# Giacinto Gelli » 7.Narrowband fading

### Outline

I-Q components
Mean, autocorrelation and cross-correlation of I-Q components
Jakes model
Envelope distributions

$r(t) = \mbox{Re} \left\{ \sum_{n=1}^{N} \alpha_n(t) e^{-j\phi_n(t)} \, u \left[ t - \tau_n(t) \right] \,e^{j 2 \pi f_ct}\right\}$

When Tm << 1/Bu one has u[t-τn(t)] ≈u(t):

$r(t) = \mbox{Re} \left\{ u(t) \, e^{j 2 \pi f_c t}\left[ \sum_{n=1}^{N} \alpha_n(t) e^{-j\phi_n(t)} \right]\right\}$

Bu = 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)

### Gaussian model

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

$r_I(t) = \sum_{n=1}^{N} \alpha_n(t) \cos \phi_n(t)\qquad$

$r_Q(t) = \sum_{n=1}^{N} \alpha_n(t) \sin \phi_n(t)$

For N >> 1, the I/Q components rI(t) and rQ(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.

### Quasi-static assumption

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

• αn(t) ≈ αn
• τn(t) = dn(t)/c ≈[dn – v cos(θn) t]/c =τn- v t cos(θn)/c
• Φn(t) = 2π fcτn(t) ≈ 2π fcτn – 2π fc vt cos(Φn)/c = 2π fcτn – 2π fDn t
• θn is the multipath arrival angle at the receiver
• fDn= fc(v/c) cos(Φn )=(v/λ) cos(Φn) is the Doppler shift

Since fcτn>> 1, Φn(t) ~ U(0,2π) (uniform distribution) for any value of the Doppler shift fDn.

Reasonable assumption for NLOS propagation, in LOS propagation the phase is not uniform since a deterministic component should be added to rI(t) and rQ(t).

### Mean of the I/Q components

The average of rI(t) for a fixed Doppler shift fDn is:

$\mbox{E}[r_I(t)] = \sum_{n=1}^{N}\mbox{E}[ \alpha_n] \underbrace{\mbox{E}[ \cos \phi_n(t)]}_{=0} = 0$

where we assumed that αn and Φn(t) ~ U(0,2π) are statistically independent.

A similar result holds for the Q-component:

$\mbox{E}[r_Q(t)] = \sum_{n=1}^{N}\mbox{E}[ \alpha_n] \underbrace{\mbox{E}[ \sin \phi_n(t)]}_{=0} = 0$

The I/Q components are zero-mean for any value of the Doppler shift fDn.

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

### I/Q autocorrelation

Recall that θn is the multipath arrival angle.
The autocorrelation of the I/Q components.

$A_{r_I}(\tau) = A_{r_Q}(\tau) =\frac{1}{2}\sum_{n=1}^{N}\mbox{E}[ \alpha_n^2] \,\mbox{E}_{\theta_n}\left[ \cos \left(2 \pi \frac{v}{\lambda} \cos \theta_n \right)\right]$

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 θn must be carried out to obtain an explicit expression → a simple model for the angular distribution of multipath is required.

### I/Q cross-correlation

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

$A_{r_I r_Q}(\tau) =\frac{1}{2}\sum_{n=1}^{N}\mbox{E}[ \alpha_n^2] \,\mbox{E}_{\theta_n}\left[ \sin \left(2 \pi \frac{v}{\lambda} \cos \theta_n \right)\right] =- A_{r_Q r_I}(\tau)$

For τ=0 we have ArI,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 θn.

The autocorrelation of the received signal is:

$A_r(\tau) =A_{r_I}(\tau) \, \cos(2 \pi f_c \tau) +A_{r_I r_Q}(\tau) \, \sin(2 \pi f_c \tau)$

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

### Jakes model

Assumption: the scattering is uniformin angle

• large number (N>>1) of scatterers uniformly distributed over the circle θn = n Δθ
• all multipath component with the same power: E[αn2]=2Pr/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).

### Jakes model (cont’d)

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

$A_{r_I}(\tau) = A_{r_Q}(\tau) =P_r J_0(2 \pi f_D \tau)$
$A_{r_I r_Q}(\tau) = 0$

where:

• J0(x) zeroth-order Bessel function
• fD 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).

### Envelope distributions

The envelope of the received signal is:

$z(t) = |r(t)| = \sqrt{r_I^2(t) + r_Q^2(t)}$

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

$p_Z(z) = \frac{z}{\sigma^2} \exp \left( - \frac{z^2}{2 \sigma^2} \right)\qquadz \ge 0$

When LOS component is present → Rice distribution:

$p_Z(z) = \frac{z}{\sigma^2} \exp \left[ - \frac{(z^2+s^2)}{2 \sigma^2} \right] \,I_0 \left( \frac{z s}{\sigma^2}\right)\qquadz \ge 0$

### Envelope distributions (cont’d)

Measurements support Nakagami distribution in some environments:

$$p_Z(z) = \frac{2 \, m^m z^{2m-1}}{\Gamma(m) \overline{P}_r^m}\exp \left( - \frac{m z^2}{\overline{P}_r} \right)\qquadz \ge 0$\end{itemize}$

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.

### Conclusions

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).
Rayleigh, Rice, Nakagami are common choices.

### I materiali di supporto della lezione

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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