UWB Channel Models

In this section, we will discuss some empirical as well as statistical channel models for the UWB systems.

1.6.1 Empirical Models

1.6.1.1 Path Loss Models

The average power loss experienced in the radio propagation at a given distance d is described by the path loss models. A log distance model is generally adopted for UWB communications [31, 32] which gives the average path loss in dB:

P L_dB(d) = P L_dB(d₀) + 10nlog₁₀(d

d₀) (1.13)

where P LdB(d0) is the mean path loss at a reference distance d0 and n represent the environment-specific path loss decay exponent. In practice, the path loss can be approximately calculated from the measured frequency response of the channel:

P L_dB(d) = 10log₁₀ 1 N

XN i=1

|H(d, f_i)|²

!

(1.14) where H(d, f_i) is the measured frequency response at a frequency f_i and N is the total number of points in the response. As mentioned previously, the path loss decay exponent is usually less than 2 in the LOS scenarios while it varies from 2 to 5 in NLOS scenarios.

1.6.1.2 Power delay profile

The average power delay profile for UWB channels has been experimentally studied by a number of researchers. [33,34,35] are few examples. Some common observations about the UWB channels are:

1. Distinct clusters of scatterers are present in the PDP.

2. The PDP decays exponentially with cluster dependent decay exponents.

3. A large number of resolvable multi paths are present in the PDP.

1.6.1.3 Delay spread

The mean excess delay spread of channel can be extracted from the PDP of a certain channel. It is defined as the first moment of the PDP and can be written as:

τ_m = PN

i=1a²_iτ_i PN

i=1a²_i (1.15)

where a_i is the i_th tap of the CIR with a time delay of τ_i and N is the total number of taps in the CIR. The instantaneous root mean square (RMS) delay spread is an important parameter which indicates the time dispersion of the channel due to the presence of multi-paths [36]. It can be calculated by the first and the second moment of the measured channel power delay profile (PDP):

τrms =

sPN

i=1a²_i(τ_i−τ_m)² PN

i=1a²_i (1.16)

Typical values of the RMS delay spread vary from 5−25 ns depending on con-figurations (LOS and NLOS) and the environments (office, home and industrial etc.) [37].

1.6.2 Statistical Models

Statistical models for the path loss, large scale fading, general shape of the CIR, path interval times are presented in [24]. In this section, we present the statistical models for the CIR and channel delays.

Generally, a tapped-delay line model with clusters is adopted for the CIR of a UWB channel. The most famous way to mathematically represent the CIR of the UWB channel is known as ‘Saleh Valenzuela (SV)’ model [38]. According to SV model, the CIR of the UWB channel can be written as:

h(t) = XL

l=1

XK k=1

a_k,lexp(jφ_k,l)δ(t−T_l−τ_k,l) (1.17) whereLis total number of clusters,K is the number of MPCs within a cluster,a_k,l is the coefficient of the k_th tap in the l_th cluster, T_l is the cluster arrival time for the lth cluster,τk,l is the delay of the kth MPC relative to theTl. The number of clusters can either be assumed fixed [38], or considered to be a stochastic variable [39]. The number of clusters is usually a function of bandwidth and the considered environment.

In UWB channels, on the basis of the results of the measurement campaigns, it can be stated that the tap amplitudes of CIR differ significantly from the famous Rayleigh and Rice distributions. Therefore, some other distributions such as Nakagami [40]-[42] and the log-normal [41] are adopted for UWB channels. The phases φk,l are uniformly distributed, i.e., for a bandpass system, the phase is taken as a uniformly distributed random variable from the range [0,2π] [24].

1.6.2.2 Statistics of channel delays

For the modeling of multiple echoes of the MPCs, two models are generally used in the literature. One is the ∆−K model and the other is Neyman-Scott model [42].

The inter arrival time of two of MPCs is modeled as a two state Markov model in

∆−K model. At a timet₀, the process changes from the state S₁ toS₂ on the arrival of a MPC. If at the end of time interval [t0+ ∆], no MPC arrives, the state changes again from S₂ toS₁.

The Neyman-Scott model is the most popular model for the arrival of MPCs. It assumes that the arrival times within a cluster is a Poisson process. It is used in the SV model [38] as well as in the IEEE 802.15.4a channel model [39]. Although the ∆−K and Neyman-Scott models are in good agreement with the experimental data but their parameters cannot be easily evaluated. For this reason, simpler stochastic models have been proposed which uses the split-Poisson model for the arrival of different MPCs [43]. This assumes the existence of only two clusters each with a different arrival frequency.

1.6.2.3 Power delay profile statistics

According to the cluster based UWB models, the power delay profile of a UWB channel is usually approximated as a double exponential. The average power of the

k_th echo in the l_th cluster of 1.17 is given by:

E{a²_k,l}= Ω₀exp(−T Γl

)exp(−τ_kl

γ ) (1.18)

where Ω₀ is the average power of the first multi-path component, and Γ and γ are the intra cluster and inter cluster decay time constants of the clusters respectively.

The inter-cluster decay time constant Γ is typically around 10−30 ns, while widely differing values (between 1−60ns) have been reported for the intra-cluster constant γ. The information about the fast fluctuations of the received power due to fading is however not provided in 1.18.

1.6.2.4 Auto-regressive models for UWB channels

The models based on the description of CIR such as given by 1.17 need a large number of parameters for a satisfactory description of the CIR. An accurate modeling of the channel frequency response can be done by making use of a second order auto-regressive model in the frequency domain [44]. Furthermore, these models closely match the experimental data even with a small number of required parameters.