## 2 UWB RADAR Setup  
The experiments are performed using the NRM400 UWB radar from TimeDomain [39], which is briefly described here. A block diagram and photography of the measurement system to reproduce a typical scenario (at home or in apnea diagnosis) is shown in Figure 1. The distance between the body and the antennas is about 1 m and the distance between transmitting antenna and receiving antennas is 65 cm. The center frequency of the radar is around 4.3 GHz and its bandwidth is 1.3 GHz. The pulse repetition frequency (PRF) is 10 MHz and a coherent integration scheme of pulses is used (4,096 pulses). Figure 2 shows the typical waveform generated by the radar (Figure 2a). Figure 2b shows the output spectrum. It can be shown that the radar complies with the FCC frequency mask if the transmit power is adjusted to \(-\)14.5 dBm by choosing the correct gain of the internal amplifier of the radar. The delay resolution is 69.7 ps. The time interval for the data transfer between waveforms is limited to 14 Hz (7 Hz per channel). For simplicity, a Vivaldi UWB antenna with frequency range from 1-10 GHz is used in transmission and two commercial UWB antennas from Geozondas model AU-3.1G10.6G-1 (with a frequency range of 3.1-10.6 GHz) are pointed directly towards the subject for reception. The AS186-302 switch from Skyworks with a typical insertion loss of 1 dB at 4 GHz is used to switch between the two receiver antennas. The switch is controlled by a USB serial port using a PIC microcontroller. Therefore, the utilization of the switch allows the use of only one radar. The transmit antenna is located pointing to the center of the bed, and the receiver antennas are located 45 degrees at each side of the bed. The horizontal half-power beam width of the transmit antennas is about 100 \({}^{\circ}\)and the horizontal half-power beam width of the receiver antennas are about 85 \({}^{\circ}\)(directivity of 9 dBi). This configuration of antennas allows illuminating the whole bed for typical distances between 1 m to 3 m.  
Figure 1: UWB radar test setup and photography.  
the range along each received waveform (\(\tau\)) is denoted as \"fast-time\", and is in the order of nanoseconds. Since the radar is composed by an only receiver, the signals collected by the two receiving antennas are interleaved. The process consists on sampling the signal received by the antenna 1, then the receiver is switched to the other antenna and the signal received by the antenna 2 is sampled. Then, the switch changes to the first antenna and the procedure is repeated. The interval between successive received waveforms is \(T_{s}=1/7\) s. The time axis along the measurement interval (\(t\)) is denoted as \"slow-time\", and is in the order of seconds. This means that the sampling frequency in slow time, \(F_{s}=1/T_{s}=7\) Hz, is greater than the Nyquist sampling rate for the breathing signal (bandwidth \(<\)0.7 Hz). In order to compare the periods of apnea with an independent system, the signal from a microphone placed near the nose is sampled simultaneously with a computer sound card to monitor breathing. The microphone can detect breathing even if the person moves but it is an invasive system that can be uncomfortable during sleep. Therefore, it will be only used for comparison purposes.