2.1. Antenna

The antenna element is the most critical component of single-element telescopes for global 21 cm signal detection. Unfortunately, the antenna response can not be calibrated for an all-sky measurement using an external calibrator on the sky. The antenna response affects the sky signal received at the antenna output in two ways. Firstly, the antenna beam $G(\theta, \nu)$ is a function of frequency $\nu$ , and $\theta$ is the angle measured from the dipole axis (length of the dipole). As a result, the antenna may receive signals from a larger part of the sky at some frequencies than others. Secondly, the antenna’s coupling efficiency at any frequency depends on the Ohmic losses within the antenna and the impedance mismatch between the antenna and the transmission line, represented by antenna voltage reflection coefficient $\Gamma_{a}(\nu)$ .

Ideally, to preserve the spectral characteristics of the 21 cm global signal, the antenna should be frequency-independent, i.e., $G_{a}(\theta, \nu)= G_{a}(\theta)$ and $\Gamma_{a}(\nu)= constant$ over the observing band. However, the theoretical predictions of the global 21 cm signal are expected to be spread over a wide frequency range between 30–200 MHz. At these low frequencies, the physical size of the antenna is large. Owing to the large manufacturing tolerances at low frequencies, the frequency-independent performance of the antenna over multiple octaves is compromised. As a result, the antenna frequency response can generate spectral features that can be confused with the global 21 cm signal. In the absence of a true frequency-independent antenna at low frequencies, ongoing global 21 cm experiments use a variety of electrically short antenna structures that have smooth spectral responses over the desired frequency range. In this case, $G_{a}, \Gamma_{a}$ is described with a polynomial of low order in frequencies as described in Mahesh et al. (Reference Mahesh2021), Raghunathan, Shankar, & Subrahmanyan (Reference Raghunathan, Shankar and Subrahmanyan2013).

For HYPERION, we adopt a variant of the electrically short fat dipole (Raghunathan et al. Reference Raghunathan, Shankar and Subrahmanyan2013) to operate between 30–120 MHz Figure 4. It is a three-dimensional structure made of Aluminium which is symmetric around the dipole axis (z-axis) with a square cross-section in the x-y plane. The key design parameter responsible for the wideband performance of this antenna is its surface profile. Raghunathan et al. (Reference Raghunathan, Shankar and Subrahmanyan2013) had shown that a smooth frequency response over an octave bandwidth is achieved between 87.5–175 MHz by shaping the edge of each surface as $x = x_{0}+A\sin^{\alpha}({2\pi z\over \lambda})$ (Figure 4) where x is the half diagonals of the square cross sections on the x-y plane. Therefore, we tune the parameters $x_{0}, A,$ and $\alpha$ to design the antenna between 30–120 MHz (Figure 3). Figure 7 shows the measurements of antenna return loss $S_{11}(\nu)$ as a function of frequencies. Details of antenna simulation and measurements are out of this paper’s scope and will be discussed elsewhere.

Figure 3. Panel 1: Antenna beam pattern measured between 30–120 MHz. The measured beam patterns overlap at all measurement frequencies up to 50 MHz. Towards the lowest end of the band at 30 MHz, the antenna gain reduces at all position angles. This is why the lower cut-off for observation is kept at 50 MHz while the antenna is designed with a lower cut-off at 30 MHz. Panel 2: Magnitude and phase of the measured antenna return loss $S_{11}$ .

Figure 4. Panel 1: The HYPEREION antenna is a fat dipole antenna that is electrically short between 30–120 MHz. The basic design is adopted from Raghunathan et al. (Reference Raghunathan, Shankar and Subrahmanyan2013). Bottom: Schematic reproduced from Raghunathan et al. (Reference Raghunathan, Shankar and Subrahmanyan2013) showing various design parameters. We tune these design parameters to obtain the optimum antenna performance between 30–120 MHz to achieve a spectral response that is non-degenerate with the Cosmic Dawn signal of the form Equation (1). The resulting antenna beam pattern $G(\theta,\nu)$ and magnitude and phase of the return loss $\Gamma_{a}(\nu)$ is shown in Figure 3.

2.2. Analogue receiver

HYPEREION analogue receiver is implemented in two modules. The frontend module is located in the field at the antenna base. The backend module is located 100 m away from the antenna in an electromagnetically shielded enclosure known as the Telstra-Hut that provides 80 dB attenuation to the RFI generated inside the hut. The separation ensures the radio frequency interference (RFI) generated in the digital system, also located in Telstra-Hut, is not coupled into the system via the antenna. The frontend box receives the antenna output and adds a broadband calibration noise into the signal path by a directional coupler. The output of the directional coupler is fed into a low-noise amplifier. The output of the amplifier is fed into the $\Sigma$ (sum) port of a four-port power splitter via a mechanical switch. A precision 50 Ohm termination is connected to the $\Delta$ (difference) port of the power splitter via the mechanical switch. The mechanical switch alternates the antenna and the reference noise between the $\Sigma$ and the $\Delta$ port. Outputs of the power splitter are fed into a pair of identical low-noise amplifiers (LNAs). Each of these RF components is so chosen that any uncalibrated spectral feature introduced in the final bandpass calibrated spectrum does not affect the detectability of the Global signal.

• Directional coupler: A coaxial directional coupler (Figure 6) is used in reverse to couple a broadband calibration noise in the antenna path. The antenna is connected to port $P_{2}$ of the directional coupler by an RF cable of length L. The calibration noise is connected to the coupled port $P_{3}$ of the directional coupler. The calibration noise is turned on-off every 0.5 s. The antenna and the calibration noise travel downstream via port $P_{1}$ and follow the identical signal path, including the multipath reflections. Therefore, the calibration noise is able to completely calibrate the complex system bandpass response to antenna noise from this point onwards. From hereon, port $P_{1}$ is referred to as our calibration plane. Any spurious spectral response imprinted on the antenna signal prior to reaching port $P_{1}$ of the directional coupler remains in the bandpass calibrated spectrum.

• Low-Noise Amplifier 1: The directional coupler output is fed to a low-noise-amplifier+attenuator assembly, referred to as stage1-LNA from hereafter. The amplifier noise figure is typically 3.8 dB between 10–500 MHz, equivalent to a receiver noise temperature of 400 K. We chose an amplifier with a relatively higher noise figure as it has a better input match than a very low-noise amplifier. It is the input match of the first stage of the amplifier that critically determines the spectral smoothness of the calibrated data as shown in the measurement equation. For this amplifier, the receiver noise becomes comparable to that of the sky at the upper end of the band at 120 MHz. However, for the purpose of corroboration of the reported Cosmic Dawn signal that extends up to 82 MHz, this amplifier is indeed the best choice. This assembly aims to prevent any signal originating from the rest of the receiver downstream from propagating upward to the antenna and radiating in the environment. Components such as the RF mechanical switch is periodically energised and de-energised. The contact stabilisation time of the switch is about 20 ms, during which a transient burst of energy is coupled to the main signal path that propagates upstream towards the antenna and radiates. The digital receiver system that is clocked over a range of rates can generate transient broadband radiation that can couple to the main signal path, propagate upstream to the antenna and radiate. While we exclude the 20 ms data during every transition to remain unaffected by this, other experiments located at the same observatory might be sensitive enough to pick up these transient broadband radiations. Any upward travelling signal is first attenuated by 20 dB first by the attenuator and then by an additional 22 dB by the reverse isolation of the LNA. Based on the mil-standard test $MIL-STD_461F$ performed in an EMC chamber, this provides sufficient isolation of any upward travelling noise from being radiated into the surroundings at the Murchison Radio Observatory.

• Mechanicalen Switch: Output of the stage1-LNA is connected to a two-input/two-output mechanical RF transfer switch (Figure 5). When the switch is de-energised, referred to state-‘0’ hereon, $J_{1}$ internally connects to $J_{3}$ , and $J_{4}$ internally connects to $J_{2}$ . In this switch state, antenna (+calibration) noise incident to port $J_{1}$ is transferred to the ‘sum’ $(\Sigma)$ port of a power splitter. A precision 50 Ohm load, termed as the ‘reference’ load from hereon, is externally connected at port $J_{4}$ of the switch. Noise from the reference incident to port $J_{4}$ is transferred to the ‘difference’ $(\Delta)$ port of the same power splitter. When the switch is energised, referred to as state ‘1’ hereon, $J_{1}$ internally connects to $J_{2}$ , and $J_{4}$ internally connects to $J_{3}$ . The antenna (+calibration) noise is now transferred to the $(\Delta)$ port of the splitter, and the reference noise is transferred to the $(\Sigma)$ port. It takes 20 ms for the RF mechanical switch to establish the electrical contact after a state transition. This sets the lowest integration time to be $>$ 20 ms. The insertion-loss between port $J_{1}-J_{3}$ , $J_{4}-J{2}$ , $J_{1}-J_{2}$ , $J_{4}-J{3}$ are 0.01 dB between 10–200 MHz. In state ‘0’, isolation between port $J_{1}-J_{2}$ , and $J_{4}-J_{3}$ is $>$ 90 dB. In state ‘1’, isolation between port $J_{1}-J_{3}$ , and $J_{4}-J_{2}$ is also $>$ 90 dB. Therefore, antenna (+calibration) and reference noise follow isolated paths with no mutual coupling within the switch. As a consequence, the calibration noise is unable to calibrate the bandpass response of the switch imprinted on the reference noise.

  

  Figure 5. Top: Schematic diagram of the switch. In the de-energised state of the switch port $J_{1}$ is electrically connected to $J_{3}$ and $J_{2}$ is connected to $J_{4}$ . In this state, the antenna is connected to the $\Sigma$ port of the power splitter, and the reference is connected to the $\Delta$ port. In the energised state, $J_{1}$ is electrically connected to $J_{2}$ and $J_{3}$ is connected to $J_{4}$ . In this state, the antenna is connected to the $\Delta$ port of the power splitter, and the reference is connected to the $\Sigma$ port. Bottom: Schematic diagram of a power splitter: Any noise connected to the $\Sigma$ port of the splitter is split into two halves ‘in-phase’. Any noise connected to the $\Delta$ port of the splitter is split into two halves ‘out of phase’.

  The switch cycles through state ‘0’, and ‘1’ every 1 s and alternately connects the antenna and the reference noise to $\Sigma$ and $\Delta$ port of the power splitter. In each position of the switch, the calibration noise is turned on and off for 0.5 s.

• Power Splitter: The output ports $J_{3}$ , and $J_{2}$ of the switch are connected to the $\Sigma$ and the $\Delta$ port of the power splitter. The complex voltage gains within the power splitter are denoted as $g_{\Sigma1}, g_{\Sigma2}, g_{\Delta_{1}}, g_{\Delta_{2}}$ (Figure 5). $|g_{\Sigma1}|^{2} = |g_{\Sigma2}|^{2} = |g_{\Delta1}|^{2} = |g_{\Delta2}|^{2} =-3\; dB$ , $<g_{\Sigma1} - <g_{\Sigma2} = 0^{\circ}$ , i.e., any signal entering the $\Sigma$ port of the splitter is split into two parts with equal amplitude and in-phase. $ <g_{\Delta1} - <g_{\Delta2} = 180^{\circ}$ , i.e., any signal entering the $\Delta$ port of the splitter is split into two parts with equal amplitude and out of phase. The reference is an additive blackbody noise at ambient temperature. Downstream from $P_{1}, P_{2}$ , the antenna, the calibration, and the reference noise follow the identical signal path. Therefore, the calibration noise fully calibrates any system bandpass response to these noises from this point onwards.

• Low noise amplifier 2,3 The power splitter outputs are directly connected to two identical Low-Noise Amplifiers that amplify the input signals by 22 dB. Amplifier outputs are sent downstream about 100 m away to the Telstra-Hut into the backend receivers module, which houses two identical two-stage filter-amplifier chains. It band limits the signal between 30–120 MHz and further amplifies it before feeding it to the digital cross-correlator. Since the two receiver noises are independent, they are uncorrelated. Therefore, the contribution of these two amplifiers to the cross-power output is ideally zero. Secondly, the mechanical switch and the power splitter assembly enable cancelling any spurious correlated noise that is added downstream to both channels after the switching state, such as the broadband correlator noise generated within the digital correlator. The digital correlators are clocked at a range of rates. Such clock signals generate broad-spectrum emissions. Common mode coupling of these broadband emissions to both analogue signal paths generates spurious correlation at the final digital correlator outputs. Alternating the position of the antenna and the reference between the sum and the delta port of the power splitter cancels such spurious additive contribution, as shown in section 3.

2.3. Digital receiver

Output voltages from the analogue backend module are fed into a digital data processing unit where these voltages are sampled and digitised by a SIGNATEC PX-1500—a two-channel 8 bit digitiser board at a sampling rate of 300 MHz. The digitised output is fed into a GPU that computes the Fourier Transform of each receiver channel voltage by an 8 192-point Fast Fourier Transform (FFT) algorithm. The complex voltages at corresponding frequencies between two channels are then multiplied to compute the cross-power spectrum across the observing band. The auto-correlation spectra of individual channels are also computed. The digital spectrometer can compute one cross-power spectrum every 13 ms. However, the switching speed of the mechanical switch determines the shortest integration time. Therefore, one spectrum is recorded every 20 ms. The broadband RFI generated within the digital system is contained within the Telstra Hut as per the mil standard requirement of the Murchison Radio Observatory. The RF switching cancels any additive noise electronically generated within the digital system during computation.

3.1. Ideal system response to antenna noise

At system state OBS0, i.e., when the antenna is connected to the $\Sigma$ port of the splitter, the antenna voltages at the input of the correlator are,

(3) \begin{equation}v_{a1} = g G G_{1} v_{a}\end{equation}

(4) \begin{equation}v_{a2} = g G G_{2} v_{a}\end{equation}

The cross-power at the output of the spectrometer due to antenna noise,

(5) \begin{eqnarray}P_{oa} & = & v_{a1}v_{a2}^{*} \nonumber \\[5pt] & = & |g|^{2}|G|^{2}G_{1}G^{\ast}_{2} P_{a}\end{eqnarray}

Similarly, at system state OBS1, i.e., when the antenna is connected to the $\Delta$ port of the splitter, the antenna voltages at the input of the correlator are,

(6) \begin{equation}v_{a1} = g G v_{a}G_{1}\end{equation}

(7) \begin{equation}v_{a2} = - g G v_{a}G_{2}\end{equation}

The cross-power at the output of the spectrometer due to antenna noise,

(8) \begin{eqnarray}P_{1a} & = & v_{a1}v_{a2}^{*} \nonumber \\[5pt] & = & - |g|^{2}|G|^{2}G_{1}G^{\ast}_{2} P_{a}\end{eqnarray}

The phase imbalance of $180^{\circ}$ between the voltages along the two arms, when the antenna noise is connected at the $\Delta$ port, is reflected in the negative voltage alone in one arm.

3.2. Ideal system response to receiver noise

Receiver noise in the HYPEREION is contributed by the first low-noise-amplifier alone. The noise figure of the low-noise amplifier (number) is 3.8 across our frequency band. Therefore, a mean receiver noise ranging 400 K is added to the antenna noise at the output of the LNA. We denote the LNA noise voltage referred to its input as $v_{lna}$ . Just like the antenna, at system state OBS0, the LNA1 is connected to the $\Sigma$ port of the splitter. Receiver voltages at the input of the correlator are,

(9) \begin{equation}v_{rec1} = g G v_{lna}G_{1}\end{equation}

(10) \begin{equation}v_{rec2} = g G v_{lna}G_{2}\end{equation}

The cross-power at the output of the spectrometer due to receiver noise,

(11) \begin{eqnarray}P_{or} & = & v_{rec1}v^{\ast}_{rec2} \nonumber \\[5pt] & = & |g|^{2}|G|^{2}G_{1}G^{\ast}_{2} P_{lna}\end{eqnarray}

In system state OBS1, the LNA1 is connected to the $\Delta$ port of the splitter. Receiver voltages at the input of the correlator are,

(12) \begin{equation}v_{rec1} = g G v_{lna}G_{1}\end{equation}

(13) \begin{equation}v_{rec2} = - g G v_{lna}G_{2}\end{equation}

The cross-power at the output of the spectrometer due to receiver noise,

(14) \begin{eqnarray}P_{1r} & = & v_{rec1}v^{\ast}_{rec2} \nonumber \\[5pt] & = & - |g|^{2}|G|^{2}G_{1}G^{\ast}_{2} P_{lna}\end{eqnarray}

3.3. Ideal system response to reference noise

At system state OBS0, the reference is connected to the $\Delta$ port of the splitter. The reference voltages at the input of the correlator are,

(15) \begin{equation}v_{ref1} = g v_{ref}G_{1}\end{equation}

(16) \begin{equation}v_{ref2} = - g v_{ref}G_{2}\end{equation}

The cross-power at the output of the spectrometer due to reference noise,

(17) \begin{eqnarray}P_{oref} & = & v_{ref1}v_{ref2}^{*} \nonumber \\[5pt] & = & -|g|^{2}G_{1}G^{\ast}_{2} P_{ref}\end{eqnarray}

At system state OBS1, the reference is connected to the $\Sigma$ port of the splitter. The reference voltages at the input of the correlator are,

(18) \begin{equation}v_{ref1} = g v_{ref}G_{1}\end{equation}

(19) \begin{equation}v_{ref2} = g v_{ref}G_{2}\end{equation}

The cross-power at the output of the spectrometer due to reference noise,

(20) \begin{eqnarray}P_{1ref} & = & v_{ref1}v_{ref2}^{*} \nonumber \\[5pt] & = & |g|^{2}G_{1}G^{\ast}_{2} P_{ref}\end{eqnarray}

3.4. Ideal system response to total noise

The total complex cross-power spectrum measured in OBS0 state

(21) \begin{eqnarray}P_{OBS0} & = & P_{oa}+P_{or}+P_{oref}+P_{cor} \nonumber \\[5pt] & = & |g|^{2}|G|^{2} G_{1}G^{\ast}_{2} \!\left(P_{a}+P_{lna} - {P_{ref}\over |G|^{2}} \right)+ P_{cor}\end{eqnarray}

Since the calibration noise is injected into the antenna path at the output of the directional coupler, when the calibration noise is ‘on’, the total cross-power output in state CAL0 is,

(22) \begin{eqnarray}P_{CAL0} & = & |g|^{2}|G|^{2} G_{1}G^{\ast}_{2} \!\left(P_{a}+P_{cal}+P_{lna}-{P_{ref} \over |G|^{2}}\right) + P_{cor}\end{eqnarray}

$P_{cor}$ represents any additive noise that is either coupled to both signal chains after the switch and results in correlated output power or any noise generated within the digital system during computation that is added to the computed cross power. Since it is added after the switching state, $P_{cor}$ doesn’t change sign with the switch position. The reference noise is injected into the system after the injection of the calibration noise. The term ${P_{ref} \over |G|^{2}}$ indicates the reference noise contribution referred to the input of the LNA1 assembly.

Similarly, when the antenna and the reference positions are switched, the total complex cross-power spectrum measured in OBS1, CAL1 state are,

(23) \begin{equation}P_{OBS1} = - |g|^{2} |G|^{2}G_{1}G^{\ast}_{2} \!\left(P_{a}+P_{lna} - {P_{ref} \over |G|^{2}} \right)+ P_{cor}\end{equation}

(24) \begin{equation}P_{CAL1} = - |g|^{2}|G|^{2}G_{1}G^{\ast}_{2} \!\left(P_{a}+P_{cal}+P_{lna} - {P_{ref} \over |G|^{2}}\right)+ P_{cor}\end{equation}

3.5. Cancellation of spurious additive response

Equations (21), (22), (23), (24) are the four successive spectra measured by the spectrometer over a period of two seconds. From these, the bandpass calibrated spectra are derived. Differencing Equations (21), (23) cancels the unwanted additive contribution $P_{cor}$ .

(25) \begin{equation}P_{OBS} = {P_{OBS0}-P_{OBS1} \over 2}= |g|^{2}|G|^{2}G_{1}G^{\ast}_{2} \!\left(P_{a}+P_{lna} - {P_{ref} \over |G|^{2}} \right)\end{equation}

(26) \begin{equation}P_{CAL} = {P_{CAL0}-P_{CAL1} \over 2} = |g|^{2}|G|^{2} G_{1}G^{\ast}_{2} \!\left(P_{a}+P_{cal}+P_{lna} - {P_{ref} \over |G|^{2}} \right)\end{equation}

3.6. Receiver bandpass calibration

The system response to the calibration noise is derived by differencing Equations (25), (26),

(27) \begin{equation}P_{CAL} - P_{OBS} = |g|^{2}|G|^{2}G_{1}G^{\ast}_{2} P_{cal}\end{equation}

Dividing the Equation (25) by the system response to the calibration noise, we derive the bandpass calibrated spectrum in temperature unit,

(28) \begin{equation}T_{meas} = {P_{OBS} \over P_{CAL} - P_{OBS} } \times T_{cal} = T_{a}+T_{lna} -{T_{ref} \over |G|^{2}}\end{equation}

4.1. Real system response to Antenna noise

• Effect of directional coupler on antenna noise: A directional coupler can imprint spurious spectral response on the antenna noise within the observing band in two ways. 1. Due to impedance mismatch between the transmission line connecting the antenna output to port $P_{2}$ of the directional coupler, only a fraction. $(1+\Gamma_{D})$ fraction is transmitted to the directional coupler output where $\Gamma_{D}$ is the voltage reflection coefficient at port $P_{2}$ . $\Gamma_{D}$ fraction of the antenna voltage $v_{a}$ is reflected back from port $P_{2}$ to the antenna as shown in Figure 6. The antenna reradiates the majority of this reflected signal, and a small fraction $\Gamma_{a}$ (antenna voltage reflection coefficient) is coupled back into the antenna path with a frequency-dependent phase delay $\Phi _{1}= {2\pi i 2L \over \lambda}$ where L is the electrical path length between $P_{2}$ and the antenna output. The net antenna voltage at the output of the directional coupler after one reflection at the input of the directional coupler,

  (29) \begin{equation}v_{a}^{\prime} = v_{a} (1+ \Gamma_{D})(1+ \Gamma_{D}\Gamma_{a} e^{i\Phi_{1}})\end{equation}

  2. A unidirectional coupler is a four-port device with the fourth port $P_{4}$ terminated internally by a 50 Ohm resistance that generates broadband noise at ambient temperature. Isolation between $P_{2}, P_{4}$ is 20 dB. For an ambient temperature of 300 K, a broadband noise of temperature 3 K is coupled to port $P_{2}$ that travels upward to the antenna. The antenna radiates the majority of this noise. $\Gamma_{a}$ fraction of this noise is reflected back and added to the antenna signal path. Once again, at the $(1+\Gamma_{D})$ fraction of this noise voltage is transmitted by the directional coupler. The total noise voltage at the output of the directional coupler is,

  (30) \begin{eqnarray}v_{a}^{\prime} = (1+ \Gamma_{D})\left [v_{a} (1+ \Gamma_{D}\Gamma_{a} e^{i\Phi_{1}})+ v_{amb} \Gamma_{iso} \Gamma_{a} \right ]\end{eqnarray}
  where, $\Gamma_{iso} = 10$ dB is the isolation between $P_{2}, P_{4}$ in voltage unit.
  (31) \begin{eqnarray}P^{\prime}_{\!\!a} = |(1+ \Gamma_{D})(1+ \Gamma_{D}\Gamma_{a} e^{i\Phi_{1}})|^{2} P_{a} \\[5pt] \nonumber + |(1+ \Gamma_{D}) \Gamma_{a}\Gamma_{iso}|^{2}P_{amb}\end{eqnarray}

  Since the calibration noise is injected at the output of the directional coupler, it only calibrates for any multiplicative path gain imprinted on $v_{a}^{\prime}$ from this point onwards. The spectral structures introduced by the directional coupler are preserved in the final bandpass calibrated data from which the Cosmic Dawn signal is expected to be detected.

• Effect of stage1-LNA assembly on antenna noise: The impedance mismatch between the directional coupler’s output and the first LNA’s input results in partial coupling of both $v_{a}^{\prime}$ and the calibration noise into the LNA. The rest is reflected back towards the antenna via the directional coupler. $\Gamma_{a}$ fraction of this noise is coupled back into the system from the antenna terminal, and the rest is radiated. The net antenna voltage incident at the LNA input after one reflection from the LNA input and successive reflection from the antenna is,

  (32) \begin{eqnarray}v_{ain} & = & v_{a}^{\prime} (1+\Gamma_{L}) (1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})\end{eqnarray}
  where $\Gamma_{L}$ is the voltage reflection co-efficient at the LNA input and $\Phi_{2}= {2\pi i 2L \over \lambda}$ is the roundtrip phase delay. The output voltage of the LNA is,
  (33) \begin{eqnarray}v_{aout} & = & v_{a}^{\prime} (1+\Gamma_{L})G (1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}}) \nonumber \\[5pt] & = & v_{a}^{\prime} G (1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})\end{eqnarray}
  Since the term $(1+\Gamma_{L})$ , and the LNA gain G both are multiplicative frequency responses of the LNA, for notational simplicity, we absorb the two terms into one single frequency-dependent amplifier gain term G.
  

  Figure 7. Electrical parameters of RF components that contribute to system response. Measurements of the magnitudes (top left) and phases (top right) of the voltage reflection coefficients at the antenna input ( $\Gamma_{a}$ ), the LNA input ( $\Gamma_{L}$ )and the directional coupler input ( $\Gamma_{D}$ ). Measurements of the magnitude and phase (bottom left) of the LNA complex gain. Phase imbalances of the power splitter path gains between $\Sigma$ , port 1,2 and $\Delta$ , port 1,2, respectively. These system parameters collectively determine the spectral shapes of $C_{sky}, C_{lna}, C_{ref}$ .

  At system state OBS0, i.e., when the antenna is connected to the $\Sigma$ port of the splitter, the antenna voltages at the input of the correlator are,

  (34) \begin{equation}v_{a1} = g_{\Sigma1} v_{aout} G_{1}\end{equation}
  (35) \begin{equation}v_{a2} = g_{\Sigma2}v_{aout} G_{2}\end{equation}

  The cross-power at the output of the spectrometer due to antenna noise,

  (36) \begin{eqnarray}P_{oa} & = & v_{a1}v_{a2}^{*} \nonumber \\[5pt] & = & g_{\Sigma1}g^{\ast}_{\Sigma2}G_{1}G^{\ast}_{2}| v_{aout}|^{2} \nonumber \\[5pt] & = & g_{\Sigma1} g^{\ast}_{\Sigma2} |G|^{2} G_{1}G^{\ast}_{2} |(1+\Gamma_{L}\Gamma_{a}e^{i\Phi_{2}})|^{2} P^{\prime}_{\!\!a}\end{eqnarray}

  Similarly, at system state OBS1, i.e., when the antenna is connected to the $\Delta$ port of the splitter, the cross-power at the output of the spectrometer due to antenna noise,

  (37) \begin{eqnarray}P_{1a} & = & - g_{\Delta1}g^{\ast}_{\Delta2}|G|^{2}G_{1}G^{\ast}_{2}|(1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2} P^{\prime}_{\!\!a}\end{eqnarray}

4.2. Real system response to Receiver noise

Ideally, noise from the LNA has a downstream component, as discussed in Section 3.2. However, in real systems, it has a second component, correlated with the first, that travels opposite to the signal flow direction towards the antenna. After reaching the antenna, it is partially radiated, and the rest is reflected by the antenna and propagates downstream. We denoted the LNA noise voltage referred to its input as $v_{lna}$ . If a fraction $\alpha$ of the total receiver noise travels upward, the total receiver noise incident at the input of the receiver in OBS0 state,

(38) \begin{equation}v_{rec1} = g_{\Sigma1} v_{lna} G (1+\Gamma_{L})G_{1} (1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})\end{equation}

(39) \begin{equation}v_{rec2} = g_{\Sigma2}v_{lna} G (1+\Gamma_{L})G_{2} (1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})\end{equation}

The cross-power at the output of the spectrometer due to receiver noise,

(40) \begin{eqnarray}P_{or} & = & v_{rec1}v^{\ast}_{rec2} \nonumber \\[5pt] & = & g_{\Sigma1}g^{\ast}_{\Sigma2} |G|^{2} G_{1}G^{\ast}_{2} |(1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2}P_{lna}\end{eqnarray}

Once again, we combine the term $(1+\Gamma_{L})$ , and the LNA gain G for notational simplicity into one frequency-dependent amplifier gain term G.

In system state OBS1, the LNA1 is connected to the $\Delta$ port of the splitter. Receiver voltages at the input of the correlator are,

The cross-power at the output of the spectrometer due to receiver noise,

(41) \begin{eqnarray}P_{1r} & = & v_{rec1}v^{\ast}_{rec2} \nonumber \\[5pt] & = & -g_{\Delta1}g^{\ast}_{\Delta2} |G|^{2} G_{1}G^{\ast}_{2} |(1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2}P_{lna}\end{eqnarray}

Notable here is that the LNAs connected to the output of the power splitter also have upward travelling components that can travel to the antenna. However, the stage 1 LNA (ZFL-500HLN)-attenuator assembly offers a 50 dB reverse isolation to these upward propagating noises (25 dB in voltage) (ref: figure ZFL-500HLN reverse isolation). The antenna attenuates them by another 15 dB. The reflected noises once again go through the stage1 LNA (ZFL-500HLN)-attenuator assembly, which now offers a 10 dB attenuation and a 22 dB forward gain. Therefore, the part of these noise voltages that propagate downstream after being reflected by the antenna has a total power attenuation of 53 dB or voltage attenuation of 27.5 dB. A typical value of $\alpha$ is $0.1$ . For ZFL-500LN, with a noise temperature of 275 K, a 27 K noise travels upward. After a 53 dB attenuation, only 0.0001 K noise is added to the downward travelling receiver noise. Even if a $\alpha=1$ , this contribution becomes 0.001 K. This is why we ignore the effects of the upward travelling component of the receiver noise from the LNAs to which the power splitter is connected.

4.3. Real system response to Reference noise

Power Splitter: In addition to the difference in path gain, the power splitter may have phase imbalances $ \Phi_{\Sigma12} = (\angle g_{\Sigma1} - \angle g_{\Sigma2}), (\Phi_{\Delta12} =\angle g_{\Delta1} - \angle g_{\Delta2})$ can deviate from their ideal values of $0^{\circ}$ , $180^{\circ}$ . This is analogous to additional electrical path length along one receiver channel with respect to the other. The calibration noise calibrates the effect of the power splitter phase imbalance on the antenna noise since they are connected at the same port of the power splitter in both switch positions. Its effect on the reference noise remains uncalibrated. Our measurements show $ \Phi_{\Sigma12} =\Phi_{\Delta12}$ (Figure 7) If $ \Phi_{\Sigma12} =\Phi_{\Delta12} = \Delta\phi$ indicates the deviations from the ideal phase imbalances of $0^{\circ}$ and $180^{\circ}$ , then, when the reference is connected to the $\Delta$ port of the splitter, voltages at the input of the correlator are,

(42) \begin{equation}v^{\prime}_{ref1} = g_{\Delta1} v_{ref}G_{1}\end{equation}

(43) \begin{equation}v^{\prime}_{ref2} = - g_{\Delta2} v_{ref}G_{2} e^{i\Delta\Phi}\end{equation}

The cross-power at the output of the spectrometer due to reference noise,

(44) \begin{eqnarray}P_{oref} & = & v^{\prime}_{ref1}v^{\prime\ast}_{ref2} \nonumber \\[5pt] & = & -g_{\Delta1}g^{\ast}_{\Delta2}G_{1}G^{\ast}_{2} P_{ref} e^{-i\Delta\Phi}\end{eqnarray}

and,

(45) \begin{eqnarray}P_{1ref} & = & v_{ref1}v^{\prime\ast}_{ref2} \nonumber \\[5pt] & = & g_{\Sigma1}g^{\ast}_{\Sigma2}G_{1}G^{\ast}_{2} P_{ref} e^{-i\Delta\Phi}\end{eqnarray}

4.4. Real system response to total noise

(46) \begin{eqnarray}P_{OBS0} & = & P_{oa}+P_{or}+P_{oref}+P_{cor} \nonumber \\[5pt] & = & g_{\Sigma1}g^{\ast}_{\Sigma2}|G|^{2}G_{1}G^{\ast}_{2}|(1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2} P^{\prime}_{\!\!a} \nonumber \\[5pt] && + g_{\Sigma1}g^{\ast}_{\Sigma2}|G|^{2}G_{1}G^{\ast}_{2} |(1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2}P_{lna} \nonumber \\[5pt] && - g_{\Delta1}g^{\ast}_{\Delta2}G_{1}G^{\ast}_{2} P_{ref} e^{-i\Delta\Phi} +P_{cor}\end{eqnarray}

Since the calibration noise is injected into the antenna path at the output of the directional coupler, when the calibration noise is ‘on’, the total cross-power output in state CAL0 is,

(47) \begin{eqnarray}P_{CAL0} & = & P_{oa}+P_{or}+P_{oref}+P_{cor} \nonumber \\[5pt] & = & g_{\Sigma1}g^{\ast}_{\Sigma2}|G|^{2}G_{1}G^{\ast}_{2}| (1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2} (P^{\prime}_{\!\!a} +P_{cal})\nonumber \\[5pt] && + g_{\Sigma1}g^{\ast}_{\Sigma2}|G|^{2}G_{1}G^{\ast}_{2} |(1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2}P_{lna} \nonumber \\[5pt] && - g_{\Delta1}g^{\ast}_{\Delta2}G_{1}G^{\ast}_{2} P_{ref} e^{-i\Delta\Phi} +P_{cor}\end{eqnarray}

Similarly, when the antenna and the reference positions are switched, the total complex cross-power spectrum measured in OBS1, CAL1 state are,

(48) \begin{eqnarray}P_{OBS1} & = & - g_{\Delta1}g^{\ast}_{\Delta2}|G|^{2}G_{1}G^{\ast}_{2}| (1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2} P^{\prime}_{\!\!a} \nonumber \\[5pt] && - g_{\Delta1}g^{\ast}_{\Delta2}|G|^{2}G_{1}G^{\ast}_{2} |(1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2}P_{lna} \nonumber \\[5pt] && + g_{\Sigma1}g^{\ast}_{\Sigma2}G_{1}G^{\ast}_{2} P_{ref} e^{-i\Delta\Phi} +P_{cor}\end{eqnarray}

Since the calibration noise is injected into the antenna path at the output of the directional coupler, when the calibration noise is ‘on’, the total cross-power output in state CAL1 is,

(49) \begin{eqnarray}P_{CAL1} & = & - g_{\Delta1}g^{\ast}_{\Delta2} G G_{1}G^{\ast}_{2}| (1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2} (P^{\prime}_{\!\!a}+P_{cal}) \nonumber \\[5pt] && - g_{\Delta1}g^{\ast}_{\Delta2} G G_{1}G^{\ast}_{2} |(1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2}P_{lna} \nonumber \\[5pt] && + g_{\Sigma1}g^{\ast}_{\Sigma2}G_{1}G^{\ast}_{2} P_{ref} e^{-i\Delta\Phi} +P_{cor}\end{eqnarray}

(50) \begin{eqnarray}P_{OBS} & = & {P_{OBS0}-P_{OBS1} \over 2} \nonumber \\[5pt] & = & (g_{\Sigma1}g^{\ast}_{\Sigma2}+ g_{\Delta1}g^{\ast}_{\Delta2})|G|^{2} G_{1}G^{\ast}_{2} \times \nonumber\\[5pt] && [ |(1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2} P^{\prime}_{\!\!a} \nonumber \\[5pt] && + |(1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2}P_{lna} - {P_{ref}\over |G|^{2}} e^{-i\Delta\Phi} ]\end{eqnarray}

(51) \begin{eqnarray}P_{CAL} & = & {P_{CAL0}-P_{CAL1} \over 2} \nonumber \\[5pt] & = & (g_{\Sigma1}g^{\ast}_{\Sigma2}+ g_{\Delta1}g^{\ast}_{\Delta2})|G|^{2} G_{1}G^{\ast}_{2} \times \nonumber\\[5pt] && [ |(1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2} (P^{\prime}_{\!\!a}+P_{cal}) \nonumber\\[5pt] && + |(1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi})|^{2}P_{lna} - {P_{ref}\over |G|^{2}} e^{-i\Delta\Phi}]\end{eqnarray}

Therefore, for real system performance, Equation (28) modifies to,

(52) \begin{eqnarray}T_{meas} &=& {P_{OBS} \over P_{CAL} - P_{OBS} } \times T_{cal} = T^{\prime}_{a} \nonumber \\[5pt] && + { |(1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2} \over |(1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2}}T_{lna} \nonumber\\[5pt] && - { e^{-i\Delta\Phi} \over |G|^{2} |(1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2}}T_{ref} \;\;\;\; \end{eqnarray}

Using Equations (32), (52) can be written as,

(53) \begin{eqnarray}T_{meas} & = &|(1+ \Gamma_{D})(1+ \Gamma_{D}\Gamma_{a} e^{i\Phi_{1}})|^{2} T_{a} \nonumber \\[5pt] && + | (1+ \Gamma_{D}) \Gamma_{a}\Gamma_{iso}|^{2}T_{amb} \nonumber\\[5pt] && + { |(1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2} \over |(1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2}}T_{lna} \nonumber\\[5pt] && - { e^{-i\Delta\Phi} \over |G|^{2} |(1+ \Gamma_{L}\Gamma_{a} e^{i\Phi_{2}})|^{2}}T_{ref}\end{eqnarray}

We note that $T_{a}= G_{a}T_{sky}$ . Also, the directional coupler and the reference 50 Ohm load are at the same ambient temperature within the frontend module box. Therefore, $T_{amb}=T_{ref}$ . We verify this by measuring the temperature of the directional coupler and the reference load over a period of 12 h of observation using a platinum thermometer. In addition, the physical distance between the input of the directional coupler and the input of the LNA is less than 0.05 m. Therefore $\Phi_{1}\approx\Phi_{2} = \Phi$ . Therefore, (53) can be written as,

(54) \begin{eqnarray}T_{meas} & = &|(1+ \Gamma_{D})(1+ \Gamma_{D}\Gamma_{a} e^{i\Phi})|^{2} G_{a} T_{sky} \nonumber \\[5pt] && + { |(1+ \alpha \Gamma_{L}\Gamma_{a} e^{i\Phi})|^{2} \over |(1+ \Gamma_{L}\Gamma_{a} e^{i\Phi})|^{2}}T_{lna} \nonumber\\[5pt] && + [ | (1+ \Gamma_{D}) \Gamma_{a} \Gamma_{iso}|^{2}- { e^{-i\Delta\Phi} \over |G|^{2} |(1+ \Gamma_{L}\Gamma_{a} e^{i\Phi})|^{2}} ] T_{ref} \nonumber\\[5pt] & = & C_{sky} T_{sky} + C_{lna} T_{lna} + C_{ref} T_{ref}\end{eqnarray}

Due to averaging the spectra measured in two switch positions, the magnitude of the gain imbalances along the paths of the power splitter is removed in the process of the bandpass calibration. However, phase imbalances result in complex cross-power spectra contributed by the reference noise. The term $\Delta \Phi$ makes the term $C_{ref}$ complex. We use the RF power splitter AMT-2+ from the mini-circuit. Phase imbalances $(\angle g_{\Sigma1} - \angle g_{\Sigma2}) $ , and $\angle g_{\Delta1} - \angle g_{\Delta2})$ are shown in Figure 7. Any spectral ripple created by the phase imbalances within the power splitter paths in the antenna noise spectrum gets fully calibrated by the calibration noise. Another notable observation here is the contribution of the first stage LNA. For an ideal amplifier, $\alpha=0$ . This will result in an uncalibrated system bandpass response of the form ${ 1 \over |(1+ \Gamma_{L}\Gamma_{a} e^{i\Phi})|^{2}}$ in the LNA noise contribution. However, if $\alpha=1$ , the LNA bandpass response is identical to that of the calibration noise and is completely calibrated. Commonly, in radio telescopes, the first stage of amplifiers is the low-noise amplifiers. However, LNAs have poor input impedance match and extremely poor reverse isolation compared to a regular power amplifier. Poor impedance match will increase the ripple amplitude of $e^{i\Phi}$ for both LNA and reference noise. Poor reverse isolation will result in contribution from any other source of noise downstream, such as the LNA1,2 that can propagate upstream and create a spectral ripple. Therefore, for the purpose of this experiment, we chose the amplifier with a higher noise figure but better input impedance match and higher reverse isolation compared to conventional LNAs as the first stage of the amplifier. ZFL-500HLN has a noise figure of 3.8 dB, i.e., a noise temperature of $405\;K<<T_{sky}$ at all frequencies. The input return loss of the amplifier is $>22.8$ dB, and the reverse isolation is 39 dB across the band of interest.

We measure input voltage reflection coefficient of the antenna $(\Gamma_{a})$ (Figure 7), the directional coupler $(\Gamma_{D})$ and the LNA $(\Gamma_{L})$ shown in the top left panel of Figure 6. We also measure the magnitude and phases of the complex voltage gains of the LNA as well as the phase imbalances $\Phi_{\Sigma12}$ and $\Phi_{\Delta12}$ between the paths of the power splitter. $C_{sky}, C_{lna}, C_{ref}$ computed from these measurements are shown in Figure 8. Given the spectral characteristics of $C_{lna}, C_{ref}$ , we assess the detectability of the reported absorption profile in Section 6.

Figure 8. The residual uncalibrated bandpass responses $C_{sky}, C_{lna}, C_{ref}$ computed using the measurements of the RF components. Blue lines show the responses When the antenna is connected to the system input. The orange lines show the responses when the antenna input is connected to an open load. In this case, both the upward travelling receiver noise and the reference noise are entirely reflected back into the system with a roundtrip phase delay. $C_{sky} = 0$ as $G_{a}=0$ when the antenna input is open. $C_{ref}(r)$ and $C_{ref}(m)$ are the real and the imaginary parts of the $C_{ref}$ .

The remaining system bandpass parameter $C_{sky}$ represents the combined frequency response of the antenna and any external chromatic effects, such as the multipath propagation of the sky noise in the surroundings that remain uncalibrated. Antenna frequency response $G_{a}$ is composed of two different effects. 1. Variation of the angular beam size with the frequency results in receiving the sky power from different areas of the sky at different frequencies. 2. Efficiency with which the antenna couples the sky power into the system—represented by the factor $(1-|\Gamma_{a}|^{2})$ that represents how much of the received power is actually coupled to the receiver. $|\Gamma_{a}|^{2}$ represents the return loss of power at the antenna terminal. In reality, when an antenna is placed over the ground and $|\Gamma_{a}|^{2}$ is measured, the measurement also includes any multipath propagation effects, including ground reflections and reflections from the immediate surroundings. For a wideband antenna with a beam that covers roughly 30% of the sky at any time, there is no calibrator source in the sky comparable to the beam that can increase the system temperature significantly and give a “Cal-on” measurement like a point source calibration. Therefore, it is impossible to be able to measure or calibrate the beam-integrated frequency response due to the absence of any calibrator source in the far field. However, measurement of the antenna return loss can be used to assess the capacity of the antenna for this detection.

5.1. Bandpass calibration

Averaging the ‘OBS’ and ‘CAL’ spectra in two switch positions results in $P_{OBS}, P_{CAL}$ as defined by (50), (51). The system’s response to the calibration noise is derived by subtracting (50) from (51). The bandpass calibrated spectrum of the 50 Ohm termination is computed as $ P_{meas}(50\,{\rm Ohm}) = {P_{OBS}\over P_{CAL}-P_{OBS}} $ . For each set of 4 spectra measured in 4 states of the system, we obtain one bandpass calibrated spectrum $P_{meas}$ .

5.2. Absolute temperature calibration

Bandpass calibrated spectrum is to be multiplied by the system calibration temperature $T_{cal}$ to obtain the spectrum in the temperature unit. To determine $T_{cal}$ , we measure the spectrum of the precision 50 Ohm termination keeping it at different physical temperatures. A platinum resistance thermometer monitors the physical temperatures of both input 50 Ohm and the reference 50 Ohm. Physical temperatures are also the noise temperatures of the 50 Ohms since they are ‘matched’ to the system. The matched termination is first immersed in ice as a ‘cold’ load corresponding to a physical temperature of $T_{cold}$ and then in ‘hot’ water as a ‘hot’ load corresponding to a physical temperature of $T_{cold}$ . From these measurements, the system calibration temperature $T_{cal}$ is given by

(55) \begin{equation}T_{cal}= {T_{hot}-T_{cold}\over P_{50hot}-P_{50cold} }\end{equation}

We multiply the $P_{meas}(50)$ by $T_{cal}$ we obtain bandpass calibrated spectrum in temperature unit as shown in Figure 10.

6.1. Antenna input terminated with a 50 Ohm

In this case, in Equation (54), $\Gamma_{a} \approx 0$ and $T_{sky} = T_{amb} = T_{ref}$ . $C_{sky} = |(1+ \Gamma_{D})|^{2}, C_{lna} =1, C_{ref} = { e^{-i\Delta\Phi} \over |G|^{2} } $

(56) \begin{eqnarray}T_{meas} & = & C_{sky} T_{sky} + C_{lna} T_{lna} + C_{ref} T_{ref}\nonumber\\[5pt] & = &|(1+ \Gamma_{D})|^{2} T_{amb}+T_{lna} - { e^{-i\Delta\Phi} \over |G|^{2} } T_{amb}\end{eqnarray}

Since $\Gamma_{a}\approx 0 $ , the contribution from the isolated port of the directional coupler is negligible. Instead, the real part of the measured spectra is contributed by the thermal noise from the 50 Ohm input termination at ambient temperature, noise from the first LNA and the reference 50 Ohm thermal load. Since $\Gamma_{a} \approx 0$ noise from the terminated port of the directional coupler does not contribute to this $T_{meas}$ . We refer to the total noise from the first LNA and the reference thermal load as the ‘receiver noise’ from hereon. The $S_{11}$ of the termination is at the level of −48 dB. Any upward travelling receiver noise will return with a 48 dB attenuation. A receiver noise of $T_{lna}=400$ K will result in a reflected signal amplitude of <6 mK. Therefore, receiver reflection from the antenna input can be neglected in this configuration. The reference noise temperature $T_{ref}=300$ K referred to the input of the first LNA is $\approx$ 2 K. The imaginary part of $T_{meas}$ is contributed by the reference noise alone. For a 300 K ambient temperature and a 400 K receiver noise, $T_{meas} \approx 700$ K. Since the reference and the input noises are thermal, any additional spectral shape in this measurement is inherently attributed to the intrinsic noise from the LNA.

6.2. Antenna input terminated with an open load

In the next step, we replace the 50 Ohm termination at the input of the antenna with an open load. Any upward travelling receiver noise will be fully reflected from the antenna input, i.e., $\Gamma_{a}=1$ . In this case, in Equation (54), $\Gamma_{a} \approx 1$ and $T_{sky} = 0$ . $T_{iso}=T_{ref}$

(57) \begin{eqnarray}T_{meas} & = & { |(1+ \alpha \Gamma_{L} e^{i\Phi})|^{2} \over |(1+ \Gamma_{L} e^{i\Phi})|^{2}}T_{lna} \nonumber\\[5pt] && + [ | (1+ \Gamma_{D}) \Gamma_{iso}|^{2}- { e^{-i\Delta\Phi} \over |G|^{2} |(1+ \Gamma_{L} e^{i\Phi})|^{2}} ] T_{ref}\end{eqnarray}

The real part of the bandpass calibrated spectrum is contributed by the receiver and reference noise, whereas the imaginary part is contributed by the reference noise alone. Once again, since the reference is a thermal noise, any additional spectral shape in this measurement is inherently attributed to the intrinsic noise from the LNA.

6.3. Principal Component Analysis

Principal Component Analysis (PCA) is a method to find the fundamental ‘shapes’ present in a dataset without prior assumptions of their exact form. Principal Components are the orthogonal basis vectors that describe a dataset. Unlike describing a data set of multiple variables with chosen functional forms such as power law or polynomials, PCA first computes and characterises the variability in the data. A comprehensive description of PCA can be found in Jolliffe & Cadima (Reference Jolliffe and Cadima2016)

We consider each channel power $T_{meas}(\nu)$ as one variable in this work. The sample variance of $T_{meas}(\nu_{i})$ at each channel ‘i’ is described by the radiometer equation and is determined by the system noise. Ideally, each channel measurement is independent of the other. Co-variability of measured $T_{meas}$ between $\nu_{i}, \nu_{j}$ shows the underlying spectral shape. Intrinsically, the covariance between two channels is contributed by $T_{sky}$ . However, system-generated chromaticity, for example, quantities like $C_{sky}, C_{lna}, C_{ref}$ generate additional co-variability between a pair of channels. We detect these variabilities in the $T_{meas}$ to assess the spectral shapes contributed by the system.

We take a set of ‘p’ number bandpass calibrated spectra $(p = 200)$ $T_{meas}(\nu,t)$ , measured at ‘n’ frequencies $(n = 4\,096)$ and compute the covariance of power between any two frequencies. $T_{meas}(\nu, t)$ is a data set of pxn dimensional vectors, i.e., a pxn matrix. We drop the variable ‘t’ in this spectral shape analysis for notational simplicity. Each $T_{meas}(\nu_{i})$ is a random variable that represents the measured power at the i’th frequency channel. A spectral shape in the measured data means $T_{meas}(\nu_{i})$ is covariant with $T_{meas}(\nu_{j})$ in a certain way for all ‘i’, ‘j’. There can be multiple orthogonal spectral shapes resulting in multiple covariances. We compute the covariance matrix of $T_{meas}(\nu)$ by,

(58) \begin{equation}C_{ij} = {\sum_{p} T_{meas}(\nu_{i}).T_{meas}(\nu_{j}) \over (p-1)}\end{equation}

where p is the number of measurements of $T_{meas}$ .

Note on mean subtraction: The covariance method is used to find how a pair of random variables vary with each other. These variables can have different scales and units of measurement. So, they are standardised prior to computing the covariances, i.e., their mean is subtracted from all sample values, and their variances weigh each sample. This ensures the computed covariance between a pair of variables is not dominated by the variable with a higher variance. When there is no correlation between two variables, their covariance reveals ‘shape/pattern/trend’ in their relationship. Standardisation is important while comparing variability trends between two variables with different units of measurement and measurement errors. In that case, the covariance matrix is defined as,

(59) \begin{equation}C_{ij} = {\sum_{p}\left [T_{meas}(\nu_{i})-{\unicode[Times]{x03BC}}_{i} \right ].\left [T_{meas}(\nu_{j})-{\unicode[Times]{x03BC}}_{j} \right ] \over (p-1)}\end{equation}

Where ${\unicode[Times]{x03BC}}_{i}$ is the sample mean of $T_{meas}(\nu_{i})$ .

However, when we compare two deterministic variables that are highly correlated with the same unit of measurement and variances, there is information in the differences between their means and variances. Standardisation will destroy this information content. PCA on non-standardised data is done in specific cases. A discussion on this can be found in Kriegsman (Reference Kriegsman2016). Another discussion on when not to use standardisation can be found in STAT505 (2022).

In our case, $T_{meas}$ at all channels has the same unit of power. We investigate how the measured power co-vary systematically between a pair of channels. When a matched load terminates the input at ambient temperature, each channel has the same mean power given by the ambient temperature, and there are no internal reflections from the antenna input. They also have identical measurement errors (determined by the radiometer equation). So, the data are automatically standardised.

When an open load terminates the input, complete reflection of the upward travelling receiver noise creates a spectral ripple. It changes the mean power at each channel by a few tens of Kelvin. The mean power at every frequency (sample mean)—vs—frequency is a shape present in the data. In this case, it is a measure of the system’s internal reflection. Subtracting the sample mean of every channel from each spectrum will destroy this information. When data is not standardised, this shape will be described by one or more Principal Components. Similarly, differences between variances of channel power are also an indicator of additional spectral shapes. Upon complete reflection from the antenna input, the upward travelling receiver noise is added to the downstream signal path. This changes the channel variances. Increased variances show which channels are more affected due to internal reflection and contribute to additional spectral patterns beyond what is described by the mean. Standardisation of channel power will force them to have unit variance and destroy some spectral structures.

Therefore, we do not subtract a sample mean from the data in our analysis to standardise it. Instead, we use the python routine ‘numpy.cov’ that computes a constant mean from the entire dataset and subtracts this value from every measurement prior to computing the covariance matrix. In the absence of higher-order spectral shapes, this only reduces the Eigenvalue of the first Principal Component.

Figure 11. First row: The covariance matrices of $T_{meas}(50\,{\rm Ohm})$ and $T_{meas}(Open)$ computed along the frequency axis. Since the input is thermal noise, the individual channel powers are uncorrelated. If the instantaneous channel power in the i’th channel is high, it is reflected in all $C_{ij}$ resulting in fine vertical lines. The occurrence of such a sample is randomly distributed across all frequencies resulting in fine vertical lines distributed across the frequencies. Second row: The diagonal element of covariance matrices, i.e., the variance of measured power at a given frequency channel. Third row: Eigenvalues of $C_{ij}$ for different standardisations. ‘nsdd’ shows the Eigenvalues that are computed from the Covariance matrix without any standardisation of the data. ‘NumPy sdd’ shows the Eigenvalues when the covariance matrix is computed using ‘numpy.cov’ python routine as shown in the first row, first panel. In this case, a constant mean is subtracted from each measurement prior to computing the covariance matrix. The ‘sdd’ show the Eigen Values computed from the Covariance matrix when the data is fully ‘standardised’, i.e., a sample mean is subtracted from each measurement and weighted by the sample variance of that channel. This reduces the amplitude of all principal components. c Fourth row: Principal Components of $T_{meas}(50\,{\rm Ohm})$ and $T_{meas}(Open)$ as functions of frequency.

Figure 11 first panel shows the $C_{ij}$ computed from a set of $T_{meas}$ when the antenna input is terminated with a 50 Ohm and an open load. The covariance matrix of measured power is symmetric, i.e., $C_{ij}=C_{ji}$ . The second panel shows the diagonal $C_{ii}$ of the covariance matrices which is the variance of measured power at a given frequency. Per channel variance in the bandpass calibrated spectrum can also be calculated as follows. From the radiometer equation, per channel variance in measured $P_{OBS0}, P_{OBS1}, P_{CAL0}, P_{CAL1}$ are given by,

(60) \begin{equation} {\unicode[Times]{x03C3}}_{OBS0} = {\unicode[Times]{x03C3}}_{OBS1} = T_{sys}/(\Delta \nu \Delta \tau)^{1/2}\end{equation}

and,

(61) \begin{equation} {\unicode[Times]{x03C3}}_{CAL0} = {\unicode[Times]{x03C3}}_{CAL1} = T'_{\!\!sys}/(\Delta \nu \Delta \tau)^{1/2}\end{equation}

where, $\Delta \nu)$ is the channel bandwidth and, $\Delta \tau$ is the integration time in each state. Therefore, when averaged over two switched positions,

(62) \begin{equation} {\unicode[Times]{x03C3}}_{OBS} = ({\unicode[Times]{x03C3}}_{OBS0}^{2}+{\unicode[Times]{x03C3}}_{OBS1}^{2})^{1/2} = \sqrt(2)T_{sys}/(\Delta \nu \Delta \tau)^{1/2}\end{equation}

and mean channel power,

(63) \begin{equation} {\unicode[Times]{x03BC}}_{OBS} = T_{sys}\end{equation}

(64) \begin{equation} {\unicode[Times]{x03C3}}_{CAL} = ({\unicode[Times]{x03C3}}_{CAL0}^{2}+{\unicode[Times]{x03C3}}_{CAL1}^{2})^{1/2} = \sqrt(2)T'_{\!\!sys}/(\Delta \nu \Delta \tau)^{1/2}\end{equation}

Per channel variance in the calibration template $P_{CAL}-P_{OBS}$ is given by,

(65) \begin{equation} {\unicode[Times]{x03C3}}_{CAL - OBS} = ({\unicode[Times]{x03C3}}_{CAL}^{2}+{\unicode[Times]{x03C3}}_{OBS}^{2})^{1/2} = \sqrt(2)[(T_{sys}^{' 2}+T_{sys}^{2})/(\Delta \nu \Delta \tau)]^{1/2}\end{equation}

and corresponding mean is,

(66) \begin{equation} {\unicode[Times]{x03BC}}_{CAL - OBS} = (T'_{\!\!sys} -T_{sys})\end{equation}

Each channel of the bandpass calibrated spectrum $Z = {P_{OBS}\over (P_{CAL}-P_{OBS})} = {X \over Y} $ is a ratio of two independent Gaussian random variables. Its distribution is known as the ratio distribution, given by Díaz-Francés & Rubio (Reference Díaz-Francés and Rubio2012),

(67) \begin{eqnarray} p_Z(z) & = & \frac{b(z) \cdot c(z)}{a^3(z)} \frac{1}{\sqrt{2 \pi} {\unicode[Times]{x03C3}}_x {\unicode[Times]{x03C3}}_y} \left[2 \Phi \left( \frac{b(z)}{a(z)}\right) - 1 \right] \nonumber \\[5pt] && + \frac{1}{a^2(z) \cdot \pi {\unicode[Times]{x03C3}}_x {\unicode[Times]{x03C3}}_y } exp{- \frac{1}{2} \left( \frac{{\unicode[Times]{x03BC}}_x^2}{{\unicode[Times]{x03C3}}_x^2} + \frac{{\unicode[Times]{x03BC}}_y^2}{{\unicode[Times]{x03C3}}_y^2} \right)}\end{eqnarray}

where,

(68) \begin{equation}a(z)= \sqrt{\frac{1}{{\unicode[Times]{x03C3}}_x^2} (z)^2 + \frac{1}{{\unicode[Times]{x03C3}}_y^2}} \end{equation}

(69) \begin{equation}b(z)= \frac{{\unicode[Times]{x03BC}}_x }{{\unicode[Times]{x03C3}}_x^2} z + \frac{{\unicode[Times]{x03BC}}_y}{{\unicode[Times]{x03C3}}_y^2} \end{equation}

(70) \begin{equation}c(z)= exp{\frac {1}{2} \frac{b^2(z)}{a^2(z)} - \frac{1}{2} \left( \frac{{\unicode[Times]{x03BC}}_x^2}{{\unicode[Times]{x03C3}}_x^2} + \frac{{\unicode[Times]{x03BC}}_y^2}{{\unicode[Times]{x03C3}}_y^2} \right)}\end{equation}

(71) \begin{equation}\Phi(z)= \int_{-\infty}^{z}\, \frac{1}{\sqrt{2 \pi}} e^{- \frac{1}{2} u^2}\ du\ \end{equation}

If the mean and variance ${\unicode[Times]{x03BC}}_x, {\unicode[Times]{x03BC}}_y$ and ${\unicode[Times]{x03C3}}_{x}$ and ${\unicode[Times]{x03C3}}_{y}$ of the variables X,Y are strictly positive, and the coefficient of variation, defined as $\delta_x ={\unicode[Times]{x03C3}}_x/{\unicode[Times]{x03BC}}_x = 1/(\Delta \mathrm{T}) $ and $\delta_y ={\unicode[Times]{x03C3}}_y/{\unicode[Times]{x03BC}}_y 1/(\Delta \mathrm{T}) $ are both less than unity, the distribution could be approximated as a normal distribution with mean ${\unicode[Times]{x03BC}}_z = {\unicode[Times]{x03BC}}_x/{\unicode[Times]{x03BC}}_y$ and standard deviation ${\unicode[Times]{x03C3}}_z= ({\unicode[Times]{x03BC}}_x / {\unicode[Times]{x03BC}}_y)\sqrt(\delta_x^2+\delta_y^2)$ . These conditions hold true for our measurements.

When a thermal load is attached to the antenna, ${\unicode[Times]{x03BC}}_{OBS} = T_{sys} \approx 700$ K. When a 3 dB ENR calibration noise is added to the system, $T'_{\!\!sys} \approx 2* T_{sys}$ K. Therefore, ${\unicode[Times]{x03BC}}_{CAL-OBS} \approx T_{sys}$ K. Using these values, when the input is terminated with a matched 50 Ohm load at 300 K temperature, accounting for the contribution from all sources of noise in $P_{OBS}$ , per channel variance is expected to be of the order of $\approx$ 340 K. We obtain a slightly higher variance from the measurement, indicating the mean ambient and receiver temperature may be slightly higher than 300, 400 K.

Therefore, when no other system-induced spectral shape is present in the data, channel variances are expected to be Gaussian and of the same order as the covariance between two different channels. When antenna input is terminated with an open, the upward travelling receiver noise is reflected back into the system and added to the signal path downstream. This increases the mean power in each channel. In addition, the systematic increase in the increased variance of power per channel indicates that the reflected wave has an additional spectral shape. Thermal variance can be reduced by averaging multiple measurements of $T_{meas}(\nu,t)$ over time, whereas systematic spectral variances resulting from the spectral shape will not be averaged to zero.

We decompose the covariance matrix into Eigenvalues and Eigenvectors. Principal Components of $T_{meas}(\nu)$ are given by the eigenvectors of the covariance matrix, while the corresponding Eigenvalues give their strength. They are orthogonal basis vectors just like, for example, Fourier Components. The key difference is that, unlike the Fourier Transform, the particular mathematical form of the basis vector is not known as a priory and is determined from the measured $T_{meas}(\nu)$ itself. Our goal is to demonstrate that HYPEREION’s chromaticity instrument does not contain a spectral response identical to the Cosmic Dawn signal detected by EDGES. The Eigenvalues have the same units as the elements of the covariance matrix, i.e., $Kelvin^{2}$ . The first eigenvector of the covariance matrix is the first Principal Component (PC) of $T_{meas}(\nu)$ . It represents the largest and the most dominant spectral shape across frequencies. The successive Principal Components represent the spectral shapes that are orthogonal to the first PC and are present in the data with lower magnitude. The third row shows the first 20 Eigenvalues of $C_{ij}$ . Once again, in a perfect system, only one Principal Component is required for describing $T_{meas}$ . $C_{ij}$ of 50 Ohm shows there is only one Eigenvalue corresponding to the one Principal Component of variation in $T_{meas}$ . All other Eigenvalues are four orders of magnitude lower than the first at a noise level. For $T_{meas}(open)$ , the spectral ripple introduced due to the complete reflection of the upward travelling receiver noise results in a prominent second Eigenvalue.

The fourth panel shows the Eigenvectors as a function of frequency weighted by the corresponding eigenvalues. Notable here is that $T_{meas}(50\,{\rm Ohm})$ spectra contain only one Principal Component that describes the smooth variation of the $T_{meas}(\nu)$ over frequency. Higher-order Principal Components are four orders of magnitude smaller than the ‘0’th Principal Component. $T_{meas}(Open)$ shows the first Principal Component similar to $T_{meas}(50\;Ohm)$ . But the second Principal Component (yellow) is no longer noise-like and separates itself from higher-order Principal Components.

6.4. Interpretation

In an ideal system, in the absence of any internal reflections of the receiver and reference noise from the antenna terminal, $T_{meas}(50\,{\rm Ohm})$ should be described by a constant temperature across all frequencies if the receiver noise is also thermal. In the context of Principal Component Analysis, this implies two things. 1. In such a system, the covariance matrix of $T_{meas}(50\,{\rm Ohm})$ should have only one Eigenvalue of significance. 2. Corresponding Principal Component of $T_{meas}(50\,{\rm Ohm})$ is described by a constant, i.e., by a polynomial of order ‘0’. Figure 11, row 3, the first panel shows, indeed, the covariance matrix has one Eigenvalue of significance corresponding to one Principal Component. Figure 12 top panel shows $T_{meas}(50\,{\rm Ohm})$ when described by the first PC and with successive addition of up to 4th PC to the first PC. The bottom panel shows residual $T_{meas}(50\,{\rm Ohm})$ composed of all higher-order PCs. The successive addition of more PCs only reduces the spectral variance. Note: since we have $n= 4\,096$ frequency channels, there are also 4 096 Principal Components. When we subtract the first and the most significant PC, the residual is composed of 4 095 PCs. When we successively subtract more PCs up to the 4th PC, the residual is composed of 4 094, 4 093, and 4 096 PCs, all of which are random noise. Since there is no information of significance in these PCs, their subtraction only changes the residual noise statistics marginally.

Figure 12. Top panel: Real parts of $T_{meas}(50\,{\rm Ohm})$ , $T_{meas}(open)$ when described by the first PC and with successive addition of up to 4th PC to the first PC. Bottom panel: Residuals after projecting successively increasing number of Principal Components onto $T_{meas}(50\,{\rm Ohm})$ , $T_{meas}(open)$ .

We compare the PCA with the polynomial modelling of $T_{meas}$ . Figure 13 shows fits with orders of polynomials 0 to 4 and the fit residuals. Corresponding histograms are shown in Figure 14. The mean and the variance of the fit residuals remain unchanged when fitted with the polynomial order above 2. This is the minimum order of the polynomial required to describe the receiver noise spectrum in HYPEREION measurements. Notable here is that a polynomial of order 2 is also a maximally smooth function (MSF). Singh et al. (Reference Singh2021), Bevins et al. (Reference Bevins2021) have demonstrated that to recover a Global 21 cm signal from a wideband radio background measurements, the residual system response in the measured data should be modelled as MSF, hardware implementation of which is the greatest challenge faced by experiments that are aiming to detect the global 21 cm signal by precision radio background measurement.

Figure 13. Top panel: Real parts of $T_{meas}(50\,{\rm Ohm})$ , $T_{meas}(open)$ fitted with polynomials of order 0 to 4.Bottom panel: Residuals of the fit. For comparison, the EDGES Cosmic Dawn signal is overlaid with the residuals in the open case.

Figure 14. Histogram of the residuals after polynomials of 0–4 (top to bottom) are fitted to the real parts of $T_{meas}(50\,{\rm Ohm})$ (left), $T_{meas}(Open)$ (right). Units of the x-axis are in Kelvin.

When the antenna input is terminated with an open load, complete reflection of the upward travelling system noise generates an additional Principal Component of significance. Figures 13, 14 show the measured spectra fitted with orders of polynomials 0–4, the fit residuals, and corresponding histograms. We overlay the EDGES’s Cosmic Dawn signal on top of the residual receiver noise after fitting the 3,4th order polynomial in the extreme case of total internal reflection of the receiver noise. Notable here is, in case of joint fitting of the sky spectrum and systematics, polynomials of order 3, 4 will also subsume a part of the Cosmic Dawn signal. This is illustrated and addressed in our foreground simulation work elsewhere. A 3rd-order polynomial is required to describe $T_{meas}(open)$ . Fitting a polynomial of higher order doesn’t improve the residuals. The presence of an additional Principal Component in $T_{meas}(open)$ requires higher-order polynomials to describe the measured data.

The 50 Ohm and an open load spectra describe the behaviour of HYPEREION system under two boundary conditions of ‘no internal reflection’ of receiver noise and ‘complete internal reflection’ of receiver noise. When HYPEREION antenna is connected, the system response will partially reflect the receiver noise from the antenna terminal. Therefore, it will require a polynomial of order between 2–3 to describe the chromatic contribution of $ C_{lna}, C_{ref}, T_{lna}, T_{ref}$ in the measured data. This is the key difference between EDGES’s instrument calibration and that of HYPEREION. In the EDGES’s detection of the Cosmic Dawn signal, the measured spectra are calibrated for receiver noise using five receiver noise parameters that are predetermined from the laboratory measurements prior to observation by terminating the antenna input with open, short and 50 Ohm. Each of these parameters is then fitted with a 7th-order polynomial (Monsalve et al. Reference Monsalve, Rogers, Bowman and Mozdzen2017). These polynomials are then used to calibrate the final spectrum from which the Cosmic Dawn signal is estimated (Bowman et al. Reference Bowman, Rogers, Monsalve, Mozdzen and Mahesh2018). The Cosmic Dawn signal of the form detected by EDGES is described as a flattened Gaussian between 60–94 MHz. The same can be described by a polynomial of order eight or higher, as shown in Figure 15.

Figure 15. Cosmic Dawn signal detected by EDGES described by polynomial basis functions.

Zheng et al. (Reference Zheng2017), de Oliveira-Costa et al. (Reference de Oliveira-Costa2008) have demonstrated that diffused Galactic radio emission, aka radio foreground that constitutes $T_{sky}$ in equation 54, has three Principal Components of variation across the frequencies. They also showed that these Principal Components are best fitted with a polynomial of order 3 in log space. Bernardi, McQuinn, & Greenhill (Reference Bernardi, McQuinn and Greenhill2015), Bowman et al. (Reference Bowman, Rogers, Monsalve, Mozdzen and Mahesh2018) have also shown that polynomials of order 5, 6 improve the foreground residuals below a few 10s of m Kelvin. EDGES measurements require polynomials of order 5th or higher to describe the antenna gain, reflection coefficient and receiver calibration solutions (refer to ‘Extended Data Figure 4 Antenna beam model’, ‘Extended Data Figure 5 Calibration parameter solutions’, ‘Extended Data Figure 6 Raw and processed spectra’, in Bowman et al. (Reference Bowman, Rogers, Monsalve, Mozdzen and Mahesh2018) and the references therein). This is higher than what is needed to describe the sky signal indicating spectral variations in the data are dominated by receiver noise internal to the instrument. Receiver noise calibration by 7th-order polynomials will only leave higher-order variations in the calibrated data. This makes the presence of the signal in the background spectrum ambiguous.

The number of Principal Components required to describe the HYPEREION instrument response is smaller than the number of Principal Components required to describe the radio foreground. These Principal Components also overlap with that of the foreground and are accounted for in the process of foreground subtraction. In reality, HYPEREION receiver will be able to detect any form of the 21 cm Global signal that can be described by polynomials of order higher than what is needed to describe the foreground. When the antenna is connected to the spectrometer, the $T_{meas}(sky)$ will be dominated by the $T_{sky}$ . In addition, $C_{sky}$ will imprint its chromaticity on $T_{meas}$ by modulating $T_{sky}$ . $C_{sky}$ can not be measured in situ or in the lab. It can be computed using electromagnetic simulation of $G_{a}$ and $\Gamma_{a}$ which is outside the scope of this paper and will be presented elsewhere. However, $C_{sky}$ can be estimated from the measured data. Since the receiver and the foreground both can be described by only three Principal Components, any additional Principal Component in $T_{meas}(sky)$ will be generated by the $C_{sky}$ . For a system like HYPEREION that is not limited by the receiver noise reflection inside or outside the system, the number of Principal Components generated by $C_{sky}$ is the factor that determines whether or not the system can detect the Cosmic Dawn signal. The antenna performance of HYPEREION for determining $C_{sky}$ will be discussed elsewhere.