Channel Fading

[Glossary]
+ Channel Fading: There can be multiple paths between the transmitter and the receiver. The major component can be the line of sight (LOS). Other components can vary with time either because the device is moving or because there are moving objects (cars) in the channel. Using deterministic model is non-realistic. Stochastic model (fading) is constructed to represent the channel property.
+ Delay spread: The time difference between LOS and the last received signal.
+ Resolvable: When the delay is less than the inverse of bandwidth (minimum sampling time), the delayed component and the major component is non-resolvable. If the delay is larger than that, the delay is resolvable. Unresolvable component is merged into the major component and expressed as the mean of the two components (or multiple components if there are multiple unresolvable delays). The merged representative component undergoes constructive and destructive interference.
+ Outage: As the device moves, the signal amplitude may stay below a certain level. This phenomena is called outage. Often at certain locations, the signals are too weak to be clear.

[Model]
Generally, fading models can be categorized into narrow band fading and broad band model.
The received signal is the summation of multiple components as,
r(t) = sum(A_n(t) u(t-t_n) exp^j(w(t-t_n)+phi_n(t))), where
A_n(t): amplitude of the nth component (nth multiple path).
t_n: the time delay of the n_th component.
phi_n(t): the phase change from Doppler frequency shift.

phi_n(t) = Integrate(fd(t) dt), where
fd(t) is the Instant Doppler frequency shift = v cos(th(t))/lambda. th(t) is the angle of motion to the direction of arrival.

The equivalent impulse response function is
c_n(t, tau) = A_n(t)exp^(-j*phi_n(t)) delta(tau-tau_n(t))

To get the received signal, the receiver needs to integrate over the arrival components with different delay time tau.

+ Gaussian In-phase fading: when there is no major component in the multiple path, the fading process is like a zero-mean Gaussian process. It's characterized by the standard deviation and the correlation time. The std represents the amplitude fluctuation. Autocorrelation can show the correlation time.
[example]: In an environment of a uniform circular reflector and a moving car, the autocorrelation function is Bessel. There waveform is periodical.
+ Rayleigh fading: both the in-phase and quadrature fading components are Gaussian, the joint amplitude is Z = sqrt(X^2+ Y^2) is a Rayleigh fading,
p(Z=z) = z/c Exp(-z^2/std^2).
The power spectrum distribution is exponential,
p(Z^2 = x) = c1 exp(-x/x0).
+ Rician fading: there is a major component (like LOS) in the multiple paths.
The amplitude distribution is p(z;s,std) = z exp(-(z^2+ s^2)/s) I0(zs/std).\
One important parameter is s, the strength of LOS.
+ Nakagami fading: a function in between Rayleigh and Rician.
+ Level crossing rate: below a certain level of the signal, the system is difficult to demodulate. This period is called outage.
+ Finite state Markov Model: use finite random walk markov process to model the fading.
+ Flat fading: the fading amplitude for different frequency component is equal.

[Wide band model]
Wide band channels causes the delay symbols to have ISI. For narrow band, the delay spread is localized within symbol time. There can be constructive and destructive interference within a symbol, but the ISI is small. On the other hand, wide band channel has shorter symbol time. The delay spread tends to be larger than the symbol time. The upshot is ISI from delay spread.
+ Frequency selective fading: the fading at different frequencies are varying. Multi-carrier modulation is also subjected to frequency selecting fading.