** 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.

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.

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

**Path Loss:**

*P _{r}=P_{t}G_{t}G_{r}(λ/4πd)^{2}*

more in general

*P _{r}=P_{t}G_{t}G_{r}(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.

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:

*n* = 0 – *LOS*

*n* ≠ 0 - *LOS*

*L(t)* – number of paths

*α _{n}(t)*- time – variant attenuation factor associated with the

*τ _{n}(t) *- propagation delay associated with the

*Φ _{Dn}* – 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 τ_{i} and τ_{j} are very different:

the components i and j are distinguishable (wideband fading)

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

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.

**If the parameters are time-invariant:**

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

** Alternative definition:**

**RMS delay spread**

*rms* of the* kth path*

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

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

In general c_{n}(t) is modeled as a complex gaussian process

Tm – multipath spread

where L is the numbers of paths, is the time resolution i.e. the duration of transmitted signal.

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.

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

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:**

The signal duration is much smaller than the temporal scale T_{C} of the channel variation.

*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

*13*. Cyclic Codes

Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion