14
CHAPTER 2. FMCW SIGNAL PROCESSING
Figure 2.2: FMCW signal 2D FFT processing
have a different phase shift based on the distance. This information can be
used to find the angle of arrival of the wave and thus angular position of
the target. To achieve that a third FFT can be taken over processed signals
from different antennas. Using a phase comparison mono-pulse technique,
see Figure 2.3, we can find the phase shift between two array antennas.
Figure 2.3: Principle of phase interferometry [1]
If antennas are located in distance d from each other, and the angle of
arrival of waves is θ, we can find the phase difference through Equation 2.21,
where λ is the wavelength of the signal:
∆ϕ =
2πd sin(θ)
λ
(2.21)
Since 2π phase shift equals to λ and the wave that reaches to antenna 1
travels d sin θ more distance, we can find the phase shift associated with
that additional travel distance which will give us the equation above. If we
consider having K number of equally spaced antennas with distance d, we
2.2. MIMO RADAR CONCEPT
15
can rewrite 2.16 as:
x
m
(t
s
, n, k) =
AB
2
cos(2π(
2αR
c
· t
s
+
2f
c
vn
c
· T +
dk sin θ
λ
) +
4πf
c
R
c
) (2.22)
where 0 ≤ k ≤ K − 1 and 1 ≤ n ≤ N , and N is the total number of chirps
per frame.
2.2
MIMO Radar Concept
Multiple input multiple output (MIMO) radar is a type of radar which uses
multiple TX and RX antennas to transmit and receive signals. Each trans-
mitting antenna in the array independently radiates a waveform signal which
is different than the signals radiated from the other antennas. The reflected
signals belonging to each transmitter antenna can be easily separated in the
receiver antennas since orthogonal waveforms are used in the transmission.
This will allow to create a virtual array that contains information from each
transmitting antenna to each receive antenna. Thus, if we have M number
of transmit antennas and K number of receive antennas, we will have M · K
independent transmit and receive antenna pairs in the virtual array by using
only M + K number physical antennas. This characteristic of the MIMO
radar systems results in number of advantages such as increased spatial reso-
lution, increased antenna aperture, higher sensitivity to detect slowly moving
objects [10, 11].
2.2.1
MIMO Signal Model
As stated above, signals transmitted from different TX antennas should be
orthogonal. Orthogonality of the transmitted waveforms can be obtained
by using time-division multiplexing (TDM), frequency-division multiplexing
and spatial coding. In the presented case, TDM method is used which allows
only a single transmitter to transmit at each time. Considering M number
of transmitting antennas and K number of receiving antennas (Figure 2.4),
the transmitting signal from i
th
antenna towards target can be defined as:
x
tx
(t, m) = A cos(µ(t) +
2πd
t
m sin θ
λ
)
(2.23)
where 0 ≤ k ≤ K − 1 and 0 ≤ m ≤ M − 1.
The corresponding received signal at j
th
antenna can be expressed by:
x
rx
(t, m, k) = B cos(µ(t − τ ) +
2πd
t
m sin θ
λ
+
2πd
r
k sin θ
λ
)
(2.24)
16
CHAPTER 2. FMCW SIGNAL PROCESSING
and consequently the difference signal can be written as:
x
m
(t
s
, n, m, k) = cos(2π(
2αR
c
· t
s
+
2f
c
vn
c
· T +
d
t
m sin θ
λ
+
d
r
k sin θ
λ
)) (2.25)
The steering vector represents the set of phase delays experienced by a
plane wave as it reaches each element in an array of sensors. By using the
equations above, we can describe the steering vector of transmitting array
as:
a
t
(θ) = [1, e
−j2πdtsinθ
λ
, e
−j2πdt2 sin θ
λ
, ..., e
−j2πdt(M−1) sin θ
λ
]
T
(2.26)
and the steering vector of receiving array as:
a
r
(θ) = [1, e
−j2πdr sinθ
λ
, e
−j2πdr 2 sin θ
λ
, ..., e
−j2πdr (K−1) sin θ
λ
]
T
(2.27)
Figure 2.4: TX and RX antennas of MIMO radar
The steering vector of the virtual array (Figure 2.5) can be found by
the Kronecker product of the steering vector of transmitting array and the
steering vector of receiving array.
Kronecker product can be thought as
multiplying each element of the first vector with all the elements of the
second vector and concatenate all the multiplication results together to form
one vector. Kronecker product of two vectors sized M × 1 and K × 1, will
result in an M × [K × 1] size vector. Thus, steering vector of the virtual
array can be expressed by:
a
v
(θ) = a
t
(θ) ⊗ a
r
(θ) = [1, e
−j2πdr sinθ
λ
, ..., e
−j2πdt sin θ
λ
, e
−j2π(dt+dr) sin θ
λ
,
..., e
−j2π(dt(M−1)+dr(K−1)) sin θ
λ
]
T
(2.28)
The vector above contains phase delays that waveform experiences in its
path from each transmitting antenna to each receiving antenna. It can be
used to find the angular position of the object which can be expressed as:
P (θ) =
L−1
l=0
X
l
(f ) · a
l
v
(θ) =
M −1
m=0
K−1
k=0
X
m,n
(f ) · e
−j2π(dtm+drk) sin θ
λ
(2.29)