# 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 Tm, which is related to frequency selectivity, and
• the Doppler spread BD 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 Tm 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 Bc of the channel is proportional to the reciprocal of Tm. Bc 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 Bc will suffer from intersymbol interference. If its bandwidth is considerably smaller than Bc 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 BD. The reciprocal of BD approximates the coherence time Tc of the channel. If we represent the channel influence as an attenuation of the signal amplitude, Tc 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 Ts and the frequency axis in multiples of 2/Ts. The dispersion of the channel is given by the corresponding normalized parameters tm = Tm / Ts and fD = BD Ts/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 tm = 0.05 and fD = 0.005. (Highly frequency selective, but not very time selective)

Figure 4: Amplitude of the channel attenuation in dB for tm = 0.005 and fD = 0.05. (Almost frequency non-selective, very time selective: rapidly fading)

## Time dispersion and frequency dispersion

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 tm = 0.05 and normalized Doppler spread fD = 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?