Part 4. Communications over Wireless Channels

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1 Part 4. Communications over Wireless Channels p. 1

2 Wireless Channels Performance of a wireless communication system is basically limited by the wireless channel wired channel: stationary and predicable wireless channel: random How to get a wireless channel model field measurement statistical model p. 2

3 Mobile Radio Propagation Characteristics p. 3

4 Large-Scale Propagation Characteristics Large-scale propagation model to predict the mean signal strength for arbitrary transmitter-receiver separation There is large variation of mean signal strength for different distances. propagation loss for car 1 < propagation loss for car 2 p. 4

5 Small-Scale Propagation Characteristics Small-scale propagation model to model signal variation within a small area or short time durations over this small area, the mean signal strength is the same. to model: multipath delay spread, signal variation due to fast fading Delay spread Due to multipath propagation, the signal via the direct path arrives at the earliest time whereas signals via reflected paths arrive with a delay relative to the one arrived via the direct path. Fast fading Due to constructive and destructive addition of signals arrived from different paths, the signal strength varies when the mobile station moves. Multipath Propagation p. 5

6 Example of Propagation Measurement Result Measured Signal Power small-scale variation Mean Signal Power (note: large-scale variation) Source: Rappaport s Wireless Communications, p. 71. p. 6

7 Large-Scale Fading p. 7

8 Free Space Propagation Model (1) Is used to predict received signal strength when the transmitter and receiver have a clear, unobstructed, line-of-sight path between them. Example: Satellite communication Example: When the base systems station is at the top of the mountain and the mobile station can view this base station. p. 8

9 Friis free space equation: Free Space Propagation Model (2) P( d) = r PG t tgrλ 2 2 ( 4π) d L where P t is the transmitted power, P r (d) is the received power which is a function of the transmitter-receiver (T-R) separation, G t is the transmitter antenna gain, G r is the receiver antenna gain, d is the T-R separation in meters, and λ is the wavelength in meters, and L ( 1) is the system loss factor. Note: Received power is inversely proportional to the square of d. 2 Path loss (PL) in db is given by Pt PL( ) 10log 10log P db = = L N M r GG t ( 4π) 2 r 2 2 λ d O QP p. 9

10 Practical Aspects The Friis equation is only useful for the far field. [Distance of interest to mobile communications is usually long enough to ensure that radiation falls onto the far field.] With a reference received power P r (d 0 ) measured at a known reference distance d 0, the received power at distance d is computed by Notes: P( d) P( d ) r F = r H 0 The reference distance d 0 is called the close-in distance. The close-in distance is usually chosen to be 1m for indoor environments and 100m or 1km in outdoor environments d d 0 I K 2 inverse square law p. 10

11 Example G t = 1 G r = 1 P P = ( d d ) {inverse square law} 10log ( P 1mW) = 10log ( P 1mW) + 20log ( d d ) Source: Rappaport s Wireless Communications, p. 74. p. 11

12 Two-Ray Ground Reflection Model One-ray communication seldom occurs. Ground reflection is a realistic phenomenon for mobile communications. A two-ray ground reflection model is useful. Transmitter line-of-sight path h t Receiver ground-reflected path hr It is known that P PG G hh t r = if d >> 4 d r t t r 2 2 d h h Ground where P t is the transmitted power and P r is the received power. t r a result from EM theory p. 12

13 Long-Distance Path-Loss Model Usually, the path-loss model is deduced from experimental data. The propagation characteristic can be described by the path loss exponent n and the path loss formula n F d PL( d ) = PL( d ) H G I d K J 0 0 PLdB( d ) = PLdB ( d ) + 10nlog d d {in db domain} where PL( d ) stands for the average path loss at a distance d away from the transmitter, and PL d is the corresponding value in db. db( ) Note that knowledge of PL( d0 ) is required, which can be obtained by field measurement. p. 13

14 Path-Loss Exponent Path-loss exponents are usually obtained through field measurement. Typical values of n: (source: Rappaport s Wireless Communications) Environment Path Loss Exponent, n Free space 2 Urban area cellular radio 2.7 to 3.5 Shadowed urban cellular radio 3 to 5 In building line-of-sight 1.6 to 1.8 Obstructed in building 4 to 6 Obstructed in factories 2 to 3 p. 14

15 Shadowing The average power loss does not take into account the effects of surrounding environmental clutter, such as trees, and the movement of people and vehicles d d These effects are summarized as shadowing. p. 15

16 Lognormal Shadowing Lognormal distribution is widely used to model the signal power variation due to shadowing. Lognormal shadowing: The power level in db is distributed as a Gaussian random variable The power level in db at a distance d is given by: ( ) P ( d) = P ( d) + σ X = P ( d ) + 10nlog d d + σ X db db db db db where X is a standard normal random variable and σ db is the standard deviation of the power level in db. For example, σ db of 8dB is generally used. p. 16

17 Overview of Propagation Models Outdoor propagation models: Longley-Rice model Durkin s model Okumura model important; empirical Hata model empirical model PCS extension to Hata model Wideband PCS microcell model Indoor propagation models: 1. Loss due to signal penetration into buildings 2. Partition loss 3. Floor attenuation p. 17

18 Small-Scale Fading p. 18

19 Characteristics of Small-Scale Fading Rapid changes in signal strength over a small travel distance or time interval. Random frequency modulation due to variant Doppler shifts on different multipath signals Time dispersion (echoes) caused by multipath propagation delays. ( c + f () t ) cos 2 π f 0 t, τ 0 received signal: ( c + f () t ) cos 2 π f 1 t, τ 1 ( f + f () t ) cos 2 π t, τ c L 1 L 1 p. 19

20 Observations of Small-Scale Fading The received signal is the sum of signals arrived via different paths. Constructive (destructive) summation of signals causes an enhancement (a reduction) of the received signal. Deep fading may occur. A vehicle moving at high speed may experience several fades in a small period of time. Due to relative motion between the mobile and base stations, each path wave experiences a shift in frequency. Doppler shift L 1 L 1 ( fl ( t) )( ) received signal: r = α cos 2π f + t τ l l c l l= 0 l= 0 Constructive example: r0 = 2 cos 2π fct r1 = 1.9 cos 2 π fct r0 + r1 = 3.9cos 2 π fct Destructive example: r0 = 2 cos 2π fct r1 = 1.9 cos 2 π fct + π r + r 0 1 = 0.1cos 2 π ft c : deep fade p. 20

21 Doppler Shift Approximations: Δl is the distance difference between SX and SY; the angle formed by SX and XY is the same as the angle formed by SY and XY Holds if S is far enough v = velocity of the vehicle distance difference: Δ = dcosθ = vδtcosθ additional phase change: 2πΔ 2πvcosθΔt Δ φ = = λ λ Doppler shift: f d 1 Δφ v = = cosθ 2π Δt λ p. 21

22 Example from Rappaport s Wireless Communications Higher observed frequency θ=0 f d v = cosθ λ Lower observed frequency θ=π p. 22

23 Multipath propagation Factors Influencing Small-Scale Fading A result of the presence of reflecting objects and scatters in the channel. Random amplitudes and phases of multipath components cause small-scale fading and/or signal distortion. Signal delay introduced by propagation via a longer reflected path causes smearing of the signal, known as intersymbol interference. Speed of the mobile Affects the Doppler shifts on each of the multipath components. Speed of surrounding objects Induces time-varying Doppler shift on multipath components. Effects become dominant only if the surrounding objects move faster than the mobile station. Signal bandwidth If the transmitted signal has a bandwidth greater than the coherence bandwidth of the channel, the received signal is distorted by the introduction of intersymbol interference p. 23

24 Multipath Fading Channels (1) Time varying multipath fading channels - an example of the discrete time impulse response model p. 24

25 Multipath Fading Channels (2) The channel can be regarded as a linear time-varying filter. The received signal is the convolution between the transmitted signal and the channel impulse response. D D D D h(t,τ 0 ) h(t,τ 1 ) h(t,τ 2 ) h(t,τ 3 ) h(t,τ L-1 ) Channel impulse response: Is used to characterize the channel. Can be measured (though not conveniently) by sending a pulse to the channel and recording the channel output by a receiver. Is time-varying in nature for mobile communication channels. p. 25

26 Channel Impulse Response Example: channel impulse responses (source: Proakis s Digital Communications) Measurement set-up (Source: S.-C. Kim, H. L. Bertoni and M. Stern, Pulse propagation characteristics at 2.4GHz inside buildings, IEEE Trans. Veh. Technol., vol. 45, pp , Aug ) p. 26

27 Characterization of Multipath Fading Channels (1) The bandpass transmitted signal, s BP (t), at a carrier frequency f c is given by s t s t e j f c () = 2π Re{() t } BP where s(t) is the complex envelope of the signal (low-pass equivalent signal model). The received bandpass signal, x BP (t), can be expressed in the form n s j2πf t j2πf t c c xbp( t) = Re x() t e = Re e c ()( t s t τ ()) t RST where x(t) is the complex envelope of the received signal, c n (t) is the time-varying, complex-valued channel gain of the nth path and τ n (t) is the nth-path delay, which can be assumed to be time-varying. n n n UVW Thus, the low-pass equivalent channel impulse response, h(τ;t), is given by n h(;) τ t = c ()( t δ τ τ ()) t = c () t e δ( τ τ ()) t where δ(.) is the Dirac delta function. n n n n n j arg c ( t ) n p. 27

28 Characterization of Multipath Fading Channels (2) Denote the time-varying, low-pass-equivalent channel impulse response as h(τ;t). The autocorrelation function of h(τ;t) is given by * φh( τ1, τ2; Δt) = En 1 2 h ( τ1; t) h( τ2; t+ Δt) Assume uncorrelated scattering, that is, different multipaths are statistically uncorrelated in properties such as gain and phase. Then φ ( τ, τ ; Δt) = φ ( τ ; Δt) δ( τ τ ) h 1 2 h where φ h (τ ; Δt) is the delay cross-power density function. s The special case that φ (;) τ 0 φ () τ h h is known as multipath intensity profile or delay power spectrum. p. 28

29 Characterization of Multipath Fading Channels (3) The delay power spectrum can be discrete or continuous. Example 1: double-spike channel φ(τ) φ ( τ) = δ( τ) + δ( τ τ ) where τ d is the delay of the delayed path. Example 2: exponentially dispersive channel where τ rms is the root-mean-square delay spread. h 1 rms φh ( τ ) = τ exp( τ τ ) rms d 0 τ d φ(τ) 0 τ τ p. 29

30 Characterization of Multipath Fading Channels (3a): Experimental Results regarding the Delay Power Spectrum Source: S.-C. Kim, H. L. Bertoni and M. Stern, Pulse propagation characteristics at 2.4GHz inside buildings, IEEE Trans. Veh. Technol., vol. 45, pp , Aug p. 30

31 Characterization of Multipath Fading Channels (4) Taking the Fourier transform of h(τ;t) on the variable τ, we get z j2πfτ H( f; t) = h( τ; t) e dτ Assume that H(f;t) is wide-sense stationary. The autocorrelation function is It follows that * ΦH ( f1, f2; Δt) = En 1 2 H ( f1; t) H( f2; t+ Δt) Φ ( f, f ; Δt) = E h ( τ ; t) h( τ ; t+ Δt) e dτ dτ H * j2π( f τ f τ ) j2π( f τ f τ ) h j2π( f1 f2) τ1 φh( τ1; Δte ) dτ 1 j2π( Δf ) τ h 1 1 H = φ ( τ ; Δt) δ( τ τ ) e dτ dτ = z z z z z z n 1 = φ ( τ ; Δte ) dτ Φ ( Δf; Δt) with Δf= f f 1 so that Φ H (Δf ; Δt), known as the spaced-frequency, spaced-time correlation function, is the Fourier transform of the delay cross-power density function. s s Based on the assumption of uncorrelated scattering p. 31

32 Let Then we have Characterization of Multipath Fading Channels (5) Φ H Φ ( Δf ; 0) Φ ( Δf ) H z j2π( Δf ) τ ( Δf ) = φh( τ) e dτ H p. 32 Source: Proakis s Digital Communications

33 Characterization of Multipath Fading Channels (6) Significance of Φ H (Δf): Since Φ H (Δf) is an autocorrelation function in the frequency domain, it provides us with a measure of the frequency coherence of the channel, i.e., the correlation of the frequency components separated by Δf (Hz) in frequency. Coherence bandwidth of the channel: The reciprocal of the multipath spread is a measure of the coherence bandwidth of the channel, (Δf) c, i.e., where T m is the multipath spread. ( Δf ) c 1 T m Note 1. If the coherence bandwidth is small compared to the signal bandwidth, the channel is said to be frequency-selective; otherwise it is frequency-nonselective. Note 2. A frequency-selective fading channel introduces intersymbol interference (ISI), which distorts the signal, but ISI can be utilized by spread-spectrum transmission to exploit multipath diversity to combat the adverse effects due to fading. p. 33

34 Characterization of Multipath Fading Channels (7) Taking the Fourier transform of Φ H (Δf ; Δt) on the variable Δt, we have The special case that S ( Δf; λ) = Φ ( Δf; Δt) e dδt H z j2πλ( Δt) H is known as the Doppler power spectrum of the channel. S H ( 0; λ) S ( λ) H p. 34 Source: Proakis s Digital Communications

35 Characterization of Multipath Fading Channels (8) Doppler spread: The range of values of λ over which S H (λ) is essentially non-zero is called the Doppler spread B d of the channel. Coherence time of the channel: The coherence time of the channel, (Δt) c, is just the reciprocal of the Doppler spread. That is, ( Δt) c 1 B d Note. If the coherence time is small compared to the symbol duration, the channel is said to be time-selective. Time selectivity complicates the receiver design because the signal is varying so fast that it is difficult to establish frequency and time synchronization at the received signal. fast fading p. 35

36 Characterization of Multipath Fading Channels (9) The scattering function is given by z j2πλ( Δt) Sh(; τλ) = φh(; τδt) e dδt It follows that and 2 ( ) (; ) ( ;) j πτ Δ S f h τλ = SH Δf τe dδf z z j2πλ( Δt) j2πτ( Δf ) Sh(; τλ) = ΦH( Δf; Δt) e e d( Δt)( d Δf ) Note: The scattering function is the most commonly used function to describe a multipath fading channel. A simplifying assumption: Sometimes the scattering function is assumed to be decomposable into a product of the delay power spectrum and the Doppler power spectrum, i.e., S ( τ, λ) = φ ( τ) S ( λ) h h H p. 36

37 Characterization of Multipath Fading Channels (Summary) Φ H (Δf): Spacedfreq. correlation function φ h (τ): Multipath intensity profile Taking Fourier transform in the variable τ Δt=0 Delay cross-power density function φ h (; τ Δt) Taking Fourier transform in the variable Δt Δt=0 Δf=0 Spaced-frequency, spaced-time correlation function Φ H ( Δf ; Δt) S h (; τ λ) Scattering function Φ H (Δt): Spacedtime correlation function Taking Fourier transform in the variable Δt SH ( Δf ; λ) Δf=0 Taking Fourier transform in the variable τ S H (λ): Doppler power spectrum p. 37

38 Mathematical model for the channel response in each path Consider an example multipath intensity profile with discrete paths: L 1 φ ( τ ) = c ( t) δ ( τ τ ) h n n n= 0 where c n (t) is a zero-mean complex-gaussian WSS random process and all c n (t) s are statistically uncorrelated. Note that uncorrelated Gaussian processes are statistically independent. Assume that the processes are ergodic, so that the time average equals the ensemble average. Then the ensemble average of the magnitude square of c n (t) is the power gain of the nth multipath of the channel. Rayleigh fading is a widely used fading model for statistically characterizing small-scale fading experienced by channels without line-ofsight paths. Rician fading occurs when there is a line-of-sight (LOS) path. p. 38

39 Origin of Rayleigh Fading Assume that the mobile station isotropically receives the diffused signals arrived from all 360 in the angle. The addition of all diffused signals gives rise to one resultant signal, whose signal amplitude and phase are random rather than deterministic. The amplitude and phase are distinguished by correlating the received signal with the sine and cosine components of the carrier frequency. By the central-limit theorem, the resultant distributions of the sine and cosine components are Gaussian. It is reasonable to assume that the two components are uncorrelated and hence statistically independent. Mobile station Signal distribution diffused signals arrived from all directions, with equal signal strength sine direction cosine direction p. 39

40 The PDFs of amplitude and phase: Mathematics of Rayleigh Fading Rayleigh distribution 2 r r pr () = exp, 2 2 σ 2σ r 0 1 p( θ ) =, 2π 0 θ < 2π uniform distribution Channel response at each path: c = j re θ p. 40

41 Rician Fading Rayleigh fading occurs frequently when there is no line-of-sight (LOS) path. When there is a LOS path, the mean of the complex-valued signal strength is not zero. The resultant distribution is known as the Rician distribution (or Nakagami-Rice distribution in some references). Let r exp(jθ) = x + jy where x and y are Gaussian distributions with means m x and m y, repectively, and a common standard deviation σ. The random variable r follows a Rician distribution with (see Proakis s Digital Communications) r F rsi pr e ( r + s ) 2σ ( ) = IH K, r σ σ where s 2 = m x2 + m y2 and I 0 (x) is the 0 th order modified bessel function of the first kind The Rician factor K is the ratio of the LOS-path power to the power of diffused components, given by 2 s K = 2 2σ p. 41

42 Statistical Modeling of Mobile Radio Channels for Outdoor and Indoor Scenarios Outdoor environments: Macrocells, having a coverage radius in the order of 10km, are likely not to have LOS paths. Rayleigh fading can be assumed. Microcells, having a coverage radius in the order of 1km, may or may not have LOS paths, depending on the distance between the base station and the mobile station. In the presence (absence) of a LOS path, Rician (Rayleigh) fading can be assumed. Indoor environments: Picocells, having a coverage in the order of 100m or 10m in radius, may or may not have LOS paths. Rician and Rayleigh distributions can be assumed for, respectively, the channels with and without LOS paths. Temporal variation. It is possible that the mobile station is a portable communication equipment, wherein the equipment does not move when making communications. In this case, the channel is relatively stable and can be modeled by a Rician-fading channel. See R. J. C. Bultitude, Measurement, characterization and modeling of indoor 800/900 MHz radio channels for digital communications, IEEE Communications Magazine, vol. 25, no. 6, pp. 5-12, Jun p. 42

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