{T}_{{\rm{L}}}(1-| {{\rm{\Gamma }}}_{{\rm{rec}}}{| }^{2})+{T}_{0}^{* }],\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{P}_{{\rm{L+NS}}}={g}^{* }[({T}_{{\rm{L}}}+{T}_{{\rm{NS}}})(1-| {{\rm{\Gamma }}}_{{\rm{rec}}}{| }^{2})+{T}_{0}^{* }].\end{eqnarray} \tag{ 6 }$$

In these equations, the system gain (${g}^{* }$) and noise offset (${T}_{0}^{* }$) are not exactly the same as in Equation (2) because the noise from the internal references is injected at SW2 instead of at the receiver input itself. This aspect, in addition to the assumption for the reflection coefficients, is accounted for below with the introduction of two new parameters fitted during calibration that are able to absorb these small effects.

Plugging the three PSD definitions into Equation (1) results in the following identity:

$$\begin{eqnarray}({T}_{{\rm{ant}}}^{* }-{T}_{{\rm{L}}}){C}_{1}+({T}_{{\rm{L}}}-{C}_{2}) & = & {T}_{{\rm{ant}}}\left[\displaystyle \frac{(1-| {{\rm{\Gamma }}}_{{\rm{ant}}}{| }^{2}| F{| }^{2}}{(1-| {{\rm{\Gamma }}}_{{\rm{rec}}}{| }^{2})}\right]\\ & & +{T}_{{\rm{unc}}}\left[\displaystyle \frac{| {{\rm{\Gamma }}}_{{\rm{ant}}}{| }^{2}| F{| }^{2}}{(1-| {{\rm{\Gamma }}}_{{\rm{rec}}}{| }^{2})}\right]\\ & & +{T}_{\cos }\left[\displaystyle \frac{| {{\rm{\Gamma }}}_{{\rm{ant}}}| | F| }{(1-| {{\rm{\Gamma }}}_{{\rm{rec}}}{| }^{2})}\cos \alpha \right]\\ & & +{T}_{\sin }\left[\displaystyle \frac{| {{\rm{\Gamma }}}_{{\rm{ant}}}| | F| }{(1-| {{\rm{\Gamma }}}_{{\rm{rec}}}{| }^{2})}\sin \alpha \right].\end{eqnarray} \tag{ 7 }$$

This equation establishes the relationship between the uncalibrated antenna temperature ${T}_{{\rm{ant}}}^{* }$, computed with Equation (1), and the calibrated antenna temperature ${T}_{{\rm{ant}}}$ at the receiver input. More generically, ${T}_{{\rm{ant}}}$ represents the calibrated noise temperature of any device under measurement at the receiver input. Any losses in the device under measurement have to be corrected for externally.

The quantities C₁ and C₂ introduced on the left-hand side of Equation (7) represent a scale and an offset that correct the first-order assumptions used for ${T}_{{\rm{L}}}$ and ${T}_{{\rm{NS}}}$ in Equation (1). They also account for the small path difference between the internal calibration position of SW2 and the receiver input. Finally, they also account to first order for the non-zero reflection coefficient of the internal load and noise source. Unaccounted higher-order effects are expected to be negligible due to the low reflection coefficients of these devices.

The reflection coefficient of the receiver input and the antenna are measured directly with a VNA. F and α (Equations (3) and (4)) are computed from these coefficients. Therefore, the remaining calibration task consists of estimating the scale C₁, offset C₂, and the noise wave parameters ${T}_{{\rm{unc}}}$, T_cos, and T_sin, in order to apply Equation (7). To solve for these five frequency-dependent quantities we conduct laboratory measurements of four absolute calibration standards connected to the input of the receiver in place of the antenna.

Following Rogers & Bowman (2012), the four calibrators are: (1) an ambient load, (2) a hot load, (3) a long ($\approx$ 8 m) open-ended coaxial cable, and (4) the same long cable but short-circuited at its far end. The ambient and hot loads provide the main temperature references, while the open and shorted cable produces ripples in its spectra that manifest the noise properties of the receiver and enable the estimation of the noise wave parameters. The specific quantities that need to be measured for receiver calibration are:

• 1.  
  The uncalibrated temperature spectra for each calibrator (${T}_{A}^{* },{T}_{H}^{* },{T}_{O}^{* },{T}_{S}^{* }$) via Equation (1),
• 2.  
  The physical or noise temperature of the calibrators (${T}_{A},{T}_{H},{T}_{O},{T}_{S}$), and
• 3.  
  The reflection coefficient of the calibrators (${{\rm{\Gamma }}}_{A},{{\rm{\Gamma }}}_{H},{{\rm{\Gamma }}}_{O},{{\rm{\Gamma }}}_{S}$).

The subscripts A, H, O, and S correspond to the ambient load, hot load, open cable, and shorted cable, respectively.

The ambient and hot loads used for receiver calibration are implemented as a single device, depicted in Figure 3. It consists of an RF termination of 50 Ω nominal impedance connected to an 8 cm semi-rigid cable inside a thermally insulated metal enclosure. When acting as an ambient load the device operates at room temperature ($\approx$ 296 K) and when used as a hot load the termination is heated up to $\approx$ 400 K by powering a resistor that is thermally connected to the termination.

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The effective noise temperature of the device when operating as a hot load (T_H) is related to the physical temperature of the termination (T_Ht) and the physical temperature of the cable (${T}_{{\rm{cab}}}$) by

$$\begin{eqnarray}&&{T}_{H}={{GT}}_{{Ht}}+(1-G){T}_{{\rm{cab}}},\end{eqnarray} \tag{ 8 }$$

where G is the available power gain of the assembly, defined as (Pozar 2004):

$$\begin{eqnarray}&&G=\displaystyle \frac{| {S}_{21}{| }^{2}(1-| {{\rm{\Gamma }}}_{{Ht}}{| }^{2})}{| 1-{S}_{11}{{\rm{\Gamma }}}_{{Ht}}{| }^{2}(1-| {{\rm{\Gamma }}}_{H}{| }^{2})}.\end{eqnarray} \tag{ 9 }$$

In this equation, S₁₁ and S₂₁ are two of the S-parameters of the cable, with port 1 attached to the RF termination and port 2 corresponding to the output connector of the device. ${{\rm{\Gamma }}}_{{Ht}}$ is the reflection coefficient of the termination alone and ${{\rm{\Gamma }}}_{H}$ is the reflection coefficient of the device as a whole.

The uncalibrated temperature spectra of the absolute calibrators are measured by connecting each calibrator to the receiver input in place of the antenna. These measurements involve the same internal three-position switching as with the antenna, where in each 39 s cycle the setup measures the PSD of the calibrator and the two internal noise references. After the measurements, Equation (1) is applied offline to produce the uncalibrated temperature spectra. The noise temperatures assumed for the internal load and noise source at this step are ${T}_{{\rm{L}}}=300$ K and ${T}_{{\rm{NS}}}=350$ K, constant across frequency. The final uncalibrated spectra, ${T}_{A}^{* }$, ${T}_{H}^{* }$, ${T}_{O}^{* }$, and ${T}_{S}^{* }$, are obtained by averaging 24 hr of data from each calibrator. This is done in order to reduce thermal noise. Residual noise is the main source of uncertainty in these quantities.

The physical temperatures, T_A, T_Ht, T_O, and T_S, are measured using thermistors attached to the termination of the ambient/hot load and to the long cable. These measurements are conducted in parallel to the spectra measurement of each calibrator and with a similar time resolution. They are averaged in time to have a direct correspondence with the spectra averages. Uncertainties in these measurements are dominated by potential inaccuracies in the resistance-to-temperature model used for the thermistors and unaccounted thermal gradients in the calibrators.

The reflection coefficients and S-parameters required in the calibration are measured with a VNA. In each measurement, hundreds of traces are averaged to reduce the noise to levels so that they are an insignificant source of uncertainty. The calibrators are passive devices and, thus, they are measured using a typical VNA power of 0 dBm (1 mW). In contrast, the receiver input has to be measured at −30 dBm to avoid saturating the active electronics designed for low-level signals. Lower VNA power results in higher measurement noise, but we compensated through averaging. We calibrate the VNA immediately before every measurement and, thus, do not assume long-term VNA stability. The main uncertainty associated with these measurements arises from imperfect VNA calibration.

The isolation between the inputs of each of the mechanical switches (SW1 and SW2 in Figure 2) is about 80 dB. However, we assume that the isolation is perfect, which has a negligible impact on the calibration. The switching repeatability is assessed through high-precision S-parameter measurements and the scatter is found to be within the measurement uncertainties.

We model and fit the uncalibrated temperature spectra, the reflection coefficients, and the S-parameters, to avoid propagating measurement noise to the fiducial calibration quantities and to bring all the measurements to the same frequency resolution. Due to the smooth frequency behavior of the ambient and hot load spectra, and of the S-parameters of the semi-rigid cable, we model these measurements as polynomials in frequency. On the other hand, we use Fourier series in frequency to model the spectra of the open and shorted cable, as well as all the reflection coefficients. Fourier series are more efficient than polynomials at capturing the higher frequency structure encountered in these measurements. The model parameters are computed through least squares using QR decomposition for better numerical stability. The fit residuals for the spectra are noise-like. They are accounted for as a source of uncertainty in the MC analysis of Section 4. For the reflection coefficients and S-parameters, the rms residuals are $\lt 0.001$ dB in magnitude and $\lt 0\buildrel{\circ}\over{.} 008$ in phase. Their effects are negligible compared to that from the uncertainty in VNA calibration, addressed in Section 4.

The frequency-dependent calibration quantities (C₁, C₂, ${T}_{{\rm{unc}}}$, T_cos, and T_sin) are computed from the modeled laboratory measurements through an iterative process. Specifically, in each iteration the scale and offset are computed as

$$\begin{eqnarray}&&{C}_{1}^{i}={C}_{1}^{i-1}\cdot \displaystyle \frac{({T}_{H}-{T}_{A})}{({T}_{H}^{i}-{T}_{A}^{i})},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{C}_{2}^{i}={C}_{2}^{i-1}+{T}_{A}^{i}-{T}_{A}.\end{eqnarray} \tag{ 11 }$$

Here, T_H is the noise temperature of the hot load from Equation (8), T_A is the physical temperature of the ambient load, and ${T}_{A}^{i}$ are ${T}_{H}^{i}$ are the calibrated temperature spectra of the ambient and hot loads from Equation (7) evaluated at the ith iteration. The initial values are ${C}_{1}^{0}=1$ and ${C}_{2}^{0}=0$. The noise wave parameters (${T}_{{\rm{unc}}}$, T_cos, and T_sin) also take an initial value of zero, and in subsequent iterations they are modeled as polynomials in frequency. The polynomial coefficients are computed through a least squares fit to Equation (7) evaluated using the measurements of the open and shorted cable and the current values of C₁ and C₂. This process converges in three iterations, after which C₁ and C₂ are also modeled as polynomials in frequency.

To find the optimum number of terms in the polynomials, the calibration quantities are modeled with increasing number of terms until the rms difference between the physical temperature and the calibrated temperature spectra of the calibrators themselves reaches a minimum. It is found that seven terms are needed to model each of the five calibration quantities.

Figure 4 presents the derived calibration quantities, computed as just described, along with the reflection coefficient of the receiver input.

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The input antenna temperature is modeled by the convolution of a sky model with an antenna beam model:

$$\begin{eqnarray}&&{T}_{{\rm{ant}}}^{{\rm{in}}}=\displaystyle \frac{{\int }_{{\rm{\Omega }}}{T}_{{\rm{sky}}}(\theta ,\phi )B(\theta ,\phi )d{\rm{\Omega }}}{{\int }_{{\rm{\Omega }}}B(\theta ,\phi )d{\rm{\Omega }}},\end{eqnarray} \tag{ 12 }$$

where ${T}_{{\rm{sky}}}$ is the sky model, B is the beam model, θ and ϕ are the zenith and azimuth angles respectively, and Ω represents coordinates above the horizon. Below 200 MHz, the sky brightness temperature is dominated by foregrounds that are more than four orders of magnitude stronger than the expected cosmological signal (Haslam et al. 1982; Rogers & Bowman 2008; Mozdzen et al. 2016b). Since the dominant effects of receiver mis-calibration and uncertainty operate on the total sky signal, it is acceptable to neglect the cosmological 21 cm signal in our sky model and only include the foreground contribution. We model the foregrounds using the GSM described in de Oliveira-Costa et al. (2008).

For the antenna beam model, we use the simple frequency-independent azimuthally symmetric expression used by Pritchard & Loeb (2010),

$$\begin{eqnarray}&&B(\theta ,\phi )={\cos }^{2}\theta .\end{eqnarray} \tag{ 13 }$$

Real beams are known to be more complex than this expression, introducing spectral structure into the antenna temperature that could limit the detection of the cosmological signal (Vedantham et al. 2014; Bernardi et al. 2016; Mozdzen et al. 2016a). However, in these simulations it is assumed that the chromatic effects of the beam are perfectly known and removed.

The sensitivity of the instrument to errors in receiver calibration is a function of the antenna temperature. Ground-based instruments such as EDGES observe the sky, and thus foregrounds, continuously drifting over the antenna instead of conducting deep integrations on a single sky region. Therefore, the strength of the antenna temperature used for science analysis will vary as different parts of the sky drift through the beam. To account for the different foreground levels, we explore two scenarios in the simulations: with the (1) lowest and (2) highest foreground contamination available at the EDGES observation latitude. These cases are labeled quiet and loud sky, respectively, and their convolution with the beam is shown in Figure 6.

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Since we are mainly interested in understanding the effect of errors in receiver calibration, we assume a simple model for the antenna reflection coefficient. Our model for the reflection magnitude is flat in frequency with a level of −15 dB. We model the phase as a linear decrease by 500° between 90 and 190 MHz. This is realistic and corresponds to a delay of $500^\circ /(360^\circ \times 100\ {\rm{MHz}})=13.88$ ns. This model reproduces, to first order, the measured EDGES antenna properties and is generally consistent with the performance achievable from a dipole-based antenna operating over an octave bandwidth.

Here, we describe the uncertainties assigned to each of the calibration measurements from Section 3, as well as the antenna reflection coefficient. These uncertainties are applied inside the MC block of the uncertainty propagation, to create slightly perturbed realizations of the calibration parameters. A summary of the nomenclature and values is presented in Table 1.

Table 1.  Nomenclature and Uncertainties of Receiver Calibration Measurements

Quantity	Device	$1\sigma$ Uncert	Note
Uncalibrated Spectrum
${T}_{A}^{* }$	Ambient Load	66 mK
${T}_{H}^{* }$	Hot Load	66 mK	(i)
${T}_{O}^{* }$	Open Cable	95 mK
${T}_{S}^{* }$	Shorted Cable	95 mK
Physical Temperature
TA	Ambient Load
THt	Hot Load Termination	100 mK	(ii)
TO	Open Cable
TS	Shorted Cable
Magnitude of Reflection Coefficient
$| {{\rm{\Gamma }}}_{A}|$	Ambient Load
$| {{\rm{\Gamma }}}_{H}|$	Hot Load
$| {{\rm{\Gamma }}}_{O}|$	Open Cable	0.0001	(iii)
$| {{\rm{\Gamma }}}_{S}|$	Shorted Cable
$| {{\rm{\Gamma }}}_{{\rm{rec}}}|$	Receiver
$| {{\rm{\Gamma }}}_{{\rm{ant}}}|$	Antennaa
Phase of Reflection Coefficient
$\angle \ {{\rm{\Gamma }}}_{A}$	Ambient Load
$\angle \ {{\rm{\Gamma }}}_{H}$	Hot Load
$\angle \ {{\rm{\Gamma }}}_{O}$	Open Cable	$0\buildrel{\circ}\over{.} 015/| {\rm{\Gamma }}|$	(iv)
$\angle \ {{\rm{\Gamma }}}_{S}$	Shorted Cable
$\angle \ {{\rm{\Gamma }}}_{{\rm{rec}}}$	Receiver
$\angle \ {{\rm{\Gamma }}}_{{\rm{ant}}}$	Antennaa
Magnitude of Transmission Coefficient
$| {S}_{21}|$	Semi-rigid cable	0.015	(v)

Notes. (i) Uncertainty from thermal noise. (ii) Same uncertainty for all. (iii) Same uncertainty for all, in linear scale. (iv) Uncertainty is a function of reflection magnitude. (v) In linear scale.

^aAntenna reflection measurements are not part of receiver calibration but their uncertainty is also accounted for in this study.

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In the top block of Table 1, we show the $1\sigma$ uncertainties associated with each of the uncalibrated spectra of the absolute calibration loads (${T}_{A}^{* }$, ${T}_{H}^{* }$, ${T}_{O}^{* }$, ${T}_{S}^{* }$). The uncertainties are set equal to the rms of the residuals from polynomial or Fourier series fits to the actual measured spectra. In all cases, the residuals are noise-like. The residual rms for the ambient and hot load measurements is 66 mK, while for the open/shorted cable measurements it is 95 mK. For our MC realizations of the perturbed spectra, we begin with the models for the fiducial spectra and add to each frequency channel realizations of Gaussian noise, uncorrelated from channel to channel, drawn from the assigned uncertainties.

The second block of Table 1 shows the uncertainties associated with the physical temperatures (T_A, T_Ht, T_O, T_S) of the calibrator sources. We measure all the physical temperatures with thermistors and a resistance-to-temperature model. The accuracy of the setup is estimated to be 100 mK, hence we use that value for the uncertainty of all physical temperature measurements. For the MC realizations of the calibrator temperatures, we draw from Gaussian distributions centered at the fiducial values with a $1\sigma$ width of 100 mK.

The third and fourth blocks of Table 1 summarize the uncertainties applied to the amplitudes and phases of reflection coefficients, respectively. Our MC modeling propagates the reflection coefficient uncertainties for: (1) the absolute calibration loads (${{\rm{\Gamma }}}_{A}$, ${{\rm{\Gamma }}}_{H}$, ${{\rm{\Gamma }}}_{O}$, ${{\rm{\Gamma }}}_{S}$), (2) the receiver input (${{\rm{\Gamma }}}_{{\rm{rec}}}$), and (3) the antenna (${{\rm{\Gamma }}}_{{\rm{ant}}}$). Although the antenna reflection coefficient is not measured as part of the receiver calibration, it is tightly coupled to final receiver performance and we account for its uncertainty here. We also consider the uncertainty (listed in the fifth block in Table 1) on the transmission coefficient, $| {S}_{21}|$, of the semi-rigid cable inside the hot load because it is relevant in the computation of the effective noise temperature from the device (Equation (9)).

The main uncertainties in reflection coefficient measurements arise from uncertainty in VNA calibration. VNA calibration involves measuring three reflection standards at the VNA measurement plane: an open standard, a short standard, and a 50 Ω standard. As part of the EDGES efforts to increase the accuracy in calibration measurements, we improved upon the manufacturer-specified VNA tolerances through more accurate modeling of the 50 Ω standard (Blackham & Wong 2005; Scott 2005; Ridler & Nazoa 2006; Monsalve et al. 2016). With the improved model for the standard, the additive uncertainty in the magnitude of measured reflection coefficients was reduced to a $1\sigma$ linear voltage ratio of ${10}^{-4}$, which is equivalent to 0.005 dB for a reference reflection of −15 dB. In addition to residual modeling errors, sub-dominant effects that contribute to this uncertainty include VNA drifts and imperfect connection repeatability. Thus, the uncertainties assumed in the MC simulation for the reflection magnitudes are modeled as an additive contribution, fully correlated across the band, with an amplitude drawn from a Gaussian with a $1\sigma$ width of ${10}^{-4}$.

Figure 7 presents a verification of the magnitude accuracy of our VNA measurements using open-ended RF attenuators. In the figure, we show reflection coefficient measurements of an attenuator compared to forecasts based on a direct DC measurement of the attenuator resistance. In particular, for −20 dB reflection, the direct agreement is better than 0.003 dB and remains better than ±0.01 dB even if we assume a pessimistic error in the resistance measurement.

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We model reflection phase uncertainty using the form $k/| {\rm{\Gamma }}|$ found in typical VNA specifications. Here, k is a constant that represents the uncertainty value for the special case of a total reflection ($| {\rm{\Gamma }}=1|$). This model captures the fact that it is more difficult to determine the phase for smaller reflections than for larger reflections. In our simulations, we use $k=0\buildrel{\circ}\over{.} 015$ and model phase uncertainty as an additive contribution with a $1\sigma$ width given by $k/| {\rm{\Gamma }}|$. For reference, this corresponds to a $\pm 3\sigma$ range of $\pm 0\buildrel{\circ}\over{.} 25$ at the −15 dB magnitude of our antenna reflection coefficient model. As was the case for the reflection magnitude, the additive contribution to the phase is modeled as fully correlated across the band. This is a realistic first-order approximation for our VNA measurements of 100 MHz bandwidth.

The uncertainty of transmission coefficient ($| {S}_{21}|$) measurements also benefits from the improvements in the measurement of reflection coefficient. We assign $1\sigma$ uncertainty of 0.015 to the transmission coefficient of the semi-rigid cable in the hot load calibrator and model it in the same way as the reflection magnitudes.

Final cosmological parameter estimation for EDGES and similar experiments is performed by simultaneously fitting a signal model with a parametrized foreground model. In this step, the foreground model consists of a low-order polynomial or set of basis functions. These functions are able to fit the foreground and beam chromaticity structure in the measured spectra, but also tend to absorb some of the expected 21 cm signal. Calibration errors that are similar to foreground structures—i.e., those that exhibit large spectral coherence—will be associated with the foreground model terms during the final parameter estimation and will have little additional impact on the 21 cm signal estimation. Calibration errors that vary relatively rapidly in frequency, on the other hand, will not be fit by the foreground model and will yield more interference with the signal estimation. In order to take this effect into account in our analysis here, we characterize the output antenna temperatures from our uncertainty propagation after fitting and removing a foreground model. We use the polynomial model given by:

$$\begin{eqnarray}&&{\rm{model}}=\displaystyle \sum _{i=0}^{N-1}{a}_{i}{\nu }^{-2.5+i}.\end{eqnarray} \tag{ 14 }$$

This model was introduced in Mozdzen et al. (2016a) as an efficient expression for removing foreground and beam effects. In the context of this paper, it shares the flexibility of a generic polynomial but is more efficient for residuals with a predominant $\beta \approx -2.5$ power-law behavior. It is chosen as a model that could help remove the foreground and residuals from calibration simultaneously with few terms.

Several trends can be identified from Table 2. First, and as expected, for most error sources and polynomial orders the residuals are larger when observing a loud sky. This motivates the preference of low-foreground observations for estimation of the cosmological signal, or the downweighting of strong-foreground regions as suggested, for instance, in Liu et al. (2013).

The table also makes evident the low sensitivity of the calibration to errors in measurements of the open and shorted cable. As stated before, the residuals for their spectra (${T}_{O}^{* }$, ${T}_{S}^{* }$) remain consistently below 1 mK and are not shown in the table. For their physical temperatures (T_O, T_S) the initial residuals are $\leqslant 2$ mK and for the reflections (${{\rm{\Gamma }}}_{O}$, ${{\rm{\Gamma }}}_{S}$) they are $\leqslant 5$ mK. At five terms they all reach 1 mK or less.

The largest initial residuals occur for errors in the physical temperatures of the ambient and hot loads (T_A, T_Ht) which, for the loud sky, reach up to $\approx 2$ K. However, as shown in Figure 8, they strongly follow a power law and can be removed with few terms. Specifically, three terms are needed to reduce the residuals to $\leqslant 1$ mK for our foreground model. The same applies to the transmission coefficient magnitude of the cable inside the ambient/hot load ($| {S}_{21}|$) since this quantity is involved in the computation of the hot load noise temperature (Section 3.1).

The initial impact of thermal noise in the ambient and hot spectra (${T}_{A}^{* }$, ${T}_{H}^{* }$) is at the 5 and $\approx 25$ mK level for the quiet and loud skies respectively. These residuals decrease slowly as terms are added. More than five terms would be needed to reach 1 mK. For the reflections of these loads (${{\rm{\Gamma }}}_{A}$, ${{\rm{\Gamma }}}_{H}$) the initial residuals are significantly higher (up to 320 mK). However, five terms are sufficient to reduce them to $\leqslant 1$ mK due to their smooth spectral shape.

For the receiver and antenna reflection coefficients (${{\rm{\Gamma }}}_{{\rm{rec}}}$, ${{\rm{\Gamma }}}_{{\rm{ant}}}$) the initial residuals are comparable to those for the ambient and hot loads. However, it takes more than seven terms to reduce them to $\leqslant 1$ mK. With three or more terms removed these residuals are the largest among all the effects. The similarity in behavior between the two sources is not surprising considering their strong interaction in the calibration equations (Section 2.2).

Finally, the last rows of the table present the residuals when all the uncertainties are combined in a comprehensive Monte Carlo analysis. As expected, initially they are larger than those from individual sources, amounting to about 0.5 and 2.8 K for the quiet and loud skies respectively. For the quiet sky they drop to 26 mK with five terms.

In the full uncertainty propagation, errors in reflection coefficients are assumed uncorrelated. However, a test was also conducted with correlated errors where, in any given MC repetition, the same error was applied to all the magnitudes/phases. This exercise simulates a case where all these measurements are affected by the same VNA bias. The results are qualitatively and quantitatively very similar to those shown in the table, thus not shown for brevity.

An alternative approach to reduce the sensitivity of EDGES to errors in reflection measurements consists of improving the impedance match between the antenna and receiver. As stated in Section 2.2, our reflection coefficients are referenced to the 50 Ω system impedance. Therefore, improvement involves bringing both impedances closer to 50 Ω or, equivalently, lowering both reflection coefficients.

Table 3 presents five simulation examples with potential for reducing sensitivity to errors in the antenna reflection coefficient (magnitude, $| {{\rm{\Gamma }}}_{{\rm{ant}}}|$, and phase, $\angle \ {{\rm{\Gamma }}}_{{\rm{ant}}}$) for different levels of reflection from the antenna and receiver. The computations assume a quiet sky and the nominal uncertainties of Table 1.

Table 3.  95% Confidence rms Residuals for a Quiet Sky due to Errors in Antenna Reflection Coefficient, in mK

		Number of Terms
Source	Case	0	1	2	3	4	5	6	7
$| {{\rm{\Gamma }}}_{{\rm{ant}}}|$	(a)	32	12	12	12	9	6	3	1
	(b)	29	7	6	6	5	3	1	1
	(c)	21	12	12	11	9	6	3	1
	(d)	34	12	11	6	3	1
	(e)	18	6	6	6	5	3	1	1
	(f)	19	7	6	3	2
$\angle \ {{\rm{\Gamma }}}_{{\rm{ant}}}$	(a)	36	33	29	28	23	16	7	4
	(b)	18	17	16	15	13	8	4	1
	(c)	35	32	29	27	23	15	7	4
	(d)	36	33	25	16	6	3
	(e)	18	17	15	15	12	7	4	1
	(f)	18	17	12	9	2	1

Note. Case (a): Nominal. Same as in Table 2. Case (b): After lowering $| {{\rm{\Gamma }}}_{{\rm{rec}}}|$ from $\approx -20$ dB to −30 dB. Case (c): After lowering $| {{\rm{\Gamma }}}_{{\rm{ant}}}|$ from −15 dB to −20 dB. Case (d): After changing antenna delay from 13.88 to 6.94 ns. Case (e): Combined (b) and (c). Case (f): Combined (b)–(d).

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Case (a) corresponds to the same residuals shown in Table 2, presented again for reference. In case (b), the magnitude of the receiver reflection coefficient is lowered from $\approx -20$ to −30 dB. This change produces a reduction in residuals from the antenna reflection magnitude and phase by a factor of about two for almost any number of terms. In case (c), the nominal settings have been modified by lowering the magnitude of the antenna reflection coefficient from −15 to −20 dB. Relative to (a), this scenario produces an improvement in the initial residuals of the magnitude, which decreases from 32 to 21 mK. However, when introducing terms in the model, the residuals become almost identical to those in the nominal case. In case (d) the delay of the antenna has been reduced from 13.88 to 6.94 ns, i.e., the linear change in phase between 90 and 190 MHz has been reduced from 500° to 250°. This change has no impact on the initial residuals, but as more terms are introduced the improvement becomes significant. With five terms, the magnitude residuals are reduced from 6 to 1 mK, and in phase they are reduced from 16 to 3 mK. Case (e) corresponds to the combination of cases (b) and (c), i.e., both reflection magnitudes have been reduced. Case (f) combines all the suggested improvements and reduces the residuals to the lowest figures. They start at 19 and 18 mK for the magnitude and phase with no terms removed, and converge to $\leqslant 1$ mK with five terms.

The improvements described above have the potential to reduce sensitivity to errors. At the same time, their implementation has to consider tradeoffs typically encountered in wideband instrument design. For instance, reducing the antenna reflection coefficient can have an impact on the spectral smoothness of the beam. Similarly for the receiver, it is necessary to balance its input reflection coefficient with its noise performance. In addition, the phase of the antenna reflection depends on the physical dimensions of the antenna and on the length of any transmission line between the antenna and the reflection measurement plane. We are currently investigating these alternatives to converge to an optimal instrumental solution.

The results presented thus far correspond to 95% residuals in the range 90–190 MHz. On top of refinements in the instrumentation, a decrease in residuals could be achieved by reducing the bandwidth used in the science analysis motivated, for instance, by the interest in probing late (${z}_{r}\lesssim 10$) reionization transitions (Robertson et al. 2015; Planck Collaboration XLVII 2016).

Here we show the improvement in performance when reducing the bandwidth to 80 MHz by computing residuals in the range 110–190 MHz ($11.9\gtrsim z\gtrsim 6.5$) from the MC simulations of Section 4. Discarding a 20 MHz section at the low-frequency end results in less spectral structure, and in structure with lower amplitude due to the lower sky temperature in the remaining band. As depicted in Figure 8, this especially occurs for key sources of uncertainty such as the antenna and receiver reflection coefficients.

With a quiet sky and after removing five terms in the range 110–190 MHz, the residuals for the ambient and hot load spectra are reduced to 1 mK. For the magnitude and phase of the receiver reflection coefficient they go down to 3 and 5 mK, respectively. For the antenna reflection they are reduced to 2 and 8 mK. Finally, when all the uncertainty sources are considered the result is 9 mK. This result for the combined case is about three times better than the 26 mK obtained for the 90–190 MHz range. This suggests that for a given signal model it may be possible to optimize the bandwidth processed to maximize the constraints on the model.