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Properties of the Mobile Radio Propagation Channel Jean-Paul M.G. Linnartz Department Head CoSiNe Nat.Lab., Philips Research

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JPL 1 Statistical Description of Multipath Fading The basic Rayleigh / Ricean model gives the PDF of envelope But: how fast does the signal fade? How wide in bandwidth are fades? Typical system engineering questions: What is an appropriate packet duration, to avoid fades? How much ISI will occur? For frequency diversity, how far should one separate carriers? How far should one separate antennas for diversity? What is good a interleaving depth? What bit rates work well? Why can't I connect an ordinary modem to a cellular phone? The models discussed in the following sheets will provide insight in these issues

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JPL 2 The Mobile Radio Propagation Channel A wireless channel exhibits severe fluctuations for small displacements of the antenna or small carrier frequency offsets. Channel Amplitude in dB versus location (= time*velocity) and frequency Time Frequency Amplitude

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JPL 3 Time Dispersion vs Frequency Dispersion Time Domain Channel variationsDelay spread Interpretation Fast FadingInterSymbol Interference Correlation DistanceChannel equalization Frequency Doppler spectrum Frequency selective fading Domain Intercarrier Interf.Coherence bandwidth Interpretation Time Dispersion Frequency Dispersion

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JPL 4 Two distinct mechanisms 1.) Time dispersion Time variations of the channel are caused by motion of the antenna Channel changes every half a wavelength Moving antenna gives Doppler spread Fast fading requires short packet durations, thus high bit rates Time dispersion poses requirements on synchronization and rate of convergence of channel estimation Interleaving may help to avoid burst errors 2.) Frequency dispersion Delayed reflections cause intersymbol interference Channel Equalization may be needed. Frequency selective fading Multipath delay spreads require long symbol times Frequency diversity or spread spectrum may help

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JPL 5 Time dispersion of narrowband signal (single frequency) Transmit: cos(2 f c t) Receive: I(t) cos(2 f c t) + Q(t) sin(2 f c t) = R(t) cos(2 f c t + ) I-Q phase trajectory As a function of time, I(t) and Q(t) follow a random trajectory through the complex plane Intuitive conclusion: Deep amplitude fades coincide with large phase rotations Animate

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JPL 6 Doppler shift and Doppler spread All reflected waves arrive from a different angle All waves have a different Doppler shift The Doppler shift of a particular wave is Maximum Doppler shift: f D = f c v / c Joint Signal Model Infinite number of waves First find distribution of angle of arrival, then compute distribution of Doppler shifts Uniform distribution of angle of arrival : f ( ) = 1 / 2 Line spectrum goes into continuous spectrum Calculate

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JPL 7 Doppler Spectrum If one transmits a sinusoid, … what are the frequency components in the received signal? Power density spectrum versus received frequency Probability density of Doppler shift versus received frequency The Doppler spectrum has a characteristic U-shape. Note the similarity with sampling a randomly-phased sinusoid No components fall outside interval [f c - f D, f c + f D ] Components of + f D or -f D appear relatively often Fades are not entirely memory-less

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JPL 8 Derivation of Doppler Spectrum

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JPL 9 Vertical Dipole

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JPL 10 How do systems handle Doppler Spreads? Analog Carrier frequency is low enough to avoid problems (random FM) GSM Channel bit rate well above Doppler spread TDMA during each bit / burst transmission the channel is fairly constant. Receiver training/updating during each transmission burst Feedback frequency correction DECT Intended to pedestrian use: only small Doppler spreads are to be anticipated for. IS95 Downlink: Pilot signal for synchronization and channel estimation Uplink: Continuous tracking of each signal

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JPL 11 Autocorrelation of the signal We now know the Doppler spectrum, but... how fast does the channel change? Wiener-Kinchine Theorem: Power density spectrum of a random signal is the Fourier Transform of its autocorrelation Inverse Fourier Transform of Doppler spectrum gives autocorrelation of I(t), or of Q(t)

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JPL 12 Derivation of Autocorrelation of I-Q components

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JPL 13 For uniform angle of arrival

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JPL 14 Relation between I and Q phase

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JPL 15 PDF of the real amplitude R For the amplitude r 1 and r 2 NB: Correlation: R 2 = I 2 + Q 2 Note: get more elegant expressions for complex amplitude H

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JPL 16 Autocorrelation of amplitude R 2 = I 2 + Q 2 The solution is known as the hypergeometric function F(a,b;c;z) or, in good approximation,..

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JPL 17 Autocorrelation of amplitude R 2 = I 2 + Q 2

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JPL 18 Amplitude r(t 0 ) and Derivative r(t 0 ) are uncorrelated J 0 () Correlation is 0 for = 0

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JPL 19 Simulation of multipath channels Jakes' Simulator for narrowband channel generate the two bandpass noise components by adding many sinusoidal signals. Their frequencies are non- uniformly distributed to approximate the typical U-shaped Doppler spectrum. N Frequency components are taken at 2 f i = f m cos -------- 2(2N+1) with i = 1, 2,.., N All amplitudes are taken equal to unity. One component at the maximum Doppler shift is also added, but at amplitude of 1/sqrt(2), i.e., at about 0.707. Jakes suggests to use 8 sinusoidal signals. Approximation (orange) of the U-Doppler spectrum (Black) ?

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JPL 20 How to handle fast multipath fading?

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Frequency Dispersion

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JPL 22 Frequency Dispersion Frequency dispersion is caused by the delay spread of the channel Frequency dispersion has no relation to the velocity of the antenna

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JPL 23 Frequency Dispersion: Delay Profile

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JPL 24 RMS Delay Spread and Maximum delay spread

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JPL 25 Typical Values of Delay Spreads

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JPL 26 Typical values of delay spread Picocells 1.. 2 GHz: T RMS < 50 nsec - 300 nsec Home 50 nsec Shopping mall 100 - 200 nsec Railway station200 - 450 nsec Office block100 - 400 nsec

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JPL 27 Typical Delay Profiles For a bandlimited signal, one may sample the delay profile. This give a tapped delay line model for the channel, with constant delay times

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JPL 28 COST 207 Typical Urban Reception (TU6) COST 207 describes typical channel characteristics for over transmit bandwidths of 10 to 20 MHz around 900MHz. TU-6 models the terrestrial propagation in an urban area. It uses 6 resolvable paths COST 207 profiles were adapted to mobile DVB-T reception in the E.U. Motivate project. Tap numberDelay (us)Power (dB)Fading model 10.0-3Rayleigh 20.20Rayleigh 30.5-2Rayleigh 41.6-6Rayleigh 52.3-8Rayleigh 65.0-10Rayleigh

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JPL 29 COST 207 A sample fixed channel Tap numberDelay ( ( s) Amplitude r Level (dB)Phase ( (rad) 10.0500.36-8.88-2.875 20.479100 30.6210.787-2.092.182 41.9070.587-4.63-0.460 52.7640.482-6.34-2.616 63.1930.451-6.922.863 COST 207 Digital land mobile radio Communications, final report, September 1988. European Project AC 318 Motivate: Deliverable 06: Reference Receiver Conditions for Mobile Reception, January 2000

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JPL 30 How do systems handle delay spreads?

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JPL 31 Correlation of Fading vs. Frequency Separation

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JPL 32 Inphase and Quadrature-Phase Components

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JPL 33 Multipath Channel Transmit signal s(t) Received signal r(t) = a n g n (t) + n(t), Channel model: I w reflected waves have the following properties: D i is the amplitude T i is the delay Signal parameters: a n is the data c is the carrier frequency TO DO make math consistent

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JPL 34 Doppler Multipath Channel Correlation between p-th and q-th derivative TO DO make math consistent

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JPL 35 Doppler Multipath Channel TO DO make math consistent

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JPL 36 Random Complex-Gaussian Amplitude Special case This defines the covariance matrix of subcarrier amplitudes at different frequencies This is used in OFDM for cahnnel estimation TO DO make math consistent

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JPL 37 Coherence Bandwidth

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JPL 38 Frequency and Time Dispersion

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JPL 39 Scatter Function of a Multipath Mobile Channel Gives power as function of Doppler Shift (derived from angle of arrival ) Excess Delay Example of a scatter plot Horizontal axes: x-axis:Excess delay time y-axis:Doppler shift Vertical axis z-axis:received power

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JPL 40 Special cases Zero displacement / motion: = 0 Zero frequency separation: f = 0 Freq. and time selective channels

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Effects of fading on modulated radio signals

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JPL Effects of Multipath (I) Frequency Time Narrowband Frequency Time OFDM Frequency Time Wideband

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JPL Effects of Multipath (II) Frequency Time + - + - - + - + DS-CDMA Frequency Time + - - Frequency Hopping Time Frequency +-+- +-+- +-+- MC-CDMA

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JPL 44 BER BER for Ricean fading calculate

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Time Dispersion Revisited The duration of fades

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JPL 46 Time Dispersion Revisited: Duration of Fades

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JPL 47 Two state model

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JPL 48 Level crossings per second Number of level crossing per sec is proportional to speed r' of crossing R (derivative r' = dr/dt) probability of r being in [R, R + dR]. This Prob = f R (r) dr

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JPL 49 Derivation of Level Crossings per Second Random process r is derivative of the envelope r w.r.t. time Note: here we need the joint PDF; not the conditional PDF f(r r=R) We derive f(r r=R) from f(r r) and f(i 1,i 2,q 1,q 2 )

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JPL 50 Covariance matrix of (I, Q, I, Q)

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JPL 51 Joint PDF of R, R,,

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JPL 52 Level Crossings per Second Calculate

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JPL 53 Average Fade / Nonfade Duration Calculate

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JPL 54 Average nonfade duration Calculate

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JPL 55 How to handle long fades when the user is stationary?

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JPL 56 Optimal Packet length

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JPL 57 Optimal Packet length

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JPL 58 Derivation of Optimal Packet length Calculate

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JPL 59 Average fade duration

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JPL 60 Conclusion The multipath channel is characterized by two effects: Time and Frequency Dispersion Time Dispersion effects are proportional to speed and carrier frequency System designer need to anticipate for channel anomalies

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JPL 61

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JPL 62 Statistical properties of a Rayleigh signal Parameter Probability Distribution I(t 1 ) and Q(t 1 ) i.i.d. Gaussian, Zero mean I(t 1 ) and Q(t 2 ) Correlated jointly Gaussian I(t 2 ) given I(t 1 ) Gaussian Amplitude r Rayleigh r given r Ricean r derivative r Gaussian, Independent of amplitude Phase Uniform Derivative of Phase Gaussian, Dependent on amplitude Power Exponential The nice thing about jointly Gaussian r.v.s is that the covariance matrix fully describes the behavior

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JPL 63 Taylor Expansion of Amplitude Rewrite the Channel Model as follows Tayler expansion of the amplitude g n (t) = g n (0) + g n (1) (t- t ) + g n (2) (t- t ) 2 /2 +... g n (q) : the q-th derivative of amplitude wrt time, at instant t = t. g n (p) is a complex Gaussian random variable To address frequency separation n s (later):

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JPL 64 Doppler Multipath Channel Simplified baseband model for narrowband linear modulation Received signal r(t) = a n g(t) + n(t), with time-varying channel amplitude g(t) Channel model: I w reflected waves, each has the following properties: D i is the amplitude I is the Doppler shift T i is the path delay Note g(t) is not an impulse response; It is the channel amplification and phase Signal parameters: linear modulation a n is the user data c is the carrier frequency n(t) is the noise

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JPL 65 Doppler Multipath Channel p-th Derivative g p (t) of the channel w.r.t. to time t All derivatives are gaussian rvs if I w and D i i.i.d. The covariance matrix fully describes the statistical properties

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JPL 66 Doppler Multipath Channel Correlation between p-th and q-th derivative

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JPL 67 Doppler Multipath Channel

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JPL 68 Random Complex-Gaussian Amplitude It can be shown that for p + q is even and 0 for p + q is odd. This defines the covariance matrix of subcarrier amplitudes and derivatives, OFDM: allows system modeling and simulation between the input of the transmit I-FFT and output of the receive FFT.

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