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Giacinto Gelli » 8.Wideband fading

Outline

Wideband fading model
WSSUS model
Measures of time dispersion:

power delay profile
average delay spread, RMS delay spread and coherence bandwidth

Measures of frequency dispersion:

doppler spectrum
coherence time and Doppler spread

Discrete channel model

Wideband fading model

Individual multipath components are resolvable.

True when time difference between components (delay spread) exceeds the reciprocal bandwidth of the signal u(t).

Fonte Stanford University

Deterministic scattering function

Fourier transform of c(τ,t) with respect to t

$S_c(\tau,\rho)=\int_{-\infty}^{+\infty}c(\tau,t)\,e^{-j 2 \pi \rho t}\,dt$

Describes the time variability of the channel (hence the Doppler effect) for each fixed value of τ (each path).

Since the channel is random, the deterministic scattering function is not always adequate → it is required to characterize c(τ,t) as a 2-D random process.

Fonte Stanford University

Statistical characterization of the channel

Under CLT, c(τ,t) is a 2-D Gaussian process -> need only to characterize mean (zero) and correlation.

Recall that τ is related to the “path”, hence can be interpreted as a spatial variable, whereas t is a temporal variable.

Compared to traditional random processes, additional complications arise due to the fact that there are two variables (t and τ).

This problem was studied for the first time by Paul Bello (1963) (see references).

Autocorrelation and WSS model

Start from the definition of the space-time autocorrelation function:

It represents the correlation between two multipath components (τ₁ and τ₂) evaluated at times t e t+Δt

$A_c(\tau_1, \tau_2; t, t + \Delta t) = \mbox{E} [c^*(\tau_1,t) \, c(\tau_2, t + \Delta t)]$

WSS model: the autocorrelation depends only on Δt and not on t:

$A_c(\tau_1, \tau_2 ; \Delta t) = \mbox{E}[ c^*(\tau_1,t) \, c(\tau_2, t + \Delta t) ]$

Uncorrelated scattering

Uncorrelated scattering (US) assume that scattering associated to path τ₁ is uncorrelated (i.e., independent due to the Gaussian assumption) with that of any other path τ₁ ≠ τ₂:

$A_c(\tau_1, \tau_2; t, t + \Delta t) = \mbox{E}[ c^*(\tau_1,t) \, c(\tau_2, t + \Delta t) ] = A_c(\tau_1; t, t+\Delta t) \, \delta(\tau_1-\tau_2)$

Reasonable model for resolvable components: non-resolvable components are instead highly correlated.

WSSUS model

Under WSS + US (WSSUS) assumptions:

$A_c(\tau_1, \tau_2; \Delta t) = \mbox{E}[ c^*(\tau_1,t) \, c(\tau_2, t + \Delta t) ]=A_c(\tau_1, \Delta t) \, \delta(\tau_1-\tau_2)$

For a fixed path τ of the multipath channel, the function A_c(τ, Δt) measures the correlation existing between two values of the channel impulse response temporally spaced by Δt.
Can be characterized also in the frequency domain -> statistical scattering function.

Statistical scattering function

Fourier transform of Ac(τ, Δt) with respect to the second variable -> $S_c(\tau,\rho) = \int_{-\infty}^{+\infty} A_c(\tau,\Delta t) \, e^{-j 2 \pi \rho \Delta t} \, d\Delta t$

For each path τ of the multipath channel, S_c(τ, ρ) describes the power distribution as a function of the Doppler frequency ρ.

Power delay profile

For Δt = 0, the function A_c(τ)= A_c(τ, Δt=0) indicates how the power is distributed over the different paths (power delay profile or multipath intensity profile):

practical measure of the temporal dispersion of the channel
can be measured experimentally

Average and rms delay spread

Since A_c(τ) ≥ 0, the power delay profile can be interpreted as the pdf of the delay spread T_m:
$p_{T_m}(\tau) = \frac{A_c(\tau)}{\displaystyle \int_{0}^{+\infty} A_c(\tau) \, d\tau}$

Mean and standard deviation of T_m represent the average delay spread and the RMS delay spread.

Effects of delay spread

Consider a digitally-modulated signal with symbol period T_s transmitted over a channel with RMS delay spread σ_Tm:

significant intersymbol interference (ISI) if σ_Tm>> T_s
if σ_Tm<< T_s the ISI will be negligible

The RMS delay spread limits the maximum transmission rate R_s=1/T_sexpressed in symbols/s (baud).
Some typical values:

Indoor -> σ_Tm=50 ns -> R_s = 1/T_s = 2 Mbaud
Outdoor -> σ_Tm=30 μs -> R_s = 1/T_s = 3.3 kbaud

Equalization techniques aimed at compensating ISI are needed!

Time-frequency autocorrelation

An equivalent channel characterization can be given in the frequency domain, by evaluating the Fourier transform of c(τ,t) with respect to τ (frequency response):

$C(f,t ) = \int_{-\infty}^{+\infty} c(\tau,t) \, e^{-j 2 \pi f \tau} \, d\tau$

Under the WSS assumption, similarly to c(τ,t), also C(f,t) is a Gaussian zero-mean random process, stationary in t, which is completely characterized by its time-frequency autocorrelation function:

$A_C(f_1, f_2; \Delta t) = \mbox{E}[ C^*(f_1,t) \, C(f_2, t + \Delta t) ]$

Relations with power delay profile

Under the WSSUS assumption, the time-frequency autocorrelation depends on Δf=f₂-f₁ (stationarity in frequency) and one has:

$A_C(\Delta f, \Delta t) = \int_{-\infty}^{+\infty} A_c(\tau, \Delta t) \, e^{-j 2 \pi \Delta f \tau} \, d\tau$

Define A_C(Δf)= A_C(Δf; Δt=0) (frequency autocorrelation), it results that:

$A_C(\Delta f) = \int_{-\infty}^{+\infty} A_c(\tau) \, e^{-j 2 \pi \Delta f \tau} \, d\tau$

A_C(Δf) is the Fourier transform of the power delay profile A_c(τ)
it measures the correlation between frequency components spaced at Δf in the same temporal instant (Δt=0)

Coherence bandwidth

The frequency value B_c for which A_C(Δf) ≈ 0 is called coherence bandwidth:

B_c is a measure of the spectral width of A_C(Δf)
values of the frequency response at frequencies spaced apart by more than B_c are roughly uncorrelated (hence independent)

Due to Fourier relationship between A_C(Δf) and A_c(τ), the coherence bandwidth is inversely proportional to delay spread:

B_c = 0.02/σ_Tm for a correlation greater than 0.9
B_c = 0.2/σ_Tm for a correlation greater than 0.5

Flat vs. frequency-selective fading

If the TX signal has bandwidth B_u, then:

if B_u << B_c , the fading is strongly correlated over the whole signal bandwidth -> flat fading
if B_u >> B_c , the signal components separated by Bc are approximately uncorrelated -> frequency-selective fading

Note: flat - narrowband; frequency-selective - wideband

Doppler power spectrum

By evaluating the Fourier transform of A_C(Δf, Δt) with respect to Δt, the correlation between the channel components spaced by Δf is expressed as a function of the Doppler shift (Doppler frequency spectrum):

$S_C(\Delta f,\rho) = \int_{-\infty}^{+\infty} A_C(\Delta f,\Delta t) \, e^{-j 2 \pi \rho \Delta t} \, d\Delta t$

Doppler power spectrum: to characterize the Doppler power at a single frequency we set Δf=0 obtaining thus S_C(ρ)= S_C(Δf=0, ρ):
$S_C(\rho) = \int_{-\infty}^{+\infty} A_C(\Delta t) \, e^{-j 2 \pi \rho \Delta t} \, d\Delta t$

S_C(ρ) is the Fourier transform of the time autocorrelation function A_C(Δt)= A_C(Δf=0, Δt).

Coherence time

The time value T_c for which A_C(Δt) ≈ 0 is called coherence time:

T_c is a measure of the temporal width of A_C(Δt)
values of the channel impulse response at times separated by T_c are approximately uncorrelated

The coherence time measures the speed of variationof the channel in the time domain:

for the NB model (Jakes model) it can be assumed equal to the decorrelation time T_c = 0.4/f_D -> inversely proportional to the maximum Doppler shift

Doppler spread

The spectral width B_D of S_C(ρ) is called Doppler spread or Doppler bandwidth.

Due to the Fourier transform relationship between A_C(Δt) and S_C(ρ), the Doppler spread B_D is inversely proportional to coherence time T_c.

Fast vs. slow fading

Comparison between the speed of variation of the channel (measured by T_c or B_D) and the speed of variation of the signal (measured by 1/B_u):

T_c << 1/B_u -> the channel exhibits strong variations during the typical time of variation of the signal -> fast fading
T_c >> 1/B_u -> the channel is roughly constant during the characteristic time of variation of the signal -> slow fading

In terrestrial communications, fast fading occurs only for very low-data rate transmissions.

Fading types: time domain

Fading types: frequency domain

Relations with the scattering function

To complete the mathematical framework, note that the statistical scattering function S_c(τ,ρ) is the 2-D Fourier transform of the time-frequency autocorrelation A_C(Δf; Δt):

$S_c(\tau,\rho) = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty}A_C(\Delta f,\Delta t) \, e^{-j 2 \pi \rho \Delta t} \, e^{-j 2 \pi \tau \Delta f} \, d\Delta t \, d\Delta f$

The many functions introduced for channel characterization and their relationships are pictorially represented in the following slide.

The WSSUS model at one glance

Discrete channel model

Discrete multipath channels are often described as tapped-delay line models with N paths, time-varying gains and constant delays:
$c(\tau,t)=\sum_{n=1}^{N} a_n(t)\,\delta(\tau-\tau_n)$

>τ_n delay of the n-th path
a_n(t) = α_n(t) exp[-jφ_n(t)] WSS Gaussian complex random process (zero-mean if NLOS) with autocorrelation r_n(τ) and PSD S_n(f)
E[|a_n(t)|²] average power of the nth path

Discrete channel model (cont’d)

The characteristic functions of the WSSUS model for the tapped-delay channel can be calculated with fairly simple math:

$A_c(\tau,\Delta t) = \sum_{n=1}^{N} r_n(\Delta t) \, \delta(\tau-\tau_n)\hspace{1,5cm}\text{space time correlation}$

$A_c(\tau) = \sum_{n=1}^{N} \mbox{E}[ |a_n(t)|^2] \, \delta(\tau-\tau_n)\hspace{1,5cm}\text{power delay profile}$

$S_C(\rho) = \sum_{n=1}^{N} S_n(\rho)\hspace{3,5cm}\text{Doppler spectrum}$

$S_c(\tau,\rho) = \sum_{n=1}^{N} S_n(\rho) \, \delta(\tau-\tau_n)\hspace{1,5cm}\text{scattering function}$

Discrete channel model (cont’d)

$A_c(\tau) = \sum_{n=1}^{N} \mbox{E}[ |a_n(t)|^2] \, \delta(\tau-\tau_n)$

In particular, the power delay profile has a very simple graphical interpretation: see figure.

ITU-R channel models

Indoor office channel

ITU-R channel models (cont’d)

Outdoor to indoor and pedestrian channel

ITU-R channel models (cont’d)

Vehicular channel

Conclusions

The impulse response of the wideband multipath channel is modeled as a 2-D Gaussian random process c(τ,t).
The WSSUS assumption allows ua to simplify the statistical description of c(τ,t).
The frequency selectivity/time dispersion of the channel is measured by the delay spread or by the coherence bandwidth.
The time variability/frequency dispersion of the channel is measured by the coherence time or by the Doppler spread.
Four fading models are possible depending on the signal bandwidth (fast/slow, flat/frequency-selective).