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SARAS 3 CD/EoR radiometer: design and performance of the receiver

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Abstract

SARAS is an ongoing experiment aiming to detect the redshifted global 21-cm signal expected from Cosmic Dawn (CD) and the Epoch of Reionization (EoR). Standard cosmological models predict the signal to be present in the redshift range \(z \sim \)6–35, corresponding to a frequency range 40–200 MHz, as a spectral distortion of amplitude 20–200 mK in the 3 K cosmic microwave background. Since the signal might span multiple octaves in frequency, and this frequency range is dominated by strong terrestrial Radio Frequency Interference (RFI) and astrophysical foregrounds of Galactic and Extragalactic origin that are several orders of magnitude greater in brightness temperature, design of a radiometer for measurement of this faint signal is a challenging task. It is critical that the instrumental systematics do not result in additive or multiplicative confusing spectral structures in the measured sky spectrum and thus preclude detection of the weak 21-cm signal. Here we present the system design of the SARAS 3 version of the receiver. New features in the evolved design include Dicke switching, double differencing and optical isolation for improved accuracy in calibration and rejection of additive and multiplicative systematics. We derive and present the measurement equations for the SARAS 3 receiver configuration and calibration scheme, and provide results of laboratory tests performed using various precision terminations that qualify the performance of the radiometer receiver for the science goal.

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Acknowledgements

We thank RRI Electronics Engineering Group, particularly Kasturi S., for their assistance in receiver assembly. We also thank the Mechanical Engineering Group (RRI), led by Mohamed Ibrahim, for construction of chassis and shielding cages for receivers. Gaddam Sindhu took an active role in implementing the receiver. We are grateful to the staff at the Gauribidanur Field Station led by H.A. Ashwathappa for providing excellent support in carrying out field tests and measurements.

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Appendix A: Derivation of the measurement equation for SARAS 3

Appendix A: Derivation of the measurement equation for SARAS 3

The Dicke switch alternately connects the radiometer receiver to the antenna and to a reference load. The reference load is a noise source followed by an attenuation, so that the reference noise temperature may be switched between ambient and high temperature states depending on whether the noise source is on or off, while maintaining the impedance of the reference constant.

We first consider the case in which the Dicke switch is connected to the antenna and define the following terms to describe the noise model:

  • ZN is the input impedance of the low noise amplifier (LNA) and ΓN is the reflection coefficient of the LNA as referred to a Z0 = 50 Ω measuring system impedance.

  • ZA is the input impedance of the antenna and ΓA is the reflection coefficient of the antenna, again assuming a Z0 = 50 Ω impedance for the measuring instrument.

  • G is the power gain of the front-end amplifier in the radiometer.

  • VA is the voltage from the antenna terminals that is coupled into the transmission line, which is assumed to be of Z0 = 50 Ω impedance. VA is a voltage waveform in the transmission line at the LNA input resulting from the coupling of antenna temperature to the line.

  • VN is the voltage of the noise wave generated in the LNA, referred to the amplifier input.

  • f is the fraction of that noise wave voltage that gets coupled in the reverse direction into the Z0 = 50 Ω transmission line connecting the antenna to the amplifier, which is of length l.

The amplifiers connected to the antenna in SARAS 3 are all in a compact module that is followed immediately by an optical modulator and hence is optically isolated from all electronics that follows. Therefore, the total amplification—that of the first low-noise amplifier, a second amplification stage that follows, and the amplifier associated with the optical modulator—may be treated as lumped, referred to as the front-end amplifier, and represented by a single noise wave VN. The analysis parameters are depicted in Fig. 15.

Fig. 15
figure 15

Simplified noise model for the SARAS radiometer, when connected to the antenna

Taking into account the first order reflections of front-end amplifier noise from the antenna, the voltage VS at the input of the amplifier can be written as

$$ \begin{array}{@{}rcl@{}} \mathrm{V}_{\mathrm{S}} &=& \mathrm{V}_{\mathrm{A}} (1+{\Gamma}_{\mathrm{N}}) + \mathrm{V}_{\mathrm{N}}(1+{\Gamma}_{\mathrm{N}}) + \text{fV}_{\mathrm{N}}{\Gamma}_{A}\mathrm{e}^\mathrm{i \phi} (1+{\Gamma}_{\mathrm{N}}) \\ &=& \mathrm{V}_{\mathrm{A}} (1+{\Gamma}_{\mathrm{N}}) + \mathrm{V}_{\mathrm{N}} (1+{\Gamma}_{\mathrm{N}}) [ 1+ \mathrm{f}{\Gamma}_{\mathrm{A}}\mathrm{e}^\mathrm{i \phi} ], \end{array} $$
(20)

where ϕ is the phase difference between the forward propagating wave and the reflected wave due to the finite length l of the transmission line connecting the amplifier and the antenna. ϕ and l are related as ϕ = (4πνl)/(vfc) where c is the speed of light in vacuum and vf is the velocity factor of the transmission line.

Taking into account the power gain of the amplifier, the time-averaged power flow out of the front-end amplifier is

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\mathrm{S}} & = & \left\langle \mathrm{G}~\text{Re}\left( \frac{\mathrm{V}_{\mathrm{S}} \textup{V}_{\mathrm{S}}^{*}}{\mathrm{Z}_{\mathrm{N}}} \right) \right\rangle \\ & = & \text{G~Re} \{\mathrm{P}_{\mathrm{A}} [1-2\text{iIm}({\Gamma}_{\mathrm{N}})-|{\Gamma}_{\mathrm{N}}|^{2}] \\ && + \mathrm{P}_{\mathrm{N}} [1-2\text{iIm}({\Gamma}_{\mathrm{N}})-|{\Gamma}_{\mathrm{N}}|^{2}] [ 1+ \mathrm{f}{\Gamma}_{\mathrm{A}}\mathrm{e}^{\mathrm{i} {\phi}} ] [ 1+ \mathrm{f}^{*}{\Gamma}_{\mathrm{A}}^{*}\mathrm{e}^{-\mathrm{i} \phi} ] \} \\ & = & \mathrm{G}~ (1-|{\Gamma}_{\mathrm{N}}|^{2})\{\mathrm{P}_{\mathrm{A}} + \mathrm{P}_{\mathrm{N}} [1 + 2\text{Re}(\mathrm{f}{\Gamma}_{\mathrm{A}}\mathrm{e}^{i \phi} ) + |\textup{f}|^{2}|{\Gamma}_{A}|^{2}] \}, \end{array} $$
(21)

where the following definitions are used

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\mathrm{A}} = \left\langle \frac{\mathrm{V}_{\mathrm{A}} \mathrm{V}_{\mathrm{A}}^{*}}{\mathrm{Z}_{0}} \right\rangle \end{array} $$
(22)

and

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\mathrm{N}} = \left\langle \frac{\mathrm{V}_{\mathrm{N}} \mathrm{V}_{\mathrm{N}}^{*}}{\mathrm{Z}_{0}} \right\rangle. \end{array} $$
(23)

The relation ZN(1 −ΓN) = Z0(1 + ΓN) is used in the above derivation to express ZN in terms of Z0. Additionally, it may be noted here that PA represents the available power from the antenna: the power corresponding to the antenna temperature that couples into the transmission line, with characteristic impedance Z0, connecting to the receiver. PN corresponds to the receiver noise, referred to the input of the LNA.

The correlation receiver response contains unwanted additives as a result of coupling of any common mode self-generated RFI or noise into the two arms of the correlation receiver. For example, the samplers on the digital receiver board that digitise the analog signals of the two arms would inevitable have common mode noise of the digital board, which results in an unwanted additive component in the response. This additive is expected to be constant in time and we denote the net unwanted common mode response as Pcm. With this power included, the measurements in each of two switch states, POBS00 and POBS11, are:

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{OBS00}} & = & -\mathrm{G}~ (1-|{\Gamma}_{\mathrm{N}}|^{2})\{\mathrm{P}_{\mathrm{A}} + \mathrm{P}_{\mathrm{N}} [1 + 2\text{Re}(\mathrm{f}{\Gamma}_{\mathrm{A}}\mathrm{e}^{i\phi} ) + |\mathrm{f}|^{2}|{\Gamma}_{\mathrm{A}}|^{2}] \} + \mathrm{P}_{\text{corr}} \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{OBS11}} & = & \mathrm{G}~ (1-|{\Gamma}_{\mathrm{N}}|^{2})\{\mathrm{P}_{\mathrm{A}} + \mathrm{P}_{\mathrm{N}} [1 + 2\text{Re}(\mathrm{f}{\Gamma}_{\mathrm{A}}\mathrm{e}^{i\phi} ) + |\mathrm{f}|^{2}|{\Gamma}_{\mathrm{A}}|^{2}] \} + \mathrm{P}_{\text{corr}}. \end{array} $$

Their difference POBS is:

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{OBS}} & = & \mathrm{P}_{\text{OBS11}} - \mathrm{P}_{\text{OBS00}} \\ & = & 2 \mathrm{G}~ (1-|{\Gamma}_{\mathrm{N}}|^{2})\{\mathrm{P}_{\mathrm{A}} + \mathrm{P}_{\mathrm{N}} [1 + 2\text{Re}(\mathrm{f}{\Gamma}_{\mathrm{A}}\mathrm{e}^{\mathrm{i} \phi} ) + |\mathrm{f}|^{2}|{\Gamma}_{\mathrm{A}}|^{2}] \}. \end{array} $$
(24)

Consider the case in which, instead of an antenna, an impedance matched Z0 = 50 Ω calibration noise source is connected. This also serves as an ambient temperature reference termination when the noise source is off. As there is no mismatch between the transmission line and noise source or reference termination, the noise wave from the amplifier that is coupled into the transmission line is absorbed at the calibration noise/reference termination. The analysis parameters in this case are depicted in Fig. 16. For this case, we may write the time-averaged power flowing out of the front-end amplifier as

$$ \begin{array}{@{}rcl@{}} \mathrm{P}^{\prime}_{\mathrm{S}} = \mathrm{G} ~(\mathrm{P}_{\text{REF}}+ \mathrm{P}_{\mathrm{N}})(1 - |{\Gamma}_{\mathrm{N}}|^{2}) \end{array} $$
(25)
Fig. 16
figure 16

Simplified noise model for the SARAS radiometer, when connected to the reference termination and calibration source

when the noise source is off and

$$ \begin{array}{@{}rcl@{}} \mathrm{P}^{\prime\prime}_{\mathrm{S}} = \mathrm{G} ~(\mathrm{P}_{\text{CAL}}+ \mathrm{P}_{\mathrm{N}})(1 - |{\Gamma}_{\mathrm{N}}|^{2}) \end{array} $$
(26)

when the noise source is on. In deriving this, we have used (21) and set ΓA = 0 and replaced PA with PREF or PCAL. PREF represents the noise power from the reference termination that couples into the transmission line, and PCAL that from the termination when the calibration source is on. We may write:

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{REF}} = \left\langle \frac{\mathrm{V}_{\text{REF}} \mathrm{V}_{\text{REF}}^{*}}{\mathrm{Z}_{0}} \right\rangle, \end{array} $$
(27)

where VREF is the noise voltage from the reference port, and

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{CAL}} = \left\langle \frac{\mathrm{V}_{\text{CAL}} \mathrm{V}_{\text{CAL}}^{*}}{Z_{0}} \right\rangle, \end{array} $$
(28)

where VCAL is the noise voltage from the reference port when the calibration noise source is on.

Taking into account the unwanted common-mode noise from the digital boards, which are inevitably added, the measurement data provided by the correlation receiver in each of the states CAL00, CAL01, CAL10 and CAL11 may be written as:

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{CAL00}} = -\mathrm{G} (\mathrm{P}_{\text{REF}}+ \mathrm{P}_{\mathrm{N}})(1 - |{\Gamma}_{\mathrm{N}}|^{2}) + \mathrm{P}_{\text{corr}}, \end{array} $$
(29)
$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{CAL01}} = \mathrm{G} (\mathrm{P}_{\text{REF}}+ \mathrm{P}_{\mathrm{N}})(1 - |{\Gamma}_{\mathrm{N}}|^{2}) + \mathrm{P}_{\text{corr}}, \end{array} $$
(30)
$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{CAL10}} = -\mathrm{G} (\mathrm{P}_{\text{CAL}}+ \mathrm{P}_{\mathrm{N}})(1 - |{\Gamma}_{\mathrm{N}}|^{2}) + \mathrm{P}_{\text{corr}} \end{array} $$
(31)

and

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{CAL11}} = \mathrm{G} (\mathrm{P}_{\text{CAL}}+ \mathrm{P}_{\mathrm{N}})(1 - |{\Gamma}_{\mathrm{N}}|^{2}) + \mathrm{P}_{\text{corr}}. \end{array} $$
(32)

Differencing the measurements recorded in the two switch positions gives

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{CAL0}} &=& \mathrm{P}_{\text{CAL01}} - \mathrm{P}_{\text{CAL00}} \\ &=&2\mathrm{G} (\mathrm{P}_{\text{REF}}+ \mathrm{P}_{\mathrm{N}})(1 - |{\Gamma}_{\mathrm{N}}|^{2}) \end{array} $$
(33)

and

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{CAL1}} &=& \mathrm{P}_{\text{CAL11}} - \mathrm{P}_{\text{CAL10}} \\ &=&2\mathrm{G} (\mathrm{P}_{\text{CAL}}+ \mathrm{P}_{\mathrm{N}})(1 - |{\Gamma}_{\mathrm{N}}|^{2}). \end{array} $$
(34)

The correlation spectrometer thus provides three differenced measurements: POBS corresponding to when the antenna is connected to the receiver, PCAL0 when the reference is connected, and PCAL1 when the calibration noise is on. Together with TSTEP, these yield a calibrated measurement of the antenna temperature:

$$ \begin{array}{@{}rcl@{}} \mathrm{T}_{\text{meas}} & = & \frac{\mathrm{P}_{\text{OBS}}- \mathrm{P}_{\text{CAL0}}}{\mathrm{P}_{\text{CAL1}}- \mathrm{P}_{\text{CAL0}}} \mathrm{T}_{\text{STEP}} \\ &=& \frac{\mathrm{T}_{\text{STEP}} \left[ \mathrm{P}_{\mathrm{A}}-\mathrm{P}_{\text{REF}} + \mathrm{P}_{\mathrm{N}} [2\text{Re}(\mathrm{f}{\Gamma}_{\mathrm{A}}\mathrm{e}^\mathrm{i} \phi) + |\mathrm{f}|^{2}|{\Gamma}_{\mathrm{A}}|^{2}]\right]}{ \left[ (\mathrm{P}_{\text{CAL}}-\mathrm{P}_{\text{REF}}) \right]}. \end{array} $$
(35)

This (35) may be written in the form

$$ \begin{array}{@{}rcl@{}} \mathrm{T}_{\text{meas}} & = & \mathrm{T}_{\text{STEP}}\left[\frac{\mathrm{P}_{\mathrm{A}}-\mathrm{P}_{\text{REF}}} {\mathrm{P}_{\text{CAL}}-\mathrm{P}_{\text{REF}} }\right] \\ &&+ \mathrm{T}_{\text{STEP}}\left[ \frac{\mathrm{P}_{\mathrm{N}}}{\mathrm{P}_{\text{CAL}}-\mathrm{P}_{\text{REF}}} \times \left\{ 2|\mathrm{f}||{\Gamma}_{\mathrm{A}}|\cos(\phi_{\mathrm{f}}+\phi_{\mathrm{A}}+\phi) + |\mathrm{f}|^{2}|{\Gamma}_{\mathrm{A}}|^{2} \right\} \right],\\ \end{array} $$
(36)

where ϕf is the phase associated with the complex f and ϕA is the phase associated with the scattering parameter S11 of the antenna.

So far, we have considered only first order reflection of the LNA noise from the antenna, which introduces sinusoidal standing waves with a single period within the transmission line and, consequently, sinusoidal modulation of the measured spectrum with a single period. However, reflections of the LNA noise as well as the antenna signal that occurs at the input of the LNAs leads to higher order reflections and standing waves in the transmission line. We now proceed to quantify these reflections and associated spectral structure.

We begin with (20) and introduce higher order reflection terms. For clarity, we split the voltage at the input of the LNA into two parts, a part originating in the antenna and a second part corresponding to the LNA noise, and superpose the responses to get the resultant. Since the antenna signal and noise from the LNA are uncorrelated, this separation can be extended to the powers as well.

The voltage due to the antenna, denoted as VSA, may be written as

$$ \begin{array}{@{}rcl@{}} \mathrm{V}_{\text{SA}} &=& \mathrm{V}_{\mathrm{A}} (1+{\Gamma}_{\mathrm{N}}) +\mathrm{V}_{\mathrm{A}}({\Gamma}_{\mathrm{N}} {\Gamma}_{\mathrm{A}}{\mathrm{e}}^{\mathrm{i} {\phi}}) (1+{\Gamma}_{\mathrm{N}}) +\mathrm{V}_{\mathrm{A}}({\Gamma}_{\mathrm{N}}^{2} {{\Gamma}_{\mathrm{A}}^{2}}\mathrm{e}^\mathrm{i {2\phi}}) (1+{\Gamma}_{\mathrm{N}}) .... \\ &&~~~~ + \mathrm{V}_{\mathrm{A}}({{\Gamma}_{\mathrm{N}}^{\mathrm{n}}} {{\Gamma}_{\mathrm{A}}^{\mathrm{n}}}\mathrm{e}^{\mathrm{i} \mathrm{n}\phi}) (1+{\Gamma}_{\mathrm{N}}) + .... \end{array} $$
(37)
$$ \begin{array}{@{}rcl@{}} &=& \mathrm{V}_{\mathrm{A}} {\sum}_{\mathrm{n}=0}^{+\infty}({\Gamma}_{\mathrm{N}}{\Gamma}_{\mathrm{A}}{\mathrm{e}}^{\mathrm{i} \phi})^{\mathrm{n}}(1+{\Gamma}_{\mathrm{N}}). \end{array} $$
(38)

The time averaged power flow out of the system, due to the signal from the antenna, can be written as

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{SA}} & =& \left\langle \mathrm{G}~\text{Re}\left( \frac{\mathrm{V}_{\text{SA}} \mathrm{V}_{\text{SA}}^{*}}{\mathrm{Z}_{\mathrm{N}}} \right) \right\rangle \\ & =& \left\langle \mathrm{G}~\text{Re}\left[ \frac{\mathrm{V}_{\mathrm{A}}\mathrm{V}_{\mathrm{A}}^{*}(1+{\Gamma}_{\mathrm{N}})(1+{\Gamma}_{\mathrm{N}}^{*}) \{ {\sum}_{\mathrm{m}=0}^{+\infty}({\Gamma}_{\mathrm{N}}{\Gamma}_{\mathrm{A}}\mathrm{e}^\mathrm{i \phi})^{\mathrm{m}}\} \{ {\sum}_{\mathrm{n}=0}^{+\infty}({\Gamma}_{\mathrm{N}}^{*}{\Gamma}_{\mathrm{A}}^{*}\mathrm{e}^{-\mathrm{i} \phi})^{\mathrm{n}}\} } {\mathrm{Z}_{\mathrm{N}}} \right] \right\rangle.\\ \end{array} $$
(39)

Using Cauchy product to evaluate the product of the two infinite series, the above expression may be simplified to

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{SA}} &= & \mathrm{G}~\text{Re}\left[ \mathrm{P}_{\mathrm{A}} \{1-2i\text{Im}({\Gamma}_{\mathrm{N}})-|{\Gamma}_{\mathrm{N}}|^{2}\} \left\{\sum\limits_{\mathrm{k}=0}^{+\infty}~|{\Gamma}_{\mathrm{N}}|^{\mathrm{k}} |{\Gamma}_{\mathrm{A}}|^{\mathrm{k}}\right\}\left\{ \sum\limits_{\mathrm{l}=0}^{\mathrm{k}} \mathrm{e}^{\mathrm{i}(2\mathrm{l}-\mathrm{k})(\phi_{\mathrm{N}}+\phi_{\mathrm{A}}+\phi)}\right\} \right] \\ &= & \mathrm{G}~\mathrm{P}_{\mathrm{A}} \sum\limits_{\mathrm{k}=0}^{+\infty}~|{\Gamma}_{\mathrm{N}}|^{\mathrm{k}} |{\Gamma}_{\mathrm{A}}|^{\mathrm{k}} \sum\limits_{l=0}^{\mathrm{k}} \left[{\cos}\{(2\mathrm{l}-\mathrm{k})(\phi_{\mathrm{N}}+\phi_{\mathrm{A}}+\phi)\}(1-|{\Gamma}_{\mathrm{N}}|^{2}) \right. \\ &&\left. +2\text{Im}({\Gamma}_{\mathrm{N}})\sin\{(2\mathrm{l}-\mathrm{k})(\phi_{\mathrm{N}}+\phi_{\mathrm{A}}+\phi)\}\right]. \end{array} $$
(40)

Since the last term containing the sine function is anti-symmetric, the summation of all of the sine terms is zero. Therefore, the expression for the power corresponding to the antenna becomes

$$ \mathrm{P}_{\text{SA}} = \mathrm{G}~\mathrm{P}_{\mathrm{A}} (1-|{\Gamma}_{\mathrm{N}}|^{2}) \sum\limits_{\mathrm{k}=0}^{+\infty}~|{\Gamma}_{\mathrm{N}}|^{\mathrm{k}} |{\Gamma}_{\mathrm{A}}|^{\mathrm{k}} \sum\limits_{\mathrm{l}=0}^{\mathrm{k}} \cos\{(2\mathrm{l}-\mathrm{k})(\phi_{\mathrm{N}}+\phi_{\mathrm{A}}+\phi)\}. $$
(41)

In a similar fashion, we may derive the voltage and power for the additive noise from the front-end amplifier. The voltage originating in the LNA is

$$ \begin{array}{@{}rcl@{}} \mathrm{V}_{\text{SN}} &=& \mathrm{V}_{\mathrm{N}}(1+{\Gamma}_{\mathrm{N}}) + \text{fV}_{\mathrm{N}}{\Gamma}_{\mathrm{A}}\mathrm{e}^{\mathrm{i} {\phi}}(1+{\Gamma}_{\mathrm{N}}) +\text{fV}_{\mathrm{N}}{{\Gamma}_{\mathrm{A}}^{2}} {\Gamma}_{\mathrm{N}}\mathrm{e}^{\mathrm{i} {2\phi}} (1+{\Gamma}_{\mathrm{N}}) .... \\ &&+ \text{fV}_{\mathrm{N}}{\Gamma}_{\mathrm{N}}^{\mathrm{n}-1} {{\Gamma}_{\mathrm{A}}^{\mathrm{n}}}\mathrm{e}^{\mathrm{i} \mathrm{n}\phi} (1+{\Gamma}_{\mathrm{N}}) + .... \end{array} $$
(42)
$$ \begin{array}{@{}rcl@{}} &=& \mathrm{V}_{\mathrm{N}}(1+{\Gamma}_{\mathrm{N}}) \left\{1 +\mathrm{f} \sum\limits_{\mathrm{n}=0}^{+\infty}({\Gamma}_{\mathrm{A}}^{\mathrm{n}+1}{{\Gamma}_{\mathrm{N}}^{\mathrm{n}}}\mathrm{e}^{\mathrm{i}(\mathrm{n}+1)\phi})\right\}. \end{array} $$
(43)

The power due to this voltage is given as

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{SN}} & =& \left\langle \mathrm{G}~\text{Re}\left( \frac{\mathrm{V}_{\text{SN}} \mathrm{V}_{\text{SN}}^{*}}{\mathrm{Z}_{\mathrm{N}}} \right) \right\rangle \\ & =& \mathrm{G}~\text{Re}\left[ \frac{\mathrm{V}_{\mathrm{N}} \mathrm{V}_{\mathrm{N}}^{*}}{\mathrm{Z}_{\mathrm{N}}} (1+{\Gamma}_{\mathrm{N}})(1+{\Gamma}_{\mathrm{N}}^{*})\left\{1 + \sum\limits_{\mathrm{m}=0}^{+\infty}(\mathrm{f}~{\Gamma}_{\mathrm{A}}^{(\mathrm{m}+1)}{{\Gamma}_{\mathrm{N}}^{\mathrm{m}}}\mathrm{e}^{\mathrm{i}(\mathrm{m}+1)\phi})\right\} \right. \\ &&\left. \times \left\{1+ \sum\limits_{\mathrm{n}=0}^{+\infty}(\mathrm{f}^{*}~{\Gamma}_{\mathrm{A}}^{*(\mathrm{n}+1)}{\Gamma}_{\mathrm{N}}^{*\mathrm{n}}\mathrm{e}^{- \mathrm{i}(\mathrm{n}+1)\phi})\right\} \right] \end{array} $$
(44)
$$ \begin{array}{@{}rcl@{}} & =& \mathrm{G}~\text{Re}\left[\mathrm{P}_{\mathrm{N}} \left\{1-2\text{iIm}({\Gamma}_{\mathrm{N}})-|{\Gamma}_{\mathrm{N}}|^{2}\right\} \left\{1 + \sum\limits_{\mathrm{m}=0}^{+\infty}(\mathrm{f}~{\Gamma}_{\mathrm{A}}^{(\mathrm{m}+1)}{{\Gamma}_{\mathrm{N}}^{\mathrm{m}}}\mathrm{e}^{\mathrm{i}(\mathrm{m}+1)\phi})\right\} \right.\\ &&\left.\times \left\{1+ \sum\limits_{\boldsymbol{n}=0}^{+\infty}(\mathrm{f}^{*}~{\Gamma}_{\mathrm{A}}^{*(\mathrm{n}+1)}{\Gamma}_{\mathrm{N}}^{*\mathrm{n}}\mathrm{e}^{-\mathrm{i}(\mathrm{n}+1)\phi})\right\} \right]. \end{array} $$
(45)

The above equation simplifies to

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{SN}} &=& \mathrm{G}~\mathrm{P}_{\mathrm{N}} ~\text{Re}\left[\left\{1-2\text{iIm}({\Gamma}_{\mathrm{N}})-|{\Gamma}_{\mathrm{N}}|^{2}\right\} \left\{1 + \sum\limits_{\mathrm{m}=0}^{+\infty}2\text{Re}(\mathrm{f}~{\Gamma}_{\mathrm{A}}^{\mathrm{m}+1}{{\Gamma}_{\mathrm{N}}^{\mathrm{m}}}\mathrm{e}^{\mathrm{i}(\mathrm{m}+1)\phi}) \right.\right. \\ &&\left.\left.+ |\mathrm{f}|^{2}|{\Gamma}_{\mathrm{A}}|^{2}\sum\limits_{\mathrm{k}=0}^{+\infty}|{\Gamma}_{\mathrm{N}}|^{\mathrm{k}}|{\Gamma}_{\mathrm{A}}|^{\mathrm{k}}\sum\limits_{l=0}^{\mathrm{k}} \mathrm{e}^{\mathrm{i}(2\mathrm{l}-\mathrm{k})(\phi_{\mathrm{N}}+\phi_{\mathrm{A}}+\phi) } \right\} \right]. \end{array} $$
(46)

Expanding terms, identifying m = k, and using arguments similar to that used in derivations above for the case of a single reflection at the antenna, we obtain

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\text{SN}} &=& \mathrm{G}~\mathrm{P}_{\mathrm{N}} (1-|{\Gamma}_{\mathrm{N}}|^{2})\left[1 + \sum\limits_{k=0}^{+\infty} (2|\mathrm{f}||{\Gamma}_{\mathrm{A}}|^{(\mathrm{k}+1)}|{\Gamma}_{\mathrm{N}}|^{\mathrm{k}} \cos\{\phi_{\mathrm{f}} +(\mathrm{k}+1)(\phi_{\mathrm{A}}+\phi) +\mathrm{k}\phi_{\mathrm{N}}\}) \right.\\ &&\left.+ |\mathrm{f}|^{2}|{\Gamma}_{\mathrm{A}}|^{2}\sum\limits_{\mathrm{k}=0}^{+\infty}~|{\Gamma}_{\mathrm{N}}|^{\mathrm{k}} |{\Gamma}_{\mathrm{A}}|^{\mathrm{k}} \sum\limits_{l=0}^{\mathrm{k}} \cos\{(2\mathrm{l}-\mathrm{k})(\phi_{\mathrm{N}}+\phi_{\mathrm{A}}+\phi)\} \right]. \end{array} $$
(47)

The total power flow out of the system can then be expressed as

$$ \begin{array}{@{}rcl@{}} \mathrm{P}_{\mathrm{S}} & =& \mathrm{P}_{\text{SA}} + \mathrm{P}_{\text{SN}} \\ & =& \mathrm{G}(1-|{\Gamma}_{\mathrm{N}}|^{2}) \left[ \mathrm{P}_{\mathrm{N}}\left[1 + \sum\limits_{\mathrm{k}=0}^{+\infty} (2|\mathrm{f}||{\Gamma}_{\mathrm{A}}|^{(\mathrm{k}+1)}|{\Gamma}_{\mathrm{N}}|^{\mathrm{k}} \cos\{\phi_{\mathrm{f}} +(\mathrm{k}+1)(\phi_{\mathrm{A}}+\phi) +\mathrm{k}\phi_{\mathrm{N}}\}) \right.\right.\\ &&\left.+ |\mathrm{f}|^{2}|{\Gamma}_{\mathrm{A}}|^{2}\sum\limits_{\mathrm{k}=0}^{+\infty}~|{\Gamma}_{\mathrm{N}}|^{\mathrm{k}} |{\Gamma}_{\mathrm{A}}|^{\mathrm{k}} \sum\limits_{\mathrm{l}=0}^{\mathrm{k}} \cos\{(2\mathrm{l}-\mathrm{k})(\phi_{\mathrm{N}}+\phi_{\mathrm{A}}+\phi)\} \right] \\ &&\left. \mathrm{P}_{\mathrm{A}} \sum\limits_{\mathrm{k}=0}^{+\infty}~|{\Gamma}_{\mathrm{N}}|^{\mathrm{k}} |{\Gamma}_{\mathrm{A}}|^{\mathrm{k}} \sum\limits_{\mathrm{l}=0}^{\mathrm{k}} \cos\{(2\mathrm{l}-\mathrm{k})(\phi_{\mathrm{N}}+\phi_{\mathrm{A}}+\phi)\} \right] \end{array} $$
(48)

It may be noted here that (33) and (34) for the calibration states remain unchanged since it is assumed here that the reference port is impedance matched to the transmission line and both have impedances Z0; there are no reflections of voltage waveforms at the reference port.

Omitting the pedagogical steps, the calibrated spectrum may thus be written as

$$ \begin{array}{@{}rcl@{}} \mathrm{T}_{\text{meas}} & =& \mathrm{T}_{\text{STEP}}\left\{\frac{\mathrm{P}_{\mathrm{A}} [{\sum}_{\mathrm{k}=0}^{+\infty}~|{\Gamma}_{\mathrm{N}}|{~}^{\mathrm{k}} |{\Gamma}_{\mathrm{A}}|{~}^{\mathrm{k}} {\sum}_{\mathrm{l}=0}^{\mathrm{k}} \cos\{(2\mathrm{l}-\mathrm{k})(\phi_{\mathrm{N}}+\phi_{\mathrm{A}}+\phi)\}]-\mathrm{P}_{\text{REF}}} {\mathrm{P}_{\text{CAL}}-\mathrm{P}_{\text{REF}} } \right.\\ && + \frac{\mathrm{P}_{\mathrm{N}}}{\mathrm{P}_{\text{CAL}}-\mathrm{P}_{\text{REF}}} \times \left[\sum\limits_{\mathrm{k}=0}^{+\infty} (2|\mathrm{f}||{\Gamma}_{\mathrm{A}}|{~}^{(\mathrm{k}+1)}|{\Gamma}_{\mathrm{N}}|{~}^{\mathrm{k}} \cos\{\phi_{\mathrm{f}} +(\mathrm{k}+1)(\phi_{\mathrm{A}}+\phi) +\mathrm{k}\phi_{\mathrm{N}}\})\right. \\ &&\left.\left. + |\mathrm{f}|{~}^{2}|{\Gamma}_{\mathrm{A}}|{~}^{2}{\sum}_{\mathrm{k}=0}^{+\infty}~|{\Gamma}_{\mathrm{N}}|{~}^{\mathrm{k}} |{\Gamma}_{\mathrm{A}}|^{\mathrm{k}} \sum\limits_{l=0}^{\mathrm{k}} \cos\{(2\mathrm{l}-\mathrm{k})(\phi_{\mathrm{N}}+\phi_{\mathrm{A}}+\phi)\} \right]\right\}. \end{array} $$
(49)

If we set k = 0 in the above equation, we recover (36) that represents the measured temperature assuming single reflection at the antenna and neglecting higher order terms.

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T., J.N., Subrahmanyan, R., Somashekar, R. et al. SARAS 3 CD/EoR radiometer: design and performance of the receiver. Exp Astron 51, 193–234 (2021). https://doi.org/10.1007/s10686-020-09697-2

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