The main criterion for the derivation of the radar equation, is the Free-Space Path Loss (FSPL) equation [91]. In telecommunication systems, the FSPL is defined as the loss in signal strength of an electromagnetic wave that would result from a line of sight path through free space, with no obstacles nearby to cause reflection or diffraction. It takes into account several principles. First of all, it is intrinsic the fact that the energy which is emitted by an isotropic radiator, propagates uniformly in all directions [92]. With this in mind, it can be understood that areas which have the same power density, form spheres that have an area of A = 4⇡R2, around the radiating antenna. However, the power density on the surface of a sphere is inversely proportional to the surface area A, or the square of the radius R, of the sphere. Therefore, considering a transmitter and a receiver, the FSPL equation considers how the power loss between the two, is proportional to their distance.
Additionally, the FSPL takes into account the fact that the loss is proportional to the square of the frequency of the radio signal and the effect of the receiving antenna’s
Chapter 2. Fundamentals of FMCW MIMO Radars 22 aperture, which describes how well an antenna can pick up power from an incoming electromagnetic wave.
From the inverse square law, the spreading out of the electromagnetic energy in free space [93], can be expressed as
S = Pt
(4⇡R)2 (2.1)
where Ptrepresents the transmitted power and R is the distance between the transmitter
and the receiver. The second aspect, with regard to an isotropic antenna and its aperture [94], is the received power, which is defined as
Pr=
Sλ2
4⇡ (2.2)
where λ is the transmitted wavelength. All things considered, the expression for the total loss, seen as the ratio between the transmitted and received power, given by the FSPL equation, is F SP L = Pt Pr = (4⇡R) 2 λ2 = (4⇡Rf )2 c2 (2.3)
where f represents the frequency of the signal and c is the speed of light.
The radar equation [95], takes on from the FSPL equation and considers additional factors, such as the directivity of an antenna and the antenna gain [94]. Since a spherical segment emits equal radiation in all direction, at constant transmit power, if the power radiated is redistributed to provide more radiation in one direction, then this eventually leads to an increase of the power density in direction of the radiation. This effect is called the antenna gain [96]. This gain is obtained by directional radiation of the power emitted by the antenna. The directional power density, can be derived from equation (2.1), as
St= S · G =
Pt
(4⇡R)2 · G (2.4)
where, here as well, R represents the distance between the transmitter and the target. In a radar system, the target detection isn’t only dependent on the power density at the target position, but also on the amount of power which is reflected back towards the radar. In order to determine the useful reflected power, it is necessary to know the so called, radar cross section (RCS) [97], defined as σ. The value of RCS for a target, depends on several elements. Most importantly, the larger the section of a reflecting object, the higher will the be the back reflected power. However, other factors, such as the type of material which the targets is made of, the design and the composition of its surface, will influence the target’s RCS value. Thus, all things considered, the reflected power from a target, can be extracted from equation (2.4) as
Chapter 2. Fundamentals of FMCW MIMO Radars 23 Ptar = Pt 4⇡R2 tx · G · σ (2.5)
where Rtxrepresents the distance from the radar’s transmitter to the target. Dually for
the receiver, it can be obtained that the power density at the radar receiver is
Sr =
Ptar
(4⇡Rrx)2
(2.6)
where Rrx represents the distance between the radar’s receiver and the target. Con-
secutively, considering the effective antenna aperture Aef f and the power density, the
received power at the radar’s receiver is
Pr= Sr· Aef f =
Ptar
(4⇡Rrx)2
· A · Ka (2.7)
where A represents the geometric antenna area and Ka denotes its efficiency. Usually,
the antennas used for the transmit and receive elements of a radar system are the same. Accordingly, it can be assumed that Rrx is equal to Rtx. Thus, considering now, both
the transmitted and received power, it can be seen that
Pr =
PT · G · σ
(4⇡)2· R4 · A · Ka (2.8)
Finally, from antenna theory, the antenna gain can be expressed as [94],
G = 4 · ⇡ · A · Ka
λ2 (2.9)
hence, obtaining the radar’s equation expressed for Pr as,
Pr=
Pt· G2· λ2· σ
R4· (4⇡)3 (2.10)
It is important to notice from this equation, the significant proportional effect of the target’s RCS value σ, together with the antenna gain G and transmit power Pt. Addi-
tionally, it can be also understood that the distance between the radar and the target, has a inversely proportional effect which evolves with the power of 4 of R. Another way to see the radar’s equation is by solving equation (2.10) for the range R, thus, obtaining
R = 4
s
Pt· G2· λ2· σ
Pr· (4⇡)3
Chapter 2. Fundamentals of FMCW MIMO Radars 24 Furthermore, in a radar system, it is important to calculate the maximum detectable range [98]. Usually, parameters such as Pt, G and λ can be regarded as constants, since
these vary in small ranges. However, the radar’s cross section plays an important role in influencing the radar’s equation, as previously mentioned. The RCS can vary from 1 m2 to 40 m2, for a human target or a corner refelctor used as a target, respectively. The lowest received power that can be detected by the radar is defined as Smin. Values
which are lower than this, are masked by the receiver’s noise. Therefore, calculating equation 2.11, with Pr = Smin, yields Rmax, which denotes the maximum detectable
range of the radar system, as shown in the graph of Fig. 2.1.
The above mentioned radar’s equation is the one used in the characterisation of almost all radar system demonstrators. However, it could be further extended to take into consideration, not only the propagation under ideal conditions, but also all the radar’s internal losses, the atmospheric losses and the influence of the earth’s surface [99].
Figure 2.1: Maximum range