Modelling and analysis of wireless fading channels Geir E. Øien 29.08.2003 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 fc [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 fc by a maximal Doppler frequency of fD = vfc/c [Hz] where c = 3•108 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 fD. 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 N0/2 [W/Hz] over the total (two-sided) bandwidth 2B. • I.e., total noise power N0.