## 2 Cloud Radar Prototype
### Digital Processing  
This section describes how we obtain a reflectivity profile using the signals acquired by our radar. This is an implementation of the pulse-pair processing algorithm. This procedure reduces the background noise implementing a coherent integration trough time, and it also allows the retrieval of Doppler profiles by comparing the phase of sequential pulses. The same signal processing step is being used in other FMCW cloud radars such as [12; 18]. For this algorithm to work, we need to have correlated pulses. The biggest challenge to this requirement is the lack of a phased-locked-loop for the local oscillator. Analyzing the oscillator's stability, we observed a variation of 8 KHz during 80 \(\upmu\)s, and this was considered acceptably low given the repetition interval of 6.6 \(\upmu\)s.  
Because we have a homodyne receiver, the back-end signal frequency (\(S_{0}\) in Figure 3) equals the frequency difference between transmitted and received signals. This information allows for the retrieval of target's range with Equation (1). The transmitter reference voltage (\(S_{1}\) in Figure 3) will be used to find the start of each repetition period. By only processing the second half of each repetition period, we reduce ambiguous range measurements, in exchange of bandwidth.  
The transmitter reference signal has the shape of a saw-tooth. The discrete derivative determines the slope, thus the saw-tooth rising edges (pulse start) are found by searching for transitions between a negative and positive slope. For each half-period, the Fast Fourier transform (FFT) of the IF signal (\(S_{0}\) in Figure 3) weighted by a Hann window is calculated to find the impulse response (\(R\)) of the IF signal. Thus,  
\[R=FFT\left(IF\cdot Hann\right). \tag{4}\]  
The spectral density function is obtained by multiplying in pairs the sequence of \(N\) radar pulses in frequency (\(R\)) following:  
\[X=\frac{1}{N}\sum_{i=1}^{N-1}conj(R_{i})\cdot R_{i+1}. \tag{5}\]  
Figure 4: The actual radar. In the image, it is possible to see the power supply and the Transmitter and Receiver chains. The container box is a conductor which is usually employed in the implementation of electrical panels. It serves as a structure to attach the components and also provides protection from the environment during field campaigns. During operation, the aperture in front of the antennas is covered with a material that is practically transparent within the Ka-band.  
Then, the module of \(X\) represents the echo received power and the complex argument of \(X\) is proportional to the Doppler velocity, as in [12]. This uncorrected Doppler velocity will be presented alongside power or reflectivity profiles for qualitative analysis.  
A detection threshold is taken as the average power coming from the penultimate range bin plus 2 dB. At that range, we predict just noise because the echo power is expected to be below the minimum-detectable-signal. For the range cells where the echo power is higher than this threshold, we find the radar reflectivity (\(Z\)) using the meteorological radar Equation (3) [17], where \(Z\) in dB is obtained by using \(|X|\) as the \(10\cdot log(P_{r})\), after subtracting the noise mask power.