Stochastic Models for System Simulations

Transcription

1 Lecture 4: Stochastic Models for System Simulations Contents: Radio channel simulation Stochastic modelling COST 7 model CODIT model Turin-Suzuki-Hashemi model Saleh-Valenzuela model Stochastic Models for System Simulations Stochastic Models for System Simulations Contents (cont d): WAND model Spencer-Jeffs-Jensen-Swindlehurst model IEEE 8. model 3GPP Stochastic Channel Model Stochastic Models for System Simulations

2 The importance of stochastic channel modelling Main application fields of stochastic channel models: Design and optimization of communication systems: - Design of the constituents of communication systems - Optimization of the behaviour and performance of these constituents - Analytical or simulation-based investigations of the performance of these constituents Monte Carlo simulations for system performance assessment - Link-level simulations (today) - System-level simulations (today) - Simulation of cooperative networks (coming soon) in complex scenarios as close as possible to real operation conditions Stochastic channel models (SCMs) are crucial and indispensable tools in the design of communication systems. Stochastic Models for System Simulations 3 Radio channel simulation The different approaches for channel simulations: Stored Channel Models (ATDMA) Measured space-variant or time-variant delay spread functions (SFs) are selected according to some specified criteria to form classes of so-called reference channels. These delay SFs are stored and can be read on demand for simulation purposes. Example: Reference space-variant delay SF in an atypical microcellular urban environments: Stochastic Models for System Simulations 4

3 Radio channel simulation The different approaches for channel simulations (cont d): Deterministic Channel Models These models employ ray optical techniques (ray tracing, ray launching combined with UTD method) to compute the delay SF or the direction-delay SF using some more or less extensive geographical information (building layout, electric properties of walls and floors, etc.) about the propagation environment under consideration. Stochastic Models for System Simulations 5 Radio channel simulation The different approaches for channel simulations (cont d): Stochastic Channel Models (COST 7, CODIT) - A parametric model for the delay SFs is derived. - Realizations of the delay SF are then generated according to specified probability distributions of the model parameters. - These probability distributions are gathered by means of statistical analyses of measurement data collected during extensive measurement campaigns. Example: COST 7 HT gtτ ( ; ) τ μ delay [mys] Time t [ms] real time [ms] Delay [ s] 5 Stochastic Models for System Simulations 6

4 Radio channel simulation Comparison of the different approaches: Channel simulation approaches Computational expense Intrinsic variability of the models Retained channel features Challenge Reference channels Deterministic channel models Stochastic channel models High (storage) High (identification of the dominant propagation paths) Low Low (=number of stored channels) Medium (=number of environments considered) High (probability distributions) All Part of them (ray optical methods are not exact) Part of them (depending on the model used) Choice of the appropriate selection criteria Identification of the dominant prop. paths + accurate method for computing the path weights Incorporate all relevant channel features Stochastic Models for System Simulations 7 Stochastic modelling Features to be incorporated into SCMs: Environment: Type of environment Frequency range Transceiver characteristics: Trajectory, velocity Bandwidth Antenna (array) types Stochastic channel model Short-term fluctuations: Fast fading Channel dispersion Delay Direction of departure Direction of incidence Doppler frequency Polarization Short range/term (fast) fading Long-term fluctuations: Path loss Shadowing Transitions Delay drift Drift in directions Birth & death of paths Stochastic Models for System Simulations 8

5 Stochastic modelling Requirements for SCMs: Completeness SCMs must reproduce all effects that impact on the performance of communication systems. => Guarantee simulation scenarios close to reality => Full basis for system comparisons Accuracy SCMs must accurately describe these effects. => Realistic results from analytical and/or simulation-based investigations Simplicity/low complexity Each effect must be described by a simple model. => Enable theoretical study of some particular system aspects and performance => Tractable computational effort to simulate the channel in Monte Carlo simulations Stochastic Models for System Simulations 9 Approach for SCM: Specification of the area type additional system parameters (array characteristics, mobile velocity, etc.) Stochastic modelling Software package generating a specific parametric system function of the channel, e.g. N gtτ ( ; ) g i () t δτ ( τ i () t ) i = Statistical processing of data obtained from extensive measurement campaigns in the identified areas Estimation of the probability density functions of the parameters occurring in the parametric system function Realizations of the system function, e.g. gtτ ( ; ) Stochastic Models for System Simulations

6 Stochastic modelling Model for g i () t : Short-term fluctuations: ( ST) g i t () = gi, c t () + g id, () t Specular part: g ic, () t = h ic, exp( jπν i t) ν i : Doppler frequency Diffuse part: g id, () t is a WSS zero-mean circular symmetric complex Gaussian process specified by its ACF R i ( Δt) or equivalently its (Doppler) spectrum P i ( ν) : Δt ν R i ( Δt) P i ( ν) Stochastic Models for System Simulations Model for g i () t (cont d): Long-term fluctuations (path loss & shadowing effect): g i () t = Lt () Lt (): time-dependent path loss computed from one of the models presented in Lecture 3. ΔL i ( d) : real zero-mean Gaussian process with ACF R ΔLi ( Δd). Usually, R ΔLi Stochastic modelling ΔL i ( vt) ( Δd) = ϒ exp( Δd κ ) ( ST) g i () t ϒ: standard deviation of ΔL i ( d) ϒ = 6 8 db κ: decorrelation length Small macrocells: κ = 5 8m Stochastic Models for System Simulations

7 Models for τ i () t : Short-term fluctuations: where { τ, τ, τ, i, } is a specified random point process. Long-term fluctuations: Stochastic modelling τ i () t = τ i τ i () t = τ i + Δτ i () t where { τ, τ, τ, i, } is the above random point process and { Δτ i () t } is a sequence of random processes describing the drift of the components in the time-variant SF on the delay axis. Stochastic Models for System Simulations 3 COST 7 model Main characteristics: Cell type Area Frequency range Time-variant SF Input Macrocell Typical non-hilly urban (TU), bad hilly urban (BU) Non-hilly rural area (RA), hilly terrain (HT) Around GHz N P gtτ ( ; ) = --- N exp{ j( πν i t + ϕ i )} δτ ( τ i ) i = Area type Vehicle velocity Number of components in gtτ ( ; ) Delay and Doppler resolution Stochastic Models for System Simulations 4

8 COST 7 model Normalized delay-doppler scattering function: P n ( ντ, ) --- P( ντ, ) P We can decompose P n ( ντ, ) as P n ( ντ, ) = P n ( τ) P n ( ντ) ( P n ( ντ, ) d νdτ = ) ( P n ( τ) P n ( ν, τ) dν) Normalized delay scattering function Delay-dependent normalized Doppler scattering function The COST 7 models are specified by the two functions: P n ( τ) P n ( ντ) Stochastic Models for System Simulations 5 COST 7 model Normalized delay scattering function: Typical urban non-hilly area (TU): P n ( τ). v MS Local scattering P n ( τ) exp ( τ) ; τ [ μs ] 7 ; elsewhere τ [μs] Stochastic Models for System Simulations 6

9 COST 7 model Normalized delay scattering function (cont d): Typical bad urban hilly area (BU): P n ( τ).7.6 Distant scattering v MS Local scattering P n ( τ) exp ( τ) ; τ [ μs ] 5.5exp ( 5 τ) ; 5 τ [ μs ] ; elsewhere τ [μs] Stochastic Models for System Simulations 7 COST 7 model Normalized delay scattering function (cont d): Typical rural non-hilly area (RA): P n ( τ) 9 v MS Local scattering P n ( τ) exp ( 9.τ) ; τ [ μs ].7 ; elsewhere τ [μs] Stochastic Models for System Simulations 8

10 COST 7 model Normalized delay scattering function (cont d): Typical hilly terrain (HT): P n ( τ) 3 v MS Local scattering.5 Distant scattering.5 exp ( 3.5τ) ; τ [ μs ] P n ( τ).exp ( 5 τ) ; 5 τ [ μs ] ; elsewhere τ [μs] Stochastic Models for System Simulations 9 COST 7 model Normalized Doppler scattering function: CLASS (Clarke s Doppler spectrum) [ τ.5 μs]: v MS Many scatterers with nearly the same features uniformly distributed around the MS C νd ( ν) ( ν D = [Hz] ) CLASS 5 P n ( ντ) = ; ν < ν πν D ( v ν D ) D ; elsewhere C νd ( ν) ( ν D v λ) ν [Hz] Stochastic Models for System Simulations

11 COST 7 model Normalized Doppler scattering function (cont d): GAUS [.5 μs τ μs]: v MS Two strong distant scatterers P n ( ντ) ( ν D = [Hz] ) GAUS ( ν ν ) G( ν; a, ν, ν ) a exp ( ν ) P n ( ντ) G( ν; a,.8ν D,.5ν D ) + G( ν; a,.4ν D,.ν D ) with a ---- = db a [db] ν [Hz] Stochastic Models for System Simulations COST 7 model Normalized Doppler scattering function (cont d): GAUS [ μs τ]: Two strong distant scatterers 4 P n ( ντ) ( ν D = [Hz] ) GAUS v MS P n ( ντ) G( ν; a,.7ν D,.ν D ) + G( ν; a,.4ν D,.5ν D ) with a ---- = 5 db a [db] ν [Hz] Stochastic Models for System Simulations

12 COST 7 model Normalized Doppler scattering function (cont d): RICE: LOS or reflected component.5 P n ντ ( ) ν = D [Hz] RICE ( ) v MS Many scatterers with nearly the same features uniformly distributed around the MS.5.9 P n ( ντ) δ( ν.7ν πν D ( v ν D ) D ) ; ν < ν D ; elsewhere ν [Hz] Stochastic Models for System Simulations 3 COST 7 model Simulation issues: Make use of the following result: If the random variables ν, τ, satisfy ϕ, ν, τ, ϕ,, ν N, τ, N ϕ N the following properties: the pairs ( ν i, τ ) are independent with probability distribution i P n ( ν, τ), ϕ i the phases are independent and uniformly distributed over [ π, ), the sequences {( ν i, τ )} and { ϕ i } are independent, i then for N sufficiently large, the scattering function of the simulated channel is close to P( ντ, ). Stochastic Models for System Simulations 4

13 COST 7 model Simulated time-variant delay SF [TU]: P n ( ντ) = C νd ( ν) gtτ ( ; ) Delay τ [ μs] 3 4 delay [mys] ν D v 5 real time [ms] = 83.3 Hz ( = km/h, f= 9 MHz) Time t [ms] 5 Source: ETHZ, CTL, R. Heddergott Stochastic Models for System Simulations 5 COST 7 model Simulated time-variant delay SF [BU]: P n ( ντ) = C νd ( ν) gtτ ( ; ) Delay τ [ μs] 6 delay [mys] 8 5 real time [ms] 5 5 Time t [ms] ν D = 83.3 Hz Source: ETHZ, CTL, R. Heddergott Stochastic Models for System Simulations 6

14 COST 7 model Simulated time-variant delay SF [RA]: gtτ ( ; ) P n ( ντ) = C νd ( ν) ν D = 83.3 Hz Delay τ [ μs] delay [mys] Time t [ms] real time [ms] 5 Source: ETHZ, CTL, R. Heddergott Stochastic Models for System Simulations 7 COST 7 model Simulated time-variant delay SF [HT]: P n ( ντ) = C νd ( ν) gtτ ( ; ) ν D = 83.3 Hz Delay τ [ μs] delay [mys] Time t [ms] real time [ms] 5 Source: ETHZ, CTL, R. Heddergott Stochastic Models for System Simulations 8

15 COST 7 model The COST 7 tables: Example: BU Tap No Delay Power μ [lin] Power [db] Doppler spectrum.. -7 CLASS..5-3 CLASS CLASS 4.8. GAUS GAUS GAUS Tap No Delay Power μ [lin] Power [db] Doppler spectrum GAUS GAUS GAUS GAUS GAUS..3-5 GAUS Delay spread: σ τ =.5 μs Stochastic Models for System Simulations 9 COST 7 model The COST 7 tables (cont d): Delay scattering function for the BU model: [ μs] Stochastic Models for System Simulations 3

16 COST 7 model The COST 7 tables (cont d): Comment:!! The delays are regularly spaced!! i.e., = m i Δτ with m i integer and Δτ =.μs. τ i -> The transfer function is periodic (with period = 5 MHz) -> The parameter settings in the COST 3 tables are not appropriate for investigating the performance of systems which exploit channel frequency selectivity, e.g. GSM with frequency hopping. Solution: Choose an irregularly spacing of the delays or randomly select them. Stochastic Models for System Simulations 3 COST 7 model The COST 7 tables (cont d): Frequency hopping: Real channel frequency transfer function: H( f ) B c =. B B B ' B ' Transfer function of the COST 7 channel: H( f ) Δτ f B B B ' B ' f Stochastic Models for System Simulations 3

17 CODIT model Main characteristics: Cell type Area Frequency band Time-variant SF Input Macro-, micro, picocell Macrocell: urban, suburban, suburban hilly rural, rural hilly Microcell: dense urban linear street, town square, industrial area Indoor: floor cell in buildings, corridor, large and very large rooms GHz range Up to MHz signal bandwidth N gtτ ( ; ) g i () t δτ ( τ i () t ) i = Area type Vehicle velocity System bandwidth Stochastic Models for System Simulations 33 CODIT model Short-term variations of g i () t : ( ST) g i t () = hi c xxxxxxx, exp jπ v λ -- cos( φ ic, ) t + xxxxxxxxxxxxxxxxxxx ν ic, ν i, j Specular part J xxxxxxx h exp i, j jπ v λ -- cos( φ i, j ) t xxxxxxxxxxxxxxxxxxxx = j Diffuse part h ic, = σ ic,, arg{ h ic, } uniformly distributed over [, π) h i, j N --σ, J i, d E g : mean power of the th component i () t [ ] = σ i, c + σ id, σ i i ( σ i, c σ id, ) : coherent to diffuse power ratio of the ith component Stochastic Models for System Simulations 34

18 CODIT model Probability distribution of the azimuths and : φ ic, φ i, j p φi ( φ), j Scatterer i φ i, j φ ic, MS v φ ic, π φ φ i, j N ( φ ic,,.5 [radian] ) modπ Stochastic Models for System Simulations 35 CODIT model Long-term variations of g i () t : ( ST) g i () t = z i () t g i () t Term describing the long-term fluctuations Shadowing effects: z i () t π z i () t + Δz i cos v λ --t + ψ i Δz i q i ψ i : Random variable uniformly distributed over [ π), q i ( v λ) t Stochastic Models for System Simulations 36

19 CODIT model Long-term variations of g i () t (cont d): Emergence of the components: z i () t ; vt z vt z λ 6 ; vt > z z i () t z v t Fading of the components: z z i () t ; vt z vt z λ 6 ; vt < z z i () t is a random variable uniformly distributed over [ 5λ, 7λ]. z v t Stochastic Models for System Simulations 37 CODIT model Short-term variations of τ i () t : Long-term variations of τ i () t : τ i () t = τ i τ i () t τ i + Δτ i + cos π v λ --t + υ i τ i () t Δτ i = τ p i i υ i : Random variable uniformly distributed over [ π), p i ( v λ) t Stochastic Models for System Simulations 38

20 CODIT model Selection of the parameters of gtτ ( ; ) : The number N of components depends on the area type but is fixed for a given area, N max =. J = The parameters,( σ ic, σ id, ), φ ic,, Δz i, q i, τ i, Δτ i, p i describing the behaviour of the components of g i () t are random variables specified by σ i probability distributions. Stochastic Models for System Simulations 39 CODIT model Setting Tables: Example: Suburban hilly environment: σ i E[ h i ] Var[ h i ] φ i τ i [μs ] Δz i q i p i Δτ i [ns] N σ i = i Source: CODIT Report: Final Propagation Model Stochastic Models for System Simulations 4

21 CODIT model Setting Tables (cont d): Example: Power delay spectrum generated with the previous table: Source: CODIT Report: Final Propagation Model Stochastic Models for System Simulations 4 CODIT model Example: Microcell LOS Area : Contour plot of a realization of gdτ ( ; ) : Evolution of the corresponding delay scattering function d = 8λ Distance d [ λ ] Power [db] d = Delay τ [ns] Delay τ [ns] Source: CODIT Report: Final Propagation Model Stochastic Models for System Simulations 4

22 CODIT model Example: Indoor Picocell LOS Area : Contour plot of a realization of gdτ ( ; ) : Evolution of the corresponding delay scattering function Distance d [ λ ] Power [db] Delay τ [ns] Delay τ [ns] Source: CODIT Report: Final Propagation Model Stochastic Models for System Simulations 43 Main characteristics: Turin-Suzuki-Hashemi model Cell type Area Frequency range Time-invariant delay SF Main features Macro-, micro- and picocell (by appropriately setting the probability densities of model parameters) Urban 5MHz -GHz gtτ ( ; ) = h( τ) = N h i δτ ( τ i ) i = Time-invariant Clustering of the components in h( τ) is reproduced by modelling the sequence { τ i } as a Poisson point process or a modification thereof. Stochastic Models for System Simulations 44

23 Turin-Suzuki-Hashemi model Modelling { } as a Poisson point process: τ i { } is a Poisson point process with rate γ( τ) : τ i γ( τ) Γ( T ) τ τ τ i τ i τ N T Γ( T ) γ( τ) dτ T τ τ N T : Number of points in the time interval T P[ N T = n] Γ( T ) n = exp( Γ( T )) [ E[ N n! T ] = Var[ N T ] = Γ( T )] Stochastic Models for System Simulations 45 Turin-Suzuki-Hashemi model Modelling { } as a homogeneous Poisson point process: τ i γ( τ) = γ In this case, the delay differences τ i τ i are independent random variables which are (identically) exponentially distributed with parameter γ: Probability density of τ i τ i : p τi τ i ( τ) = γexp( γτ) p τi τ ( τ) i γ γ τ Drawback of the Poisson model: It does not describe sufficiently accurately the clustering of the components as observed in measured delay SFs. Stochastic Models for System Simulations 46

24 Turin-Suzuki-Hashemi model Estimated rates in outdoor environments: ft ns Source: Paper by Turin at al. Stochastic Models for System Simulations 47 Turin-Suzuki-Hashemi model Modelling { τ i } as a modified Poisson point process: The rate γ( τ) depends on the random sequence { τ i } in the following way: γ( τ) h( τ) τ s γ ( τ) τ s τs τ s c s γ ( τ) τ The basic rate γ ( τ) is multiplied by a factor c s during a predetermined relaxation interval τ s at each τ i : c s > : Clustering effect c s = : Poisson point process c s < : Isolated delay points τ τ τ 3 τ 4 τ Clustering effect ( c s > ) Values for τ s and τ i (macrocell): τ s = ns (arbitrarily selected). < c s < 3 (estimated from measurements) Stochastic Models for System Simulations 48

25 Turin-Suzuki-Hashemi model Probability distribution of the coefficients h i : The distribution of the amplitude h i is log-normal, i.e. log( h i ) is a Gaussian random variable with a mean μ h and a standard deviation. Both quantities depend on. σ h The phase arg{ } is uniformly distributed over [, π). h i τ i Stochastic Models for System Simulations 49 Turin-Suzuki-Hashemi model Probability distribution of the coefficients h i (cont d): Two examples showing the behaviour of μ h ( τ) and σ h ( τ) for macrocells: μ h ( τ) + σ h ( τ) μ h ( τ) σ h ( τ) μ h ( τ) μ h ( τ) + σ h ( τ) μ h ( τ) μ h ( τ) σ h ( τ) Excess delay τ [ns] Excess delay τ [ns] Source: Paper by Turin at al. Stochastic Models for System Simulations 5

26 Main characteristics:b Saleh-Valenzuela model Cell type Area Frequency range Time-invariant delay SF Picocell Indoor Around GHz N J() i gtτ ( ; ) = h( τ) = h i, j δτ ( τ i, j ) i = j = Cluster index Index for the components within the clusters Main features Static model The clustering of the components in h( τ) is reproduced by modelling the sequence { τ i, j } as a concatenation of two Poisson point processes Stochastic Models for System Simulations 5 Saleh-Valenzuela model Modelling { } as a concatenation of two Poisson point processes: h( τ) τ i, j Cluster # Cluster # Cluster #i xxxxxxx xxxxxxxxxxx xxxxxxx h, h h,, h i, hi h, = τ, τ i,, h τ τ, 3, i τ τ τ τ = τ τ, h i, τ, τ, = τ i, τ 3, i, h,, h, = h h i, i, τ i, τ i h i, τ i, τ i, Δτ i, Δτ i, Stochastic Models for System Simulations 5

27 Modelling { } as a concatenation of two Poisson point processes: τ i, j Cluster delays { }: (i) τ = τ i Saleh-Valenzuela model (ii){ τ i ; i =,,, N }: Poisson process with rate Λ. Delays within cluster i: (i) τ i, = τ i,..., τ i, j τ i + Δτ i, j ; j =,,, J() i (ii) {,, } : Poisson process with rate λ. Δτ i, Δτ i, Experimentally obtained values for Λ and λ: Λ = 3 ns λ = 5 ns Stochastic Models for System Simulations 53 Saleh-Valenzuela model Probability distribution of the coefficients h i, j : Conditioned on τ i, j, h i, j is a complex zero-mean circular-symmetric Gaussian random variable with variance: σ h ( τ i, j ) E [ h i, j τi ], j = σ h exp( τ i η) exp( Δτ i, j η) σ h σ h exp( τ η) σ h exp( τ i η) exp( ( τ τ i ) η) σ h ( τ i, j ) Experimental values for η and η: η = 6 ns η = 5 ns τ i Δτ i, j τi, j τ Stochastic Models for System Simulations 54

28 WAND model Include before --- Main characteristics: Cell type Area Application range Time-variant azimuthdelay SFSF Input Picocell Indoor, 5, and 7 GHz bands N t gtφ ( ;, τ) g i ()δφφ t ( i )δ( τ τ i () t ) i = N t gtτ ( ; ) = g i ()δτ t ( τ i () t ) i = Area type: small, large rooms, factory halls, corridors Velocity of the mobile station Stochastic Models for System Simulations 55 WAND model Local dispersion: Azimuth-delay spread function: N h( φτ, ) h i δφφ ( i )δ( τ τ i ) i = where N is a Poisson distributed random variable. { φ i }: sequence of independent, uniformly distributed random azimuths { τ i }: sequence of independent, exponentially distributed random delays with common expectation E[ τ i ] = σ d. { h i }: sequence of independent, zero-mean complex Gaussian random amplitudes with individual variance E[ h i φi = φ, τ i = τ] exp{ τ σ a } Stochastic Models for System Simulations 56

29 WAND model Local dispersion (cont d): Short term time-variant delay SF: ( ) ( t; τ) h i exp{ jπν i t} δτ ( τ i ) g ST i = v with ν i -- cos( φ denoting the Doppler frequency. λ i ) N g i () t Stochastic Models for System Simulations 57 WAND model Local scattering function (cont d): Local azimuth-delay scattering function: P L ( φτ, ) E L [ h( φτ, ) ] Local delay scattering function: = N i = h i δφφi ( )δ( τ τ i ) P L ( τ) E L [ g ( ST) ( t; τ) ] = N i = h i δτ ( τi ) Stochastic Models for System Simulations 58

30 WAND model Global scattering function: Global azimuth-delay scattering function: Global delay scattering function: P( φτ, ) E[ P L ( φτ, )] exp{ τ σ τ } P( τ) E[ P L ( τ) ] exp{ τ σ τ } with delay spread σ τ = ( σ a + σ d ) Stochastic Models for System Simulations 59 WAND model Local versus global delay scattering function: 5 local PDS (one realization) average of local PDS global PDS 5 Power Delay [ns] Stochastic Models for System Simulations 6

31 WAND model Fluctuations of the number of impinging waves: τ τ ( 3) τ ( ) τ ( n) τ ( ) Δt Δt Δt 3 Δt n Δt n Δt n + t t t 3 t 4 t n t n t n + t Birth times: { t n }: Uniform Poisson process with rate ( Λ b ). Live times: { Δt n }: Independent and identically, exponentially distributed random variables with expectation E[ Δt n ] = Λ l. Stochastic Models for System Simulations 6 WAND model Fluctuations of the number of impinging waves (cont d): Let N t number of active components in gtτ ( ; ) at time t. N t τ t t Stochastic Models for System Simulations 6

32 WAND model Fluctuations of the number of impinging waves (cont d): N t is a Poisson distributed random variable with expectation E[ N t ] = Λ l Λ b Stochastic Models for System Simulations 63 WAND model Fluctuations of the number of impinging waves (cont d): Example of a realization of N t : ( E[ N t ] = ) 4 8 one realization average of 5 realizations 6 Time (s) 8 6 N t Number of paths M Delay (ns) 5 5 Time (s) Stochastic Models for System Simulations 64

33 WAND model Long-term variations of g i () t : ( ST) g i () t = z i () t g i () t Transition function: z i () t t i Δti t i + Δt i t Stochastic Models for System Simulations 65 WAND model Long-term variations of g i () t (cont d): Selected shape for z i () t : z i () t z t t i Δt i with the pattern function zt () Cosine shape t Stochastic Models for System Simulations 66

34 WAND model Long-term variations of g i () t (cont d): Example of a realization of the long-term fluctuations of the instantaneous power of the components g i () t : Power relative to maximum (db) Instantaneous power of g i () t : z i () t h i Time (s) Delay (ns) Stochastic Models for System Simulations 67 WAND model Long-term variations of τ i () t : ν i τ i () t = τ i --- [ t ( t f i + Δt i ) ] v where ν i = -- cos( φ is the Doppler shift of the th component. λ i ) i Stochastic Models for System Simulations 68

35 WAND model Long-term variations of τ i ( t ) (cont d): Example of a realization of the long-term fluctuations of the relative delays: 4 Time (s) Delay (ns) Stochastic Models for System Simulations 69 WAND model Example of a realization of a time-variant delay SF: Relative amplitude (lin) Time (s) Delay (ns) Stochastic Models for System Simulations 7

36 Spencer-Jeffs-Jensen-Swindlehurst Model Main characteristics: Cell type Area Time-invariant azimuthdelay SF Main features Picocell Indoor N J() i h( φτ, ) = h i, j δφφ i, j ( i, ) j Cluster index i = j = Index of the components within the clusters Component azimuth of incidence Component delay Time-invariant The model is an extension of the model by Saleh-Valenzuela to include dispersion in azimuth of arrival. The concept of double Poisson process is maintained. Stochastic Models for System Simulations 7 Spencer-Jeffs-Jensen-Swindlehurst Model Stochastic model for { }: φ i, j Cluster delays { }: (i) φ = φ i (ii){ φ i ; i =,,, N }: Independent uniformly distributed over [ π, ). Delays within cluster i: (i) φ i, j φ i + Δφ i, j, j =,,,, J()i i, =,, N ; (ii) { Δφ i, j ; j =,, J() i }: Independent, Laplace( σ φ () i ); (iii){{ Δφ i, j ; j =,, J() i }; i =,, N }: independent sets Stochastic Models for System Simulations 7

37 Spencer-Jeffs-Jensen-Swindlehurst Model Probability density function of Δφ i, j [ σ φ = 6 ]: Δφ [ ] Stochastic Models for System Simulations 73 Spencer-Jeffs-Jensen-Swindlehurst Model Additional independence assumption: The following three sets of random variables { τ i, Δτ i, j ; j =,,, J()i i, =,, N } { φ i ; i =,, N } { Δφ i, j ; j =,,, J()i i, =,, N } are independent. Estimates the model parameters [Building, Building ]: η = 34ns, 78ns η = 9ns, 8 ns Λ = 7ns, 7 ns λ = 5ns, 7 ns σ φ = 6,, Stochastic Models for System Simulations 74

38 IEEE 8. Model Main characteristics: Cell type Area Application range Biazimuth-delay SF Input Feature Picocell (indoor and outdoor) Model (M) B (delay spread: 5ns): residential LOS/NLOS M C (3ns): Small office NLOS, typical offices LOS M D (5ns): Typical office NLOS, large office LOS M E (ns): Large office NLOS, large spaces (indoor & outdoor) LOS M F (5ns): Large space (indoor & outdoor), NLOS and 5 GHz bands N J() i h( φ, φ, τ) = h i, j ( φ, φ, τ) i = j = Area type Velocity of the mobile station The model is inspired from the Spencer-Jeffs-Jensen-Swindlehurst model Stochastic Models for System Simulations 75 IEEE 8. Model Stochastic properties of the biazimuth delay SF: h i j h( φ, φ, τ) = h i j i = j =, ( φ, φ, τ), ( φ, φ, τ) : complex uncorrelated process E[ h i, j ( φ, φ, τ) ] = σ( τ i, j ) f σφ () i ( φ φ )f, i σφ () i ( φ φ, i ) J() i τ i, j = n i, j Δτ Δτ = ns n i, j integer f σφ ( φ) exp( φ σ φ ) 8 φ < 8 (Laplace) N δτ ( ) τ i, j Stochastic Models for System Simulations 76

39 IEEE 8. Model Stochastic properties of the biazimuth-delay SF (cont d): The area-dependent parameters N, στ ( ), σ φ ()σ i, φ ()J i () i, n ij, j =,,, J()i i, =,, N are provided in tables. Some illustrative figures: Number of clusters I: B & C: ; D: 3, E: 4; F: 6 Azimuth spread (departure) σ φ : deg Azimuth spread (incidence) σ φ : deg Number of paths/cluster J: Cluster : 5-6; Cl. : 7-; Cl. 3: 4-7; Cl. 4: 3-4; Cl. 5 & 6: Comment: The Kronecker factorization applies to the proposed MIMO (narrow-band) transfer matrix, which is inconsistent with the above model Stochastic Models for System Simulations 77 3GPP SCM Main characteristics: Cell type Area Frequency range Biazimuth-Dopplerdelay SF Macrocell (BS-BS spacing 3km), Microcell (< km) Urban macro- (U-Ma) and microcell (U-Mi), suburban macrocell (S-Ma) GHz-band N h( φ, φ, υ, τ) = h n ( φ, φ, υ, τ) = n = Cluster index h n ( φ, φ, υ, τ) = Subpath index P n γ M s G M ( φ, n, m ) G ( φ, nm, ) exp( jϑ n, m )x m = δφ ( φ, nm, )δ ( φ φ )δ( ν ν, nm, )δ n, ( τ τ m nm, ) v ν n, m = cos( φ λ, n, m φ ) v Stochastic Models for System Simulations 78

40 3GPP SCM Main features N = 6 clusters Uplink-downlink reciprocity Site-to-site correlated shadowing with.5 correlation coefficient Further options: - Per-path polarization - Bad urban scenario with 5th and 6th paths allocated as far clusters - LOS scenario (microcell only) specified by a K-factor: K( d) = 3.3d [db] ( d: distance base station - mobile station) - Urban canyon (modification of the angles of arrival) Stochastic Models for System Simulations 79 3GPP SCM Characteristics of one-bounce clusters: Cluster n σ φ,n Subpath (n, m) σ φ,n φ,n,m φ,n,m φ,n φ v v φ Mobile station φ,n φ d Base station Stochastic Models for System Simulations 8

41 3GPP SCM Parameters: Environment parameters: Distance BS-MS d and azimuths φ, φ from one station to the other [determined from the cell layout] MS velocity direction: φ v uniformly distributed over [ π), Path loss and shadowing Path loss: - Macrocell: Hata model - Microcell: COST3-Walfish-Ikegami model Shadowing γ s : - Macrocell: log( γ s) Gauss(, ς γ s ), with ς γ s = 8dB [ ] - Microcell: log( γ s ) = 4dB (LOS); db (NLOS) Stochastic Models for System Simulations 8 3GPP SCM Global spread factors: Delay spread: E[ σ τ ]: S-Ma: 7ns; U-Ma: 65ns; U-Mi: 5ns Macrocell: log( σ τ ) Gauss( μ τ, ς τ ) - Suburban: μ τ = 6.8, ς τ =.88 - Urban: μ τ = 6.8, ς τ =.8 Stochastic Models for System Simulations 8

42 3GPP SCM Global spread factors (cont d): Azimuth spread at BS site E[ σ φ ]: S-Ma: 5 ; U-Ma: 8, 5 ; U-Mi: 9 Macrocell: log( σ φ ) Gauss( μ φ, ς φ ) - Suburban: μ φ =.69, ς φ =.3 - Urban, E[ σ φ ] = 8 : μ φ =.8, ς φ =.34 - Urban, E[ σ φ ] = 5 : μ φ =.8, ς φ =. Azimuth spread at MS site: E[ σ φ ] = 68 (all areas) Stochastic Models for System Simulations 83 3GPP SCM Correlation between the log- Global -factors in macrocells: Intrasite correlation coefficients: - log( σ τ ) and log( σ φ ): log( γ s ) and log( σ φ ): log( γ s ) and log( σ τ ): -.6 Intersite correlation of shadowing:.5 correlation coefficient The four above quantities are random with a joint Gaussian law specified by - their expectations μ τ, μ φ, μ γ s = - their variances, ςφ, ς τ ς γ s - their correlation coefficients as above specified - non-specified correlation coefficients are zero Stochastic Models for System Simulations 84

43 3GPP SCM Clustering: Number of clusters: N = 6 Number of subpaths per cluster: M = Path weights: Unnormalized mean-squared weights: P n = exp( τ n σ τ ) ξ n ξ n Gaussian(, 3 ) per-path shadowing) Normalized mean-squared weights: P n = P' n P' n' (lognormal Stochastic Models for System Simulations 85 3GPP SCM Path delays: Macrocell - τ', τ', τ', N Exp( σ' τ ), independent, with σ' τ = η τ σ τ - η τ =.4 (S-Ma);.7 (U-Ma) Microcell - τ', τ', τ', N Uniform( [,.]μs), independent τ' ( ) τ' ( ) τ' ( N ) (ordering) τ n = τ' ( n) τ' ( ), n =,, N Subpath characteristics: Phases: ϑ nm, Uniform( [, π) ) φ, nm, φ, nm, Azimuths of departure and of incidence : Fixed, tabulated. Stochastic Models for System Simulations 86

44 3GPP SCM Azimuth dispersion at the BS: Per-path nominal azimuth: -Macrocell: φ, n Gauss(, σ φ ) with σ φ = η φ σ φ - Suburban: η φ =. - Urban: - Microcell: φ n η φ =.3, Uniform( 4, 4 ) Per-path azimuth spread: - Macrocell: σ φ, n = - Microcell: σ φ, n = 5 Stochastic Models for System Simulations 87 3GPP SCM Azimuth dispersion at the MS: Per-path nominal azimuth: φ, n Gauss(, σ φ ), n The per-path azimuth spread σ φ is a monotone increasing function of, n the path weight: σ φ = 4.( exp(.75 log( P, n n ) ) Per-path azimuth spread: σ φ, n = 35 (all areas) Stochastic Models for System Simulations 88