## 3 Instrument Characteristics and Tests
### Time bandwidth product  
The uncertainty \(\sigma U_{\rm RM}\) of the radiometer output voltage \(U_{\rm RM}\), depends on the product \(B\tau\) of the radiometer effective bandwidth \(B\) and the effective integration time \(\tau\) used to measure \(T_{\rm RM,\;in}\). The low frequency noise \(\sigma U_{\rm PDA}\) of the PDA may also contribute to \(\sigma U_{\rm RM}\). Assuming these two voltage noise contributions are quasi-Gaussian and uncorrelated, the variance \(\sigma U_{\rm RM}\)\({}^{2}\) can then be expressed as:  
\[\sigma U_{\rm RM}^{2}=\frac{G_{\rm RM}^{2}\left(T_{\rm RM,\;in}+T_{\rm RM,\;0 }\right)^{2}}{B\tau}+\sigma U_{\rm PDA}^{2} \tag{12}\]Provided that measurements \(\sigma U_{\rm RM}=\sigma U_{\rm ACS}\) and \(\sigma U_{\rm RM}=\sigma U_{\rm RS}\) for the two different \(T_{\rm RM,in}=T_{\rm ACS}\) and \(T_{\rm RM,in}=T_{\rm RS}\) are available, \(B\tau\) and \(\sigma U_{\rm PDA}\) can be derived by solving the corresponding two equations of the form (12):  
\[B\tau=\frac{G_{\rm RM}^{2}\left[\left(T_{\rm RS}+T_{\rm RM,\,0}\right)^{2}- \left(T_{\rm ACS}+T_{\rm RM,\,0}\right)^{2}\right]}{\sigma U_{\rm RS}^{2}- \sigma U_{\rm ACS}^{2}}\ \ \ {\rm and}\ \ \sigma U_{\rm PDA}^{2}=\sigma U_{\rm RS}^{2}-\frac{G_{\rm RM}^{2}\left(T_{ \rm RS}+T_{\rm RM,\,0}\right)^{2}}{B\tau} \tag{13}\]  
As can be seen in Table 2, the relative difference between \(B\tau\) found for the LSB and the USB are very small (\(\approx\)0.5%), whereas the \(\sigma U_{\rm PDA}\) of the two PDA channels differ significantly. Measurements on the PDA alone revealed \(\sigma U_{\rm PDA}<\) 1 mV, mostly generated by the instrumentation amplifiers (\(\leq\)0.6 mV), but also by noise leakage coupled e.g., through the power supply (\(\leq\)0.3 mV). These measurements are in accordance with \(\sigma U_{\rm PDA}\) given in Table 2. They also explain the significant difference between the USB and the LSB frequency channels. Considering the nominal values for the bandwidth (11 MHz) and the integration time (2.5 ms), the nominal time bandwidth product would be \(B\tau\) = 27,500 Hz s, which is significantly larger than \(B\tau\) actually measured (Table 2). However, this is to be expected as neither the frequency transfer function of a channel (Figure 5a) nor the post detection frequency cut-off of \(f_{\rm LP}\) = 400 Hz (LPa/b) are ideal, which means that the real filter characteristics are not step functions at their band edges, but rather -3 dB values. This implies that the effective channel bandwidth, as well as the effective integration time, are both smaller than the nominal values. However, it is not critical to know the effective bandwidth and integration time precisely since the measured \(B\tau\) determine the measurement uncertainty. Table 3 shows uncertainties \(\sigma U_{\rm RM}\) for three noise temperatures \(T_{\rm in,\,RM}\) = 10 K, 41 K, 313 K at the MA input, corresponding to the approximate values for a sky measurement, the ACS and the RS noise. The \(\sigma U_{\rm RM}\) for \(\tau_{\rm rec}\) = 2.5 ms are uncertainties of single measurements \(U_{\rm RM}\) with the shortest possible integration time (\(<\)2.5 ms), limited by the applied post-process low-pass filtering (\(f_{\rm LP}\) = 400 Hz). The corresponding values are computed with (12) using the parameters \(G_{\rm RM}\), \(T_{\rm RM\,0}\), \(B\tau\), \(\sigma U_{\rm PDA}\) measured (right column in Table 2). The values shown in Table 3 agree well with the standard deviations of all the voltages, \(U_{\rm RM}\) and \(U_{\rm ACS}\), measured (\(\sigma U_{\rm RM}\) = 6.912 mV and \(\sigma U_{\rm ACS}\) = 2.924 mV). Furthermore, the distribution of these voltages closely follows a Gaussian distribution. The \(\sigma U_{\rm RM}\) for \(\tau_{\rm rec}\) = 1 s, 3 s, 10 s given in Table 3 are computed from \(\sigma U_{\rm RM}\) for \(\tau_{\rm rec}\) = 2.5 ms by considering that the standard deviation decays with \(N^{1/2}\), where \(N=f_{\rm LP}\cdot\tau_{\rm rec}\) is approximately the number of independent measurements available in a data record.