JPL's Wireless Communication Reference Website

Chapter: Wireless Channels
Section: Multipath Fading

Scatter Function

Multipath fading and user mobility lead to a time and frequency dependent channel. The Transfer function of a particular sample channel does not necessarily provide enough details about the stochastic behavior of the radio channel. Such stochastic properties are captured in the scatter function. The scatter function combines information about The scatter function provides a statistical model for the channel.

  Figure: the basic idea behind the scatter function is that it plots the expected power per Doppler shift and per excess delay bin. Sometimes, angle of incidence (bearing) is plotted in stead of the Doppler shift.  

Each path can be described by its

Thus we can plot the received energy in a two dimensional plane, with Doppler shift on one horizontal axis and delay on the other horizontal axis.

EnvironmentDelay SpreadAngle spreadMax. Doppler shift
at 1800 MHz
Macrocellular: Rural flat0.5 ms1 degree 200 Hz
Macrocellular: Urban5 ms20 degrees 120 Hz
Macrocellular: Hilly20 ms30 degrees 200 Hz
Microcellular: Factory, Mall0.3 ms120 degrees 10 Hz
Microcellular: Indoors, Office0.1 ms360 degrees 2..6 Hz
  Table Source: A.J. Paulray and C.B. Papadias, "Space-Time Processing for Wireless Communications", Signal Processing Magazine, November 1997, pp. 49-83.

Audio commentary: Peter M. Grant, Distinguished IEEE Lecturer 1997, discusses the table parameters (MPEG audio). See also: Full talk, MPEG plug-in on

A Practical Example from Germany

Figure: measured scatter plot for DCS 1800 MHz system.
Doppler spread = 60.3 Hz; Coherence time = 5.9 msec.
Delay Spread = 1.2 msec; coherence BW = 1.3 MHz

Source: Research group of Prof. Paul Walter Baier, U. of Kaiserslautern, Germany.

Figure: Doppler spread corresponding to above scatter plot.
Note that the Doppler spread is the projection of the scatter plot on the Doppler frequency axis.
Figure: Distribution of angle of arrival corresponding to above scatter plot.
Figure: Delay spread profile corresponding to above scatter plot.
Note that the Doppler spread is the projection of the scatter plot on the time delay axis.

An example from Edinburgh

Received power (according to color) versus time of arrival (horizontal axis) and angle of incidence (vertical axis). Source credit: Nortel.

Audio commentary: Peter M. Grant, Distinguished IEEE Lecturer 1997. MPEG audio:

  • MP2 ** Power versus delay and angle
  • MP2 Measurement artifacts in delay-angle map
See also: Full talk, MPEG plug-in on

An indoor example from Zurich

A realization of the local Power Delay-Direction Profile (PDDP). Carrier frequency 5.2 GHz, MT velocity 1 m/s, delay spread TRMS = 50 ns. Such a situation is typical for a small room environment.

See the PDDP evaluation by Peter E. Leuthold and Pascal Truffer.

Theoretical Example

Let's consider a Moreover, we assume that the delay spread and Doppler spread are separable. Then the amount of (scatter power) per frequency and time bin can be expressed as

            Plocal-mean      1               1           t 
  p(f,t) = ------- --------------------  ---- exp(- -----)
            4 p  fm            (f-fc)2     Trms        Trms
                     sqrt( 1 - -------)
                                fm2 
The integral over p(f,t) gives to total received local mean power Plocal-mean.


Figure: Scatter function. Received power per unit of frequency shift and per unit of excess time delay.
Frequency shift normalized to the maximum Doppler shift.
Delay time normalized to the delay spread.


Figure: Scatter function projected to frequency axis. This gives the Doppler spread. Received power per unit of frequency shift.
Frequency shift normalized to the maximum Doppler shift.


Figure: Scatter function project to delay time axis: This gives the delay profile. Received power per unit of excess time delay.
Delay time normalized to the delay spread.

Channel simulations based on this theoretical model have been contributed by Ralph Haas.



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