9.2 Statistical Modeling of Multipath Fading Channels

We first describe the statistical modeling of mobile wireless channels. We follow [396] closely. For a typical terrestrial wireless channel, we can assume the existence of multiple propagation paths between the transmitter and the receiver. With each transmission path we can associate a propagation delay and an attenuation factor, which are usually time-varying due to changes in propagation conditions resulting primarily from transceiver mobility. In the absence of additive noise, the received complex baseband signal in such a channel is given by

Equation 9.1

where x(t) is the transmitted baseband signal; a _n ( t ) and t _n ( t ) are, respectively, the path attenuation and the propagation delay for the signal received on the n th path; and f _c is the carrier frequency. By inspecting (9.1), we can see that we can model the multipath fading channel by a time-varying linear filter with impulse response g ( t , t ) given by

Equation 9.2

For some mobile channels, we can further assume that the received signal consists of a continuum of multipath components . Accordingly, for these channels, (9.1) is modified as follows :

Equation 9.3

where a ( t , t ) denotes the attenuation factor associated with a path delayed by t at time instant t . The corresponding baseband time-varying impulse response of the channel is then

Equation 9.4

By the central limit theorem, assuming a large enough number of paths between the transmitter and the receiver, and by further assuming that the associated attenuations per path are independent and identically distributed, the impulse response g ( t , t ) can be modeled by a complex-valued Gaussian random process. If the received signal r(t) has only a diffuse multipath component, g ( t , t ) is characterized by a zero-mean complex Gaussian random variable (i.e., g ( t , t ) has a Rayleigh distribution). In this case the channel is called a Rayleigh fading channel . Alternatively, if there are fixed scatterers or signal reflections in the medium, g ( t , t ) has a nonzero mean value and therefore g ( t , t ) has a Rician distribution. In this case the channel is a Rician fading channel .

We will assume that the fading process g ( t , t ) is wide-sense stationary in t , and define its corresponding autocorrelation function as

Equation 9.5

A further reasonable assumption for most mobile communication channels is that the attenuation and phase shift associated with path delay t ₁ are uncorrelated with the corresponding attenuation and phase shift associated with a different path delay t ₂ . This situation is known as uncorrelated scattering . Thus (9.5) can be expressed as

Equation 9.6

where R _g ( t , D t ) represents the average channel power as a function of the time delay t and the difference D t in observation time. The multipath spread of the channel, T _m , is the range of values of the path delay t for which R _g ( t , 0) is essentially constant. Let S _g ( f , D t ) = F _t { R _g ( t , D t )} [i.e., the Fourier transform of R _g ( t , D t ) with respect to t ]. Then S _g ( f , D t ) is essentially the frequency response function of the linear time-varying channel. The coherence bandwidth of the channel, B _c , is the range of values of frequency f for which S _g ( f , 0) is essentially constant. Hence the multipath delay spread T _m and the coherence bandwidth B _c are related reciprocally (i.e., B _c 1/ T _m ). Roughly speaking, the channel frequency response remains the same within the coherence bandwidth B _c . Let W be the bandwidth of the transmitted signal. When W > B _c , the channel is called frequency-selective fading ; and when W < B _c , the channel is called frequency nonselective fading or flat fading .

We can also take the Fourier transform of R _g ( t , D t ) with respect to D t to obtain the scattering function S _g ( t , l ) = F _{D t} { R _g ( t , D t )}. The Doppler spread of the channel, B _d , is the range of values of frequency l for which S _g (0, l ) is essentially constant. The channel coherence time is given by T _c 1/ B _d . Roughly speaking, the channel time response remains the same within the coherence time T _c . Let T be the symbol interval of the transmitted signal. When T < T _c (i.e., small Doppler), the channel is said to be time-nonselective fading or slow fading ; and when T > T _c (i.e., large Doppler), the channel is said to be time-selective fading or fast fading .

9.2.1 Frequency-Nonselective Fading Channels

Note from (9.3) and (9.4) that we have

Equation 9.7

where X ( f ) = F { x ( t )} and G ( f,t ) = F _t { g ( t , t )}. Assume that the channel fading is frequency-nonselective (flat) (i.e., W < B _c ); then the channel frequency response G ( f,t ) is approximately constant over the signal bandwidth [i.e., G ( f,t ) = G ( t )]. In this case, (9.7) can be written as

Equation 9.8

Hence the effect of a flat-fading channel can be modeled as a time-varying multiplicative distortion. Note that since g ( t , t ) is assumed to be a complex Gaussian process, G ( t ) is also a complex Gaussian process. When the fading is Rayleigh, we have E { G ( t )} = 0. For mobile communications, the autocorrelation function of G ( t ) is typically modeled by the Jakes model [216]:

Equation 9.9

where P is the average power of the fading process (i.e., P = E { G ( t ) ² }) J ( ·) is the Bessel function of the first kind and zeroth order. The corresponding Doppler power spectrum of the channel is then given by

Equation 9.10

9.2.2 Frequency-Selective Fading Channels

Now assume that the transmitted baseband signal has a bandwidth of W and that W > B _c (i.e., the channel exhibits frequency-selective fading). By the sampling theorem, we have

Equation 9.11

Equation 9.12

Hence the noiseless received signal is given by

Equation 9.13

Let ; then for practical purposes we can use the following truncated tapped-delay-line model to describe the frequency-selective fading channel [396]:

Equation 9.14

where , and conprises independent complex Gaussian processes.