Statistical Channel Characterization Overview

Wireless Channel Characterization for the 5 GHz Band Airport Surface Area ∗

3.2 Statistical Channel Characterization Overview

have a completely obstructed LOS to the ATCT. These regions are near airport gates or behind large airport buildings.

Aircraft and ground vehicles may traverse all three types of regions as they move about the ASA. This has consequences for statistical channel models that will be addressed subsequently. Afinal comment on the ASA channel regards the spatial distribution of MPCs: scattering is almost never isotropic about the mobile terminal.

resolved Doppler frequency ωDk ˆ 2πf_D;k with f_D;k… † ˆ v t…†ft _c cos‰θk… †t Š=c. Here, v(t) is relative velocity between Tx and Rx, θk(t) is the aggregate phase angle of all components arriving in the kth delay“bin,” and c is the propagation velocity, well approximated by the speed of light. The delay bin width is approximately equal to the reciprocal of the signal bandwidth– components separated in delay by an amount smaller than the bin width are“unresolvable.” The kth resolved component in (3.1) thus often consists of multiple terms“subcomponents” from potentially different spatial anglesθk;i. We do not address“spatial” channel modeling, for exam-ple Refs [29,30], here, other than brief comments in a subsequent section.

The CTF corresponding to Equation 3.1 is H…f ; t† ˆX_{L…t† 1}

kˆ0 αk…t†expfj‰ωD;k…t†ft τk…t†gŠgexp‰ jωcτk…t†Še ^{j2πf τk…t†}

(3.2) where the frequency dependence is expressed by the final exponential term. The second exponential can change significantly with small changes in delayτk… † when ft cis large. This second term typically dominates the small-scale fading variation, as fcis usually much larger than f_D;k. As an example, for ASA applications, if the carrier frequency is 5 GHz, and relative velocity is 60 m/s (roughly 135 miles/h), f_D;maxˆ 1 kHz  f_c.

In Figure 3.3, we illustrate a conceptual CIR (magnitude). This diagram illustrates variation (fading) in time t and variation of impulse energy (∼α²_k) with delayτ. This type of CIR is useful in analysis, simulations, or hardware in terms of the common tapped-delay line (TDL) model. The TDL is a linear,finite IR filter, as shown in Figure 3.4.³In Figure 3.4, the input signal is s(t) the output signal is y(t), and other parameters are as defined for Equation 3.1.

3.2.2 Statistical Channel Characteristics

Essentially, any of the CIR parameters, other than the carrier frequency (unless it is“randomized” by transmission) and the propagation velocity, can be modeled statistically. Typically, the MPC amplitudes αk, phase shiftsϕk(and spatial angles of arrivalθkembedded within these phases), and delaysτkare modeled as random. Also, modeled as random are more

“composite” channel features such as attenuation, delay spread, and

3 Implicit in Figure 3.3 is the concept of multiple timescales: the short-term“delay” scale (τ) and the longer-term timescale over which the channel’s parameters evolve. For most channels considered to be“slowly fading,” the CIR is viewed as “decaying” or ending in delayτ long before any of the components change in time t appreciably. This will be true for ASA channels.

Doppler spread, which arise from statistics of the CIR. We briefly address these next.

Attenuation is typically quantified as the loss in power of a transmitted signal. This can be expressed, in dB, as 20 log10[|h(τ,t)|], and the models are known as path loss models. The well-known, deterministic, free-space model (the simplest possible wireless channel path loss model) is given by L_{f s}ˆ 20 log…4πd=λ†, with d the link distance and λ the wavelength.

Modern path loss models are most often specified, in dB, as

L…d† ˆ L0‡ 10v log…d=dmin† ‡ X; dmin  d  dmax (3.3) Figure 3.3 Conceptual illustration of time-varying CIR magnitude |h(τ,t)|.

Figure 3.4 General form of wideband TDL model corresponding to CIR in Equation 3.1.

where L₀is a constant value of attenuation at distance dmin,ν is the path loss exponent, and X is a zero mean Gaussian random variable with standard deviationσX. Parameters L₀,ν, and σXare most often determined empirically for a given setting. In the model of (3.3), the only statistical parameter isσX.

Delay spread is a measure of the width, in delayτ, of the CIR. There are multiple measures for this width, with the most common being the root mean square delay spread (RMS-DS), denoted στ [6]. Other metrics include the delay window and delay interval [31]. The RMS-DS can be computed as follows

στ ˆ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P_{L 1}

kˆ1α²_kτ²_k P μ²_τ s

σ_τ ˆ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P_{L 1}

kˆ0α²_kτ²_k P μ²_τ

s (3.4)

whereαkandτkare as defined in (3.1), and μτis the mean energy delay, given by

μ_τˆ PL 1

kˆ1α²_kτk

P (3.5)

The term PˆP_{L 1}

kˆ0α²_k in (3.4) and (3.5) is the total power in the power delay profile (PDP). Both (3.4) and (3.5) can be computed for either an average CIR over a set of data, or for a single CIR. In the latter case, the parameters in (3.4) and (3.5) are termed the instantaneous RMS-DS [32]

and instantaneous mean energy delay, respectively.

The computation of the average CIR or average PDP wasfirst addressed in the classic paper by Bello [33]. In this work, the CIR (and consequently, the CTF as well) was treated as a random process, and multiple correla-tion funccorrela-tions were defined to characterize its second-order statistics.

Details of this stochastic treatment are beyond the scope of this chapter, and can be found in Refs [6] and [33], for example. Here, we briefly describe the correlation functions of the CIR and CTF, and how these give rise to common statistical parameters such as the RMS-DS.

The CIR correlation function⁴is

Rhh…τ1; τ2; t1; t2† ˆ E‰h…τ1; t1†h^∗…τ2; t2†Š (3.6)

4 Bello focused on correlation and not covariance functions because of many cases of interest, for example, NLOS conditions, the channel is considered to be zero mean. Also, for stationary or wide-sense stationary assumptions, the correlation and covariance differ only by a constant.

where the asterisk denotes complex conjugation. The CTF correlation function is analogously defined:

RHH…f₁; f₂; t1; t2† ˆ E‰H…f₁; t1†H^∗…f₂; t2†Š (3.7) The functions Rhh and RHH are related through the double Fourier transform, that is,

R_HH…f₁; f₂; t1; t2† ˆ ∫∫R^hh…τ1; τ2; t1; t2†e ^j2πf1τ1e ^{j2πf2τ2dτ1dτ2} (3.8) Since four-dimensional functions are complicated to work with, sim-plifying assumptions are often made. Thefirst is uncorrelated scattering (US), which means that MPCs at different values of delay are uncorre-lated. This reduces RhhtoRhh…τ; t1; t2†δ τ τ… 1†, and RHHto RHH…Δf ; t1; t2†, with Δf ˆ f ₁ f₂. The second assumption is wide-sense stationarity (WSS) in time, which results in replacing t₁and t₂in both R_hh and RHH

withΔt ˆ tj 1 t₂j. The resulting WSSUS correlation functions are then R_hh…τ; Δt†δ τ τ… 1† and RHH…Δf ; Δt†, with the latter termed the spaced-frequency/spaced-time correlation function. WhenΔt ˆ 0, Rhhbecomes R_hh… †, and this is the average PDP. The RMS-DS is standard deviation ofτ Rhh… †.τ

From the Fourier relationships, withΔt ˆ 0, the standard deviation of RHH… † is termed the correlation or coherence bandwidth, BΔf co. An approximate relationship isσ_τ∼ 1=Bco and more conservatively, as dis-cussed in Chapter 2, σ_τ∼ 1=…5Bco† [34]. These two parameters – the RMS-DS and correlation bandwidth– are widely used measures of delay dispersion and frequency correlation, respectively.

The correlation functions also allow us to quantify the channel’s time variation. Specifically, setting Δf ˆ 0, the standard deviation of RHH… †Δt is termed the coherence time T_c, roughly the time over which the channel’s statistics remain constant. From the Fourier relations again, the coherence time is reciprocally related to the RMS Doppler spread B_ds: T_c∼ 1=Bds. The Doppler spread’s physical interpretation is the range of Doppler shifts impressed upon the transmitted signal due to platform velocity and the various angles of arrival of the MPCs.

3.2.3 Common Channel Parameters and Statistics

Recall from our characterization of mobile radio channel that four related parameters,στ, B_co, T_cand B_ds, are widely used to quantify the channel’s effect upon signals. Along with the path loss model parameters, particu-larly the path loss exponent ν and the standard deviation σX, these parameters provide the communication engineer with vital information for assessing channel effects upon transmitted signals.

Since attenuation can differ significantly with the presence or absence of a LOS component in the channel, the probability of LOS present [35], in a given environment, is sometimes provided as an additional channel param-eter. This can be of use in assessing the statistics of attenuation over an area.

Other channel statistics often estimated and reported are the number of MPCs, MPC fading distributions, correlations among MPCs, and proba-bilities of MPC occurrence and duration. For estimation of all these MPC statistics, one generally collects measurement data (or, sometimes, simu-lation data). The corresimu-lation functions R_hh… † and Rτ HH…Δf ; Δt† can be estimated, but strictly these require the WSSUS assumption, which will not pertain for long durations or very wide frequency spans. Hence, recent work (see, for example, Ref. [36] and references therein) has investigated the estimation of the channel stationarity time or stationarity distance (SD). With knowledge of the approximate value of SD, one can proceed to estimate the correlation functions and parameters. For the NLOS-S and NLOS ASA regions, for typical vehicle velocities, the channel can be statistically nonstationary over fairly short durations (tens to hundreds of milliseconds).

Finally, spatial parameters of wireless channels are also of interest in modern communication systems that employ multiple antennas at either Tx or Rx or both (MIMO). With antenna arrays at one or both link ends, spatial correlations among antenna elements can be estimated. The spatial distribution of received power, often expressed as a function of azimuth angle, is also often reported. Additional spatial statistics can be found in the literature, for example, Ref. [30].