4.7 Statistical Models for Multipath Fading Channels Ossana[Oss64] of multipath channel
Download ReportTranscript 4.7 Statistical Models for Multipath Fading Channels Ossana[Oss64] of multipath channel
4.7 Statistical Models for Multipath Fading Channels several models have been suggested to explain observed statistical nature of multipath channel Ossana[Oss64] presented 1st model • based on interference of waves incident & reflected from flat sides of randomly located buildings • assumes existence of LOS path • predicts flat fading spectra that agrees with measurements in suburban areas • limited to restricted range of reflection angles • inflexible & inappropriate for urban areas – there is ususaly not not an LOS path 5/25/2016 1 4.7.1 Clarke’s Model for Flat Fading • statistical characteristics of electromagnetic fields of received signal are deduced from scattering • model assumes fixed transmitter with vertically polarized antenna • field incident on mobile antenna is assumed to consist of N azimuthal plane waves with - arbitrary carrier phases - arbitrary angles of arrival - equal average amplitude • with no LOS path scattered components arriving at a receiver will experience similar attenuation over small scale distances 5/25/2016 2 e.g. mobile receiver moving with velocity = v along x-axis • signal’s angle of arrival in x-y plane measured with respect to mobile’s direction • every wave incident on the mobile undergoes Doppler shift & arrives at receiver at the same time flat fading assumption: no excess delay due to MPCs z y v x Doppler Shift for the nth wave arriving at angle n to the x-axis is given by v cos n (Hz) ( 4.57) fn = = wavelength of incident wave 5/25/2016 3 Electromagnetic fields of vertically polarized plane waves arriving at the mobile given by N z Ez = E0 Cn cos( 2f c n ) (4.58) n 1 y x E0 N v Hx = Cn sin n cos( 2f c n ) (4.59) Hy = E0 n 1 N C n 1 n cos n cos( 2f c n ) (4.60) • E0 = real amplitude of local average E-field (assumed constant) • Cn = real random variable representing the amplitude of each wave • = intrinsic impedence of free space (377) • fc = carrier frequency • n = random phase of nth arriving component that includes Doppler Shift n = 2fn + n 5/25/2016 (4.61) 4 Amplitude of electric & magnetic fields are normalized such that ensemble average of Cn’s is N 2 C n 1 n 1 (4.62) • since fn << fc field components Ez, Hx, Hy can be approximated as Gaussian random variables if N is sufficiently large • phase angles are assumed to have uniform PDF on interval (0,2] 5/25/2016 5 Received E-field, Ez(t) can be expressed in terms of in-phase and quadrature components [Rice48] Ez = Tc(t) cos(2fct) – Ts(t)sin(2fct) (4.63) N Tc(t) = E0 Cn cos( 2f nt n ) n 1 (4.64) N Ts(t) = E0 Cn sin( 2f nt n ) (4.65) n 1 • Tc(t) & Ts(t) are Gaussian RPs denoted by Tc & Ts at time t • Tc & Ts are uncorrelated 0-mean Gaussian RVs with equal variance given by: Tc2 Ts2 E z 2 E02 2 (4.66) - overbar denotes ensemble average 5/25/2016 6 r(t) is envelope of Ez(t) given by: r(t) = |Ez(t)| = Tc2 (t ) Ts2 (t ) (4.67) • since Tc & Ts are Gaussian random variables Jacobean transform [Pap91] shows that r has has a Rayleigh Distribution - r = random received signal envelope p(r) = r r2 exp 2 2 2 0 ( 0 r ) (4.68) (r 0 ) where 2 = E02/2 5/25/2016 7 4.7.1.1 Spectral Shape Due to Doppler Spread in Clarke’s Model Spectrum Analysis for Clarkes model [Gans72] Let: • p()d = fractional part of total incoming power in d of angle • A = average receive power for isotropic antenna • G() = azimuthal gain pattern of mobile antenna as a function of As N → then p()d goes from discrete distribution to continuous distribution Total received power can be expressed as: 2 Pr = AG( ) p ( )d (4.69) 0 AG()p()d = differential variation of received power with angle 5/25/2016 8 PSD of CW signal with frequency fc that is scattered • instantaneous frequency of received signal component arriving at angle is obtained from 4-57 and given by f() = f = f mcos + fc • f m= v (4.70) = maximum Doppler shift • f() is an even function of (e.g. f() = f(-)) Let S(f) = PSD of received signal differential variation of received power with frequency is S(f)|df | 5/25/2016 (4.71) 9 equate S(f)|df | with AG()p()d, where •AG()p()d = differential variation of received power with angle •S(f)|df | = differential variation of received power with frequency S(f)|df | = A[p()G() + p(-)G(-)] |d| (4.72) differentiate 4.70 with respect to yields |df |=|d ||-sin()| fm solve 4.70 for 4.74 implies sin = 5/25/2016 = cos-1 f fc fm f fc 1 fm (4.73) (4.74) 2 (4.75) 10 PSD, S(f), found by substitution of (4.73), (4.75) into (4.72) S(f) = A p( )G ( ) p( )G ( ) f fc f m 1 fm (4.76) 2 where S(f) = 0 for | f - fc | > fm (4.77) • S(f) is centered on fc and = 0 outside of limits of fc fm • each arriving wave has carrier slightly offset from fc due to angle of arrival 5/25/2016 11 Assume vertical /4 antenna (G() = 1.5) and incoming power p() = 1/2, uniformly distributed over [0,2] output spectrum from 4.76: 1.5 SE z ( f ) f m f fc 1 fm (4.78) 2 output at fc fm = • Doppler components arriving at exactly 0° & 180° have PSD • since is continuously distributed probability of components arriving at 0° & 180° = 0 S (f) Ez 5/25/2016 fc- fm fc fc+ fm 12 Baseband signal recovered after envelope detection of Doppler shifted signal • resulting baseband spectrum has maximum frequency of 2fm • [Jak 74] showed that electric field produces baseband PSD of SbbE z = 2 f 1 K 1 8f m 2 fm (4.79) • (4.79) is result of temporal correlation of received signal when passed through nonlinear envelope detector • K(•) = complete elliptical integral of 1st kind Spectral shape of Doppler spread • determines time domain fading waveform • dictates temporal correlation & fade slope behaviors • Rayleigh fading simulators must use fading spectrum (e.g. 4.78) to produce realistic fading waveforms 5/25/2016 13 Baseband PSD of CW received signal after envelope detection 10log[8fmSbbEz(f)] 0dB -1dB -2dB -3dB -4dB -5dB -6dB -7dB -8dB 10-3 5/25/2016 10-2 10-1 1 2 10 f/fm 14 Mobile Terrestrial Channel • Clarke Model for fast fading in assumes all rays are arriving from horizontal direction • more sophisticated models account for possible vertical components conclusions are similar • empirical measurements tend to support Clarke model with a Doppler bandwidth related to transmission frequency & velocity • measured spectra usually show peaks near Doppler Frequencies 5/25/2016 15 Mobile Satellite Communications • Empirical measurements indicate Clarkes Model is not always valid • For aeronautical terminals & satellites Gaussian spectrum is a better model of spectrum fading process - maximum fD is not proportional to aircraft speed, but ranges between 20Hz & 100Hz - factors include nearby reflections from slowly vibrating fuselage and wings • For maritime mobile terminal & satellite Gaussian fading spectrum with Doppler bandwidth < 1Hz more accurately reflect empirical results - due to slower motion of ship - distant reflections from sea surface tend to be directional (not omni directional) - effects of ocean waves as reflective surfaces 5/25/2016 16 4.7.2 Simulation of Clarke’s & Gans Fading Model • design process includes simulation of multipath fading channels • simulate in-phase & quadrature modulation paths to represent Ez as given in (4.63) - in-phase & quadrature fading branches produced by 2 independent Guassian low-pass noise sources - each noise source formed by summing 2 independent, orthogonal Guassian random variables e.g. g = a+jb a & b are Gaussian random variables g is complex Gaussian - spectral filter in (4.78) used to shape random signals in frequency domain - allows production of accurate time domain waveforms of Doppler fading using IFFT at last stage of simulator 5/25/2016 17 Simulator using quadrature amplitude modulation independent 4.22a RF Doppler Filter cos2fct Baseband Gaussian Noise Source balanced mixers Baseband Gaussian Noise Source Doppler Filter s0(t) sin2fct 4.22b Baseband Doppler Filter independent cos2fct 5/25/2016 Baseband Gaussian Noise Source Baseband Gaussian Noise Source Baseband Doppler Filter Baseband Doppler Filter balanced mixers s (t) 0 sin2fct 18 [Smith 75] demonstrated simple computer program that implements baseband Doppler filter (figure4.22b) (i) complex Gaussian random number generator (noise source) produces baseband line spectrum with complex weights in positive frequency band - fm = maximum frequency component of line spectrum (ii) from properties of real signals negative frequency components obtained as complex conjugate of Gaussian values for positive frequencies (iii) IFFT of each complex Gaussian signal should be purely real Gaussian RP in time domain - used in each of the quadrature arms in figure 5.24 (iv) random valued line spectrum is then multiplied with discrete frequency representation of S E ( f ) (4.63) z - noise source and S E ( f ) have same number of points z (v) truncate SEz(fm) at passband edge ( fc fm = ) • compute function’s slope at sampling frequency just before passband edge & increase slope to passband edge 5/25/2016 19 Simulations usually implemented in frequency domain using complex Gaussian line spectra - leverages easy implementation of 4.78 - implies low pass Gaussian noise components are a series of frequency components (line spectrum from –fm to fm ) - equally spaced and each with complex Gaussian weight 5/25/2016 20 figure 4.24 Frequency Domain Implementation of Rayleigh fading simulator at baseband * g*N/2 g (N/2)-1 -fm 0 g(N/2)-1 gN/2 fm S Ez ( f ) IFFT (•)2 -fm fm * g*N/2 g (N/2)-1 -fm 0 g(N/2)-1 gN/2 fm r(t) S Ez ( f ) IFFT (•)2 -fm fm independent complex Guassian samples form line spectra 5/25/2016 21 Steps to implement simulator shown in figure 4.24 1. Specify N = number of frequency points used to represent S E ( f ) z 2. Compute frequency spacing between adjacent spectral lines f = 2fm/(N-1) defines T = time duration of fading waveform T = 1/ f 3. Generate complex Gaussian random variable for each N/2 positive frequency components of noise source 4. Construct negative frequency components of noise source by conjugate of positive frequency values 5. Multiply in-phase & quadrature noise sources by fading spectrum, S E ( f ) yields real frequency domain signal z 5/25/2016 22 6a. Perform IFFT on resulting frequency domain signals in (5) yields two N-length time series 6b Add squares of each signal point in time to create N-point time series under radical 5.67 7. Take square root of sum in 6 obtain N- point time series of simulated Rayleigh fading signal with Doppler Spread and Time Correlation 5/25/2016 23 To Produce Frequency Selective fading effects use several Rayleigh fading simulators and variable gains and time delays s(t) (signal under test) 1 Rayleigh Fading Simulator a0 Rayleigh Fading Simulator a1 Rayleigh Fading Simulator a2 gains 2 delays r(t) Fig 4.25: Determine performance range for wide range of channel conditions depending on gain and time-delay settings 5/25/2016 24 To create Ricen Fading channel • make single frequency component dominant in amplitude within fading spectrum at f = 0 To create multipath fading simulator with many resolvable MPCs • alter probability distribution of individual multipath components in simulator IFFT must be implemented to produce real time-domain signal given by Tc(t) and Ts(t) (5.64 and 5.65) To determine impact of flat fading on s(t) compute s(t) r(t) • s(t) = applied signal • r(t) = output of fading simulator To determine impact of several MPCs use convolution (figure 4.25) 5/25/2016 25 4.7.3 Level Crossing & Fading Statistics 2 important statistics for designing error control codes and diversity schemes for Rayleigh fading signal in mobile channel (1) Level Crossing Rate (LCR) = average number of level crossings (2) Average Fade Duration (AFD) = mean duration of fades Makes it possible to relate received signals time rate of change to • received signal level • velocity of mobile Rice computed joint statistics for fading model similar to Clarke’s – that provided simple expressions for computing LCR & AFD 5/25/2016 26 LCR = expected rate at which Rayleigh Fading envelope crosses specified level in a positive-going direction • Rayleigh Fading normalized to local rms signal level • NR = level crossings per second at specified threshold level of R NR = 2 r p ( R , r ) d r 2 f exp m (4.80) 0 fm= maximum Doppler Frequency R = specified threshold for level crossing r = time derivative (slope) of received signal r(t) p(R, r ) = joint density function of r and r at r = R = R/Rrms normalized threshold, the value of R normalized to local rms amplitude of fading amplitude 5/25/2016 27 e.g. 4.7: Rayleigh fading signal with R = Rrms = 1 fm = 20 Hz (maximum doppler frequency) compute LCR: NR= 2 f m e 2 2 20e 1 = 18.44 crossings/second compute maximum velocity of mobile for fc = 900MHz: fm = v/ vmax = fm = 20Hz (0.333m) = 6.66 m/s (24 km/hr) 5/25/2016 28 (2) Average Fade Duration (AFD) average time period for which r < R 1 Pr[ r R] AFD For a Rayleigh fading signal: = NR 1 i Probability that r R is given by: Pr[r R] = T i i = duration of the fade T = observation interval of fading signal r = received signal R = specified threshold level (4.81) (4.82) From Rayleigh distribution probability that r R is given by: 2 p ( r ) dr 1 exp R Pr[r R] = (4.83) 0 p(r) = pdf of Rayleigh distribution 5/25/2016 29 AFD as function of & fm is derived from 4.80 & 4.83 as 1 exp 2 = 2 2 f m e = exp 2 1 2 f m (4.84) AFD of a signal helps determine likely number of signaling bits lost during a fade • primarily depends on speed of the mobile • decreases as fm becomes large if fade margin is built into a mobile system – it is appropriate to evaluate receiver performance by determination of (i) NR, the rate at which input falls below R (ii) , how long it remains below R, on average useful for relating SNR to resulting instantaneous BER during a fade 5/25/2016 30 e.g. for fm = 200Hz find AFD for given normalized threshold levels 2 = e 1 2 f m R << Rrms R < Rrms R = Rrms 5/25/2016 ρ = 0.01 = ρ = 0.1 = ρ=1 = 0.012 e 1 2 0.01(200) = 19.9us 0.12 e 1 2 0.1(200) = 200us e1 1 2 (200) = 3430us 31 e.g. for ρ = 0.707 and fm = 20Hz 2 0.707 2 e 1 e 1 18.3ms 2 f m 2 0.707 20 1. AFD: = 2. For binary digitial modulation with Rb = 50bps then Tb = 20ms Tb > signal undergoes fast Rayleigh fading 3. Assume bit errors occurs when portion of bit encounters fade for which ρ < 0.1, what is average number of bit errors/second = 0.12 e 1 200us 2 0.1 20 Tb > only one bit will be lost on average during a fade NR= 2 20 0.1e 0.12 4.96 crossings/second • if 1 bit error occurs during a fade 5 bits errors occur per second • BER = bit errors per second/Rb = 5/50 = 10-1 5/25/2016 32 4.7.4 2-Ray Rayleigh Fading Model • Common multipath model & specific implementation of fig 4.25 • Clarkes model and Rayleigh fading statistics are for flat fading - don’t consider multipath time delay • Impulse response model given as: hb(t) = α1exp(j1) δ(t) + α2exp(j2) δ(t-) 4.85 • αi = independent, Rayleigh distributed, generated from 2 independent waveforms (e.g. IFFT in 5.7.2) • i= independent and uniformly distributed over [0,2] 5/25/2016 33 α1exp(j1) input output α2exp(j2) Fig 5.26: 2-Ray Rayleigh Fading Model • set α2 = 0 special case of Rayleigh flat fading channel hb(t) = α1exp(j1) δ(t) 4.86 • vary to create wide range of FSF effects 5/25/2016 34 Other Small Scale Fading Models 4.7.5 Saleh & Valenzuela Indoor Statistical Model: wideband model where resolvable MPCs arrive in clusters 4.7.6 SIRCIM & SMRICM indoor and outdoor statistical models • based on empirical measurements in 5 factory buildings at 1.3GHz • statistical model generates measured channel based on discrete impulse response of channel model • developed computer programs that generate small scale channel impulse response (i) Simulation of Mobile Radio Channel Impulse Response Model (SMRICM) (ii) Simulation of Indoor Radio Channel Impulse Response Model (SIRCIM) 5/25/2016 35 4.8.1.2 Fading Rate Variance Relationships ~ Complex Received Voltage, V ( r ) baseband representation of summation of multipath waves impinging on receiver ~ 2 Received Power, P(r) = V ( r ) differs by constant related to receiver input impedance result is independent of receiver ~ Receive Envelop R(r) = V (r ) 5/25/2016 36