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 delays 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 quadrature 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

∗ 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. Thus we have the{ } probability density function of the recd. signal given by: ∗ Φrr(τ)= E r (t).r(t + τ) (28) 2 { } x −x /Ωp Pz(x)= .e ; x 0 (34) Ωp/2 ≥

=ΦrI rI (τ)+ jΦrI rQ (τ) The above is called Rayleigh fading and is derived from Clarke’s fading model, wherein the PSD of Therefore: the received signal has the U-shape shown above. Rayleigh fading is generally applicable when there is no line-of-sight component. This is a good model for Φ (τ)=Φ (τ) rr rI rI cellular mobile radio. Also note that the squared envelope r(t) 2 is exponentially distributed at any time Further: t: | |

Φ (τ)= Re Φ (τ).ej2πfc t (29) xx rr 1 −x/Ωp Pz2 (x)= .e ; x 0 (35) Ωp ≥

j2πfc t = Re ΦrI rI (τ).e

2It is known that the random variable obtained by ﬁnding the square-root of the sum of the squares of two independent S (f)= Re Φ (τ).ej2πfc t (30) gaussian random variables has a Rayleigh distribution xx F rI rI 4 simple AWGN model with no fading. The avegrage power is given by E[z2]=Ωp = s2 +2σ2. Also:

Kω Ω s2 = p , 2σ2 = p (37) K +1 K +1

III. NAKAGAMI FADING MODEL

The Nakagami Fading model is a purely emperical model and is not based on results derived from phys-

Figure 3: Power spectral density of the received signal, Sxx(f). ical consideration of radio propagation. It uses a chi- This is called the U-shaped PSD characteristic of Rayleigh fading square distrbution with m degrees of freedom. The modeled by the clarke’s model distribution of the received signal’s envelope is given by:

m 2m−1 2m x 2 1 II. RICIAN FADING −mx /Ωp pz(x)= m .e ; m (38) Γ(m)Ωp ≥ 2

We now consider the situation that arises when there where Ω = E[z2] = average power, and m is a model is a line-of-sight component in the received signal. p parameter. By varying the value of this parame- This is common in microcellular systems. The prob- ter, the model can capture various distributions. For ability dstribution for the envelope of the received m = 1 the model converges to the Rayleigh fading signal is then given by: model, setting m = 1/2 makes it a one sided gaussian distribution, while setting m = transforms 2 2 x −(x +s ) xs it into a ’no-fading’ model. Finally, the∞ Rician dis- pz(x)= .e 2σ2 .I0 ; x 0 (36) σ2 σ2 ≥ tribution can be approximated though the Nakagami n model using the followingrelationships: 2 2 2 2 2 2 where s = α0 cos θ0 + α0 sin θ0 = α0 = ’non cen- trality parameter’. It denotes the power in the line- √m2 m of-sight component. 3 I(.) is the modified bessel K = − ; m 1 (39) 4 m √m2 m ≥ function of the zeroth order . − − 2 or, s 2 The quantity K = 2σ2 is called the Rice factor. Note (K + 1) m = (40) that setting K = 0 transforms this model into the 2K +1 Rayleigh fading model discussed in the preceeding section and setting K = would transform it into a The Nakagami model is favoured because it has a ∞ closed form analytical expression. 3 α0 denotes the amplitude gain of the zeroth wave, (ref: previous section) which in this case is the line-of-sight component. All the small-scale fading models considered above 4 The modified Bessel functions In(x) are defined as the so- assume that all frequencies in the transmitted sig- 2 x2 d y nal are affected similarly by the channel, i.e. by the lutions to the ’modified’ Bessel differential equation: dx2 + dy − 2 2 fading. This is called flat-fading or frequency non- x dx (x + n )y = 0 and can be expressed in terms of the −n Bessel functions as: In(x) = (j) Jn(jx) selective fading.

5 Figure 4: The power delay proﬁle gives the the power received as Figure 5: Since received power can only be measured on a dis- a function of time when an impulse is transmitted over the wireless crete time scale, we can only have a discrete power delay proﬁle, channel. which indicates the power received at discrete instants of time when an impulse is transmitted on the wireless channel.

IV. FREQUENCY SELECTIVE FADING CHANNELS The RMS delay spread is a way of quantifying the multipath nature of the channel. It is of the order Consider only WSSUS (Wide Sense Stationary of µs in outdoor situation and of the order of ns in Uncorrelated Scattering). Recall that the channel indoor situations. Note that the absolute transmit response is given by c(t, τ) and represents the power level does not affect the definition of στ and response of the channel at time t to an input impulse µτ . Instead the above two definitions only depend at time t τ. − on the relative amplitudes of multipath components.

Definition: The power delay profile or multipath in- As against the power delay profile shown above, in tensity profile is defined as: reality we can only have a discrete power delay profile. Corresponding to this discrete delay profile, we 1 ∗ Φ (τ)= E[c(t, τ)c (t, τ)] (41) have the following definitions: c 2 P (τ )τ It gives the average power at the channel output as τ = k k k (44) a function of time delay. P (τ ) P k k P Definition: Average delay is defined as: σ = τ¯2 (¯τ)2 (45) τ − ∞ q 0 τφc(τ)dτ where µτ = ∞ (42) φc(τ)dτ R 0 2 P (τk)τ R τ¯2 = k k (46) Definition: RMS Delay spread is defined as: P (τ ) P k k ∞ 2 P 0 (τ µT ) φc(τ)dτ στ = −∞ (43) sR 0 φc(τ)dτ R 6 V. CHARACTERIZATION OF FADING CHANNELS defined as a measure of spectral broadening caused by the time-rate of change of the channel (related to Fading radio channels have been classified in two the doppler frequency). 6. The coherence time is a ways. 5 The first type of classification discusses statistical measure of the time duration over which whether the fading is flat (frequency non-selective) two received signals have a strong potential for am- or frequency selective, while the second classification plitude correlation. Thus if the inverse bandwidth is based on the rate at which the wireless channel of the basebad signal is greater than the coherence is changing (or in other words, the rate of change time of the channel then the channel changes during of the impulse response of channel), i.e. whether transmission of he baseband message. This will cause the fading is fast or slow. In connection with these a distortion at the receiver. It is shown that: characterizations of fading channels, it is useful to 1 note the following quantities: Tc (48) ≈ BD Coherence bandwidth: Coherence bandwidth is a If the coherence time is defined as the duration of statistical measure of the range of frequencies over time over which the time correlation function is > 0.5, which the channel can be considered ”flat” (i.e. fre- then: quency non-selective, or in other words a channel which passes all spectral components with equal gain 9 Tc 2 (49) and phase). It may also be defined as the range of fre- ≈ s16πfm quencies over which any two frequency components have a strong potential for amplitude correlation. It where fm is the maximum doppler frequency = v/λ. has been shown that: 1 Example - Consider a vehicle travelling at 60 mi. Bc (47) per hour and communicating with a stationary base ∝ στ station using a carrier frquency fc = 900 Mhz. This where στ is the RMS delay spread. Also, if we define would give a channel coherence time of Tc 6.77 the coherence bandwidth as that bandwidth over msec. Therefore if the symbol rate of transmission≈ which the frequency correlationfunction is above is greater than 150 samples per second then the 0.5 (i.e. the normalized cross-correlation coefficient fading nature of the channel doesn’t really affect the 1 > 0.5 for all frquencies) then Bc . Note that ≈ 5στ transmitted signal being received by the receiver in a if the signal bandwidth is > BC , then the different harmful way. For a smaller symbol rate, the symbol frequency components in the signal will not be faded width is so large that the channel changes (symbol the same way. The channel then appears to be duration >Tc) within a single symbol. ’frequency-selective’ to the transmitted signal. Flat fading: If a channel has a constant response Doppler spread and Coherence time: While στ for a bandwidth > the transmitted signal bandwidth, and Bc describe the time dispersive nature of the then the channel is said to be a flat fading channel. channel in an area local to the receiver, they do not The conditions for a flat fading channel are: offer any information about the time-variations of the channel due to relative motion between the trans- Bs Bc (50) mitter and the receiver. The doppler spread BD,

5An excellent treatment of the characterization of fading Ts Tc (51) channels is found in an article in the Sept. 1997 issue of the IEEE communications magazine: ’Rayleigh Fading Channels 6If the baseband signal frequency is much greater than the in Mobile Digital Communication Systems: Part I: Caharac- doppler spread BD then the eﬀects of doppler spread are neg- terization’, by Bernard Sklar. ligible

7 where Bs and Ts are the signal bandwidth and the Slow fading: In a slow fading channel, the chan- symbol duration respectively. nel impulse response changes at a rate much slower than the transmitted baseband signal S(t). In the fre- Frequency selective fading: A channel is said quency domain, this implies that the Doppler spread to be frequency selective if the signal bandwidth is of the channel is much less than the bandwidth of the greater than the coherence bandwidth of the chan- baseband signal. There fore, a signal undergoes slow nel. In such a case, diﬀerent frequency components fading if: of the transmit signal undergo fading to diﬀerent ex- tents. For a frequency-selective fading situation: T T (57) s c B B (58) Bs >Bc (52) s D Key Channel Parameters and Symbol Typical Ts

στ > 0.1Ts (54)

Fast fading: In a fast fading channel, the chan- It must be noted that the wireless channel is func- nel impulse response changes rapidly within the sym- tion of what is transmitted over it. In order to bol duration, i.e. the coherence time of the channel determine whether fading will affect communication is smaller that the symbol period of the transmit- on a wireless channel, we must compare the symbol ted signal. Viewed in the frequency domain, signal duration of data transmission with the coherence distortion due to fast fading increases with increas- time and the bandwidth of the baseband signal (fast ing Doppler spread relative to the bandwidth of the / slow fading) with the coherence bandwidth of the transmitted signal. Therefore, a signal undergoes fast channel (flat / frequency selective nature). fading if: It should also be clear that when a channel is specified Ts >Tc (55) as a fast or slow fading channel, it does not specify

8 deal with weather the relationship between the signal bandwidth and the range of frequencies over which the fading behaviour of the channel is uniform.

References

[1] Wireless communication technologies, lecture notes, Spring 2005, Dr. Narayan Mandayam, Rutgers Univer- sity

[2] Rayleigh fading channels in mobile digital communication systems, Part I: Characterization, Bernard Sklar, IEEE Communications Magazine, Sept. 1997

[3] Wireless Communications, Andrea Goldsmith, Stanford University 2004

[4] Fundamentals of Wireless Communication, David Tse, University of California Berkely, Promod Vishwanath, University of Illinios Urbana-champaign