Читать книгу Inverse Synthetic Aperture Radar Imaging With MATLAB Algorithms - Caner Ozdemir - Страница 61

A conceptual sketch of the above events for the bistatic radar configuration is illustrated in Figure 2.6. First, the microwave generator (usually with a magnetron, klystron, or a traveling‐wave‐tube amplifier) produces the radar signal of P_in watts. This signal is transferred to a transmitting antenna through a transmission line or a waveguide. Because of the finite conductivity of the lines and other microwave components, some of the power is lost during the transmission. Therefore, the power delivered to the transmitting antenna is given by

(2.13)

where L₁ is the transmit loss mainly due to finite conductivity of the transmission line or dielectric losses. Once the power is delivered to the antenna, the power radiated by the antenna is

(2.14)

where Γ_tx is the reflection coefficient at the terminal of the antenna and is given by

(2.15)

In the above equation, Z_a is the antenna radiation impedance and Z_o is the characteristic impedance of the transmission line connected to the terminals of the antenna. The loss, 1/(1 − |Γ_tx|²) ≜ L_tx, associated by the impedance mismatch at the transmitter, is called the transmitter loss. Of course, if the antenna is matched, the radiated power will be equal to the power transmitted to the antenna as

Figure 2.6 Geometry for obtaining bistatic radar range equation.

(2.16)

When the power is radiated by the transmitting antenna at a particular direction with the antenna gain of G₁, power density at the range of R₁ (where the target is located) can be found as

(2.17)

The power incident to the target is scattered with an amount of the equivalent echoing area, or simply the RCS of the target. Therefore, the scattered power is then equal to

(2.18)

Then, the power is reradiated by the target as it reaches the receiver antenna located at R₂ distance away from the target. The power density of the scattered power around the receiver becomes

(2.19)

At this stage, a very small amount of power is available to the receiving antenna. Therefore, it is crucial to catch as much power as possible by using a large aperture antenna. This implies that it is preferable to use largest practical aperture or reflector to collect as much of the incident power as possible. Antennas capture the incident signals with their effective apertures, A_eff, but not their actual apertures (Balanis 1982). In most antennas, effective aperture sizes are smaller than the actual aperture sizes. The power captured by the receiver antenna is then equal to

(2.20)

Antenna effective aperture, A_eff, can be written in terms of its gain, G₂ as (Balanis 1982)

(2.21)

Replacing effective aperture A_eff with the receiver gain G₂ in Eq. 2.22 using the relationship in the above equation, one can easily obtain the following equation for the power captured by the receiver as

(2.22)

Some of the captured power is delivered to the transmission line of the receiver if the antenna is not perfectly matched. Then, the received power at the front end of the transmission line is given by

(2.23)

Here, Γ_rx is the reflection coefficient at the terminal of the receiving antenna and is equal to

(2.24)

In the above equation, Z_b is the receiving antenna's radiation impedance, and Z_o is the characteristic impedance of the transmission line connected to the terminals of the antenna. The loss, 1/(1 − |Γ_rx|²) ≜ L_rx, associated by the impedance mismatch at the transmitter, is known as the receiver loss. Of course, if the antenna is matched, the received power will be equal to the power captured by the antenna as

(2.25)

Rewriting Eq. 2.23 in terms of the received power, one can easily obtain

(2.26)

This power is then transmitted to radar receiver by using a transmission line. If there are some electric and/or dielectric losses, L₂, within the line, the power output to the radar receiver will be equal to

(2.27)

If we insert the input power to the above equation with the help of Eqs. 2.13 and 2.14, we can get the famous radar range equation as below:

(2.28)

In this equation, L_tot is the total loss accounted for all the losses and is given by

(2.29)

If both the transmitter and the receiver antenna are perfectly matched and there are no losses inside the transmission lines, then L_tot = 1; therefore, Eq. 2.28 can be simplified to give

(2.30)