# Samples of a Rayleigh-Fading Multipath Channel

*contributed by Ralf Haas, ENST, Paris*
The phenomenon of multipath fading on a mobile radio channel is characterized by two parameters:

- the multipath delay spread
*T*_{m}, which is related to frequency selectivity, and
- the Doppler spread
* B*_{D} which is related to time selectivity.

They are due to multipath propagation delays and the motion of the transmit and/or receive antenna, respectively.

### Frequency Selectivity

The multipath delay spread *T*_{m} represents the time interval for which the
the impulse response of the channel is considerably greater than zero.
Figure 1: Delay profile. Expected received power as a function of excess delay time
The coherence bandwidth *B*_{c} of the channel is
proportional to the reciprocal of *T*_{m}. *B*_{c} denotes the maximum frequency separation
of two sinusoidal signals, for which the channel affects these waves still in a highly correlated
manner.
This implies that a signal with a bandwidth larger than *B*_{c} will suffer from intersymbol interference.
If its bandwidth is considerably smaller than *B*_{c} the channel can be considered as frequency-nonselective or "flat" fading.

### Time Selectivity

Time variations of the channel due to a relative motion between transmitter and receiver lead to a broadening of the signal spectrum. This frequency dispersion can be characterized by the U-shaped power spectrum of isotropic scattering.

The spectral line of a pure sine wave will have a power spectrum as shown in figure 2 after transmission over the channel. The frequency range where the power spectrum is nonzero defines the Doppler spread *B*_{D}. The reciprocal of *B*_{D} approximates the coherence time *T*_{c} of the channel. If we represent the channel influence as an attenuation of the signal amplitude,* T*_{c} denotes the minimum time interval between two decorrelated attenuation factors.

Figure 2: Power density spectrum of a sine wave suffering from a Doppler spread.

## Influence on the channel attenuation

Figures 3 to 5 show how the channel attenuation depends on time and frequency
of the signal for various values of the delay spread and Doppler spread.
The time axis is scaled in multiples of the symbol duration* T*_{s} and the frequency axis in multiples of 2/*T*_{s}. The dispersion of the channel is given by the corresponding normalized parameters* t*_{m} = T_{m} / T_{s} and *f*_{D} = B_{D} T_{s}/2.
It can be observed that the rate of variations of the channel attenuation increases if the Doppler spread increases. This coincides with a decrease of the coherence time of the channel.
Similarly, the effect of changing the signal frequency increases if the channel has a smaller coherence bandwidth, corresponding to a longer delay spread.

Figure 3: Amplitude of the channel attenuation in dB for* t*_{m} = 0.05 and* f*_{D} = 0.005.
(Highly frequency selective, but not very time selective)

Figure 4: Amplitude of the channel attenuation in dB for* t*_{m} = 0.005 and* f*_{D} = 0.05.
(Almost frequency non-selective, very time selective: rapidly fading)

In most practical mobile radio systems, the received signal amplitude depends on both frequency and location (or time).
Figure 5: Amplitude of the channel attenuation in dB
versus carrier frequency (in MHz) and time (in milliseconds).

Normalized delay spread * t*_{m} = 0.05 and normalized Doppler spread *f*_{D} = 0.05.

A time selective and frequency selective Rayleigh fading channel; "The real thing."
While the above figures give a good deterministic description for a particular
channel, a designer or researcher usually relies more on the scatter function. The scatter function gives a full statistical description of the channel, so it allows
the development of simulation tools and statistical analysis, with broader validity
than a specific channel.

### Exercise

Discuss how BPESK-type of radio signals with various bit rates
would be affected by the channel sampled on this page.
Which countermeasures would be useful to combat fading?

Answer: see digital modulation page

Source: R. Haas, "Applications of multicarrier modulation in mobile radio communications",
PhD thesis, Ecole Nationale Superieure des Telecommunications, Paris, 1996.