carrier frequency of the first pulse is f0, then the nth pulse frequency is: fn - 1 = f0 + ^n + 1 h Tf (5) Frequency (f) In SFCW radars, the time interval between adjacent pulses is called x, while the time interval between two groups of N pulses is Nx, with each group is called a burst. The burst time (Nx) is called the coherent processing interval (CPI). Its concepts are illustrated in Figure 2 [2]. The receiving antenna captures part of the reflected signal, which is then compared with the transmitted signal to extract useful information about the target. Typically, the following four signal parameters are expected to differ between the transmitted and received signal: amplitude, frequency, phase, and polarization. Another major determinant in ensuring radars can extract useful information about the target, is the amount of reflected power captured by the receiver. This factor also determines the maximum radar operating range-the distance below where the radar can correctly detect the target and extract information. The power reflected from the target can be expressed as follows [8]: f1 B Ts Figure 1. Transmitted FMCW signal with varying frequency in the duration, Ts [8]. (6) where Pt is the transmitted signal power, Pref is the reflected power, G t is the gain of the transmitting antenna, v is the radar cross section (RCS) of the target, and R is the distance between the radar and the target. It should be noted that the aforementioned equation is a simplified version which assumes no attenuation exist between the radar and the target due to precipitation, cloud or gases. It also assumes that the angular extent of the target is greater than the radar beam width in both azimuth and elevation planes. The received power is: v Pr = Pt G t G r 2 2A e ^ 4 rR h (7) where A e is the effective area of receiving antenna and G r is its gain. Based on the previous equation, the maximum radar detectable range, R max can be calculated as follows [8]: 1 4 R max = ; Pt G t G2 r vA e E 16r S min (8) where S min is the minimum detectable signal power. If the reflection is received from a moving target, the wave is modulated by the target motion based on the Doppler effect. The phase of the received signal, i, can be written as i= f0 Time (t) 4 rd m (9) where d is the distance to the target and m is the wavelength of the radar signal. The phase noise is significant in the principle of radar-based detection. It is a characteristic of the signal source and is due to the phase fluctuation within the oscillator. Assuming equation (1) is transmitted, the received signal could be written as [1]: d R ^ t h = A r cos c ~ 0 t + 2r ^2d 0 + 2x ^ t hh + z c t - 2 0 mm (10) m c f fN-1 f2 f2 ∆f f1 f1 f0 f0 τ t Nτ Figure 2. Time frequency representation of SFCW waves [2]. 44 Pref = Pt G t v 4 rR 2 IEEE CIRCUITS AND SYSTEMS MAGAZINE where A r is amplitude of received signal, ~ 0 is the oscillation frequency, t is elapsed time, m is the signal wavelength, d 0 is the nominal distance between the target and the radar, x ^ t h is the time varying chest displacement of the target, the term z (t - (2d 0 /c)) is the delayed version of the transmitted phase noise and c is the speed of light. This equation indicates that the phase has been modulated by the chest motion to some extent, and phase demodulation is needed to detect this motion. Moreover, this motion is buried in the phase noise, which may affect the actual phase of the target and, hence, the chest displacement accuracy. When FIRST QUARTER 2021

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