# Giacinto Gelli » 8.Wideband fading

### Outline

WSSUS model
Measures of time dispersion:

• power delay profile

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).

### 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.

### 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 Ac(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, Sc(τ, ρ) describes the power distribution as a function of the Doppler frequency ρ.

### Power delay profile

For Δt = 0, the function Ac(τ)= Ac(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 Ac(τ) ≥ 0, the power delay profile can be interpreted as the pdf of the delay spread Tm: $$p_{T_m}(\tau) = \frac{A_c(\tau)}\int_{0}^{+\infty} A_c(\tau) \, d\tau}\$ Mean and standard deviation of Tm represent the average delay spread and the RMS delay spread. ### Effects of delay spread Consider a digitally-modulated signal with symbol period Ts transmitted over a channel with RMS delay spread σTm: • significant intersymbol interference (ISI) if σTm>> Ts • if σTm<< Ts the ISI will be negligible The RMS delay spread limits the maximum transmission rate Rs=1/Tsexpressed in symbols/s (baud). Some typical values: • Indoor -> σTm=50 ns -> Rs = 1/Ts = 2 Mbaud • Outdoor -> σTm=30 μs -> Rs = 1/Ts = 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 AC(Δf)= AC(Δf; Δt=0) (frequency autocorrelation), it results that: $$A_C(\Delta f) = \int_{-\infty}^{+\infty} A_c(\tau) \, e^{-j 2 \pi \Delta f \tau} \, d\tau$$

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

### Coherence bandwidth

The frequency value Bc for which AC(Δf) ≈ 0 is called coherence bandwidth:

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

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

• Bc = 0.02/σTm for a correlation greater than 0.9
• Bc = 0.2/σTm for a correlation greater than 0.5

If the TX signal has bandwidth Bu, then:

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

### Doppler power spectrum

By evaluating the Fourier transform of AC(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 SC(ρ)= SC(Δf=0, ρ): $$S_C(\rho) = \int_{-\infty}^{+\infty} A_C(\Delta t) \, e^{-j 2 \pi \rho \Delta t} \, d\Delta t$$

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

### Coherence time

The time value Tc for which AC(Δt) ≈ 0 is called coherence time:

• Tc is a measure of the temporal width of AC(Δt)
• values of the channel impulse response at times separated by Tc 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 Tc = 0.4/fD -> inversely proportional to the maximum Doppler shift

The spectral width BD of SC(ρ) is called Doppler spread or Doppler bandwidth.

Due to the Fourier transform relationship between AC(Δt) and SC(ρ), the Doppler spread BD is inversely proportional to coherence time Tc.

Comparison between the speed of variation of the channel (measured by Tc or BD) and the speed of variation of the signal (measured by 1/Bu):

• Tc << 1/Bu -> the channel exhibits strong variations during the typical time of variation of the signal -> fast fading
• Tc >> 1/Bu -> 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.

### Relations with the scattering function

To complete the mathematical framework, note that the statistical scattering function Sc(τ,ρ) is the 2-D Fourier transform of the time-frequency autocorrelation AC(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.

### Discrete channel model

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

• n delay of the n-th path
• an(t) = αn(t) exp[-jφn(t)] WSS Gaussian complex random process (zero-mean if NLOS) with autocorrelation rn(τ) and PSD Sn(f)
• E[|an(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.

### 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).

### I materiali di supporto della lezione

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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