Page 93 - Fundamentals of Radar Signal Processing

(2.11)

Equation (2.11) is one simple form of the radar range equation. It relates

received echo power to fundamental radar system and target parameters such as
transmitted power, operating frequency, and antenna gain; radar cross section;
and range. Because the power of the radar signal is proportional to the square of
the electric field amplitude, the range equation also serves as a model of the
amplitude of the target and clutter components of the signal. Note that all

variables in Eq. (2.11) are in linear units, not decibels, even though several of
the parameters are often specified in decibels; frequent examples include the
atmospheric losses, antenna gain, and RCS. Also note that P is instantaneous,
r
not average, received power. Finally, realize that for a scatterer at range R, the
backscattered EM wave will be received with a time delay of 2R/c seconds
after transmission.
As an example, consider an X-band (10-GHz) radar with a peak
transmitted power of 1 kW and a pencil beam antenna with a 1° beamwidth, and

2
suppose an echo is received from a jumbo jet aircraft with an RCS of 100 m at
a range of 10 km. The received power can be determined using Eq. (2.11). The
antenna gain can be estimated from Eq. (1.10) to be G = 26,000/(1)(1) = 26,000
–2
8
9
= 44 dB. The wavelength is λ = c/F = 3 × 10 /10 × 10 = 3 × 10 m = 3 cm.
Assuming atmospheric and system losses are negligible, the received power is

(2.12)

Even though this example is a large target at short range, the received power is
only 3.07 nW, nearly 12 orders of magnitude less than the transmitted power!
Nonetheless, this signal level is adequate for reliable detection in many cases.
This example illustrates the huge dynamic ranges observed in radar between

transmitted and received signal powers.
An important consequence of Eq. (2.11) is that for a point target, the
received power decreases as the fourth power of range from the radar to the
target. Thus, the ability to detect a target of a given radar cross section
decreases rapidly with range. Range can be increased by increasing transmitted
4
power, but because of the R dependence, the power must be raised by a factor
of 16 (12 dB) just to double the detection range. Alternatively, the antenna gain
can be increased by a factor of 4 (6 dB), implying an increase in antenna area by
a factor of 4. On the other hand, designers of “stealth” aircraft and other target
vehicles must reduce the RCS σ by a factor of 16 in order to halve the range at
which they can be detected by a given radar system.
The range equation is a fundamental radar system design and analysis tool.
More elaborate or specialized versions of the equation can be formulated to