\end{array}$$ where the explicit frequency dependence of each term has been dropped and $p_0$ is the spectrum for the ambient load, $p_1$ is the spectrum for the ambient load plus calibration noise, and $p_2$ is the spectrum for the antenna. In this terminology, $g$ is the gain, $T_L$ is the ambient load temperature, $T_R$ is the receiver noise temperature, $T_{cal}$ is the calibration noise temperature, and $T_A$ is the antenna temperature. Thermal uncertainty in the measurements is explicitly included in the Gaussian random variables $n_0$, $n_1$, and $n_2$, the magnitudes of which are given by $n_i =
(\epsilon~b~\tau_i)^{-1/2}$, where $\epsilon=0.5$ is the efficiency for the Blackman-Harris window function (which could be improved to 0.93 by processing two sets of overlapping windows), $b =
122\times10^3$ Hz is the resolution bandwidth, and $\tau_i$ is the integration time in seconds (for each switch position, $i$). Temporarily setting the noise terms to zero, $n_i\rightarrow0$, and solving for the antenna temperature yields $$T_A = T_{cal}\frac{p_2 - p_0}{p_1 - p_0} + T_L.\label{eqn_ta}$$ In practice, the impedance match between the antenna and receiver is not perfect and some of the incident sky noise may be reflected back out of the system. This produces deviations between the derived 