Remote-keyless-entry (RKE) systems unlock cars from a distance by transmitting a coded radio signal over the air from a key fob to a receiverin the car. The receiver decodes the signal and controls an actuator that opens the door. An important performance benchmark of an RKE system is its useful range. This range is determined by a link-budget calculation. The most crucial factors of this calculation are the power transmitted from the key fob, receiver sensitivity, and path loss. Transmitted power can be improved by carefully matching the transmitter to the small antenna in the key fob. Sensitivity can be improved by using phase-locked-loop (PLL) transmitters, such as the Maxim MAX1479, in conjunction with PLL RF integratedcircuit (RF IC) receivers like the MAX1471.

Path loss depends on the distance between the transmitter and the receiver, the radio transmission's frequency, and the height of the transmitter relative to the receiver. In an "empty parking lot" environment, the most significant characteristic of path loss over more than a few meters is that it varies as the fourth power of the distance instead of the square of the distance in free-space transmission. Path loss is, in fact, independent of frequency. It obeys a very simple equation for small antennas with unity antenna gain:

where R is the horizontal distance between transmitter and receiver, h1 is the transmitter height, and h2 is the receiver height.

"Ground bounce" is responsible for this compact, easy-to-remember equation for path loss. In any location near the ground, the radio transmission takes both a direct path and a ground-bounce path from the transmitter to the receiver (Fig. 1). The ground-bounce contribution can be thought of as a reflection from a mirror. It is reflected with a 180° phase shift for conventional terrain. In addition, it travels a longer distance than the direct contribution. The two contributions recombine at the receiver, where they would cancel completely if not for the path-length difference.

The direct and ground-bounce distances are given by Eqs. 2 and 3:

For R, R₁, R₂>> h₁, h₂, these expressions are approximated by Eqs. 4 and 5:

The difference between the two distances is given by Eq. 6:

Ground bounce is a simple example of multipath transmission. A transmitted radio wave reflects from multiple surfaces. As a result, multiple signals with different amplitudes and delays arrive at the receiver.

In free space, under line-of-sight conditions, there is only one transmission path. The signal power at the receiver is given by Eq. 7 as:

where:

P_R = the received power,
P_T = the transmitted power,
G_T = the transmitter antenna gain,
G_R = the receive antenna gain, and
λ = the wavelength.

Recall that when the ground is present, the transmitted power takes two paths: direct and ground bounce. There are many ways to model this transmission. Most of them are worthy of a graduate thesis. A reasonable and intuitive way to show the effect of the second path is to assume that half of the power goes into the direct path while the other half goes into the groundbounce path. Consequently, two voltages with slightly different phases subtract at the receive antenna. (Remember the 180-deg. phase reversal of the reflection). Equation 8 shows the complex number representation for the combination of these two voltages:

The two voltages, V₁ and V₂, are virtually the same in magnitude for most flat-ground conditions. We can consider V to be a "voltage" (in this case, V/Ω ^{1/ 2}) that is equal to the square root of half the received power or as in Eq. 9: