- 1. 1 EE 463 - Wireless CommunicationsCommunications Dr. Ahmed Iyanda Sulyman Associate Professor Electrical Engineering DepartmentElectrical Engineering Department King Saud University Chapter 4: Mobile radio propagation: Small-scale fading 1. Introduction: small-scale fading & multipath 2. Impulse response model of a multipath channel 3. Multipath channel parameters 4. Types of small-scale fading 5. Statistical models 2 6. Fading channel Simulations
- 2. 2 Chapter 4: Mobile radio propagation: Small-scale fading Direct path or line of sight (LOS) wave Reflected wave Diffracted wave 3 Mobile userBase station Scattered wave Small-scale fading (or simply fading) describes the rapid fluctuations of a radio signal over a short period of time (few seconds) or a short travel distance (few 1 – Introductions ( ) ( wavelength). Path 1 Path 2 Path 3 4 The radio waves from the transmitter arrive at the mobile from different directions, each with different amplitude and propagation delays (multipath comp.). Path 4
- 3. 3 1 – Introductions • Multipath components may combine (vectorial addition) either constructively or destructively at the mobiles,y y , and may thus cause the signal received by the mobile to distort or fade (small-scale fading). Factors causing small-scale fading: Multipath propagation: presence of reflectors, scatterers, etc., in the environment. Speed of the mobile: movement of transmitter or receiver or both cause frequency shift in the transmitted signal (called Doppler shift) 5 transmitted signal (called Doppler shift). Speed of surrounding object: movements of surrounding objects induce time-varying Doppler shift. Transmission bandwidth of the signal: If the transmitted radio signal has bandwidth greater than the bandwidth of the multipath channel. Doppler shift (revision): Consider a mobile moving at a constant velocity v 1 – Introductions Consider a mobile moving at a constant velocity, v, along a path segment of length d between points A and B, while it receives signals from a remote source S. The phase shift in the received signal due to the difference in path length as the mobile moves from points A to B is given by: )( 22 dL 6 )cos( 2 )cos( tv
- 4. 4 Source, S Illustration: Doppler Shift L )cos( 2 )cos( 22 tv dL • Phase shift between A and B: 7 d v )cos( 2 1 v t fDoppler A B • Frequency shift between A and B (Doppler frequency): Doppler shift (revision): The apparent change in frequency (Doppler shift or 1 – Introductions The apparent change in frequency (Doppler shift or Doppler frequency) is. Positive Doppler shift (apparent received frequency is increased): receiver moves towards transmitter. Negative Doppler shift (apparent received frequency d d) f )cos( 2 1 v t f D 8 is decreased): receiver moves away from transmitter. If source transmits fc, the received frequency is fc ± fD
- 5. 5 Example 4.1: 1 – Introductions Consider a transmitter which radiates a sinusoidal carrier frequency of 1850MHz. For a vehicle moving at 28m/s, compute the received carrier frequency if the mobile is moving: (a) directly toward the transmitter. (b) directly away from the transmitter. (c) in a direction perpendicular to the direction of 9 (c) in a direction perpendicular to the direction of arrival of the transmitted signal. Modeling the impulse response of wireless channels allows numerical performance evaluation of different 2 – Impulse response model of a multipath channel p mobile communication systems. For a mobile at a fixed position, d, the received signal can be expressed as. d v Spatial position 10 p For a causal system ( ), dtdhxtdhtxtdy ),()(),(*)(),( 0,0)( tth t dtdhxtdy ),()(),(
- 6. 6 Assume the mobile moves at a constant velocity v, then d = vt, thus. 2 – Impulse response model of a multipath channel or where x(t) = transmitted bandpass signal (modulated signal at d v Spatial position t dtvthxtvty ,)(),( ),(*)(,)()( thtxdthxty t 11 where x(t) = transmitted bandpass signal (modulated signal at carrier frequency fc) y(t)=received signal, h (t -) = impulse response of the multi-path radio channel, t= time variable, =channel multipath delay for fixed t. The impulse response of a multipath channel can be expressed as. 2 – Impulse response model of a multipath channel 1N p ak(t, ) is the amplitude of the kth multipath component at time t. k(t) is the excess delay of the kth multipath component at time t. k(t,) is the phase shift of the kth multipath component, which is a function of the delay and time t 1 0 ))(()),(exp(),(),( N k kkk ttjtath 12 is a function of the delay and time t. N is the number of paths. If the channel is assumed to be time invariant, or stationary over a small-scale time or distance interval (quasi-static fading), then 1 0 )()exp(),( N k kkk jath
- 7. 7 ),( th t4 Illustration: time-varying impulse response t2 t3 4 10 2 3 10 2 3 1 0 2222 ))(()),(exp(),(),(E.g. N k kkk ttjtath 13 t1 1 0 0 2 0k Cellular operators develop channel impulse response for different environments to allow prior planning and design. ~ typical entry-level duties given wireless Engineers 2 – Impulse response Measurements ~ typical entry level duties given wireless Engineers Small-scale Multipath channel Measurements techniques: Small scale channel state for an environment is typically d v Spatial position 14 Small-scale channel state for an environment is typically recorded in the form of average power delay profiles, defined as P() Ek[|h(tk, )|2 ], where Ek[] denotes ensemble average over samples taken at different times tk. Different channel sounding techniques for estimating the power delay profiles of wireless systems are:
- 8. 8 Direct RF pulse system: transmits a repetitive pulse of width Tb , and uses a receiver 2 – Impulse response Measurements cf Tx RF b , with a wide bandpass filter to collect all multipath signal received from each of these pulses and take the ensemble average. Advantage: low complexity (easily implemented). Disadvantage: it is subject Pulse Generator Tb Trep Pulse width = Tb, Pulse period = Trep BPF Envelope Detector Ampl. Oscillos- cope RF 15 Disadvantage: it is subject to interference and noise in the environment during the test which may not represent the actual channel being measured. Spread Spectrum based channel sounding: rather than transmitting ordinary pulses, the transmitted l i thi h d d 2 – Impulse response Measurements cf Tx RF pulses in this approach are encoded using binary PN (pseudo-noise) sequence, which spreads the pulse over a wide band causing noise & interference rejections. Advantage: moderate complexity, and noise/interference rejection. Processing gain of the SS system allows much lower power than the PN sequence Generator Sequence length RF PN sequence Generator BPF 16 allows much lower power than the direct RF pulse system. Widely used for indoor and outdoor channel sounding (3G and 4G). Disadvantage: instantaneous measurements (real time) are not made, but over PN sequence length BPF (wide) Ampl. Oscillos- cope Envelope Detector BPF (narrow)
- 9. 9 Frequency domain channel sounding: this method measures channel impulse response in f d i d 2 – Impulse response Measurements frequency-domain and use Inverse Discrete Fourier Transform (IDFT) to convert from frequency to time domain measurements. The S parameter measured in frequency domain is proportional to H (j)= FT{h(t)}. Advantage: provides amplitude & phase information of the time- RF Inverse DFT processor S parameter Test set Vector Network Analyzer with Swept Freq. Oscillator RF )( jX )()(21 jHjS )( )( )( jX jY jH 1 )( jY Port1 Port2 17 phase information of the time- domain channel (complex). Disadvantage: hard wired sync. needed between transmitter & receiver, making it useful only for very close measurement. Used in experiments on indoor channel measurements. )}({ 1 )( jHFTth Quantitative multipath channel parameters have been developed in order to compare different multipath channels and for receiver design purposes. 3 – Parameters of mobile multipath channel g p p Time dispersion parameters: Mean Excess delay: 1 0 2 1 0 2 N k k N k kk 1 22 N kk Pt (t) t Mobile channel max TT Channel span (max. excess delay) t Prec (t) 18 RMS delay spread: where Maximum excess delay max (x dB): Is the time delay during which multipath energy falls to x dB below the maximum. These parameters are obtained directly from power delay profile. 22 1 0 2 02 N k k k
- 10. 10 Example 4.3 (Ex5.4 in Ref1) 3 – Parameters of mobile multipath channel Compute the RMS delay spread for the power delay profile of a multipath channel shown below. What is the maximum symbol rate that can be transmitted through the channel without needing an Equalizer?. )(recP 0 dB 0 dB 19 Ans: = 0.5sec, and Rs = 200k symb./sec. For BPSK, Rb=Rs. 0 1s Coherence bandwidth, Bc. 3 – Parameters of mobile multipath channel Is the range of frequencies over which the channel can be considered “flat” (i.e., the channel’s frequency response stays correlated). Bc is inversely proportional to the RMS delay spread, but its exact value depends on how it is defined: → If the coherence bandwidth is defined as the bandwidth over which the frequency correlation function is above 0.9, then 1 B 20 0.9, then → If the coherence bandwidth is defined as the bandwidth over which the frequency correlation function is above 0.5, then 50 cB 5 1 cB
- 11. 11 Example 4.4 (Ex 5.5 in Ref1) Calculate the mean excess delay and the rms delay spread for the 3 – Parameters of mobile multipath channel Calculate the mean excess delay, and the rms delay spread for the multipath power delay profile shown below. Estimate the 50% coherence bandwidth of the channel. Would this channel be suitable for AMPS or GSM cellular service without the use of an Equalizer ?. Ans: =4.38sec, = 1.37sec, and )(recP 0 dB 21 Bc (50%)= 146KHz. Bc>30kHz used in AMPS (no Equalizer required) But Bc<200kHz used in GSM (Equalizer required for GSM). -20 dB 0 1s -10 dB 2s 5s Doppler spread (BD) and coherence time (TC) ,. D l d d h b d idth B d ib th 3 – Parameters of mobile multipath channel Delay spread, στ, and coherence bandwidth, BC, describe the time dispersion of the channel in a short time window. However, they do not give information about the time-varying nature of the channel. Doppler spread BD and coherence time TC describe the effect of time-varying nature of the channel on the received signal. Doppler spread BD is a measure of the spectral broadening caused by the time rate of change of the mobile radio 22 y g channel: If the baseband signal bandwidth is much greater than BD, then the effects of the Doppler spread are negligible. v fB DD max
- 12. 12 Coherence time is a statistical measure of time duration over which the channel impulse response is essentially invariant: 3 – Parameters of mobile multipath channel p p y If is defined as the time over which the time correlation function is above 0.5, then Example 4.5 A measurement team traveling at 50 m/s uses a 900MHz max 11 DD c fB T max 16 9 D c f T 23 g carrier to estimate the small-scale propagation parameter for an urban environment: (a) What is the Doppler spread BD for the mobile channel, (b) What is the coherence time Tc for the mobile channel. Ans: BD =(3x50)Hz, Tc = 3/(50x16) sec. Depending on the relation between the signal parameters (such as bandwidth, symbol period, etc) and the propagation 4 – Types of small-scale fading ( , y p , ) p p g channel parameters (such as RMS delay spread and Doppler spread), transmitted signals will undergo different types of small-scale fading. Basically there are two fading mechanisms, one independent of the other: Small-scale fading based on multipath delay spread 24 Small-scale fading based on Doppler spread
- 13. 13 Small-scale fading based on multipath delay spread: 4 – Types of small-scale fading Time dispersion of the multipath causes the transmitted signal to undergo either flat or frequency selective fading; Flat fading: Signal BW (Bs) < Channel BW (Bc) Symbol period (Ts) > Delay spread (στ) [note: if one is satisfied, the other will be satisfied as well] 25 Frequency selective fading: Signal BW > Channel BW Symbol period < Delay spread Illustration of flat fading: )(tx ),( th )(ty 0 sT )(tx ),( th 0 max t 0 maxsT )(*),()( txthty t sTmax X(f) H(f) Y(f)= H(f).X(f) 26 cf f cf f cf f X(f) H(f) Y(f) H(f).X(f) sc BB
- 14. 14 Illustration of frequency selective fading: )(tx ),( th )(ty 0 sT )(tx ),( th 0 max t 0 maxsT )(ty t sTmax X(f) H(f) 27 cf f cf f cf f X(f) H(f) Y(f) sc BB Small-scale fading based on Doppler spread: F di i d t D l d B ( l ll d 4 – Types of small-scale fading Frequency dispersion due to Doppler spread BD, (also called time selective fading since ), leads to signal distortion (either slow or fast fading): Fast fading: Symbol period (Ts) > Coherence time (Tc) Large Doppler spread Channel variations faster than base-band signal variations Sl f di D c B T 1 28 Slow fading: Symbol period < Coherence time Small Doppler spread Channel variations slower than base-band signal variations
- 15. 15 Rayleigh fading model: Baseband received signal can be written as y(t)= yI(t)+ jyQ(t). 5 – Statistical Models Q The I-Q (In-Phase and Quad.) components of the received signal follow zero-mean Gaussian distribution. Thus the envelope of the received signal can be modeled as a zero-mean Rayleigh random variable, when LOS is absent. Prob. density function (PDF): 0.4 0.5 0.6 0.7 )(rp 10|)(|,) 2 exp()( 2 2 2 tyr rr rp 29 The cum. distribution function (cdf), is the probability that received signal envelope does not exceed a value R: 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.1 0.2 0.3 r 2 2 0 2 exp1)(Pr R drrpRr R Rayleigh fading model: Baseband received signal power is given b ( ) | ( )|2 5 – Statistical Models by Prec(t)=|y(t)|2. Thus the received signal power for a Rayleigh fading model, is exponentially distributed with mean E[Prec(t)]=2σ2. Prob. density function (PDF) of received power: 0|)(|,) 2 exp( 2 1 )( 222 tyr x xpr t rec P P 30 Note that E[Prec(t)] is the received signal power based on path loss and shadowing alone, while the pdf above models the random variations around E[Prec(t)], based on small-scale fading (see illustrations). )log(d
- 16. 16 Example 4.6 Gi th t th i d i l lit d i l 5 – Statistical Models Given that the received signal amplitude r, over a wireless channel, is a Rayleigh distributed random variable. Compute the mean value, and the variance of r. Calculate also the median value, what does this signify?. Ans: 0 2 )(][ drrrprErmean ]2[])[()var( 222 meanmeanmean rrrrErrEr 2 31 Median value represents the value of r for which the cdf is 0.5 Solve (for Rayleigh ch.) 22 0 2 4292.0) 2 2( 2 )( drrpr ,)(5.0 0 medianr drrp 177.1 medianr Example 4.6b C id h l ith R l i h f di l d 5 – Statistical Models Consider a channel with Rayleigh fading envelope and average received power E[Prec(t)]=20dBm. Find the probability that the received power is below 10dBm. Ans: note that E[Prec(t)]=20dBm=100mW. We want to find the prob. that r 2 < 10dBm =10mW. ,095.0) 100 exp( 100 1 ]10Pr[ 10 2 dx x r 32 ,) 100 p( 100 ][ 0
- 17. 17 Ricean fading model: Baseband received signal can be written as y(t)= yI(t)+ jyQ(t). 5 – Statistical Models Q The I-Q (In-Phase and Quad.) components of the received signal follow Gaussian distribution with E[y(t)]=A , when LOS is present. Thus the envelope of the received signal can be modeled as a Ricean random variable with E[y(t)]=A, when LOS is present. Prob. density function (PDF): 0,)() 2 )( exp()( 202 2 2 r Ar I Arr rp )(rp 0.4 0.5 0.6 1,1 A 33 A= amplitude of the LOS signal, I0 =Bessel function of the first kind , of order 0. Becomes a Rayleigh fading model if A=0 (no LOS). Rice factor: dB 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.1 0.2 0.3 r 2 2 2 log10 A K Ricean fading model: The average received power in Ricean fading model is given by 5 – Statistical Models Where 2σ2 is the average power in the non-LOS multipath components, and A2 is the power in the LOS component. Thus the Rice factor K is the ratio of power in the LOS to the power in the non-LOS components. 22 0 22 2)(][ AdrrprrEPrec )2( 2 22 22 PA 34 Note: Using these relations, the Ricean pdf can also be written as: )1/(2and, )1( . )1 2 ( )2( 2 2 2 2 22 KP K P K A A KKA rec rec 0,) )1( 2() )1( exp )1(2 )( 0 2 r P KK rI P rK K P Kr rp recrecrec
- 18. 18 Nakagami fading model: Gives more generalized fading statistics than Rayleigh and Ricean 5 – Statistical Models g g y g models. Practical measurements show that Nakagami distribution models signal received over mobile channels, more accurately. The Nakagami pdf is given by is the gamma function, and 0 1 exp dyyym m 0,5.0,exp )( 2 )( 2 )12( rm P mr r P m m rp rec m m rec ][ 2 rEPrec 35 m is called the Nakagami fading parameter (usually m ≥0.5) m =1, Nakagami fading becomes Rayleigh fading model. m ≥2, Less severe fading can be modeled. As m → ∞, non-fading or AWGN channel, can be modeled. For m=(K+1)2/(2K+1), Ricean fading with factor K can be modeled. Nakagami fading model: Thus the Nakagami distribution can model Rayleigh, Ricean, AWGN 5 – Statistical Models g y g , , (no fading), and more general fading in more or less severer than Rayleigh and Ricean. Thus measurements in different environments can fit the Nakagami distribution by choosing appropriate m. The power distribution for Nakagami fading, obtained by making change of variable x=r2 in above pdf, is given by 050)( 1 mxxm m m 36 0,5.0,exp )( )( xm PmP xp recrec
- 19. 19 5 – Level Crossing and Average Fade durations • Two important fading parameters are the average # of level crossings or fades and the duration of theseg fades. Level crossing rate (LCR): the expected rate at which the Rayleigh fading envelope crosses a specified level R, when envelope is in a positive going direction. (i.e. r1 – r2 = +ve). The number of level crossing per second is given by )exp(2)( 2 frdrRprN 37 Where , and is the joint pdf of at r =R. Also , where Rrms is the local rms amplitude of the fading envelope [Ref 5]. Note also: where P0 is a target received power (receiver sensitivity). )exp(2),( 0 max DR frdrRprN rr ofslopetheis ),( rRp rr and rmsRR / recPP /0 5 – Level Crossing and Average Fade durations • Example: Rayleigh fading simulation, fc=900MHz, and Mobile Speed = 120km/hrp l id2/ 38 ncorrelatio-de2/ Exercise: at threshold level -20dB about RMS, estimate NR for this simulation.
- 20. 20 5 – Level Crossing and Average Fade durations • Example (5.7 in Ref1): For a Rayleigh fading signal, compute the positive-going level crossing rate for =1,p p g g g when the maximum Doppler frequency (fm) is 20Hz. What is the maximum velocity of the mobile for this Doppler frequency if the carrier frequency is 900MHz?. Ans: crossings per sec. The maximum velocity of the mobile can be obtained using the Doppler relation, . 44.18)1exp()1)(20(2 2 RN /max vfD 39 Thus v =20Hz(1/3m)=6.66m/s=24 km/hr. max 5 – Level Crossing and Average Fade durations Average fade duration (AFD): is the average period of time for which the received signal is below aof time for which the received signal is below a specified level R. For a Rayleigh fading signal, this is given by The average fade duration helps to determine the likely number of signaling bits that may be lost when fading occurs 2 }1){exp( ]Pr[ 1 max 2 DR AFD f Rr N 40 fading occurs.
- 21. 21 5 – Level Crossing and Average Fade durations • Example (5.9 in Ref1): (a) Find the average fade duration for a threshold level of =0.707 when the Doppler frequency i 20H F bi di it l d l ti ith d t t f 50is 20Hz. For a binary digital modulation with data rate of 50 bps, is this Rayleigh fading scenario slow or fast?. (b) What is the average number of bit errors per second for the given data rate for the case = 0.1 [assuming that a bit error occurs whenever any portion of a bit encounters fading]?. Ans: (a) For Rb=50bps, Tb= 20ms, which is > . Thus the signal undergoes fast Rayleigh fading. ms3.18 2)20)(707.0( 1)707.0exp( 2 FD AFD 41 u de goes ast ay e g ad g (b) For = 0.1, we have . This is less than the duration of one bit. Therefore, only one bit on average will be lost when a fading event occur. For = 0.1, crossings per second. Thus there are total of 5 bits in error per sec., resulting in BER=(5/50)=0.1 ms2sec002.0 AFD 96.4RN Frequency Flat /Freq selective fading models: Two-ray frequency-selective fading model: Models 5 – Fading Channel Simulations the effect of multipath delay spread as well as fading. Delay Input (transmitted signal) Output (received signal) 11 exp j 42 The impulse response of the channel model is y 22 exp j tjtjth expexp, 2211
- 22. 22 Two-ray Rayleigh fading model: α1 and α2 are independent and Rayleigh distributed. 5 – Fading Channel Simulations Input (transmitted Output (received 11 exp j p y g θ1 and θ2 are independent and uniformly distributed over [0,2π]. τ is the time delay between the two rays. Setting α = 0, a flat Rayleigh fading channel is obtained. By varying τ, it is 43 Delay signal) signal) 22 exp j y y g possible to create a wide range of frequency selective fading effects. N-ray Rayleigh fading model: Example 4.7 5 – Fading Channel Simulations p Develop an (N+1) -ray mobile channel simulator as shown below: Input (transmitted signal) Output (received signal) 11 exp j 1 44 Delay 1 22 exp j Delay N NN j exp 1 N
- 23. 23 Clarke’s model for flat fading (freq domain): While stat. models (Rayleigh, Ricean, etc.) directly predict received signal envelope in time domain Clarke’s model predicts 5 – Fading Channel Simulations received signal envelope in time domain, Clarke’s model predicts the power spectrum of the received signal first in frequency domain. This is then used to produce a time-domain waveform. The model is widely used to model narrow-band channel in the IS-54 (US digital cellular system). Given that the source, S, transmits a continuous wave signal of frequency fc. Source, S (Base Station) 45 Then the instantaneous frequency of the received signal component arriving at an angle θ at the mobile is: v(mobile) cDc fff v f coscos max Clarke’s model for flat fading: Thus, ff 5 – Fading Channel Simulations Clarke’s model uses statistical characteristic of electromagnetic waves to show that the received power spectral density (PSD), S(f), is proportional to [Ref3]. Therefore, max 1 cos D c f ff f 1 1 cos c f ff ff fS Doppler power Spectrum fcfc -fDmax fc +fDmax 46 i.e., Ref1, pages 214-218, show that K = 1.5/fDmax . 2 max max 1 D c D f ff fff fS 2 max 1 )( D c f ff K fS
- 24. 24 Simulation of Clarke’s fading model using Doppler filter: Baseband received signal: y(t)= yI(t)+ jyQ(t), and r (t)=|y(t)|. 5 – Fading Channel Simulations Q Generate two independent, random complex Gaussian source. Then use spectral filter defined by S(f) in the previous slide, to shape the random signals in frequency domain. Time domain waveform of the resulting fading generated can be produced by using an inverse fast Fourier transform (IFFT) at the last stage of the simulator. Baseband Baseband Mixer 47 Gaussian Noise Source Doppler Filter Baseband Gaussian Noise Source Baseband Doppler Filter Independent Cos(2fct) Sin(2fct) Mixer To IFFT Note: for baseband waveform simulations, mixers will be omitted. Simulation of Clarke’s fading model using Doppler filter: Ref 4 provide steps for computer program to implements this: 5 – Fading Channel Simulations 1. Specify # of frequency points N to represent , and the . Usually N is to power of 2, i.e. N=2k , where k=1,2,3,… 2. Calculate frequency spacing between adjacent spectral lines as f=2fDmax /(N-1). Time duration of the fading waveform is T=1/f. 3. Generate complex Gaussian random variables for each of the N/2 positive frequency component of the noise source. 4 C t t th ti f t f th i )( fS maxDf 48 4. Construct the negative frequency components of the noise source by conjugating positive frequency values & assigning these at the negative frequency values. 5. Multiply the in-phase and quadrature noise sources by the fading spectrum . (Freq domain operation ends here))( fS
- 25. 25 5 – Fading Channel Simulations Simulation of Clarke’s fading model: 6. Perform an IFFT on the resulting frequency6. Perform an IFFT on the resulting frequency domain signals from the I-Q parts to get two N- length time series, & add the squares of each signal point in time to create an N-point time series. 7. Take the square root of the sum obtained in step 6 to obtain an N-point time series of a simulated Rayleigh fading signal, to model the expression: r (t)=|y(t)|=sqrt(yI(t)2+ yQ(t)2), with proper Doppler d d ti l ti 49 spread and time correlation. By making a freq component dominant in amplitude within , and at f=0, the fading is changed from Rayleigh to Ricean. )( fS Simulation of Clarke’s fading model: Block diagram: 5 – Fading Channel Simulations 50
- 26. 26 Simulation of Clarke’s fading model: Example: Fig 5.15 in [Ref 1, 6] – typical Rayleigh fading envelope at 900MHz and 120km /hr 5 – Fading Channel Simulations at 900MHz, and v =120km /hr. 51 Other models: Jake’s model [Ref 5], Young and Beaulieu’s method for computational efficient Rayleigh fading sim.[Ref7]. Jakes’ fading model [5]: Clarke’s model (time domain): defines the complex channel gain for non LOS frequency flat and 2 D isotropic scattering 5 – Fading Channel Simulations gain, for non-LOS, frequency flat, and 2-D isotropic scattering assumptions as [3] N= number of multipaths, nUniform(-,), and nUniform(-,) are the phase and amplitudes of the nth multipath component. Jakes’ model: the high degree of randomness in equation above N n tfj nnD e N th 1 ])cos(2[ max 2 )( 52 is not desirable for efficient simulation. Thus, Jakes proposed the following sum of sinusoid model to simulate h(t), & is widely used. M n DDI t M n f M n tfth 1 max ] 24 2 cos[2cos) 2 cos(2)2cos(2)( max M n DQ t M n f M n th 1 ] 24 2 cos[2cos) 2 sin(2)( max
- 27. 27 Jake’s fading model [5]: 5 – Fading Channel Simulations 0 4 0.6 se 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -0.2 0 0.2 0.4 Time ImpulseRespons 1 1.5 x 10 -3 53 Other recent models: Zheng et al. [8], Young and Beaulieu’s method for computational efficient Rayleigh fading sim.[Ref7]. -800 -600 -400 -200 0 200 400 600 800 0 0.5 Frequency PSD References 1. T.S. Rappaport, “Wireless Communications: principles and practice,” Second Ed., Prentice HallPTR, Upper Saddle River, NJ 07458, USA. 2 A Goldsmith “Wireless Communications ” Cambridge University Press New2. A. Goldsmith, Wireless Communications, Cambridge University Press, New York, NY 10013-2473, USA. 3. R. H. Clarke, “A Statistical Theory of Mobile-Radio Reception,” Bell Systems Technical Journal, vol. 47, pp. 957-1000, 1968. 4. J. I. Smith, “A Computer Generated Multipath Fading Simulation for Mobile Radio,” IEEE Trans. Vehicular Technology, vol. VT-24, no.3, pp.39-40, Aug 1975. 5. W. C. Jakes, “Microwave Mobile Communications,” Wiley-IEEE Press, May 1994. 6. V. Fun, T.S. Rappaport, and B. Thoma, “Bit Error Simulation of /4-DQPSK 54 Mobile Radio Communication using Two-ray and Measurement-based Impulse Response Models,” IEEE Journal on Selected Areas in Commun., Apr. 1993. 7. D. J. Young and N. Beaulieu, “The Generation of Correlated Rayleigh Random Variates by Inverse Discrete Fourier Transform,” IEEE Transactions on Communications, vol 48, no.7, July 2000. 8. Zheng and Xiao, “Simulation Models With Correct Statistical Properties for Rayleigh Fading Channels,” IEEE Trans. Commun. June 2003.