Measurements

Assume we have a system that measures the total power $P$ received at the output terminal and that input power is measured in temperature scale $T$ (see eq. 2) with an unknown gain ($K_{p}$ ) i.e. we have:

$$P = K_{p} T$$ (44)

Similarly, let's assume that the autocorrelations are on a power scale with a gain $K_{c}$ , different from $K_{p}$ , but the same for auto- and cross- correlations (this point is checked with measurement on a noise source, which is fully correlated for different antennas, so that autocorrelations match the crosscorrelation amplitudes). Then we have:

$$\widetilde{A_{ij}} = K_p T_{ij}$$ (45)

where $T_{ij}$ is the source power at the input port.

In using the chopper wheel method, we place loads with known effective temperature in front of the receiver. For NOEMA, one uses a hot load made of absorber which is at the temperature of the receiver cabin and a cold load located inside the cryostat on the 15K stage ; regular observations are made on the sky:

$$P_{hot} = K_p \left(T_{hot}+T_{rec}\right)$$ (46)

$$P_{cold} = K_p \left(T_{cold}+T_{rec}\right)$$ (47)

$$P_{sky} = K_p \left(T_{sky}+T_{spill}+T_{rec} + T_a \right)$$ (48)

$$C_{ii} = K_c \left(T_{sky}+T_{spill}+T_{rec} + T_a \right)$$ (49)

$$\vert C_{ij}\vert = K_c T_a$$ (50)

The difference between the cross-correlation amplitudes $\vert C_{ij}\vert$ and the autocorrelations $C_{ii}$ is that the "noise" (i.e. $T_{sky}+T_{spill}+T_{rec}$ ) is uncorrelated between different antennas and vanishes from the cross-products.