# 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).