Page 95 - Fundamentals of Radar Signal Processing

(2.15)

Again, this power is received 2R/c seconds after transmission. The total
received power is obtained by integrating over all space to obtain a generalized
radar range equation

(2.16)

In Eq. (2.16), the volume of integration V is all of three-dimensional space.
However, the backscattered energy from all ranges does not arrive
simultaneously at the radar. As discussed in Sec. 1.4.2, only scatterers within a
single range resolution cell of extent ΔR contribute significantly to the radar
receiver output at any given instant. Thus, a more appropriate form of the
generalized radar range equation gives the received power as a function of time,

(2.17)

where ΔR is the range interval of the resolution cell centered at range R and Ω
0
represents integration over the angular coordinates.
By integrating power, it is being assumed that the backscatter from each
volume element adds noncoherently rather than coherently. This means that the
power of the composite electromagnetic wave formed from the backscatter of
two or more scattering centers is the sum of the individual powers, as opposed
to the voltage (electric field amplitude) being the sum of the individual
amplitudes, in which case the power would be the square of the voltage sum.

Noncoherent addition occurs when the phases of the individual contributors are
random and uncorrelated with one another, as opposed to the coherent case
when they are in phase. This issue will be revisited in Sec. 2.7.
The general result of Eq. (2.17) is more useful if evaluated for the special
cases of point, volume, and area scatterers. Beginning with the point scatterer,
the differential RCS in the resolution cell volume is represented by a Dirac

impulse function of weight σ :

(2.18)

Using Eq. (2.18) in Eq. (2.17) gives the range equation for a point target at