Mobile Radio Systems Small-Scale Channel Modeling
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1 Mobile Radio Systems Small-Scale Channel Modeling Klaus Witrisal Signal Processing and Speech Communication Laboratory Graz University of Technology October 28, 2015 Mobile Radio Systems Small-Scale Channel Modeling p. 1/43 Outline 3-1 Introduction Mathematical models for communications channels [Molisch 6.2.2; Proakis 1-3] 3-2 Stochastic Modeling of Fading Multipath Channels Multipath channel [Proakis 14-1] Fading amplitude distribution (Rayleigh, Rice) [Molisch 5.4, 5.5] Time-selective fading [Molisch 5.6] Frequency-selective fading WSSUS stochastic channel description [Molisch , Proakis 14] 3-3 Classification of Small-Scale Fading [Molisch 6.5] Mobile Radio Systems Small-Scale Channel Modeling p. 2/43
2 References A. F. Molisch: Wireless Communications, 2005, Wiley J. G. Proakis: Digital Communications, 3rd ed., 1995, McGraw Hill J. R. Barry, E. A. Lee, D. G. Messerschmitt: Digital Communication, 3rd ed., 2004, Kluwer A. Paulraj, R. Nabar, and D. Gore: Introduction to Space-Time Wireless Communications, 2003, Cambridge T. S. Rappaport: Wireless Communications Principles and Practice, 2nd ed., 2002, Prentice Hall M. Pätzold: Mobile Fading Channels, 2002, Wiley Figures (partly) extracted from these references Mobile Radio Systems Small-Scale Channel Modeling p. 3/43 Signal Models Signal processing channel models can be described for different interfaces Application/design objective determines choice of appropriate model 011 tream baseband modulation baseband signal (analog) carrier modulation propagation channel carrier demodulation baseband signal (analog) (matched) filter sampling threshold detector (detection) bit stream radio channel (e.g. c(t) or c( ;t)) baseband equivalent radio channel (e.g. c l (t) or c l ( ;t)) sampled (bandlimited) baseband equivalent radio channel (e.g. h or h[k]) Mobile Radio Systems Small-Scale Channel Modeling p. 4/43
3 Additive Noise Channel Channel s frequency response is flat over signal bandwidth Simplest model transmitted (TX) signal corrupted by additive noise r(t) =αs(t)+n (t) s(t)... TX signal is a bandpass signal s(t) = 2 R{s l (t)e j2πfct } r(t)... received (RX) signal for (lowpass equivalent) baseband signals (i.e. complex envelopes of s(t),r(t),n (t)) r l (t) =hs l (t)+n l (t), with h C Mobile Radio Systems Small-Scale Channel Modeling p. 5/43 Additive Noise Channel (cont d) Noise is usually modeled as white, Gaussian (additive white Gaussian noise AWGN) φ n (τ) =E{n (t)n (t + τ)} = N 0 2 δ(τ) F S n (f) = N 0 2 Mobile Radio Systems Small-Scale Channel Modeling p. 6/43
4 Additive Noise Channel (cont d) Sampled AWGN model (lowpass equivalent model) Noise characterization r[k] =hs[k]+n[k] (all are C) E{n[k]n [l]} = σn 2 δ[k l] n[k] is zero-mean circularly symmetric complex Gaussian (ZMCSCG) Real and imaginary components are i.i.d. (independent, identically distributed) σ 2 n depends on (matched) filter at receiver front-end Real and imaginary components have σ 2 n/2 Mobile Radio Systems Small-Scale Channel Modeling p. 7/43 Linear filter channel Channel s frequency response is frequency-selective (i.e. non-flat), leading to (linear) signal distortions For time-invariant channels r(t) =s(t) c(t) +n(t) = c(τ)s(t τ)dτ + n(t) c(t)... impulse response of linear filter Mobile Radio Systems Small-Scale Channel Modeling p. 8/43
5 Linear filter channel (cont d) Sampled case (lowpass equivalent model) r[k] = L 1 l=0 h[l]s[k l]+n[k] h[k] incorporates TX pulse shape RX (matched) filter; ADC filter (thus bandwidth corresponds to signal bandwidth) physical channel n[k]... AWGN (ZMCSCG) This is actually an equivalent, whitened matched filter (WMF) channel model [Barry/Lee/Messerschmitt] Mobile Radio Systems Small-Scale Channel Modeling p. 9/43 Linear time-variant filter channel Characterized by time-variant channel impulse response (CIR) c(τ; t) response of channel at time t to an impulse transmitted at time t τ τ... elapsed time, age variable r(t) =s(t) c(τ; t)+n(t) = c(τ; t)s(t τ)dτ + n(t) model for multipath propagation c(τ; t) = α i (t)δ(τ τ i (t)) (1) i=0 Mobile Radio Systems Small-Scale Channel Modeling p. 10/43
6 Linear time-variant filter ch. [fig 5-4 Rap] Mobile Radio Systems Small-Scale Channel Modeling p. 11/43 Stochastic modeling of fading multipath channels Motivated by their randomly time-variant nature and large number of multipath components Derivation of lowpass equivalent CIR from (1) c l (τ; t) = α i (t)e j2πf cτ i (t) δ(τ τ i (t)) = i=0 α i (t)e jϕi(t) δ(τ τ i (t)) (2) i=0 considering discrete multipath components phase term ϕ i (t) = 2πf c τ i (t) varies dramatically Mobile Radio Systems Small-Scale Channel Modeling p. 12/43
7 Fading of an unmodulated carrier TX signal is unmodulated carrier (CW) s l (t) =1 RX signal w/o noise: y l (t) =c l (τ; t) 1=c l (t) 1 c l (t) = α i (t)e jϕi(t) = α i (t)e j2πf cτ i (t) i=0 i=0 sum of vectors (phasors) amplitudes α i (t) change slowly phases ϕ i (t) change by 2π if: τ i (t) changes by 1/f c i.e.: path length changes by wavelength λ large number of multipath components model c l (t) as a random process! Mobile Radio Systems Small-Scale Channel Modeling p. 13/43 Fading of an unmodulated carrier (cont d) Modeling c l (t) as a random process: large number of multipath components are added by central limit theorem (CLT): c l (t) is complex Gaussian (CIR c l (τ; t) is complex Gaussian) c l (t) has random phase and amplitude in absence of dominant component: c l (t) is zero-mean complex Gaussian its envelope c l (t) is Rayleigh distributed Rayleigh fading channel Mobile Radio Systems Small-Scale Channel Modeling p. 14/43
8 Fading of an unmodulated carrier (cont d) Mobile Radio Systems Small-Scale Channel Modeling p. 15/43 Fading of an unmodulated carrier (cont d) Rayleigh distribution: f R (r) = r σ 2 e r2 /(2σ 2 ) for r 0 characterized by σ 2 : variance of underlying Gaussian processes X 1,X 2 N(0,σ 2 ), where X 1 = R{c l (t)} and X 2 = I{c l (t)} derivation of Rayleigh distribution... Y = X X has χ2 -PDF of 2 degrees of freedom R = X X amplitude c l(t) has Rayleigh PDF Mobile Radio Systems Small-Scale Channel Modeling p. 16/43
9 Fading of an unmodulated carrier (cont d) central chi-square PDF of n degrees of freedom Rayleigh PDF characterized by σ Mobile Radio Systems Small-Scale Channel Modeling p. 17/43 Fading of an unmodulated carrier (cont d) in presence of a dominant component: c l (t) is non-zero-mean complex Gaussian its envelope c l (t) is Ricean distributed Ricean fading channel Ricean distribution: f R (r) = r σ 2 e r2 +s 2 2σ 2 I 0 ( rs σ 2 ) for r 0 I 0 (x)... zero-order modified Bessel function of first kind characterized by σ 2... variance of underlying Gaussian processes and s 2 = m m power of mean (i.e. s2 = E{c l (t)} 2 ) Mobile Radio Systems Small-Scale Channel Modeling p. 18/43
10 Fading of an unmodulated carrier (cont d) Shape of Ricean distribution defined by K = s2 2σ 2 K [db] = 10 log s2 2σ 2 Ricean K-factor Ratio of deterministic signal power (mean) and variance of multipath (scattered components) For K =0= db: Ricean distribution equivalent to Rayleigh Mobile Radio Systems Small-Scale Channel Modeling p. 19/43 Fading of an unmodulated carrier (cont d) Ricean PDFs with normalized mean power (2nd moment) K = 0; Rayleigh K=2=3dB K=4=6dB K=10=10dB 10 0 Ricean CDFs with normalized mean power (2nd moment) K = 0; Rayleigh K=2=3dB K=4=6dB K=10=10dB R amplitude r Ricean (and Rayleigh) PDFs CDF F R (r) = Pr[R = r] amplitude r [db] Ricean (and Rayleigh) CDFs Mobile Radio Systems Small-Scale Channel Modeling p. 20/43
11 Time-selective fading Characterization of the time variability s l (t) =1 y l (t) =c l (τ; t) 1=c l (t) = α i (t)e jϕ i(t) i=0 Characterize autocorrelation function of c l (t) assume c l (t) is complex Gaussian assume c l (t) is wide-sense stationary (WSS) Define: spaced-time correlation function φ c (Δt) =E{c l (t)c l(t +Δt)} F S c (ν) Doppler power spectrum S c (ν) = φ c(δt)e j2πνδt dδt Mobile Radio Systems Small-Scale Channel Modeling p. 21/43 Time-selective fading (cont d) Doppler power spectrum: average power output of channel as a function of Doppler frequency Mobile Radio Systems Small-Scale Channel Modeling p. 22/43
12 Time-selective fading (cont d) [Rappaport fig. 5-1] Mobile Radio Systems Small-Scale Channel Modeling p. 23/43 Time-selective fading (cont d) Jakes model for Doppler power spectrum assumes mobile moving at const. velocity v uniformly distributed scattering around mobile Jakes Doppler spectrum: S c (ν) = 1 π 1 ν 2 max ν 2 (for normalized power) Doppler power spectral density normalized Doppler frequency ν/ν max Mobile Radio Systems Small-Scale Channel Modeling p. 24/43
13 Time-selective fading (cont d) Characterization of time-selective fading by parameters RMS Doppler spread ν rms = ν 2 ν 2 second centralized moment of normalized Doppler PSD mean and mean squared Doppler spread νsc (ν)dν ν ν = 2 ν 2 S c (ν)dν = Sc (ν)dν Sc (ν)dν Coherence time T c 1 ν rms Mobile Radio Systems Small-Scale Channel Modeling p. 25/43 Frequency-selective fading fa(time-invariant) multipath channel Characterization of the time dispersion: CIR c l (τ) s l (t) =δ(t) y l (t) =c l (τ; t) δ(t) = α i (t)e jϕi(t) δ(t τ i (t)) c l (τ) = i=0 α i e jϕ i δ(τ τ i ) i=0 ACF: Uncorrelated scattering assumption: E{c l (τ 1)c l (τ 2 )} = S c (τ 1 )δ(τ 1 τ 2 ) S c (τ)... multipath intensity profile (= delay power spectrum; = average power delay profile) Mobile Radio Systems Small-Scale Channel Modeling p. 26/43
14 Frequency-selective fading (cont d) Time-dispersion implies frequency-selectivity Equivalent channel characterization by channel transfer function (TF) C l (f) c l (τ) F C l (f) = c l (τ)e j2πfτ dτ ACF of channel TF S c (τ) F φ C (Δf) =E{C l (f)c l(f +Δf)} φ C (Δf)... spaced-frequency correlation function TF C l (f) is wide-sense stationary (WSS in f) ifcir c l (τ) fulfills uncorrelated scattering (US in τ) Mobile Radio Systems Small-Scale Channel Modeling p. 27/43 Frequency-selective fading (cont d) Channel IR vs. channel frequency response IDFT of transfer function after correction for linear phase shift excess delay time [ns] amplitude H(f) [db] phase arg(h(f)) [rad] frequency [MHz] phase transfer function (corrected for linear phase shift) amplitude transfer function frequency [MHz] Mobile Radio Systems Small-Scale Channel Modeling p. 28/43
15 Frequency-selective fading (cont d) Multipath intensity profile: average power output of channel as a function of delay Mobile Radio Systems Small-Scale Channel Modeling p. 29/43 Frequency-selective fading (cont d) Characterization by parameters RMS delay spread τ rms = τ 2 τ 2 second centralized moment of normalized multipath intensity profile mean and mean squared delay spread τsc (τ)dτ τ τ = 2 τ 2 S c (τ)dτ = Sc (τ)dτ Sc (τ)dτ Coherence bandwidth B c 1 τ rms Mobile Radio Systems Small-Scale Channel Modeling p. 30/43
16 Frequency-selective fading (cont d) Characterization of multipath intensity profile (simplified; suitable for indoor channels) Exponentially decaying part Line-of-sight (LOS) component Defined by channel parameters P h (t) [db] Channel parameters: total power P 0 K-factor (rel. strength of LOS) RMS delay spread (duration) t Mobile Radio Systems Small-Scale Channel Modeling p. 31/43 The WSSUS channel joint modeling of time dispersion (= frequency selectivity) and time variability (= Doppler spread) Define: ACF of time-variant CIR c l (τ; t) E{c l (τ 1; t)c l (τ 2 ; t +Δt)} = φ c (τ 1 ;Δt)δ(τ 1 τ 2 ) assumes: time-variations are wide-sense stationary (WSS) attenuation and phase shifts are independent at τ 1 and τ 2 : uncorrelated scattering (US) for Δt =0: φ c (τ;δt) =S c (τ) multpath intensity profile φ c (τ;δt)... lagged-time correlation function Mobile Radio Systems Small-Scale Channel Modeling p. 32/43
17 The WSSUS channel (cont d) An equivalent representation of the t-var. CIR c l (τ; t): Time-variant channel transfer function (TF) C l (f; t) c l (τ; t) F τ Cl (f; t) = c l (τ; t)e j2πfτ dτ from US property follows WSS in f-domain equivalent characterization (ACF) φ C (Δf;Δt) =E{C l (f; t)c l(f +Δf; t +Δt)} spaced-frequency spaced-time correlation function (WSSWSS!) Mobile Radio Systems Small-Scale Channel Modeling p. 33/43 The WSSUS channel (cont d) time- and frequency-selective transfer function 30 received power [db] frequency 0 time Mobile Radio Systems Small-Scale Channel Modeling p. 34/43
18 The WSSUS channel (cont d) Equivalent representations: time-variant system functions Bello functions [Bello63] c l (τ; t) and C l (f; t) and two more Fourier transformed functions w.r.t. t ν and f τ Equivalent (2-nd order) characterizations: correlation functions of Bello functions φ c (τ;δt) and φ C (Δf;Δt) and two more Fourier transformed functions w.r.t. Δt ν and Δf τ overview shown on next slide Mobile Radio Systems Small-Scale Channel Modeling p. 35/43 The WSSUS channel (cont d) φ c (τ) multipath intensity profile delay power spectrum τ max max. delay spread FT (τ Δf) φ C (Δf) spaced-frequency correlation function B c coherence bandwidth Δt = 0 Δt = 0 wide-band characterization (time-invariant channel) characterization of time variations (flat fading) φ c (τ;δt) delay cross-power spectral density FT (τ Δf) φ C (Δf;Δt) spaced-frequency, spacedtime correlation function Δf = 0 φ C (Δt) spaced-time correlation function T c coherence time ACF (WSSUS) ACF (WSSWSS) c l (τ;t) equivalent lowpass time-variant impulse response FT (τ f) C l (f;t) time-variant transfer function FT (Δt ν) FT (Δt ν) FT (Δt ν) S(τ;ν) Scattering function FT (τ Δf) S C (Δf;ν) Doppler cross-power spectral density Δf = 0 S C (ν) Doppler power spectrum f m max. Doppler freq.; Doppler spread Mobile Radio Systems Small-Scale Channel Modeling p. 36/43
19 The WSSUS channel (cont d) Doppler-delay scattering function [Paulraj; fig 2-9] Mobile Radio Systems Small-Scale Channel Modeling p. 37/43 Channel as a space-time random field Homogenous (HO) channel is (locally) stationary in space characterization: E{c l (τ; t; d)c l(τ; t; d +Δd)} = φ d (τ; t;δd) agrees with discrete scattering model: each scatterer has discrete ToA τ i and AoA θ i space-angle transform: assume d lies on x-axis; parameterized by x (and dropping t) c l (τ; x) = c l (τ; θ)e j2π sin(θ) x λ dθ we may define the angle-delay scattering function S c (τ; θ) Mobile Radio Systems Small-Scale Channel Modeling p. 38/43
20 Channel as a space-time random field (cont d) Angle-delay scattering function [Paulraj; fig 2-10] Mobile Radio Systems Small-Scale Channel Modeling p. 39/43 Channel as a space-time random field (cont d) S c (θ)... angle power spectrum: average power vs. angle of arrival Characterization by parameters: RMS angle spread: θ rms = θ 2 θ 2 second centralized moment of normalized angle power spectrum mean and mean squared angle spread θsc (θ)dθ θ θ = 2 θ 2 S c (θ)dθ = Sc (θ)dθ Sc (θ)dθ Coherence distance D c 1 θ rms Mobile Radio Systems Small-Scale Channel Modeling p. 40/43
21 3-3 Classification of Small-Scale Fading Compares system and channel parameters Classification w.r.t. symbol period w.r.t. bandwidth T s B s 1/T s dispersiveness flat fading T s τ rms B s B c frequency selective T s <τ rms B s >B c time variations slow fading T s T c B s ν rms fast fading T s >T c B s <ν rms Mobile Radio Systems Small-Scale Channel Modeling p. 41/43 Classification example GSM Key air-interface parameters: Carrier frequency MHz, 1.8 GHz Bandwidth khz Frame; slot length ms; 0.6 ms Time dispersiveness τ rms (typical urban and suburban) ns corresponds to B c MHz flat fading Time variability assume v =50m/s at f c =1GHz ν max = 167 Hz corresponds to T c 6ms Time-invariant during slot Mobile Radio Systems Small-Scale Channel Modeling p. 42/43
22 Classification example WLAN Key air-interface parameters: Carrier frequency ; 5 GHz Bandwidth MHz (sampling f: f s =20MHz) OFDM symbol length... 4 μs Time dispersiveness τ rms (indoor) ns corresponds to B c MHz frequency selective Time variability assume v =2m/s at f c =5GHz ν max =33Hz corresponds to T c 30 ms; several 1000 symbols Time-invariant during packet Mobile Radio Systems Small-Scale Channel Modeling p. 43/43
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