ECE6604 PERSONAL & MOBILE COMMUNICATIONS. Week 3. Flat Fading Channels Envelope Distribution Autocorrelation of a Random Process


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1 1 ECE6604 PERSONAL & MOBILE COMMUNICATIONS Week 3 Flat Fading Channels Envelope Distribution Autocorrelation of a Random Process
2 2 MultipathFading Mechanism local scatterers mobile subscriber base station A typical macrocellular mobile radio environment.
3 3 MultipathFading Mechanism local scatterers local scatterers mobile station mobile station Typical mobiletomobile radio propagation environment.
4 4 local mean Ω (db) area mean µ (db) envelope fading Path loss, shadowing, envelope fading.
5 5 Doppler Shift y n th incoming wave mobile station v θ n x 2D Model of a typical wave component incident on a mobile station (MS). Assuming2Dpropagation,theDoppler shiftisf D,n = f m cosθ n,wheref m = v/λ c (λ c is the carrier wavelength, v is the mobile station velocity).
6 Multipath Propagation Consider the transmission of the bandpass signal s(t) = Re { s(t)e j2πf ct } At the receiver antenna, the nth plane wave arrives at angle θ n and experiences Doppler shift f D,n = f m cosθ n and propagation delay τ n. If there are N propagation paths, the received bandpass signal is r(t) = Re N n=1 C n e jφ n j2πcτ n /λ c +j2π(f c +f D,n )t s(t τ n ), where C n, φ n, f D,n and τ n are the amplitude, phase, Doppler shift and time delay, respectively, associated with the nth propagation path, and c is the speed of light. The delay τ n = d n /c is the propagation delay associated with the nth propagation path, where d n is the length of the path. The path lengths, d n, will depend on the physical scattering geometry which we have not specified at this point. 6
7 7 Multipath Propagation The received bandpass signal r(t) has the form where the received complex envelope is and r(t) = Re [ r(t)e j2πf ct ] r(t) = N n=1 C n e jφ n(t) s(t τ n ) φ n (t) = φ n 2πcτ n /λ c +2πf D,n t is the timevariant phase associated with the nth propagation path. Note that the nth component varies with the Doppler frequency f D,n. The term cτ n /λ c is the propagation distance cτ n normalized by the carrier wavelength λ c. For cellular frequencies (900 MHz), λ c is on the order of a foot. The phase φ n is a random introduced by the nth scatterer and can be assumed to be uniformly distributed on [ π, π). The received phase at any time t, φ n (t) is uniformly distributed on [ π,π).
8 8 Flat Fading  impulse response The received waveform is given by the convolution r(t) = t 0 g(t,τ) s(t τ)dτ It follows that the channel can be modeled by a linear timevariant filter having the timevariant impulse response g(t,τ) = N n=1 C n e jφ n(t) δ(τ τ n ) If the differential path delays τ i τ j are all very small compared to the modulation symbol period, T, then the τ n can be replaced by the mean delay µ τ inside the delta function. Note that this approximation is not applied to the channel phases φ n (t), since small changes in τ n result in large changes in φ n (t). The channel impulse response has the approximate form g(t,τ) = g(t)δ(τ µ τ ), The received complex envelope is g(t) = N n=1 C n e jφ n(t). r(t) = g(t) s(t µ τ ) (1) which experiences fading due to the timevarying complex channel gain g(t).
9 9 Flat Fading  frequency domain By taking Fourier transforms of both sides of (1), the received complex envelope in the frequency domain is R(f) = G(f) S(f)e j2πfµ τ Since the channel component g(t) changes with time, it follows that G(f) has a finite nonzero width in the frequency domain. Due to the convolution operation, the output spectrum R(f) will be wider than the input spectrum S(f). This broadening of the transmitted signal spectrum is caused by the channel time variations and is called frequency spreading or Doppler spreading. If the maximum Doppler frequency f m is much less than the signal bandwidth W c, then the Doppler spreading will not distort S(f). Fortunately, this is often the case.
10 10 Channel Transfer Function  Flat Fading The timevariant channel transfer function is obtained by taking the Fourier transform of the timevariant channel impulse response g(t, τ) with respect to the delay variable τ, i.e., T(t,f) = g(t)e j2πfµ τ. Since the magnitude response is T(t,f) = g(t), all frequency components in the received signal are subject to the same timevariant amplitude gain g(t) and phase response g(t) = 2πfµ τ. The received signal is said to exhibit flat fading, because the magnitude of the timevariant channel transfer function T(t, f) is constant (or flat) with respect to frequency variable f. The phase response g(t) = 2πfµ τ is linear in f meaning that the channel simply delays the input signal and attenuates it.
11 11 Invoking the Central Limit Theorem Consider the transmission of an unmodulated carrier, s(t) = 1. For flat fading channels, the received bandpass signal has the quadrature representation r(t) = g I (t)cos2πf c t g Q (t)sin2πf c t where g I (t) = N n=1 g Q (t) = N n=1 and where φ n (t) = φ n 2πcτ n /λ c +2πf D,n t. C n cosφ n (t) C n sinφ n (t) The phases φ n are independent and uniform on the interval [ π,π), and the path delays τ n are all independent with f c τ n 1. Therefore, the phases φ n (t) at any time t can be treated as being independent and uniformly distributed on the interval [ π,π). In the limit N, the central limit theorem can be invoked and g I (t) and g Q (t) can be treated as Gaussian random processes, i.e., at any time t, g I (t) and g Q (t) are Gaussian random variables. The complex faded envelope is g(t) = g I (t)+jg Q (t)
12 12 Rayleigh Fading Forsometypesofscatteringenvironments,g I (t)andg Q (t)atanytimet 1 areindependent identically distributed Gaussian random variables with zero mean and identical variance b 0 = E[g 2 I (t 1)] = E[g 2 Q (t 1)]. This typically occurs in a rich scattering environment where there is no lineofsight or strong specular component in the received signal (i.e., there is no dominant C n ) and isotropic antennas are used. Under such conditions, the channel exhibits Rayleigh fading. The probability density function the envelope α = g(t 1 ) = g 2 I(t 1 )+g 2 Q(t 1 ) can be obtained by using a bivariate transformation of random variables (see Appendix in textbook). The envelope α = g(t 1 ) = gi(t 2 1 )+gq(t 2 1 ) is Rayleigh distributed at any time t 1, i.e., p α (x) = x exp b x2 0 2b = 2x exp 0 Ω x2 p Ω, x 0, p where Ω p = E[α 2 ] = E[gI 2(t 1)]+E[gQ 2(t 1)] = 2b 0 is the average envelope power. The squaredenvelope α 2 at any time t 1 has the exponential distribution p α 2(x) = 1 exp Ω x p Ω, x 0. p
13 13 Ricean Fading θ 0 mobile direction of motion A lineofsight (LoS) or specular (strong reflected) component arrives at angle θ 0.
14 14 For scattering environments that have a specular or LoS component, g I (t) and g Q (t) are Gaussian random processes with nonzero means m I (t) and m Q (t), respectively. If we again assume that g I (t 1 ) and g Q (t 1 ) at any time t 1 are independent random variables with variance b 0 = E[(g I (t 1 ) m I (t 1 )) 2 ] = E[(g Q (t 1 ) m Q (t 1 )) 2 ], then the magnitude of the envelope α = g(t 1 ) at any time t 1 has a Rice distribution. With Aulin s Ricean fading model m I (t) = E[g I (t)] = s cos(2πf m cos(θ 0 )t+φ 0 ) m Q (t) = E[g Q (t)] = s sin(2πf m cos(θ 0 )t+φ 0 ) where f m cos(θ 0 ) and φ 0 are the Doppler shift and random phase offset associated with the LoS or specular component, respectively. The envelope α(t) = g(t) = g 2 I(t)+g 2 Q(t) has the Rice distribution p α (x) = x exp { x2 +s 2 } xs b 2b Io 0, x 0 0 b 0 s 2 = m I (t) 2 +m 2 Q (t) is the specular power. 2b 0 is the scatter power. The Rice factor, K = s 2 /2b 0, is the ratio of the power in the specular and scatter components.
15 15 The average envelope power is E[α 2 ] = Ω p = s 2 +2b 0 and Hence, p α (x) = 2x(K +1) Ω p exp s 2 = KΩ p K +1, K (K +1)x2 Ω p 2b 0 = Ω p K +1 I o 2x K(K +1) Ω p, x 0 The squaredenvelope α 2 (t) has noncentral chisquare distribution with two degrees of freedom p α 2(x) = (K +1) Ω p exp K (K +1)x Ω p I o 2 K(K +1)x Ω p, x 0 The squaredenvelope is important for the performance analysis of digital communication systems because it is proportional to the received signal power and, hence, the received signaltonoise ratio.
16 K = 0 K = 3 K = 7 K = 15 p α (x) x The Rice distribution for several values of K with Ω p = 1.
17 17 Nakagami Fading Nakagami fading describes the magnitude of the received complex envelope by the distribution p α (x) = 2mm x 2m 1 exp Γ(m)Ω m mx2 p Ω m 1 p 2 When m = 1, the Nakagami distribution becomes the Rayleigh distribution, when m = 1/2 it becomes a onesided Gaussian distribution, and when m the distribution approaches an impulse (no fading). The Rice distribution can be closely approximated with a Nakagami distribution by using the following relation between the Rice factor K and the Nakagami shape factor m K m 2 m+m 1 (K +1)2 m (2K +1). The squaredenvelope has the Gamma distribution p α 2(x) = m 2Ω p m x m 1 Γ(m) exp mx 2Ω p.
18 m = 1 m = 2 m = 4 m = 8 m = p α (x) x The Nakagami pdf for several values of m with Ω p = 1.
19 F α 2 (x) m=2 ~ (K=3.8 db) m=4 ~ (K=8.1 db) m=8 ~ (K=11.6 db) m=16 ~ (k=14.8 db) x Comparison of the cdf of the squaredenvelope with Ricean and Nakagami fading.
20 20 X 1 ( t) sample space S s s 1 2 s ξ X ( t ) 2 t t X ( ) ξ t t Ensemble of Sample Functions for a Random Process.
21 21 Autocorrelation Functions Let X(t) denote a random process. At any time t, X(t) is a random variable with probability density function f X(t) (x). The ensemble mean of X(t) is µ X (t) = E[X(t)] = xf X(t) (x)dx The autocorrelation of X(t) is obtained from the random variables X(t 1 ) and X(t 2 ) as φ XX (t 1,t 2 ) = E[X(t 1 )X(t 2 )] = x 1 x 2 f X(t1 ),X(t 2 )(x 1,x 2 )dx 1 dx 2 Often the same random variable will appear in both X(t 1 ) and X(t 2 ), meaning that X(t 1 ) and X(t 2 ) are correlated. For a wide sense stationary random process where τ = t 2 t 1. µ X (t) µ X φ XX (t 1,t 2 ) = φ XX (t 2 t 1 ) φ XX (τ)
22 22 Properties of φ XX (τ) The autocorrelation function, φ XX (τ), of a stationary random process satisfies the following properties. φ XX (0) = E[X 2 (t)]. This is the total power in the random process. φ XX (τ) = φ XX ( τ). The autocorrelation function must be an even function. φ XX (τ) φ XX (0). This is a variant of the CauchySchwartz inequality. φ XX ( ) = E 2 [X(t)] = µ 2 X. This holds if X(t) contains no periodic components and is equal to the d.c. power.
23 Complexvalued Random Process A complexvalued random process is given by Z(t) = X(t) + jy(t) where X(t) and Y(t) are realvalued random processes. The autocorrelation function of a wide sense stationary complexvalued random process is φ ZZ (τ) = 1 2 E[Z(t 1)Z (t 2 )] = 1 2 (φ XX(τ)+φ YY (τ)+j(φ YX (τ) φ XY (τ))). The factor of 1/2 in included for convenience, when Z(t) is a complexvalued Gaussian random process. The crosscorrelation function between two complexvalued wide sense stationary random process satisfies the following properties φ ZW(τ) = φ WZ ( τ) φ ZZ (τ) = φ ZZ( τ) 23
24 24 Power Spectral Density The power spectral density (psd) of a widesense stationary random process X(t) is the Fourier transform of the autocorrelation function, i.e., S XX (f) = = φ XX(τ)e j2πfτ dτ φ XX (τ) = S XX(f)e j2πfτ df. In our case, we are interested in the bandpass process where r(t) = g I (t)cos2πf c t g Q (t)sin2πf c t g(t) = g Q (t)+jg Q (t) is the complex fading envelope of the channel. For g(t), we are interested in its autocorrelation function and Doppler or power spectrum φ gg (τ) = φ gi g I (τ)+jφ gi g Q (τ) S gg (f) = S gi g I (f)+js gi g Q (f)
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