EE6604 Personal & Mobile Communications. Week 5. Level Crossing Rate and Average Fade Duration. Statistical Channel Modeling.

Transcription

1 EE6604 Personal & Mobile Communications Week 5 Level Crossing Rate and Average Fade Duration Statistical Channel Modeling Fading Simulators 1

2 Level Crossing Rate and Average Fade Duration The level crossing rate (LCR) at a specified envelope level R, L R, is defined as the rate (in crossings per second) at which the envelope α crosses the level R in the positive going direction. The LCR can be used to estimate velocity, and velocity can be used for radio resource management. The average fade duration (AFD) is the average duration that the envelope remains below a specified level R. An outage occurs when the envelope fades below a critical level for a long enough period such that receiver synchronization is lost. Longer fades are usually the problem. The probability distribution of fade durations, if it exists, would allow us to calculate probability of outage. Both the LCR and AFD are second-order statistics that depend on the mobile station velocity, as well as the scattering environment. The LCR and AFD have been derived by Rice (1948) in the context of a sinusoid in narrowband Gaussian noise. 2

3 Envelope Level (db) Time, t/t Rayleigh faded envelope with 2-D isotropic scattering. 3

4 Level Crossing Rate Obtaining the level crossing rate requires the joint pdf, p(α, α), of the envelope level α = g(t 1 ) and the envelope slope α = d g(t 1 ) /dt at any time instant t 1. Note we drop the time index t for convenience. Intermsof p(α, α), the expected amountof timethe envelope liesintheinterval(r,r+dα) for a given envelope slope α and time increment dt is p(r, α)dαd αdt The time required for the envelope α to traverse the interval (R,R+dα) once for a given envelope slope α is dα/ α The ratio of the above two quantities is the expected number of crossings of the envelope α within the interval (R,R+dα) for a given envelope slope α and time duration dt, i.e., αp(r, α)d αdt 4

5 The expected number of crossings of the envelope level R for a given envelope slope α in a time interval of duration T is T 0 αp(r, α)d αdt = αp(r, α)d αt The expected number of crossings of the envelope level R with a positive slope in the time interval T is N R = T αp(r, α)d α. 0 Finally, the expected number of crossings of the envelope level R per second, or the level crossing rate, is obtained by dividing by the length of the interval T as L R = 0 αp(r, α)d α This is a general result that applies to any random process characterized by the joint pdf p(α, α). 5

6 Rice (BSTJ, 1948) derived the joint pdf p(α, α) for a sine wave plus Gaussian noise. A Rician fading channel can be thought of LoS or specular (sine wave) component plus a scatter (Gaussian noise) component. For the case of a Rician fading channel, p(α, α) = α(2π) 3/2 Bb0 exp π π dθ 1 [ ( B α 2 2αscosθ+s 2) +(b 0 α+b 1 ssinθ) 2] 2Bb 0 where s is the non-centrality parameter in the Rice distribution, and B = b 0 b 2 b 2 1, where b 0, b 1, and b 2 are constants that depend on the scattering environment. Suppose that the specular or LoS component of the complex envelope g(t) has a Doppler frequency equal f q = f m cosθ 0, where 0 f q f m. Then b n = (2π) n f m S c f m gg (f)(f f q) n df = (2π) n b 2π 0 ˆp(θ)G(θ)(f 0 m cosθ f q ) n dθ where ˆp(θ) is the azimuth distribution (pdf) of the scatter component, G(θ) is the antenna gain pattern, and S c gg(f) is the corresponding continuous portion of the Doppler power spectrum. Note that the pdf ˆp(θ) in this case integrates to unity. 6

7 Note that S c gg(f) is given by the Fourier transform of φ c gg(τ) = φ c g I g I (τ)+jφ c g I g Q (τ) where φ c g I g I (τ) = Ω p 2 φ c g I g Q (τ) = Ω p 2 2π 0 2π 0 cos(2πf m τ cosθ)ˆp(θ)g(θ)dθ sin(2πf m τ cosθ)ˆp(θ)g(θ)dθ Insomespecialcases,thepsdS c gg(f)issymmetricalaboutthefrequencyf q = f m cosθ 0. This condition occurs, for example, when f q = 0 (θ 0 = 90 o ) and ˆp(θ) = 1/(2π), π θ π. Specular component arrives perpendicular to direction of motion and scatter component is characterized by 2-D isotropic scattering. In this case, b n = 0 for all odd values of n (and in particular b 1 = 0) so that the joint pdf p(α, α) reduces to the convenient product form 1 p(α, α) = exp 2πb 2 = p( α) p(α). α2 2b 2 α +s 2 ) exp b (α2 0 2b 0 Since p(α, α) = p( α) p(α), it follows that α and α are independent for this special case. I 0 αs b 0 7

8 When f q = 0 and ˆp(θ) = 1/(2π), a closed form expression can be obtained for the envelope level crossing rate. We have that b n = b 0 (2πf m ) n (n 1) n n even 0 n odd. Therefore, b 1 = 0 and b 2 = b 0 (2πf m ) 2 /2, and ( ) L(R) = 2π(K +1)f m ρe K (K+1)ρ2 I 0 2ρ K(K +1) where and R rms = E[α 2 ] is the rms envelope level. ρ = R Ωp = R R rms Under the further condition that K = 0 (Rayleigh fading) L(R) = 2πf m ρe ρ2 Notice that the level crossing rate is directly proportional to the maximum Doppler frequency f m and, hence, the MS speed v = f m λ c. 8

9 10 0 Normalized Level Crossing Rate, L R /f m K=0 K=1 K=3 K= Level, ρ (db) Normalized level crossing rate for Rician fading. A specular component arrives with angle θ 0 = 90 o and there is 2-D isotropic scattering of the scatter component. 9

10 Average Fade Duration No known probability distribution exists for the duration of fades; this is a long standing open problem! Therefore, we consider the average fade duration. Consider a very long time interval of length T, and let t i be the duration of the ith fade below the level R. The probability that the received envelope α is less than R is Pr[α R] = 1 T t i i The average fade duration is equal to t = total length of time in duraton T that the envelope is below level R average number of crossings in duration T = = i t i TL(R) Pr[α R] L(R) 10

11 If the envelope is Rician distributed, then Pr[α R] = R ( p(α)dα = 1 Q ) 0 2K, 2(K +1)ρ 2 where Q(a,b) is the Marcum Q function. If we again assume that f q = 0 and ˆp(θ) = 1/(2π), we have t = 1 Q ( 2K, 2(K +1)ρ 2 ) 2π(K +1)fm ρe K (K+1)ρ2 I 0 ( 2ρ K(K +1) ) If we further assume that K = 0 (Rayleigh fading), then and P[α R] = R 0 p(α)dα = 1 e ρ2 t = eρ2 1 ρf m 2π. 11

12 10 0 Envelope Fade Duration, tf m K = 0 K = 1 K = 3 K = 7 K = Level, ρ (db) Normalized average fade duration with Ricean fading. 12

13 Scattering Mechanism for Wideband Channels τ 2 τ 1 transmitter receiver Concentric ellipses model for frequency-selective fading channels. Frequency-selective (wide-band) channels have strong scatterers that are located on several ellipsessuchthatthecorrespondingdifferentialpathdelaysτ i τ j forsomei,j,aresignificant compared to the modulated symbol period T. 13

14 Transmission Functions Multipath fading channels are time-variant linear filters, whose inputs and outputs can be described in the time and frequency domains. There are four possible transmission functions Time-variant channel impulse response g(t, τ) Output Doppler spread function H(f, ν) Time-variant transfer function T(f, t) Doppler-spread function S(τ, ν) 14

15 Time-variant channel impulse response, g(t, τ) Also known as the input delay spread function. The time varying complex channel impulse response relates the input and output time domain waveforms r(t) = t 0 g(t,τ) s(t τ)dτ In physical terms, g(t,τ) can be interpreted as the channel response at time t due to an impulse applied at time t τ. Since a physical channel is causal, g(t,τ) = 0 for τ < 0 and, therefore, the lower limit of integration in the convolution integral is zero. The convolution integral can be approximated in the discrete form ~ s ( t ) r(t) = n m=0 g(t,m τ) s(t m τ) τ τ τ τ τ g (t,0) g(t, τ) g(t, 2 τ) g(t, 3 τ) g(t, 4 τ) ~ r ( t) Discrete-time tapped delay line model for a multipath-fading channel. 15

16 Transfer Function, T(f, t) The transfer function relates the input and output frequencies: R(f) = S(f)T(f,t) By using an inverse Fourier transform, we can also write r(t) = S(f)T(f,t)e j2πft df The time-varying channel impulse response and time-varying channel transfer function are related through the Fourier transform: g(t,τ) T(f,t) Note: thefouriertransformpairiswithrespecttothetime-delayvariableτ. TheFourier transform of g(t,τ) with respect to the time variable t gives the Doppler spread function S(τ, ν), i.e, g(t,τ) S(τ,ν) 16

17 Fourier Transforms F g ( t, τ) F S ( τ, ν) F -1 F -1 F F T ( f, t) F -1 H ( f, ν) F -1 Fourier transform relations between the system functions. 17

18 Statistical Correlation Functions Similar to flat fading channels, the channel impulse response g(t,τ) = g I (t,τ)+jg Q (t,τ) of frequency-selective fading channels can be modelled as a complex Gaussian random process, where the quadrature components g I (t,τ) and g Q (t,τ) are Gaussian random processes. The transmission functions are all random processes. Since the underlying process is Gaussian, a complete statistical description of these transmission functions is provided by their means and autocorrelation functions. Four autocorrelation functions can be defined φ g (t,s;τ,η) = E[g (t,τ)g(s,η)] φ T (f,m;t,s) = E[T (f,t)t(m,s)] φ H (f,m;ν,µ) = E[H (f,ν)h(m,µ)] φ S (τ,η;ν,µ) = E[S (τ,ν)s(η,µ)]. Related through double Fourier transform pairs φ S (τ,η;ν,µ) = φ g (t,s;τ,η) = φ g(t,s;τ,η)e j2π(νt µs) dtds φ S(τ,η;ν,µ)e j2π(νt µs) dνdµ 18

19 Fourier Transforms and Correlation Functions φ S ( τ, η; νµ, ) DF DF -1 φ g (t,s; τ, η) DF -1 DF -1 DF φ (f,m; νµ, ) H DF DF DF -1 φ T (f,m;t,s) Double Fourier transform relations between the channel correlation functions. 19

20 WSSUS Channels Uncorrelated scattering in both the time-delay and Doppler shift domains. Practical land mobile radio channels are characterized by this behavior. Due to uncorrelated scattering in time-delay and Doppler shift, the channel correlation functions become: φ g (t,t+ t;τ,η) = ψ g ( t;τ)δ(η τ) φ T (f,f + f;t,t+ t) = φ T ( f; t) φ H (f,f + f;ν,µ) = ψ H ( f;ν)δ(ν µ) φ S (τ,η;ν,µ) = ψ S (τ,ν)δ(η τ)δ(ν µ). Note the singularities δ(η τ) and δ(ν µ) with respect to the time-delay and Doppler shift variables, respectively. Some correlation functions are more useful than others. The most useful functions: ψ g ( t;τ): channel correlation function φ T ( f; t): spaced-time spaced-frequency correlation function ψ S (τ,ν): scattering function 20

21 Fourier Transforms for WSSUS Channels F ψ ( t ; τ ) g F -1 ψ ( ν ; τ ) S F -1 F -1 F F φ ( t; f) T F ψ ( ν ; f) H F -1 21

22 Power Delay Profile The autocorrelation function of the time varying impulse response is Note the WSS assumption. φ g (t,t+ t,τ,η) = E[g (t,τ)g(t+ t,η)] = ψ g ( t;τ)δ(η τ) Thefunctionψ g (0;τ) ψ g (τ)iscalledthemultipath intensity profileorpower delay profile. The average delay µ τ is the mean value of ψ g (τ), i.e., µ τ = 0 τψ g (τ)dτ 0 ψ g (τ)dτ The rms delay spread σ τ is defined as the variance of ψ g (τ), i.e., σ τ = 0 (τ µ τ ) 2 ψ g (τ)dτ 0 ψ g (τ)dτ 22

23 Simulation of Multipath-Fading Channels Computer simulation models are needed to generate the faded envelope with the statistical properties of a chosen reference model, i.e., a specified Doppler spectrum. Generally there are two categories of fading channel simulation models Filtered-White-Noise models that pass white noise through an appropriate filter Sum-of-Sinusoids models that sum together sinusoids having different amplitudes, frequencies and phases. Model accuracy vs. complexity is of concern It is desirable to generate the faded envelope with low computational complexity while still maintaining high accuracy with respect to the chosen reference model. 23

24 Filtered White Noise Since the complex faded envelope can be modelled as a complex Gaussian random process, one approach for generating the complex faded envelope is to filter a white noise process with appropriately chosen low pass filters white Gaussian noise LPF H( f ) g ( t ) Q white Gaussian noise LPF H( f ) g I ( t ) g( t ) = g( t ) +j g ( t ) I Q IftheGaussiannoisesourcesareuncorrelatedandhavepowerspectraldensitiesofΩ p /2watts/Hz, and the low-pass filters have transfer function H(f), then S gi g I (f) = S gq g Q (f) = Ω p 2 H(f) 2 S gi g Q (f) = 0 Two approaches: IIR filtering method and IFFT filtering method 24

25 IIR Filtering Method implement the filters in the time domain as finite impulse response (FIR) or infinite impulse response (IIR) filters. There are two main challenges with this approach. the normalized Doppler frequency, ˆfm = f m T s, where T s is the simulation step size, is very small. Thiscan beovercome withaninfinite impulseresponse(iir)filter designedat a lower sampling frequency followed by an interpolator to increase the sampling frequency. The second main challenge is that the square-root of the target Doppler spectrum for 2-D isotropic scattering and an isotropic antenna is irrational and, therefore, none of the straightforward filter design methods can be applied. One possibility is to use the MATLAB function iirlpnorm to design the required filter. 25

26 IIR Filtering Method Here we consider an IIR filter of order 2K that is synthesized as the the cascade of K Direct-Form II second-order (two poles and two zeroes) sections (biquads) having the form H(z) = A K k=1 1+a k z 1 +b k z 2 1+c k z 1 +d k z 2. For example, for f m T s = 0.4, K = 5, and an ellipsoidal accuracy of 0.01, we obtain the coefficients tabulated below Coefficients for K = 5 biquad stage elliptical filter, f m T s = 0.4, K = 5 Stage Filter Coefficients k a k b k c k d k A

27 IIR Filtering Method 20 Designed Theoretical 0 Magnitude Response (db) Normalized Radian Frequency, ω/π Magnitude response of the designed shaping filter, f m T s = 0.4, K = 5. 27

28 IFFT Filtering Method N i.i.d. Gaussian random variables {A[k]} Multiply by the filter coefficients {H[k]} {G[k]} N-point IDFT {g(n)} N i.i.d. Gaussian random variables {B[k]} Multiply by the filter coefficients {H[k]} -j IDFT-based fading simulator. 28

29 To implement 2-D isotropic scattering, the filter H[k] can be specified as follows: 0, k = 0 1 2πf m 1 (k/(n ˆf m )) 2, k = 1,2,...,k m 1 H[k] = [ k π m 2 arctan ( )] k m 1 2km 1, k = k m 0, k = k m +1,...,N k m 1 [ k π m 2 arctan ( )] k m 1 2km 1, k = N k m 1 2πf m 1 (N k/(n ˆf m )) 2, N k m +1,...,N 1 One problem with the IFFT method is that the faded envelope is discontinuous from one block of N samples to the next. 29

30 Sum of Sinusoids (SoS) Methods - Clarke s Model With N equal strength (C n = 1/N) arriving plane waves g(t) = g I (t)+jg Q (t) = 1/N N n=1 cos(2πf m tcosθ n + ˆφ n )+j 1/N N The normalization C n = 1/N makes Ω p = 1. The phases ˆφ n are independent and uniform on [ π,π). n=1 sin(2πf m tcosθ n + ˆφ n ). (1) With 2-D isotropic scattering, the θ n are also independent and uniform on [ π,π), and are independent of the ˆφ n. Types of SoS simulators deterministic - {θ n } and {ˆφ n } are fixed for all simulation runs. statistical - either {θ n } or {ˆφ n }, or both, are random for each simulation run. ergodicstatistical-either{θ n }or{ˆφ n }, orboth, arerandom,butonlyasinglesimulation run is required. 30

31 Clarke s Model - Ensemble Averages The statistical properties of Clarke s model in for finite N are φ gi g I (τ) = φ gq g Q (τ) = 1 2 J 0(2πf m τ) φ gi g Q (τ) = φ gq g I (τ) = 0 φ gg (τ) = 1 2 J 0(2πf m τ) φ g 2 g 2(τ) = E[ g 2 (t) g 2 (t+τ)] = 1+ N 1 N J2 0(2πf m τ) For finite N, the ensemble averaged auto- and cross-correlation of the quadrature components match those of the 2-D isotropic scattering reference model. Thesquaredenvelopeautocorrelationreaches thedesiredform1+j 2 0 (2πf mτ)asymptotically as N. 31

32 Clarke s Model - Time Averages In simulations, time averaging is often used in place of ensemble averaging. The corresponding time average correlation functions ˆφ( ) (all time averaged quantities are distinguished from the statistical averages with a ˆ ) are random and depend on the specific realization of the random parameters in a given simulation trial. The variances of the time average correlation functions, defined as Var[ˆφ( )] = E [ ˆφ( ) lim φ( ) N characterizestheclosenessofasimulationtrialwithfiniten andtheidealcasewithn. These variances can be derived as follows: Var[ˆφ gi g I (τ)] = Var[ˆφ gq g Q (τ)] 2 ], = 1+J 0(4πf m τ) 2J 2 0 (2πf mτ) 8N Var[ˆφ gi g Q (τ)] = Var[ˆφ gq g I (τ)] = 1 J 0(4πf m τ) 8N Var[ˆφ gg (τ)] = 1 J2 0(2πf m τ) 4N 32

33 Jakes Deterministic Method To approximate an isotropic scattering channel, it is assumed that the N arriving plane waves uniformly distributed in angle of incidence: θ n = 2πn/N, n = 1, 2,..., N By choosing N/2 to be an odd integer, the sum in (1) can be rearranged into the form g(t) = 1 N/2 1 [e j(2πf mtcosθ n +ˆφ n ) +e j(2πf mtcosθ n +ˆφ n ) ] N n=1 +e j(2πf mt+ˆφ N ) +e j(2πf mt+ˆφ N ) (2) The Doppler shifts progress from 2πf m cos(2π/n) to +2πf m cos(2π/n) as n progresses from1ton/2 1inthefirstsum,whileinthesecondsumtheyprogressfrom+2πf m cos(2π/n) to 2πf m cos(2π/n). Jakes uses nonoverlapping frequencies to write g(t) as g(t) = 2 1 M [e j(ˆφ n +2πf m tcosθ n ) +e j(ˆφ n +2πf m tcosθ n ) ] N where n=1 +e j(ˆφ N +2πf m t) +e j(ˆφ N +2πf m t) M = 1 2 N 2 1 (3) and the factor 2 is included so that the total power remains unchanged. 33

34 Note that (2) and (3) are not equal. In (2) all phases are independent. However, (3) implies that ˆφ n = ˆφ N/2+n and ˆφ n = ˆφ N/2 n for n = 1,...,M. This introduces correlation into the phases Jakes further imposes the constraint ˆφ n = ˆφ n and ˆφ N = ˆφ N (but with further correlation introduced in the phases) to give where g(t) = 2 N +j 2 M n=1 2 M n=1 cosβ n cos2πf n t+ 2cosαcos2πf m t sinβ n cos2πf n t+ 2sinαcos2πf m t α = ˆφ N = β n = ˆφ n 34

35 Time averages: < g 2 I(t) > = 2 N = 2 N 2 M n=1 cos 2 β n +cos 2 α M +cos 2 α+ M n=1 cos2β n < g 2 Q(t) > = 2 N = 2 N < g I (t)g Q (t) > = 2 N 2 M n=1 sin 2 β n +sin 2 α M +sin 2 α M 2 M n=1 n=1 cos2β n sinβ n cosβ n +sinαcosα. Choose the β n and α so that g I (t) and g Q (t) have zero-mean, equal variance, and zero cross-correlation. The choices α = 0 and β n = πn/m will yield < gq 2(t) >= M/(2M + 1), < g2 I (t) >= (M +1)/(2M +1), and < g I (t)g Q (t) >= 0. Note the small imbalance in the values of < g 2 Q(t) > and < g 2 I(t) >. The envelope power is < g 2 I (t) > + < g2 Q (t) >= Ω p = 1. The envelope power can be changed to any other desired value by scaling g(t), i.e., Ω p g(t) will have envelope power Ω p. 35

36 Envelope Level (db) Time, t/t Typical faded envelope generated with 8 oscillators and f m T = 0.1, where T seconds is the simulation step size. 36

37 Auto- and Cross-correlations The normalized autocorrelation function is With 2-D isotropic scattering Therefore, φ n gg(τ) = E[g (t)g(t+τ)] E[ g(t) 2 ] φ gi g I (τ) = φ gq g Q (τ) = Ω p 2 J 0(2πf m τ) φ gi g Q (τ) = φ gq g I (τ) = 0 φ n gg (τ) = E[g (t)g(t+τ)] E[ g(t) 2 ] = J 0 (2πf m τ) 37

38 Auto- and Cross-correlations For Clarke s model with angles θ n that are independent and uniform on [ π,π), the normalized autocorrelation function is φ n gg (τ) = E[g (t)g(t+τ)] E[ g(t) 2 ] = J 0 (2πf m τ). Clark s model with even N and the restriction θ n = 2πn N, yields the normalized ensemble averaged autocorrelation function φ n gg (τ) = 1 2N Clark s model with θ n = 2πn N desired values at large lags. N n=1 cos 2πf m τ cos 2πn N. yields an autocorrelation function that deviates from the Finally, the normalized time averaged autocorrelation function for Jakes method is φ n gg(t,t+τ) = 1 2N (cos2πf mτ +cos2πf m (2t+τ)) + 1 N M n=1 (cos2πf n τ +cos2πf n (2t+τ)) Jakes fading simulator is not stationary or even wide-sense stationary. 38

39 Autocorrelation, φ rr (τ) Simulation Ideal Time Delay, f m τ Autocorrelation of inphase and quadrature components obtained with Clarke s method, using θ n = 2πn N and N = 8 oscillators. 39

40 1.0 Autocorrelation, φ rr (τ) Simulation Ideal Time Delay, f m τ Autocorrelation of inphase and quadrature components obtained with Clarke s method, using θ n = 2πn N and N = 16 oscillators. 40