Chapter 2
FMCW Signal Processing
This section consists of two main parts; the first part explains the FMCW
signal processing scheme and the second part introduces the MIMO radar
concept.
2.1
FMCW Signal Analysis
There are several different modulations that are used in FMCW signals such
as sawtooth, triangle and sinusoidal. In our case, we will consider a sawtooth
model of the FMCW signal, seen in the Figure 2.1;
Figure 2.1: FMCW sawtooth signal model
As it can be seen, transmitted frequency increases linearly as a function of
time during Sweep Repetition Period or Sweep Time (T). Starting frequency
is f
c
, which is 79 GHz in our calculations. Frequency at any given time t can
9
10
CHAPTER 2. FMCW SIGNAL PROCESSING
be found by:
f (t) = f
c
+
B
T
t
(2.1)
Here,
B
T
is a chirp rate and can be thought as a “speed” of the frequency
change. We can substitute it with α:
α =
B
T
(2.2)
By using frequency change over time, we can find the instantaneous phase:
µ(t) = 2π
t
0
f (t)dt + µ
0
= 2π(f
c
t +
αt
2
2
) + ϕ
0
(2.3)
Therefore, the transmitted signal in the first sweep, considering ϕ
0
to be the
initial phase of the signal, can be written as:
x
tx
(t) = A cos(µ(t)) = A cos(2π(f
c
t +
αt
2
2
) + ϕ
0
)
(2.4)
The equation above only describes the transmitted signal in the first sweep.
If we want to describe the transmitted signal in the n
th
sweep, a modification
should be made. We can consider t
s
as a time from the start of n
th
sweep
and define t as:
t = nT + t
s
where 0 < t
s
< T
(2.5)
Therefore, our signal form for the transmitted signal in the n
th
sweep be-
comes:
x
tx
(t) = A cos(µ(t)) = A cos(2π(f
c
(nT + t
s
) +
αt
2
s
2
) + ϕ
0
)
(2.6)
Let’s consider an object located at an initial distance of R which is moving
with a relative velocity of v. The returned signal from the object will have
the same form, but with some delay τ which can be defined as:
τ =
2(R + vt)
c
=
2(R + v(nT + t
s
))
c
(2.7)
Considering the delay τ , we can describe the returned signal as:
x
rx
(t) = B cos(µ(t−τ )) = B cos(2π(f
c
(nT +t
s
−τ )+
α(t
s
− τ )
2
2
)+ϕ
0
) (2.8)
According to the FMCW radar principle, the returned signal is mixed with
the transmitted signal:
x
m
(t) = x
tx
(t)x
rx
(t)
(2.9)
2.1. FMCW SIGNAL ANALYSIS
11
The equation above will include cosine multiplication which can be trans-
formed using the trigonometric formula below:
cos(α) cos(β) = (cos(α + β) + cos(α − β))/2
(2.10)
The sum term in our case will have a very high frequency (2 · f
c
= 158GHz)
which will be filtered out. Therefore, the resulting signal will only include
the subtraction term:
x
m
(t) =
AB
2
cos(2π(f
c
(nT + t
s
) +
αt
2
s
2
− f
c
(nT + t
s
− τ ) −
α(t
s
− τ )
2
2
) (2.11)
After simplification we get:
x
m
(t) =
AB
2
cos(2π(f
c
τ + ατ t
s
−
ατ
2
2
))
(2.12)
If we replace τ with its equivalent from Equation 2.7, we will get:
x
m
(t) =
AB
2
cos(2π(f
c
2(R + v(nT + t
s
))
c
+ αt
s
2(R + v(nT + t
s
))
c
−α
4(R + v(nT + t
s
))
2
2c
2
))
(2.13)
We can simplify and write the equation as:
x
m
(t) =
AB
2
cos(2π((
2αR
c
+
2f
c
v
c
+
2αvnT
c
−
4αRv
c
2
−
4αnT v
2
c
2
)t
s
+(
2f
c
v
c
−
4αRv
c
2
)nT +
2f
c
R
c
+
2αvt
2
s
c
−
2αR
2
c
2
−
2αv
2
n
2
T
2
c
2
−
2αv
2
t
2
s
c
2
))
(2.14)
If we look at the Equation 2.14, we see that there is a frequency and a
phase that influences how the signal changes over time. In the literature,
the frequency is usually named as a ”beat frequency”. The difference in
frequency between the transmitted and the received signals is denoted by
f
B
in the Figure 2.1. The above equation shows that the ”beat frequency”
is affected by number of terms such as initial range to the object, object’s
velocity and the chirp number.
According to the Matlab model provided, the following values are used
for the parameters: