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

Individual multipath components are resolvable.

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

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

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

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).

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

- It represents the correlation between two multipath components (τ
_{1}and τ_{2}) evaluated at times t e t+Δt

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

Uncorrelated scattering (US) assume that scattering associated to path τ_{1} is **uncorrelated** (i.e., independent due to the Gaussian assumption) with that of any other path τ_{1} ≠ τ_{2}:

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

Under WSS + US (WSSUS) assumptions:

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.**

Fourier transform of Ac(τ, Δt) with respect to the second variable ->

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

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

Since A_{c}(τ) ≥ 0, the power delay profile can be interpreted as the pdf of the delay spread T_{m}:

Mean and standard deviation of T_{m} represent the **average delay** **spread** and the **RMS 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_{s}expressed 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!

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**):

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**:

Under the WSSUS assumption, the time-frequency autocorrelation depends on Δf=f_{2}-f_{1} (stationarity in frequency) and one has:

Define A_{C}(Δf)= A_{C}(Δf; Δt=0) (**frequency autocorrelation**), it results that:

- 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)

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

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**

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**):

**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}(ρ) is the Fourier transform of the time autocorrelation function A_{C}(Δt)= A_{C}(Δf=0, Δt).

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 variation**of 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

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.}

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.

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):

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

Discrete multipath channels are often described as **tapped-delay line** models with N paths, time-varying gains and constant delays:

- >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

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

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

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).

*3*. Current and emerging wireless systems

*5*. Shadowing

A. Goldsmith. Wireless Communications. Cambridge University Press, 2005 (chap. 3)

P. A. Bello, “Characterization of randomly time-variant linear channels”, IEEE Trans. Commun. Syst, pp. 360-393, December 1963

ITU-R Recommendation M.1225, “Guidelines for the evaluation of radio transmission technologies for IMT-2000”, 1997

Supplementary material eventually available on the website

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