Narrowband fading model

I-Q components

Mean, autocorrelation and cross-correlation of I-Q components

Jakes model

Envelope distributions

When T_{m }<< 1/B_{u} one has u[t-τ_{n}(t)] ≈u(t):

B_{u} = bandwidth of u(t).

The RX signal differs from the TX one for the multiplicative effect of the term in square brackets:

- narrowband fading does not introduce temporal dispersion but acts as time-varying
**multiplicative noise**→ it introduces both amplitude variations (fading) and frequency dispersion - simple characterization of NB fading can be obtained by assuming u(t)=1 (tone transmission)

Due to the random nature of the channel, the received signal r(t) when u(t)=1 is a **random process**:

For N >> 1, the I/Q components r_{I}(t) and r_{Q}(t) are approximately **jointly Gaussian** by the CLT (sum of a large number of independent random variables).

They can be characterized by their **mean, autocorrelation and cross-correlation**.

Assume that the amplitudes, the delays and the Doppler shifts are **quasi-static **(i.e., slowly varying):

- α
_{n}(t) ≈ α_{n} - τ
_{n}(t) = d_{n}(t)/c ≈[d_{n}– v cos(θ_{n}) t]/c =τ_{n}- v t cos(θ_{n})/c - Φ
_{n}(t) = 2π f_{c}τ_{n}(t) ≈ 2π f_{c}τ_{n}– 2π f_{c}vt cos(Φ_{n})/c = 2π f_{c}τ_{n}– 2π f_{Dn }t - θ
_{n}is the multipath arrival angle at the receiver - f
_{Dn}= f_{c}(v/c) cos(Φ_{n})=(v/λ) cos(Φ_{n}) is the Doppler shift

Since f_{c}τ_{n}>> 1, Φ_{n}(t) ~ U(0,2π) (uniform distribution) for any value of the Doppler shift f_{Dn}_{.}

Reasonable assumption for NLOS propagation, in LOS propagation the phase is not uniform since a deterministic component should be added to r_{I}(t) and r_{Q}(t).

The average of r_{I}(t) for a fixed Doppler shift f_{Dn} is:

where we assumed that α_{n} and Φ_{n}(t) ~ U(0,2π) are statistically independent.

A similar result holds for the Q-component:

The I/Q components are **zero-mean** for any value of the Doppler shift f_{Dn}_{.}

In LOS propagation the I/Q components are not zero-mean.

Recall that θ_{n} is the multipath arrival angle.

The autocorrelation of the I/Q components.

The I/Q components have the same autocorrelation.

The autocorrelation does not depend on time t → the I/Q components are **wide-sense stationary (WSS)** and, since they are Gaussian, they are also **strict-sense stationary.**

The average with respect to θ_{n} must be carried out to obtain an explicit expression → a simple model for the **angular distribution** of multipath is required.

The cross-correlation of the I/Q components is:

For τ=0 we have A_{rI},_{rQ}(0)=0 → the I/Q components are **uncorrelated** for τ=0 →since they are Gaussian, they are independent for τ=0.

The obtained expression must be averaged with respect to θ_{n}.

The autocorrelation of the received signal is:

The received signal is WSS Gaussian and zero-mean (remember that the TX signal was u(t)=1).

Assumption: the scattering is **uniform**in angle

- large number (N>>1) of scatterers uniformly distributed over the circle θ
_{n}= n Δθ - all multipath component with the same power: E[α
_{n}^{2}]=2P_{r}/N

Suited for urban NLOS propagation, often utilized in simulations.

The discrete angle distribution can be approximated by a continuous one θ ~ U(0,2π).

By averaging with respect to the uniform distribution for θ we get the **autocorrelation** and **cross-correlation** functions:

where:

- J
_{0}(x) zeroth-order Bessel function - f
_{D}max Doppler shift - decorrelates over roughly half a wavelength

Jakes model (cont’d)

By taking the Fourier transform of the autocorrelation function we obtain the **power spectral density (PSD).**

The envelope of the received signal is:

Gaussian approximation based on CLT leads to **Rayleigh** distribution (power is exponential):

When LOS component is present → **Rice** distribution:

Measurements support **Nakagami** distribution in some environments:

where:

- m ≥ 1/2 fading parameter
- Γ(.) Gamma function

Reduces to Rayleigh for m=1, no fading for m →∞.

Approximates Rice case for m=(K+1)^{2}/(2K+1), can model “worse than Rayleigh” cases for m<1.

Lends itself to closed form expressions for BER.

Narrowband model has I/Q components that are zero-mean stationary Gaussian processes.

Auto and cross-correlation depend on angular distribution of multipath.

Uniform scattering assumption makes autocorrelation of I/Q components follow Bessel function (Jakes model).

Fading components decorrelated over roughly half-wavelength.

Cross-correlation is zero (I/Q components independent).

Fading distribution depends on environment.

Rayleigh, Rice, Nakagami are common choices.

*3*. Current and emerging wireless systems

*5*. Shadowing

A. Goldsmith. Wireless Communications. Cambridge University Press, 2005 (chap. 3)

W.C. Jakes. Microwave Mobile Communications. Wiley, New York, 1974

Supplementary material eventually available on the website

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