In this chapter we will show how the basic radar equation is derived, that describes the conditions under which a target can be detected by a radar. We say “basic” because in the next chapters more advanced techniques to increase the performance of a radar will be introduced. After discussing various forms of the radar equation, we will focus on the losses that appear in the equation, in particular concerning thermal noise. Lastly, we will see how also the earth's horizon contributes to the maximum range at which a radar can work.

To reveal a target, usually, a radar compares the power of the received signal with a certain threshold. If the echo is greater than the threshold, the receiver assumes that there is a target with a certain given probability. The amplitude of the echo of a target strictly depends on its distance from the radar antenna. We can define the Radar Range as the maximum distance at which a given target can be detected with a certain reliability.

We consider a monostatic radar, having $P_I$ radiated peak power (in absence of losses it is equal to $P_T$, peak power in input to the antenna). In case of an ideal isotropic antenna, the power density $D_{ISO}$ at a distance $R$ is:

\begin{equation} D_{ISO}(R) = \frac{P_I}{4\pi R^2} \end{equation}

For a real radar system using a directive antenna, the directivity fo the antenna $G_D$ on the $(\theta,\phi)$ direction is:

\begin{equation} G_D(\theta,\phi) = \lim_{R\to\infty} \frac{D_T(R,\theta,\varphi)}{D_{ISO}(R)} \end{equation}

Where $D_T(R,\theta,\varphi)$ is the power density (computed without losses) measured on the direction of propagation.

We want to calculate the power density starting from the power in input to the antenna, the actual radiated power will be attenuated by a factor $L_A$ (loss coefficient, $L_A>1$).
The antenna gain can thus be defined as:

\begin{equation} G_T(\theta,\phi) = \frac{1}{L_A}G_D(\theta,\phi) \end{equation}

Writing $G_D(\theta,\phi)$ in explicit form:

\begin{equation} G_T(\theta,\phi) = \lim_{R\to\infty} 4\pi R^2\frac{D_T(\theta,\phi)}{L_AP_I} \end{equation}

$L_AP_I$ is the power given to the antenna. The radiated power is $\frac{P_T}{L_A}$.
The limit shows that the formulas hold when we are in the far field, so the distance between the antenna and the target should be greater than:

\begin{equation} R_L = \frac{l^2}{\lambda} \end{equation}

Where $l$ is the characteristic dimension of the antenna (i.e. its diameter) and $\lambda$ the wavelength.

The antenna directivity is the ratio between the power density measured in a certain point in space and the power density that an isotropic antenna would generate in the same point. $G_T$ can be written as:

\begin{equation} G_T = G_T(\theta,\phi) \qquad for \qquad \theta = \theta_{MAX},\phi = \phi_{MAX} \end{equation}

If the target is observed on the direction of maximum directivity, the power density that reaches the target is:

\begin{equation} \frac{D_T}{L_A} = \frac{P_T}{4\pi R^2}G_T \end{equation}

A part of the incident power on the target is backscattered towards the radar antenna. To characterize the capacity of a target to reflect the signal back to the receiver the Radar Cross Section (RCS) is defined. The RCS is measured in $m^2$. In the case of a monostatic radar we consider the Back-Scatter RCS (also $\sigma$), it can be considered as the surface of an equivalent lossless reflector.
In general, a target that reflects power can be seen as an antenna characterized by a given radiation pattern. The Back-Scatter RCS depends on the power reradiated towards the radar:

\begin{equation} P_0 = \frac{D_T}{L_A}\sigma \end{equation}

Substituting (7) in (8), the reradiated power is:

\begin{equation} P_0 = \frac{P_TG_T}{4\pi R^2}\sigma \end{equation}

The RCS depends on the direction of the incident wave on the target:

\begin{equation} \sigma = \sigma(\theta_{incident},\phi_{incident},\theta_{reflected-towards-RX}, \phi_{reflected-towards-RX}) \end{equation}

To calculate the RCS the incident and reflected fields' amplitude must be known:

\begin{equation} \sigma = \lim_{R\to\infty} 4\pi R^2\frac{|E_r|^2}{|E_i|^2} \end{equation}

$|E_i|$ is the modulus of the incident field, $|E_r|$ is the modulus of the reflected field. The limit tells that the formula holds only in the far field.

Once the reradiated power has been defined, we can also compute the power density reaching the receiver. In case of a monostatic radar at a distance $R$ from the target:

\begin{equation} D_R = \frac{P_0}{4\pi R^2} \end{equation}

To describe the power received from the radar antenna the equivalent area ($A_E$) is defined. It relates the incident power on the antenna with the power that is absorbed by the receiver. A typical value of the equivalent area is about 0.4-0.7 times the real physical area of the antenna.
The received power can be then expressed depending on $A_E$:

\begin{equation} P_R = D_RA_E = \frac{P_TG_TA_E\sigma}{(4\pi R^2)^2} \end{equation}

Remembering that $A_E$ is a function of the gain $G_R$:

\begin{equation} A_E = \frac{\lambda^2}{4\pi}G_R \end{equation}

The received power of a monostatic radar (Distance $R$ and gain $G$ same for transmitter and receiver, since we are using the same antenna for both transmission and reception) can be rewritten as:

\begin{equation} P_R = \frac{P_TG^2\lambda^2\sigma}{(4\pi)^3R^4} \end{equation}

This formula is known as the “deterministic radar equation”.
From this equation, it is possible to define a maximum range $R_{MAX}$ that depends on a fixed minimum received power $S_{min}$. The noise that affects the transmission is random, in this way the received signal is characterized statistically: the detection of the target at a given distance is not deterministic, but the target is detected with a certain probability. Considering the noise in our equation we can express $S_{min}$ in this way:

\begin{equation} S_{min} = \frac{S_{min}}{P_n}P_n = (SNR)_{min}P_n \end{equation}

Where $(SNR)_{min}$ is the signal to noise ratio related to the minimum detectable signal, $P_n$ is the noise power, expressed as:

\begin{equation} P_n = K_BBT_S \end{equation}

$K_B$ is the Boltzmann constant, $B$ is the equivalent noise band of the receiving chain, $T_S$ is the system noise temperature.
By inverting the deterministic radar equation (15) we can compute the radar range $R_{MAX}$:

\begin{equation} R_{MAX} = \Bigg(\frac{P_TG^2\lambda^2\sigma}{(4\pi)^3S_{min}}\Bigg)^{\frac{1}{4}} \end{equation}

This equation is valid if the signal propagates in free space. In all other cases an attenuation factor $A$ should be considered, usually lower than 1, but not always. Also, all the losses of the radar chain elements can be expressed with another factor $L_{tot} > 1$, the losses on the antenna are already considered in the gain $G$. The radar range equation becomes:

\begin{equation} R_{MAX} = \Bigg(\frac{P_TG^2\lambda^2\sigma A}{(4\pi)^3L_{tot}S_{min}}\Bigg)^{\frac{1}{4}} \end{equation}

If considering $A = L_{tot} = 1$

\begin{equation} R_{MAX}^4 = \frac{P_TG^2\lambda^2\sigma}{(4\pi)^3S_{min}} \end{equation}

$R_{MAX}$ is proportional to the square root of $\lambda$. If the antenna gain $G$ is fixed, it is then convenient to use low frequencies. In an avionic radar, the physical area of the antenna is usually fixed, assuming the dimension of the antenna much greater than the wavelength:

\begin{equation} A_E = \eta A_G \end{equation}

$A_E$ is the equivalent area, $\eta$ is the antenna efficiency, $A_G$ the geometric area.
Since $A_E$ can be expressed as:

\begin{equation} A_E = \frac{\lambda^2}{4\pi}G \end{equation}

The radar equation for an avionic radar becomes:

\begin{equation} R_{MAX}^4 = \frac{P_TA_E^2\sigma}{(4\pi)\lambda^2S_{min}} \end{equation}

In this other case $R_{MAX}$ is proportional to the inverse of the square root of $\lambda$, it is then convenient to use high frequencies.

By fixing the solid angle $\Omega_s$ to be scanned by the radar, it is possible to delete the wavelength from the radar equation. Once $\Omega_s$ is fixed, also the scanning time $t_s$ and the dwell time $t_D$ are to be defined. Meaning that the radar needs to scan the entire $\Omega_s$ in the time $t_s$ and point at a single target for the time $t_D$. We can say with a certain approximation that $\Omega_s = \Theta_{az}\Theta_{el}$, where $\Theta_{az},\Theta_{el}$ are the azimuth and elevation angles.
If the radiation pattern of the radar antenna has a beamwidth at -3dB of amplitude $\Omega_0$, the number of positions occupied by the beam in a time $t_s$ is:

\begin{equation} N_{pos} = \frac{t_s}{t_D} = \frac{\Omega_s}{\Omega_0} \end{equation}

Since the antenna gain is defined with respect to that one of an isotropic antenna, assuming the radiation intensity constant inside $\Omega_0$:

\begin{equation} \Omega_0 = \frac{4\pi}{G} \cong \theta_B\varphi_B \end{equation}

The $4\pi$ coefficient is used if the radiation pattern is rectangular, for a gaussian radiation pattern the coefficient becomes $\pi^2$. To increase the gain, $\Omega_0$ shall be reduced, having an increase in the number of positions $N_{pos}$.

Not considering losses, the gain is:

\begin{equation} G = \frac{4\pi}{\Omega_0} = \frac{4\pi}{\Omega_0}\frac{\Omega_s}{\Omega_s} = \frac{4\pi}{\Omega_s}\frac{t_s}{t_D} \end{equation}

Transmitted power is:

\begin{equation} P_T = P_{avg}\frac{t_D}{t_{ill}} \end{equation}

Where $t_{ill}$ is the effective illumination time on the target, see Figure 2.

Being $T$ the pulse repetition time and $\tau$ their duration:

\begin{equation} t_{ill} = N\tau \end{equation}

$t_{ill}$ is the time in which the target receives power emitted from the radar at each scan, while $N$ is the number of pulses during the dwell time $t_D$. $N$ is:

\begin{equation} N = \frac{t_D}{T} \end{equation}

We call duty cycle $d$ the ratio between average power and peak power, it is computed as:

\begin{equation} d = \frac{N\tau}{t_D} \end{equation}

The peak power:

\begin{equation} P_T = P_{avg}\frac{t_D}{N\tau} \end{equation}

If the transmitted pulses are rectangular of duration $\tau$, we can say that $B = 1/\tau$. Having the radar range in free space as:

\begin{equation} R_{MAX}^4 = \frac{P_TG_TA_E\sigma}{(4\pi)^2S_{min}} \end{equation}

Substituting (26), (27), (31), we obtain:

\begin{equation} R_{MAX}^4 = \frac{P_{avg}}{(4\pi)^2}\frac{t_DA_E4\pi t_s\sigma}{N\tau\Omega_st_DSNR_{min}k_BBT_s} \end{equation}

Considering $B\tau = 1$ (matched filter):

\begin{equation} R_{MAX}^4 = \frac{P_{avg}}{(4\pi)}\frac{A_E\sigma t_s}{N\Omega_sSNR_{min}k_BT_s} \end{equation}

This is called the “surveillance radar equation”. $P_{avg}A_E$ is called power factor of the antenna aperture. $R_{MAX}$ does not depend on $\lambda$.

Some parameters that appear inside the radar equation (like the RCS) are characterized in a statistical way. Since the decision on whether the target is present or not is taken by comparing the received signal's amplitude with a certain threshold, the decision cannot be taken with absolute certainty. The radar range shall thus be defined statistically.
Given an object capable of reflecting the signal generated by the radar, we should define the probability of detection of the target ($p_D$) and the probability of false alarm ($p_{fa}$) to fully characterize the radar. The radar range is then defined as the maximum distance under which given requirements of $p_{fa}$ and probability of error ($1 - p_D$) are satisfied.
Inside the radar chain, the decision is taken by comparing the signal's amplitude ($V$) with a threshold ($V_T$). If $V > V_T$ the target is assumed to be present, if $V < V_T$ the target is assumed to not be present. So doing there can be two kinds of errors:

1. Missed detection: $V < V_T$, but the target is present;
2. False alarm: $V > V_T$, but the target is not present.

We define $H_0$ the hypothesis of absent target and $H_1$ the hypothesis of present target, the probability of false alarm is: $p_{fa} = P[V > V_T|H_0]$; the probability of detection is: $p_D = P[V > V_T|H_1]$ (see Table 1).

In the $H_0$ hypothesis (only noise received) and without pulse integration, the decision is taken on a pulse of duration $\tau$, then the average duration of a false alarm is equal to $\tau$, being $B$ the band of the receiving chain adapted to the pulse duration, the average duration of a false alarm is:

\begin{equation} t_{fa} \approx \frac{1}{B} \end{equation}

From Figure 3, for an ergodic case, the relation between $p_{fa}$ and the mean time between two false alarms ($T_{fa}$) is:

\begin{equation} p_{fa} = \frac{t_{fa}}{T_{fa}} \cong \frac{1}{T_{fa}B} \end{equation}

This is realistic when only a single pulse is present inside the beam ($N = PRFt_D \approx 1$).
When there is the integration of more than one pulse ($N > 1$), the average duration of a false alarm increase of $N$ times for the correlation introduced by the integrator:

\begin{equation} t_{fa} = \frac{N}{B} \end{equation}

$T_{fa}$ becomes $N$ times greater than before:

\begin{equation} T_{fa} = \frac{N}{P_{fa}B} \end{equation}

For a radar having $N_R$ resolution cells in distance (sweep of duration $T = N_R\tau = N_R/B$) and $N_A$ resolution cells in azimuth (width of the main lobe $\theta_B = 2\pi/N_A$), the average number of false alarms per second $\frac{1}{T_{fa}}$ is:

\begin{equation} \frac{1}{T_{fa}} = P_{fa}\frac{B}{N} = P_{fa}\frac{1}{N\tau} \end{equation}

In an entire scanning time $t_s$, the total average number of false alarms is:

\begin{equation} P_{fa}N_RN_A \quad \textrm{or} \quad P_{fa}\frac{T}{\tau}\frac{2\pi}{\theta_B} \end{equation}

Knowing that the number of pulses inside the beam is:

\begin{equation} N = \frac{\theta_B}{\dot{\theta}}\frac{1}{T} \end{equation}

and the scanning time:

\begin{equation} t_s = \frac{2\pi}{\dot{\theta}} \end{equation}

The false alarms in time can be represented as Poisson points, the probability of having $k$ false alarms at time $t$:

\begin{equation} p(k,t) = \frac{(\lambda t)^k}{k!}e^{-\lambda t} \end{equation}

With frequency of the events $\lambda$:

\begin{equation} \lambda = \frac{1}{T_{fa}} = \frac{P_{fa}B}{N} \end{equation}

We recall the radar equation for the radar range in free space:

\begin{equation} R_{MAX}^4 = \frac{P_TG^2\lambda^2\sigma}{(4\pi)^3S_{min}} \end{equation}

$S_{min}$ is the minimum signal needed to detect the target with given $p_D$ and $p_{fa}$. $S_{min}$ can be expressed as the product $SNR_{min}P_N$, where $P_N$ is the noise power generated in the receiving subsystem. Figure 4 shows the typical scheme of a radar receiving subsystem.

To calculate the noise power we must take into account the external phenomena like the sun, rain, atmosphere, ground, sea, etc… Inside the radar chain, there are many components that will generate losses. Those introduced by the antenna are quantified by the factor $L_a \geq 1$ and are already taken into account in the radar equation when using the gain instead of the directivity. The transmission line introduces losses quantified by another factor $L_r$. The rotary joint, the duplexer and the transmission line between the duplexer and the amplifier of the RF converter also introduce losses.
The radiofrequency (RF) subsystem of the receiver is composed of a low noise amplifier (LNA) or of a simple mixer. After the LNA there is a mixer that converts the received signal to an intermediate frequency (IF), at that point, there is a high gain amplifier that works in the IF. To calculate the noise power, usually this is referred to the first active component (point A in figure 4), all other upstream elements only introduce losses. The noise is most sensitive to the first element of the chain, that should hence introduce a noise as small as possible, this is why it is likely to use an LNA in that point of the chain.
The signal to noise ratio can be expressed as:

\begin{equation} SNR = \frac{P_TG^2\lambda^2\sigma}{(4\pi)^3R^4P_{NA}} \end{equation}

If it is calculated at the first active element of the chain, losses introduced by the upstream elements of the chain ($L_A$) appear in the equation:

\begin{equation} SNR_A = \frac{P_TG^2\lambda^2\sigma}{(4\pi)^3R^4L_AP_{NA}} \end{equation}

$P_{NA}$ is the noise power computed at the input of the LNA (point A). If the SNR is computed on the point B the term $P_{NB}L_B$ is different and also the losses of the element $L_r$ are to be considered. Typical values of the gain of the LNA are around 20dB, so the noise introduced by the subsequent stages can be neglected.

Blake method's equation is referred to the output of the antenna and is mostly used today. The equation used is:

\begin{equation} R^4_{MAX} = \frac{P_TG^2\lambda^2\sigma}{(4\pi)^3SNR_{min}k_BT_SB_nL} \end{equation}

The loss factor $L$ is:

\begin{equation} L = L_TL_p^2L_{PROP}^2 \end{equation}

Where $L_T$ represents transmission losses, $L_p^2$ 2-way antenna radiation pattern losses, $L_{PROP}^2$ 2-way propagation losses.
The loss $L_r$ does not appear because it is already included in the computation of $T_s$.

The Hall method computation is referred to the first active element of the chain, it uses the formula:

\begin{equation} R_{MAX} = 239.474\sqrt[4]{\frac{P_TG^2\sigma\tau}{f^2T_nSNR_{min}L_TL_rL_p^2L_{PROP}^2}} \end{equation}

$T_n$ is the equivalent noise temperature. The equivalent noise band $B_n$ from the Blake formula is written as $1/\tau$ in this case.
The two methods can be considered equivalent if the antenna noise is negligible.

Figure 5 shows how a real antenna pattern behaves compared with an ideal one. When in the radar equation the gain $G_T$ is considered equal to the max gain $G_{MAX}$ an error is introduced. In fact, during the dwell time, the different pulses on the target will be weighted according to the shape of the real radiation pattern. For a gaussian antenna beam, the losses are of the order of 1.6 dB.

The next figures show the behavior of the attenuation generated by the atmosphere. Figure 6 shows how the attenuation strongly depends on the frequencies where gases present in the atmosphere ($H_2O$ and $O_2$) absorb the energy. Figure 7 shows how the attenuation in free space changes with frequency. To compute atmospheric attenuation depending on the distance we use curves as in Figure 8.

If the range is not known, there is an iterative procedure that can be used to compute the attenuation.

1. If we have a constant specific attenuation $\alpha(R)$ (dB/Km), the attenuation factor is $A(R) = 10^{-\alpha R}$;
2. The range is first computed using the radar equation without attenuation, the obtained range $R_0$ will be greater than the real range;
3. The 2-way attenuation $A_0 = 10^{-2\alpha R_0}$ is computed considering the range $R_0$ (in this case the range is too great, so the attenuation will be higher than reality too), $R_1$ is then computed by substituting the new attenuation factor $A_0$ in the radar equation;
4. $A_1$ is computed again using the new range $R_1$;
5. iterate until $R_i \approx R_{i+1}$.

The convergence of the results to an acceptable value is usually fast, after 2 or 3 iterations the obtained values can be considered optimum.
The noise captured by the antenna also depends on other phenomena in the atmosphere and on cosmic sources. In general, this noise is characterized by simply giving an antenna noise temperature $T_a$, from which the noise power can be computed as $kT_a$ (being $k$ the Boltzmann constant). This under the assumptions that the antenna is ideal, with no losses, and that the radiation pattern has no lobes pointing on the ground. Figure 9 shows how the antenna temperature changes with frequency and elevation angle.

Thermal noise is caused by the Brownian motion of electrons inside a conductor. Thermal noise can be described as a white spectrum process. The tension of thermal noise has a gaussian probability density function with zero mean and standard deviation given by $\sigma^2_n = 4kTRB$. Being $k$ the Boltzmann constant, $R$ the resistance, $T$ the temperature, $B$ the measured bandwidth. The equivalent Thevenin circuit (10) is a noisy voltage generator with an ideal resistor. When attached to another resistor, the transferred power in the frequencies interval given by $B$ is:

\begin{equation} P = \frac{\sigma^2_n}{4R} = kTB \end{equation}

For further reading see: ^[4]

The noise figure represents how noisy a device is.

Figure 11 shows a generic 2-port device, characterised by certain parameters $F,G,B$. $P_{is}$ is the power of the input signal, $P_{in}$ is the power of the input noise, $P_{os}$ and $P_{on}$ are the powers of the output signal and noise. Note that the output noise power is not only increased by the gain of the device, there is also an additive $\Delta P_{on}$ generated by the noisy device. The input noise $P_{in}$ is computed at temperature $T_0 = 290K$. The noise figure $F$ measures how the signal to noise ratio changes between the input port and the output port of the device:

\begin{equation} F = \frac{P_{is}/P_{in}}{P_{os}/P_{on}} \end{equation}

Substituting the expressions of the powers:

\begin{equation} F = 1 + \frac{\Delta P_{on}}{GkT_0B} \end{equation}

$F$ is always greater than 1 (equal to 1 if the device is ideal) and it can be expressed in dB as: $F_{dB} = 10log_{10}(F)$.
Computing $\Delta P_{on}$:

\begin{equation} \Delta P_{on} = (F-1)GkT_0B \end{equation}

The output noise power becomes:

\begin{equation} P_{on} = GkT_0B + (F-1)GkT_0B = FGkT_0B \end{equation}

We can say that the noisy device just analyzed is equivalent to an ideal one with a different temperature of the input noise that is equal to $FT_0$.
The noise power generated internally by a device $\Delta P_{on}$ does not depend on the temperature of the input noise. The equation (54) shows how $\Delta P_{on}$ depends on $T_0$, but the product $(F-1)T_0$ is constant since the noise figure $F$ is computed with respect to a given temperature $T_0$. In fact, we should always specify at which temperature $T_0$ the noise figure is computed. The standard value is $T_0 = 290K$.
If the input noise is at temperature $T_s$, the noise power generated by the device still depends on $T_0$ (at which $F$ is computed), the output noise power is:

\begin{equation} P_{on} = GkT_sB + (F-1)GkT_0B \end{equation}

To characterize the noise introduced by a device, we can define the equivalent noise temperature $T_E$, that depends on the noise figure through the relation:

\begin{equation} T_E = (F-1)T_0 \end{equation}

$T_E$ is the noise produced by the device, comparing the equations (54) and (57) such noise can becomes: $\Delta P_{on} = GkT_EB$. This is the temperature referred to the input terminals of the device, if the noise power in input has a temperature $T_1$, the overall device noise temperature will be $T_1+T_E$. Hence the noise power:

\begin{equation} P_{on} = G(P_{in}P_{E}) = Gk(T_1+T_E)B \end{equation}

When considering two (or more) subsystems in cascade having the same equivalent band $B$, each one of them can be represented as an ideal quadrupole with the same characteristics (but the fact that it is now ideal, having $F=1$) with a source of equivalent noise at the input, see Figure 12.

The noise power in output can be computed as:

\begin{equation} P_{on} = G_1G_2k\Big(T_1+T_{E1}+\frac{T_{E2}}{G_1}\Big)B \end{equation}

$T_1$ is the noise temperature in input to the chain. Considering the equivalent scheme in Figure 13:

\begin{equation} P_{on} = G_1G_2(P_{in}+P_{E}) = G_1G_2k(T_1+T_E)B \end{equation}

$T_E$ is the equivalent system temperature. By putting (59) equal to (60):

\begin{equation} T_E = T_{E1}+\frac{T_{E2}}{G_1} \end{equation}

Hence, the overall noise figure:

\begin{equation} F = 1+\frac{T_E}{T_0} = F_1+\frac{F_2-1}{G_1} \end{equation}

In the case of $N$ subsystems:

\begin{equation} T_E = T_{E1} + \frac{T_{E2}}{G_1} + \frac{T_{E3}}{G_1G_2} + \cdots + \frac{T_{EN}}{G_1G_2\cdots G_{N-1}} \end{equation}

\begin{equation} F = F_1 + \frac{F_2-1}{G_1} + \frac{F_3-1}{G_1G_2} + \cdots + \frac{F_N-1}{G_1G_2\cdots G_{N-1}} \end{equation}

In the case of an attenuator, that can be a transmission line as well as a duplexer, rotary joint or other, we also have a generation of noise if those components are at a temperature $T_p$ greater than zero (and they are for sure). By defining the attenuation introduced by such attenuator as $L=P_{is}/P_{os}$, with $L>1$, the noise power in output is:

\begin{equation} P_{on} = kT_pB\frac{L-1}{L}+\frac{P_{in}}{L} \end{equation}

If the attenuator is adapted in input and in output, the effective input noise temperature is:

\begin{equation} T_E = (L_1)T_p \end{equation}

Hence, the noise figure:

\begin{equation} F = 1+(L-1)\frac{T_p}{T_0} \end{equation}

Also in this case the real attenuator can be represented as an ideal one with in input another noise source at temperature $T_E$.
Note that if $T_p=T_0$ we have that $F=L$. This because while the signal is reduced by a factor $L$, the noise remains the same.

In general, we use the Blake Method, the computation of the chain equivalent temperature $T_E$ must be then referred to the point right after the antenna, see Figure 4. The total system temperature $T_{sys}$ will be equal to $T_A+T_E$, being $T_A$ the antenna noise temperature and $T_E$ the equivalent noise temperature of all the receiving chain after the antenna.

Multipath is the phenomenon by which the propagating waves can reach the same target (or the radar antenna when coming back from the target) via different paths. A typical case is when the waves are reflected by the ground or sea, making the detection of a target at low altitude difficult. A propagation factor is defined to take into account the effects of multipath, it depends on target and radar antenna height, on the elevation angle and the wavelength at which the radar operates. Depending on the phase difference between the waves coming from the direct path and those coming from the reflected path, there can be constructive or destructive interference.

From the antenna, if the reflected signal has a higher power than the direct one, the signal detected will be the image of the target (Figure 15). This can be a problem in the case of a tracking radar that leads to the “nodding” of the radar antenna: the antenna will recognise alternately first the real target and then the image target, causing a fast mechanical movement of the antenna (that can also be quite heavy) in the two directions, that can damage the radar. To overcome this issue tracking radars usually use a vertical polarisation, this makes the direct echo greater and the reflection coefficient lower.

Using a flat earth model (Figure 16a) can be useful to minimize the computation when the radar is working at relatively short distances. When the target is at a great distance we can use the curved earth model (Figure 16b), which can give as result both the slant range (direct distance between antenna and target) and the ground range (curve arc between the points A and B). To compute such distances using the curved earth model also the heights $h_1$ and $h_2$ are to be known.

To take into account the reflected fields when computing the radar range $R_{MAX}$, the propagation factor $F$ is defined. $R_{MAX}^4$ is multiplied by $F^4$, $F^2$ because it is a 2-way, $F^4$ because it is referred to a power. If there are not reflecting surfaces $F = 1$. Here follows an equation to compute $F$:

\begin{equation} F = f(\theta)\Bigg|\sqrt{1+\rho^2+2\rho cos\bigg(\varphi+\frac{2\pi\delta}{\lambda}\bigg)}\Bigg| \end{equation}

$f(\theta)$ is the 1-way antenna gain in voltage, $\rho$ is the reflection coefficient module, $\varphi$ is the reflection coefficient phase. If the target is far the antenna can be approximated with a point, the scheme in Figure 17 that introduces an image of the antenna can be used.

The phase difference between the two paths is:

\begin{equation} \Delta\phi = \frac{2\pi\delta}{\lambda}+\varphi \end{equation}

Where the phase of the ground reflection coefficient $\varphi$ is assumed equal to $-\pi$.
Assuming that the radar is working with plane waves, the path difference is:

\begin{equation} \delta = 2h_a\sin\theta \end{equation}

There is destructive interference when:

\begin{equation} \frac{2\pi}{\lambda}2h_a\sin(\theta) = \pi2n, \quad n \textrm{ integer} \end{equation}

And constructive interference when:

\begin{equation} \frac{4\pi}{\lambda}2h_a\sin(\theta) = (2n+1)\pi, \quad n \textrm{ integer} \end{equation}

The minimum $\theta_0$ that gives $\Delta\phi=0$ (constructive interference) is obtained as:

\begin{equation} \sin(\theta_0) = \frac{\lambda}{4h_a} \end{equation}

Hence:

\begin{equation} \theta_0 = \arcsin\bigg(\frac{\lambda}{4h_a}\bigg) \end{equation}

Figure 18 shows how the interference between the direct and reflected paths depends on the angle $\theta$.

This is called a daisy diagram, the phenomenon is called lobing. It is most important where the reflected signals are stronger, usually for targets at low altitude. In particular, a target that is below $\theta_0$ can be hard to detect and even disappear from the radar, this is why a low wavelength should be used in order to decrease $\theta_0$ in some applications.

The atmosphere has a refractive index that varies with elevation, so that the electromagnetic rays are bent and the usual curved earth model cannot be used. In order to use it considering straight electromagnetic rays the equivalent earth radius $r_e$ is defined:

\begin{equation} r_e = \frac{4}{3}r_{earth} \approx 8500km \end{equation}

$h_A$ is the antenna altitude, $h_B$ is the altitude of a target located at the horizon. Obviously a target located below the horizon cannot be detected even if the computed radar range is greater.
Considering a target on the ground ($h_B=0$), to compute the distance $d$ after which the target cannot be detected we write this equation from Figure 19:

\begin{equation} (r_e + h_A)^2 = d^2 + r_e^2 \end{equation}

If $\quad\quad h_A<<r_e$:

\begin{equation} \textrm{if} \quad h_A << r_e \quad \Rightarrow \quad d \approx \sqrt{2h_Ar_e} \end{equation}

If the target is not on the ground ($h_B \ne 0$):

\begin{equation} d \approx \sqrt{2r_e}(\sqrt{h_A} + \sqrt{h_B}) \end{equation}

Using this model the only way to increase the range of the radar is to place the antenna at a greater altitude. There are other ways to overcome this issue, by reaching somehow targets beyond the horizon using specific wavelengths (usually lower than usual radars) and technologies. The two main technologies used are the Over The Horizon Backscattering (OTH-B), that sends signals to the ionosphere to be reflected towards the ground, and the Over The Horizon Surface Wave (OTH-SW), that uses ground waves to follow the curvature of the Earth thanks to diffraction.

At low frequencies (HF band), signals directed to the sky with a certain angle are reflected by the ionosphere (where the refractive index changes) towards the ground The OTH-B radar can detect signals backscattered from the target back to the ionosphere and reflected again towards the receiver (see Figure 20). The frequencies used for this kind of radar are strictly dependant on the meteorological conditions of the atmosphere. The OTH-B radar then needs a continuous monitoring of such conditions to adapt dynamically to changes.

Adapting the frequencies used by the OTH-B radar as needed, the signals can be reflected by different layers of the ionosphere in order to reach different distances (Figure 21).

Since the wavelengths used by this kind of radar are long, the dimensions of the antennas can be of the order of kilometers and can be achieved by building arrays of smaller antennas, Figure 22 shows the Duga radar array, built by the URSS during the cold war, to track missile launches.

Note that there are two different arrays, one is used as a transmitter and the other as a receiver. The power used by OTH-B radars can reach the order of few MegaWatts (10 MW in the case of the Duga). Those radars could only be built when the technology permitted the development of Doppler radars (see Doppler Radar), since the power backscattered from the target is much lower than the clutter (often given by the sea). These Doppler radars are more complex to design with respect to the usual ones because the movement of the ionosphere generates a shift in the Doppler frequency of the clutter too.

One important characteristic of OTH-B radars is the poor resolution. Nowadays these radars are only used when precision is not needed, being less expensive than satellites. OTH-B radars built during the cold war, instead, were built because there was no better technology at the time, now for such military purposes, the use of satellites has made radars like the Duga obsolete.

Another way of exploiting the change of refractive index id the conduct effect (Figure 23). In particular conditions, in fact, there is an inversion of the behavior of the refractive index, usually decreasing at higher altitudes, that instead starts increasing with the altitude. if the radar antenna is located inside this particular substrate of the atmosphere, the signals transmitted are also contained inside the substrate and can reach distances much greater than usual. In fact, the attenuation is directly proportional to the distance $R$, and not to $R^2$. The conduct effect can only be exploited when particular conditions arise, that depend on the aleatory behavior of the atmosphere. For this, the effect is not reliable.

When using waves in frequencies even lower (between 2 and 20MHz), the diffraction of such waves over the surface of the earth can be exploited in order to build another kind of Over The Horizon radar: the OTH Surface Wave. In fact, the diffracting ground waves follow the curve of the earth's surface and can reach targets located behind the horizon.

This radar is mainly used for marine applications, such as surveillance from the coast. It can be used to detect ships as well as marine currents (see the Bragg effect), winds and other meteorological conditions of the sea.