Narrowband fading model
Mean, autocorrelation and cross-correlation of I-Q components
When Tm << 1/Bu one has u[t-τn(t)] ≈u(t):
Bu = bandwidth of u(t).
The RX signal differs from the TX one for the multiplicative effect of the term in square brackets:
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 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.
Assume that the amplitudes, the delays and the Doppler shifts are quasi-static (i.e., slowly varying):
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).
The average of rI(t) for a fixed Doppler shift fDn is:
where we assumed that αn and Φn(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 fDn.
In LOS propagation the I/Q components are not zero-mean.
Recall that θn 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 θn 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 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:
The received signal is WSS Gaussian and zero-mean (remember that the TX signal was u(t)=1).
Assumption: the scattering is uniformin angle
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:
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:
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
9. Introduction to digital communications
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