Small Scale Fading in Radio Propagation 16:332:546 Wireless Communication Technologies Spring 2005 ∗ Department of Electrical Engineering, Rutgers University, Piscataway, NJ 08904 Suhas Mathur ([email protected]) Abstract - One of the many impairments inherently present in any wireless communication system, that must be recognised and often effectively mitigated for a system to function well, is fading. Fading itself has been studied and classified into a number of different types. Here we present a detailed mathematical analysis and some useful models for capturing the effect of small scale fading. Further we discuss the types of fading as per the behaviour of the wireless channel with respect to the transmit signal. I. RAYLEIGH FADING Figure 1: The figure shows a mobile station moving along the positive x-axis moving at a velocity of v m/s and the nth incoming wave at an angle of θn(t). Small scale fading is a characteristic of radio propagation resulting from the presence reflectors and scatterers that cause multiple versions of the transmitted signal to arrive at the receiver, each distorted in amplitude, phase and angle of arrival. fD,n(t)= fm cos(θn(t)) (1) Consider the situation shown in Fig. 1 wherein a mobile receiver (mobile station or MS) is assumed to where fm = maximum Doppler frequency = v/λ, λ be travelling along the positive x axis with a velocity being the wavelength of the radiowave. Waves ar- v m/s. The figure shows one of the many waves riving from the direction of motion cause a positive arriving at the mobile station. Let us call this the doppler shift, while those coming from the opposite nth incoming wave. Let it be incident at an angle diection cause a negative doppler shift. We wish to derive a mathematical framework to characterize the θn(t), where the dependence on t stems from the fact that the receiver is not stationary. effects of small scale fading. Consider the transmit bandpass signal: The motion of the MS produces a Doppler shift in the received frequency as compared to the carrier fre- s(t)= Re u(t).ej2πfct (2) { } quency. This doppler offset is given by: where u(t) is the complex baseband equaivalent of ∗Taught by Dr. Narayan Mandayam, Rutgers University. the bandpass transmit signal. If N waves arrive at 1 the MS, the received bandpass signal can be written shall see later. as: Further continuing our modeling of the received sig- x(t)= Re r(t)ej2πfct (3) nal, we can neglect the baseband modulating signal { } for narrowband signals (i.e. signals in which the base- with band signal bandwidth is very small compared to the N carrier frequency, which is true of most communica- − r(t)= α (t).e j2πφn(t)u(t τ (t)) (4) tion systems) and consider the unmodulated carrier n − n n=1 alone. X where N −jφn(t) r(t)= αn(t)e (7) φn(t) = (fc + fD,n(t))τn(t) fD,n(t).t (5) − n=1 X is the phase associated with the nth wave. The above N expression for r(t) looks like the output of a linear − x(t)= Re α (t)e jφn(t)ej2πfct (8) time-varying system. Therefore the channel can be { N } n=1 modeled as a linear filter with a time varying impulse X response given by: N = rI (t) cos(2πfct) rQ(t) sin(2πfct) − − c(τ,t)= α (t)e jφn(t)δ(τ τ (t)) (6) n − n n=1 X where c(τ,t) is the channel response at time t to an input N at time t τ. Typically the quantity f + f (t) c D,n r (t)= α (t) cos(2πf t) (9) is large. This− means that a small change in delay I n c n=1 τn(t) causes a large change in the phase φn(t). The X delays themselves are random. This implies that N the phases of the incoming waves are random. The rQ(t)= αn(t) sin(2πfct) (10) αn(t)’s are not very different from one another, i.e. n=1 the αn(t)’s do not change much over a small time X scale. Therefore the received signal is a sum of a r(t)= r (t)+ r (t) (11) large number of waves with random phases. The I Q random phases imply that sometime these waves r (t) and r (t) are respectively the in-phase and the add constructively producing a received signal with I Q quadrature-phase components of the complex base- large amplitude, while at other times they add band equivalent of the received signal. Now we invoke destructively, resulting in a very low amplitude. the Central Limit Theorem for large N. This makes This precise effect is termed small-scale fading, and r (t) and r (t) independent gaussian random pro- the time scale at which the resulting fluctuation of I Q cesses. Further, assuming all the random processes amplitude occurs is of the order of one wave-cycle involved are WSS, we have: of the carrier frequency. The range of amplitude variation that can result can be upto 60 to 70 dB. Small scale fading is therefore pramarily due to the fD,n(t)= fD,n (12) random variations in phase φn(t) and also because of the doppler frequency fD,n(t). The effect of fading is even more important at higher data rates, as we αn(t)= αn (13) 2 τn(t)= τn (14) Ωp = Eθ cos(2πfmτ cos(θ)) We also assume that x(t) is WSS. 2 { } If the 2-D isotropic scattering assumption is used in Φxx(τ)= E x(t).x(t + τ) (15) the above analysis (i.e. the incoming angle θ is uni- { } formly distributed over ( π, π)) then the above is called the Clarke’s Model.− Using the uniform distri- =Φ (τ). cos(2πf t) Φ (τ). sin(2πf t) rI rI c − rQrI c bution for θ in the above, we get: Now, +π Ωp 1 ΦrI rI (τ)= . cos(2πfmτ. cos(θ))dθ (23) Φ (τ)= E r (t)r (t + τ) (16) 2 2π − rI rI { I I } Z π which with a change of variable gives us: N N = E α cos(φ t) . α cos(φ (t + τ)) Ω 1 +π {{ i i } { j j }} = p . cos(2πf τ. sin(θ))dθ (24) i=1 j=1 2 π m X X Z0 J0(2πfmτ) We can assume the φj ’s are independent because de- lays and doppler shifts are independent from path to | Ω {z } = p .J (2πf τ) path. 2 0 m φn(t)= U( π, π) (17) − where J0(.) is the Bessel function of the zeroth order On evaluating the expectations, we get: and first kind.1 Ω Similarly, using the uniform pdf for θ in the expres- Φ (τ)= p E cos(2πf τ) (18) rI rI 2 { D,n } sion for cross correlation of the in-phase and quadra- ture phase components of r(t) gives: where N Ω 1 Φ = 0 (25) p = E α2 (19) rI rQ 2 2 i i=1 We are now in a position to talk about the PSD of X which is the total average received power from all rI (t): multipath components. Now, in the expression above S (f)= Φ (τ) (26) rI rI F{ rI rI } (18), we have Ωp 1 f <f fD,n = fm cos(θn) (20) 4πfm − 2 m = √1 (f/fm) | | 0 otherwise Therefore, we have the auto correlation function of ( 1 the in-phase component rI (t): The Bessel functions of the first kind Jn(x) are defined 2 2 d y as the solutions to the Bessel differential equation: x 2 + Ωp dx Φ (τ)= E cos(2πf τ cos(θ) (21) x dy + (x2 − n2)y = 0. The bessel function J (x) can rI rI 2 θ m dx n { } also be defined in terms of the contour integral: Jn(x) = 1 e(x/2)(t−1/t)t−n−1dt Going through a similar series of steps for the 2πj where the contour encloses the ori- cross-correlation function between the in-phase and gin and is traversed in a counter-clockwise direction. For the specialR case of n = 0 a closed form expression due to Frobe- quadrature-phase component, we get: ∞ ( 1 x2)2 nius is J (x) = (−1)K 4 or the integral J (x) = o k=0 (k!)2 0 π Φ (τ)= E r (t)r (t + τ) (22) 1 ejx cos(θ)dθ rI rQ { I Q } π 0 P R 3 j2πfc t ∗ −j2πfc t Φr r (τ).e +Φ (τ).e = I I rI rI F ( 2 ) ∗ Note that ΦrI rI (τ)=ΦrI rI ( τ), and so for real rI (t), Φ (τ)=Φ ( τ). Thus− we have: rI rI rI rI − − Φ (τ).ej2πfc t +Φ (τ).e j2πfc t S (f)= rI rI rI rI xx F 2 (31) 1 Sxx(f)= SrI rI (f fc)+ SrI rI ( f fc) (32) Figure 2: Bessel function of the zeroth order and the first type. 2 { − − − } This is the shape of the autocorrelation function ΦrI rI (τ) of the in-phase component of the complex baseband equivalent of the Now we shall make use of the knowledge that that received signal. r(t) = rI (t)+ j.rQ(t) is a complex Gaussian process for large N. Therefore the envelope z(t) = r(t) = | | 2 2 2 rI (t)+ rQ(t) has a Rayleigh distribution Having obtained the PSD of r (t), we can now pro- I q x − 2 2 ceed to derive the PSD x(t) as follows: P (x)= .e x /2σ ; x 0 (33) z σ2 ≥ 2 2 r(t)= rI (t)+ jrQ(t) (27) where E z = Ωp = 2σ = average power.