## 2 Passband and Equivalent Baseband Transfer Functions  
We review the classical theory of passband (radio frequency) and baseband equivalent transfer functions of direct and orthogonal channels in DSB-SC modulation, applied to orthogonal carriers (QPSK). Because for the baseband (modulating) signal and the modulation is linear, we can assume that a sinusoidal signal \(V\cos(2\pi ft)\) modulates the (direct) carrier \(\cos(2\pi f_{c}t)\), i.e., \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Site** & **Latitude N (\({}^{\circ}\))** & **Longitude E (\({}^{\circ}\))** & **Altitude \(H_{S}\)** & **Precipitation Height \(H_{R}\)** & **Rain Rate Data Bank (Years)** \\ \hline Spino d’Adda (Italy) & 45.4 & 9.5 & 0.084 & 3.341 & 8 \\ Madrid (Spain) & 40.4 & 356.3 & 0.630 & 3.001 & 8 \\ Prague (Czech Republic) & 50.0 & 14.5 & 0.250 & 3.051 & 5 \\ Tampa (Florida) & 28.1 & 277.6 & 0.050 & 4.528 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geographical coordinates, altitude (km), rain height \(H_{R}\) (km), and number of years of continuous rain rate measurements at the indicated sites. we can study the transfer function of the equivalent linear direct and orthogonal channels by using the frequency response. Therefore, the modulated signal \(m(t)\) is given by [18]:  
\[m(t)=V\cos(2\pi ft)\times\cos(2\pi f_{c}t)=\frac{V}{2}\cos(\omega_{c}-\omega)t+ \frac{V}{2}\cos(\omega_{c}+\omega)t \tag{1}\]  
In Equation (1), \(\omega_{c}=2\pi f_{c}t\) and \(\omega=2\pi ft\) are the angular frequencies (radians per second). The two addends are the two side-bands at the angular frequency \(\omega_{c}-\omega\) (lower side-band) and \(\omega_{c}+\omega\) (upper side-band). A similar representation holds for the quadrature (orthogonal) carrier \(\sin(2\pi f_{c}t)\) modulated by a sinusoidal signal. In a Cartesian plane in which the phasors (vectors) at angular speed \(\omega_{c}\) are fixed (non-rotating), the two side-bands of Equation (1) can be represented by two vectors rotating in phase counterclockwise (upper side-band, positive rotation) and clockwise (lower side-band, negative rotation) with angular speed \(+\omega\) and \(-\omega\), respectively, as seen in Figure 1[18; 19]. The theory of coherent demodulation followed by a low-pass (base-band) filter (a matched filter) shows that the demodulated signal (direct channel) can be calculated as the sum of the projections of the two side-band vectors on the cosine axis (\(\cos(2\pi f_{c}t)\)), and that the projections on the sine axis give the interfering signal that the cosine channel produces in the orthogonal channel (\(\sin(2\pi f_{c}t)\)) baseband. Of course, the reverse holds also for a signal modulating \(\sin(2\pi f_{c}t)\). If the channel is ideal (e.g., radio frequency transfer function is a complex constant), from Figure 1 we can see that the projections on the cosine axis give the undistorted modulating signal \(\frac{V}{2}\cos(2\pi ft)+\frac{V}{2}\cos(2\pi ft)=V\cos(2\pi ft)\). In the orthogonal channel, the coherent demodulator using \(\sin(2\pi f_{c}t)\) gives a zero-output signal because the two projections on the sine axis of the vectors shown in Figure 1 are equal in amplitude and opposite in sign. Of course, this is the reason why we can use the sine channel to transmit another signal because in these ideal conditions there is no cross-interference. Interference can arise when the radio frequency transfer function is not ideal, as we show next. Let the radio frequency transfer function be:  
\[H(\psi)=H_{r}(\psi)+jH_{i}(\psi) \tag{2}\]  
In Equation (2) \(H_{r}(\psi)\) and \(H_{i}(\psi)\) are the real and imaginary parts of \(H(\psi)\), and \(\psi\) is the generic radio frequency. Figure 1: Cartesian plane in which the phasors (vectors) at angular speed \(\omega_{c}\) are fixed (non-rotating). The two side-bands of Equation (1) can be represented by two vectors rotating in phase counterclockwise (upper side-band, positive rotation) and clockwise (lower side-band, negative rotation) with angular speed \(+\omega\) and \(-\omega\), respectively.