# Drone SAR Imaging for Monitoring an Active Landslide Adjacent to the M25 at Flint Hall Farm
## 3 Materials and Methods
### Software
#### 3.3.2 Frequency Modulated Continuous Wave (FMCW) Radar  
GNU Radio Companion (GRC) is pre-installed on the E312 and is a freely available and open-source software development toolkit for DSP for implementing SDR through C++ and Python. GRC provides a range of DSP functions as discrete blocks, including filters, math operators, and waveform generators. FMCW is implemented through the DSP functionality of GRC. The lower transmission power of FMCW compared to pulsed Continuous Wave (CW) SAR enables a more efficient radar system for drone deployment where power supplies are limited. Furthermore, with CW radar there is no procedure to accurately time the transmission (Tx) and reception (Rx) signals and convert these into range measurements. Instead, Linear Frequency Modulation (LFM) is used for range compression. This allows measurements to within very small ranges of the target, with the minimal range comparable to the transmitted wavelength. In LFM FMCW, the frequency of the sinusoidal signal increases within the chirp between a given bandwidth. Figure 12 illustrates this modulation technique in the amplitude and frequency domains, with the latter producing a distinctive sawtooth pattern. Figure 12: FMCW modulation: (**a**) amplitude domain; (**b**) frequency domain, where transmission (Tx) is red, and reception (Rx) is green. The distance between the antenna and the target (\(R\)) (m) is determined by the frequency shift of the received signal (Rx) compared to the transmission signal (Tx), which occurs due to the two-way travel delay (\(\Delta t\)) between them:  
\[R=\frac{c_{0|\Delta t|}}{2}=\frac{c_{0|\Delta t|}}{2\Big{(}\frac{d(f)}{d(t)} \Big{)}} \tag{1}\]  
where \(c_{0}\) is the speed of light of \(3.00\times 10^{8}\) m/s, \(\Delta t\) is the delay time (s), \(\Delta f\) is the measured frequency difference (Hz), and \(d(f)/d(t)\) is the frequency shift per unit of time, or the slope (S) in Figure 12b. Figure 13 presents a radar block diagram which underpins the FMCW DSP theory and implementation in GRC. Firstly, a carrier waveform is generated and converted to an analogue signal using a digital to analogue (D/A) converter. In GRC, the Signal Source block is used to define and produce the baseband signal. The frequency is modulated using a Voltage-Controlled Oscillator (VCO), whereby a complex sinusoid of frequencies is produced from a float input stream of control voltages. A bandpass filter controls the desired frequency range. In a traditional hardware system, a physical coupler would be used to branch off some of the transmission signal for mixing with the received signal at a later stage; this is achieved internally within the SDRs and by using the multiply math operator in GRC. Before transmitting to the target, the signal is amplified using the gain and RF settings of the UHD: USRP Sink block, which represents the E312 Tx antenna. Similarly, the low-noise amplifier is encompassed within the UHD: USRP Source block for the E312 Rx antenna. The received and amplified signal is mixed with the transmitted signal, with the mixer output being the frequency difference of the transmit and receive chirp (\(\Delta t\)). A low-pass filter blocks any unwanted frequencies, noise and interference from the system or environment before another amplification stage, if necessary. A Fourier transform converts the time domain signal into the frequency domain. Additional Throttle blocks were incorporated to prevent overrunning and CPU congestion. The FMCW output is an Intermediate Frequency (\(IF\)) (Hz) that the carrier wave has shifted to:  
\[IF=\frac{S2R}{c} \tag{2}\]  
where \(S\) is the signal slope (\(d(f)/d(t)\)), \(R\) is the antenna to target distance (m), and \(c\) is the speed of light. If there are multiple targets, multiple IF tones and therefore multiple frequency peaks will be produced corresponding to the range of each target. An appropriate range resolution is therefore important for discerning closely spaced targets that may otherwise show as one peak on the frequency spectrum.