IEEE Circuits and Systems Magazine - Q1 2021 - 44

```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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IEEE Circuits and Systems Magazine - Q1 2021

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