JPL's Wireless Communication Reference Website

Chapter: Wireless Channels. Section: Multipath fading; Rayleigh fading.

Rate of Rayleigh Fading

A moving antenna in a multipath propagation environment experiences time-dispersion: the channel changes rapidly over time.

  Figure: Phasor diagram of a set of scattered waves (in blue), resulting in a Rayleigh-fading envelope (in black)  
We address a large set of reflected waves. Let the n-th reflected wave with amplitude cn and phase fn arrive from an angle an relative to the direction of the motion of the antenna.

If the mobile antenna moves a small distance e, the n-th incident wave, arriving from the angle an with respect to the instantaneous direction of motion, experiences a phase shift of

  2p  e
  -----  cos(an)
    l

Thus all waves experience their own phase rotation. The resulting vector may significantly change in amplitude if individual components undergo different phase shifts.

 

Java animation
Figure: Phasor diagram of a set of scattered waves after antenna displacement (in blue) and before motion (in light blue), resulting in a Rayleigh-fading envelope (in black)  

In mobile radio channels with high terminal speeds, such changes occur rapidly. Rayleigh fading then causes the signal amplitude and phase to fluctuate rapidly.

If e is in the order of half a wave length (l/2) or more, the phases of all incident waves become mutually uncorrelated, thus also the amplitude of the total received signal becomes uncorrelated with the amplitude at the point of departure.

Doppler Shifts

Each reflected wave experiences its own Doppler shift. If an unmodulated carrier is being transmitted, a spectrum of different components is received.

Autocovariance

The normalised covariance L(e) of the electric field strength for an antenna displacement e is of the form

          2    2p e
L(e)  =  J   (-----)
          0     l

with J0(.) the zero-order Bessel function of the first kind.
The signal remains almost entirely correlated for a small displacement, say e< l/8, but becomes rapidly independent for larger displacements, say for e > l /2.

Figure: Auto-covariance L(e) of the electric field strength in a Rayleigh-fading channel versus the normalised antenna displacement e/ l (or T fm) in horizontal direction.  

The antenna displacement can also be expressed in the terminal velocity v and the time difference T between the two samples (e = v T). So with fm the maximum Doppler shift (fm = v fc / c).

How do systems handle fast multipath fading?

  System Countermeasure  
  Analog
  • User must speak slowly
  GSM
  • Error correction
  • Interleaving to avoid burst errors
  • Error detection and speech decoding
  • Fade margins in cell planning
  DECT
  IS95 Cellular CDMA
  • Wideband transmission averages channel behaviour
  • This avoids burst errors and deep fades
  • Error correction
  Mobile data networks, Wireless LANs



JPL's Wireless Communication Reference Website Jean-Paul M.G. Linnartz, 1993, 1995.