# Luigi Paura » 14.Digital transmission on fading multipath channels

### Digital transmission on fading multipath channels

Outline:

• Introduction to wireless channels.
• Linear Time-Variant model: multipath and doppler spread.
• Tapped Delay Line Channel Model.
• Coherence Bandwidth and Coherence Time.
• Introduction to statistical model.

### Introduction to Wireless Channels

AWGN channel model and Bandlimited AWGN one are not appropriate for modelling wireless propagation mechanisms:

• Ionospheric propagation in the HF band.
• Mobile Cellular Transmission.
• Underwater Acoustic Signal Transmission.

There are three kinds of power losses in a radio communication link

Path Loss:

Pr=PtGtGr(λ/4πd)2

more in general

Pr=PtGtGr(λ/4πd)α

Loss due to slow fading effects (distance scale is in the order of 50-100 m – shadow fading).

Loss due to short range fading: distance scale is in the order of λ/2 (<10m).

Short range fading is due to two phenomenons:

Time dispersion due to multipath.

Frequency dispersion due to radial mobility between TX and RX and/or to variability of the propagation channel properties (time-variant) → linear time-variant model.

$h(t, \pi) ~~\leftrightarrow \widetilde h (t, \pi) ~~\text{complex envelope}$

$s(t)=Re\\u(t)e^{j2\pi f_c^t}\}$

s(t): transmitted signal with u(t) equivalent lowpass with band Bu

The received signal in the absence of noise can be expressed as:

$r(t)=Re\Biggl\{\sum_{n=0}^{L(t)}\alpha_n(t)u(t-\tau_n(t))e^{j(2\pi f_c(t-\tau(t))+\phi_{D_n})}\Biggr\}$

n = 0 – LOS

n ≠ 0 - LOS

L(t) – number of paths

αn(t)- time – variant attenuation factor associated with the nth path

τn(t) - propagation delay associated with the nth path

ΦDn – phase shift due to doppler

The nth distinguishable path can be generated by a single scatterer or by a cluster of no-distinguishable scatterers.

If the delays τi and τj are very different:

$\tau_i,\tau_j :|\tau_i-\tau_j|>>1/B_u~~~~~T~~(B_u \equiv \text{band of u(t)}) \Rightarrow$

the components i and j are distinguishable (wideband fading)

$\text{if} ~~\tau_1 \approx \tau_2 \approx \tau \Rightarrow u(t-\tau_1)\approx (t-\tau_2)$

The i-th and j-th replica overlap → multipath non-distinguishable

If the τi and τj of two components are similar →

The paths are not distinguishable and the contributions combine in a single path (cluster of scatterers non- distinguishable).

Wideband channels (Bu>>1) have distinguishable paths  → each term in the sum corresponds to a reflection or to a cluster of non-distinguishable paths.

If each term is due to a cluster of scatterers, αn(t) changes noticeably with the distance because of the phase changes of the single non-distinguishable contributions.

### Linear Time-Invariant Model – LTI (2/3)

If the parameters are time-invariant:

$h(t, \tau)=\sum_{n=0}^L \alpha_n e^{-j\phi_n}\delta (\tau-\tau_n)=h(\tau)$

The channel introduces only a time-dispersion and if L,τn are deterministic, the time-dispersion can be measured as the maximum delay with respect to the LOS contribution or to mean delay

### Linear Time-Invariant Model – LTI (3/3)

Alternative definition:

$\tau_{rms}\sqrt{\frac{{\sum_{k=1}^{L}}\tau_k^2\sigma_k^2}{\sum_{k=1}^L \sigma_k^2}- \Biggl(\frac{\sum_{k=1}^{L}\tau_k\sigma_k^2}{\sum_{k=1}^L \sigma _k^2}}\Biggr)^2$

$\sigma_k^2~~~E[\alpha_k^2]$ rms of the kth path

Narrowband Fading : TM << 1/Bu ≅ T  → non- distinguishable paths.

### Tapped Delay Line Channel Model (2/3)

$x(t)=Re \Biggl\{ A \sum_n \apha_n (t)e^{j2\pi f_c(t-\tau_n(t))}\Biggl\}=$

$=Re\{e^{j2\pi f_c^{t} z(t)}\}~~~~ \text{where z(t) is the complex envelope}$

$z(t)~~~~ A\sum_n \alpha_n (t)e^{-j2\pi f_c^{\tau}_n}{(t)}}=~~~~~A\sum_n \alpha_n(t)e^{-j\phi_n{(t)}}$

$e^{j2\pi f_c^t} \rightarrow z(t)e^{j2\pi f_ct}$

At the output we do not have a phasor because z(t) is not constant → Frequency dispersion

### Tapped Delay Line Channel Model (3/3)

$c_n (t)~~~~\alpha_n(t)e^{j\phi_n{(t)}}$

In general cn(t) is modeled as a complex gaussian process

$T_m=L\text{x}\frac 1 {B_u}$ where L is the numbers of paths, $\frac 1 {B_u}$ is the time resolution i.e. the duration of transmitted signal.

### Doppler bandwidth (1/3)

$z(t)=A\sum_n \alpha_n(t)e^{-j2\pi f_c \tau_n (t)}=A\sum_n \alpha_n (t)e^{-j\phi_n (t)}$

The phases Φn can give rise to constructive or destructive  contributions that weaken or strengthening the amplitude of the transmitted signal, namely the presence of fading effects.

|z(t)| models the variation law of the attenuation.

### Doppler bandwidth (2/3)

The variability measure of z(t) allows us to valuate the amount of spectral dispersion.

z(t) is a random signal, hence let us consider the mean square value of the bandwidth of z(t) namely,Bd

Bd = doppler Bandwidth

If Bd >>1 → the channel “varies” rapidly
If Bd <<1 ← the channel “varies” slowly

### Doppler bandwidth (3/3)

Let us denote with Bu the bandwidth of the transmitted signal:

If Bd <<Bu → the channel can be assumed to be  stationary since the spectral dispersion is  negligible with  respect to  the signal bandwidth Bu.

In the time domain:

Bd<<Bu → 1/Bd >>1/Bu → TC>>TS Temporal Flat Fading, namely:

$e^{j2\pi f_c t}\rightarrow x(t)= A\sum_n \alpha_n e^{j2\pi f_c(t-\tau_n)}=ze^{j2\pi f_c t}$

The signal duration is much smaller than the temporal scale TC of the channel variation.

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