The range of α_{n}(t) is generally large.

Since *f _{c}>>1 → Φ_{n}(t)=2πfcτ(t)mod 2 π (0,2π)* can be modeled as a uniform random variable, namely,

Since z(t) is a sum of independent contributions, with roughly the same strength, it can be modeled as a complex **Gaussian process** thanks to the **Central Limit Theorem.**

|z(t)| can be modeled (for any fixed instant t) as a Rayleigh RV if the components.

z_{c}(t) |z(t)| cos θ (t) and z_{s}(t) |z(t)| sin θ (t) are zero mean, independent and with the same variance σ^{2}

*θ(t) ≡ U(0,2π)*

If the mean of z_{c} (t) and / or z_{s}(t) are ≠ 0 → |z(t)| is a rice RV.

h(t,τ) is a random process (τ is a parameter).

The statistical characterization in a wide sense is obtained by evaluating the mean and the autocorrelation of h(t, τ ). Alternatively, we can evaluate the mean and the correlations of the complex envelope:

by assuming stationary and the scatterers τ_{1} and τ_{2 }uncorrelated

mean power associated with the scatters which introduces the delay τ.

If B_{u}<<B_{c }the channel is frequency non – selective namely, flat in frequency.

if T_{s}<<T_{c} the channel is temporal non-selective namely, is flat in time.

If both conditions are verified, then there isn’t Doppler (temporal variability) as well as Multipath (temporal dispersion):

The scattering function describes the distribution of the spectral components associated with any path of delay Τ.

**Doppler Power spectrum**

**Power Doppler spectrum**

** Ionospheric Channel with two paths:**

T_{M}= 1 msec

B_{u} = 10 KHz

1/Bu =0.1 msec 1/B_{u} << T_{M} → L =1msec/0.1 msec =10 delays

B_{D}T_{M}=T_{M}/T_{c} *spread factor*

*B _{D}T_{M }< 1 → *underspread channel

*B _{D}T_{M <} 1 → *overspread channel

Ionospheric propagation with short waves:

T_{M }=10^{-3}-10^{-2} ; B_{D}=10-1 ;SF=10^{-4}-10^{-3}

Mobile Telephony :

T_{M} =10^{-5} ; B_{D}=100 ;SF=10^{-3}

How to counteract the fading effects ?

By introducing redundancy:

- Channel Coding
- Diversity namely utilizing multiple channels

Kinds of diversity:

- Temporal Diversity
- Polarization Diversity
- Frequency Diversity
- Spatial Diversity

Determine for a channel with doppler spread B_{D} =80 Hz the minimum temporal separation toensure that the samples of the received signalare roughly independent.

The coherence time is 1/B_{D} =1/80 =12,5 msec

If the received signal is Gaussian , the samples separated by at least 12,5 msec are roughly independent.

An HF channel with nominal bandwidth of 3200 Hz is utilized to transmitt digital information with bit rate of (1) 4800 bps or (2) 20bps.

Assuming that T_{M} = 5msec, select a modulation technique for both bit rates and moreover state if an equalizer is necessary to counteract ISI.

Let us exclude in the presence of fading PAM or QAM the signalling since they carry information also on the amplitude.

If we consider MPSK one has:

A transmission filter with spectral characteristics of the raised cosine can be utilized to obtain the desired shape.

If the bit-rate is 20bps, by utilizing again a 4PSK 1/T =20/2 → T= 100 msec >> 5 msec → ISI is negligible

If we want to use all the available bandwidth we can resort to a Spread Spectrum technique.

Let us consider an HF channel with nominal bandwidth 3200 Hz e T_{M} =1ms. Determine the minimum number N of subcarriers operating to assure 4800bps and verify the condition: *T*_{sc}*>>T*_{M}

T_{sc} = symbol duration for each subcarrier

Since T_{M} = 1ms we choose T_{sc} =100·T_{M} =100ms → T_{sc}>>T_{M} namely:

If we utilize again a 4PSK for any subcarrier → R_{b} =2/100ms = =20bps →N = 4800/20 =240 carriers.

If we utilize 16PSK → 40bps for carrier →N = 120

Determine the P(e) for a BPSK operating over a fading flat-flat channel Rayleigh distributed. We assume that the phase of the received signal is estimated and the optimum AWGN RX is utilized.

The output of the demodulator is:

For a fixed value of α:

An antipodal signalling ± s(t) is utilized over a channel with only amplitude fading: r(t) ± αs(t)+ n(t) 0≤t≤T with n(t) AWGN and

*f(α)=0.1δ(α)+0.9δ(α-2)*

Determine the P(e) for the demodulator which utilizes the matched filter to s(t).

P(e) → ? When E/N0 →∞

Assume that the signal is transmitted over two channels statistically indipendent with

*f(α _{k})=0.1δ(α_{k})+0.9δ(α_{k}-2) k=1,2*

and noises are statistically independent.

The demodulator employ two matched filters and summ the two outputs to obtain the decision variable. Evaluate the P(e)

*with* n a Gaussian RV *with* variance N_{0}E

A binary comunication system transmits the same information over two channels utilizing an antipodal signalling. The received signals are:

With n_{1} and n_{2} statistically independent, zero mean and variances *σ*_{1}^{2}* e σ*_{2}^{2}

The detector performs the decision based on: *r r*_{1}*+kr*_{2}

Determine the value of k that minimizes the P(e)

r is Gaussian RV with

If it has been transmitted

where:

*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

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