12
CHAPTER 2. FMCW SIGNAL PROCESSING
Parameter
Value
B
1GHz
T
35.6 µs
f
c
79 GHz
c
3 ·10
8
m/s
Number of chirps
96
Number of samples per chirp
1024
Number of Tx antennas
3
Number of Rx antennas
4
Table 2.1: Parameter table
If we assume an object at a distance of 15 m (R = 15) which is moving
with a velocity of 10 m/s (v = 10), and assuming t
s
equal to T and n to be
50, we can find how the individual expressions in the equation affect the final
value of x
m
(t):
x
m
(t) =
AB
2
cos(2π((2.81 · 10
6
+ 5.26 · 10
3
+ 3.33 · 10
3
− 0.1873
−2.22 · 10
−4
)t
s
+ (5260 − 0.19)nT + 7.9 · 10
3
+0.0024 − 0.1404 − 1.97 · 10
−7
− 7.9 · 10
−11
))
(2.15)
Few observations can be made based on the equation above; first, we see
that the values of the expressions
4αRv
c
2
and
4αnT v
2
c
2
are very small and can
easily be neglected. Apart from that, the terms
2f
c
v
c
and
2αvnT
c
are relatively
small and their effect to the main frequency component
2αR
c
can be considered
negligible. Second, other terms which have c
2
in their denominators are also
very small and can be neglected too. Third, the term with t
2
s
,
2αvt
2
s
c
is also
very small (0.0024) and can be neglected as well.
Consequently, x
m
(t) equation can be approximated as:
x
m
(t
s
, n) =
AB
2
cos(2π(
2αR
c
t
s
+
2f
c
vn
c
T ) +
4πf
c
R
c
)
(2.16)
where the term
4πf
c
R
c
is a constant phase term, since R is an initial distance
at which the object is located.
The frequency spectrum of the signal computed over one modulation
period will give us
2αR
c
as a main frequency component which is the beat
frequency. The derivation of the beat frequency is usually based on the Fast
Fourier Transform (FFT) algorithm which efficiently computes the Discrete
Fourier Transform (DFT) of the digital sequence. Consequently, by apply-
ing the FFT algorithm over one signal period, we can easily find the beat
2.1. FMCW SIGNAL ANALYSIS
13
frequency (2.17) and thus the range to the target:
f
b
=
2αR
c
and R =
f
b
c
2α
(2.17)
Range resolution of a radar is the minimum range that the radar can
distinguish two targets on the same bearing [9]. Based on the above equation
and substituting α with Equation 2.2, we can find the range resolution of a
radar. It is based on the fact that the frequency resolution ∆f
b
of the mixed
signal is bounded by the chirp frequency (∆f
b
≥
1
T
) which means that in
order to be able to detect two different objects, the frequency difference of
the mixed signal returned from that objects cannot be smaller than the chirp
frequency. This intuition gives the range resolution which can be found as:
∆f
b
=
2B∆R
c
·
1
T
and ∆R =
c
2B
(2.18)
On the other hand, there is also a phase (
2f
c
v
c
· nT ) associated with the beat
frequency which changes linearly with the number of sweeps. The change of
the phase indicates how the frequency of the signal changes over consequent
number of periods. This change is based on the Doppler frequency shift
which is the shift in frequency that appears as a result of the relative motion
of two objects. The Doppler shift can be used to find the velocity of the
moving object:
f
d
=
2f
c
v
c
and v =
f
d
c
2f
c
(2.19)
The Doppler shift of the signal can be found by looking at the frequency
spectrum of the signal over n consecutive periods (n · T ). In this case, the
FFT algorithm is applied on the outputs of the first FFT. Figure 2.2 describes
this process; first, the row-wise FFT is taken on the time samples, second,
the column-wise FFT is taken on the output of the first FFT. After two
dimensional FFT processing, we have a range-Doppler map which contains
range and velocity information of the target.
Velocity resolution of a radar is the minimum velocity difference between
two targets travelling at the same range of which the radar can distinguish. It
can be found in a similar way as the range resolution. Here, the Doppler fre-
quency change over n chirp durations is bounded by the frequency resolution
(∆f
d
≥
1
nT
). Thus, the velocity resolution can be expressed as:
∆v =
c
2f
c
·
1
nT
(2.20)
Another conclusion that can be drawn from the equation is that if we have
multiple antennas which are separated by some distance, each of them will