Modelling and analysis of wireless fading channels Geir E. Øien

Wireless and mobile communication channels Will initially study single-link wireless communications (one transmitter, one receiver). For example, transmitter may be a mobile terminal and receiver a base station (uplink), or vice versa (downlink). Communication typically exposed to several kinds of impairments, some of which are unique to the wireless environment.

Wireless and mobile channel impairments Impairments result mainly from: –Multipath transmission due to reflections from (possibly moving) objects in the surroundings  multipath fading and inter-symbol interference (frequency-selectivity) –Relative transmitter-receiver motion  Doppler effect (  time-varying, correlated random fading) –Attenuation of signal power from large objects  (slow) shadowing –Interference between different wireless carriers, transmitters, and systems  inter-channel/inter-cell/inter-system interference –Spreading of radiated electromagnetic power in space as function of distance  path loss –Thermal noise and background noise  additive noise

Assumptions Bandwidth B [Hz] available for communications, on a carrier frequency f c [Hz]. Digital communications with linear modulation (e.g. QAM, QPSK) is used. I.e., transmitted waveform represents a sequence of complex-valued modulation symbols, modulated onto a complex sinusoidal carrier. Communications take place at Nyquist rate, i.e. 2B channel symbols are transmitted per second (highest possible rate where intersymbol-interference can be compensated for). Perfect synchronization in time and frequency is available [no timing errors or oscillator drift].

Assumptions, cont’d Thus, the complex baseband representation of transmitted signals can be used: Transmitted waveform x(t) is represented by a sequence of complex-valued discrete samples x(k), where sampling has taken place at Nyquist rate. Real part corresponds to in-phase (I) component of modulation symbol/waveform, and imaginary part corresponds to quadrature (Q) component.

Relative transmitter-receiver motion Assume: Transmitter and receiver move relative to each other with a constant effective velocity v [m/s]. Results in a Doppler shift in the carrier frequency f c by a maximal Doppler frequency of f D = vf c /c [Hz] where c = m/s is the speed of light. Also results in randomly time-varying fading envelope as reflection and scattering conditions change with time as transmitter and receiver move. Random fading models used to describe this phenomenon. How fast the fading varies depends on f D. The faster the motion, the more rapid the fading variations.

Random flat-fading models The complex baseband model of a flat-fading channel becomes y(k) =  (k)x(k) + w(k), k  Z, where y(k) is received symbol at discrete time instant k,  (k) is the fading envelope, x(k) is the transmitted information channel symbol, and w(k) is (complex-valued) AWGN.  (k) is modelled as a temporally correlated random variable. Distribution (pdf) given by multipath model. Correlation properties given by multipath model and transmitter-receiver motion assumptions.

Random flat-fading models, cont’d Rayleigh fading: Assumes isotropic scattering conditions, no line-of-sight [most common model] –I- and Q-components of complex fading gain are complex, zero- mean gaussian processes –thus the fading envelope follows a Rayleigh distribution Ricean (Rice) fading: Assumes line-of-sight component is also present. –I- and Q-components of complex fading gain are still complex gaussian, but not zero-mean –thus the fading envelope follows a Rice distribution Nakagami-m fading: More general statistical model which encompasses Rayleigh fading as a special case, and can also approximate Ricean fading very well.

Multipath transmission As waves are radiated from a transmitter antenna, they will be reflected from reflecting objects. Waves are also scattered from objects with rough surfaces. Thus a transmitted signal will typically travel through many different transmission paths, and arrive at the receiver as a sum of different paths, coming in at various spatial angles. Typical assumption in mobile systems (at mobile side): Isotropic scattering  Transmitted energy arrives equally distributed over all possible spatial angles, with uniformly distributed phases. In addition, a stronger line-of-sight (LOS) component may be present.

Mathematical modeling of multipath transmission Signal components from different incoming paths to receiver have different delays (phases) and amplitude gains. Thus, mutual interference between paths results in a channel response which is a weighted sum of complex numbers, in general time- and frequency-dependent. A multipath transmission channel can then be modelled by a time-variant, complex-valued channel frequency response. Here: Will only consider frequency-independent (flat) channel responses [channel impulse response has only one tap; thus no inter-symbol interference]

Attenuation of signal power from large objects The mean received power attenuation depends strongly (and relatively deterministically) on the path length undergone the transmitted signal [cf. Path loss]. However, slow stochastic variations may also be experienced in the mean received power attenuation, due to shadowing imposed by large terrain features between transmitter and receiver (e.g. hills and buildings). Empirical studies that these variations can be modelled by a log-normal probability distribution. This means that the mean received power attenuation in dB has a normal (i.e., gaussian) distribution. Cf. Stüber, Ch. 2.4 for details [self-study].

Interference Electromagnetic disturbances from different sources within a frequency band may interfer with the desired information signal. These disturbances may come from other users (intra- or inter-cell), or from other systems sharing the same frequency band (may be problem in unlicensed bands). In our discussion we shall either disregard such interference, or model it simply as an increased noise floor (appropriate if there are many independent interference sources). I.e., our “additive noise” term may encompass certain types of interference (e.g., inter-user interference in a fully loaded cellular network).

Path loss In free space, received signal power typically decays with the square of the path length d [m] experienced by the signal during transmission. However, real-life environments are not “free space”, since the earth acts as a reflecting surface: Other (maybe even more severe) models may apply. Power may decay even faster with increased d. Transmit and receive antenna gains (and heights above ground) and carrier frequency will also influence the path loss. Several analytical and empirical models developed for different environments (macro-/microcell, urban/rural…). We refer to [Stüber, Ch. 2.5] for details [self-study]. In our presentation, path loss will manifest itself as a (constant) expected power attenuation G [-].

System noise Noise in a communication system typically comes from a variety of mutually independent sources: –thermal noise in receiver equipment –atmospheric noise –various kinds of random interference Noise is typically independent of the information signal, and of the fading characteristics of the channel. Thus it usually is modelled as Additive White Gaussian Noise (AWGN). [NB: law of large numbers!] Constant power spectral density N 0 /2 [W/Hz] over the total (two-sided) bandwidth 2B. I.e., total noise power N 0.