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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.
Fading classification.

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.
Line-of-sight Microwave Radio Transmission.
Airplane-to-Airplane radio communications.
Underwater Acoustic Signal Transmission.

Fading Classification

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

Path Loss:

P_r=P_tG_tG_r(λ/4πd)²

Short Range Fading (1/4)

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.

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

Short-Range Fading (2/4)

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

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

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

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

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

Φ_Dn – phase shift due to doppler

Short-Range Fading (3/4)

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

If the delays τ_i and τ_j 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

Short-Range Fading (4/4)

If the τ_i and τ_j 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 (B_u>>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, α_n(t) changes noticeably with the distance because of the phase changes of the single non-distinguishable contributions.

Linear Time-Variant Model -LTV (1/3)

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,τ_n 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:

RMS delay spread

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

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

Narrowband Fading

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

Long-range and narrowband short-range fading effects

Narrowband Fading

Tapped Delay Line Channel Model (1/3)

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 c_n(t) is modeled as a complex gaussian process

Tm – multipath spread

$T_m=L\text{x}\frac 1 {B_u}$ 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 Φ_n 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,B_d

B_d = doppler Bandwidth

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

Doppler bandwidth (3/3)

Let us denote with B_u 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 B_u.

In the time domain:

B_d<<B_u → 1/B_d >>1/B_u → T_C>>T_S 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 T_C of the channel variation.

Le lezioni del Corso

1. Presentation of the course

2. Data Compression

3. Source Coding

4. Digital Trasmission

5. Digital Transmission over AWGN chanel

6. Evaluation of P(e) for optimum RX in AWGN

7. Error probability for M-PSK

8. Noncoherent ML Demodulation of FSK signals in AWGN

9. Transmission through bandlimited AWGN channels

10. Block Channel Coding

11. Linear block codes

12. Decoder for Linear Codes

13. Cyclic Codes

14. Digital transmission on fading multipath channels

15. Statistical Channel Model