Chapter 5 Small-scale Multipath Propagation
Multi-path Propagation 10 0 -10
dB
time or wavelength
Wireless Communication
Small-scale Fading and Multipath
Rapid fluctuation of the amplitude of a radio signal over a short period of time or travel distance Fading is caused by multipath waves • Transmitted signal which arrive at the receiver at slightly different times Effects: factors influencing small-scale fading • Rapid changes in signal strength over a small travel distance or time interval – Random frequency modulation varying Doppler shift – Speed of the mobile or speed of surrounding objects. • Time dispersion Multipath delay : depends on bandwidth of the signal. Fading • No single line-of-sight (LOS): mobile antennas are below the height of surround structures • With LOS, multipath still occurs • Multipath random distributed amplitude, phases and angles. • A mobile is stationary, the signal may fade due to movement of surrounding objects. • A receiver moving at high speed can pass through several fades in a small of time. • Doppler shift
Chapter 5 –Small-scale multipath propagation 1 Dr. Sheng-Chou Lin
Page 1 Wireless Communication
Multipath Fading
Slow Fading •over large distances, due to gross changes in path •also called shadowing, log-normal fading Fast Fading •over distances on the order of a wavelength •also called Rayleigh fading Assumptions for above types: •many waves of roughly equal amplitude arrive •Rayleigh distributed amplitude •uniformly distributed phase •spatial angle of arrival •azimuth is uniformly distributed •elevation: PDF has mean of 0o, biased towards small angles, does not extend to infinity, and has no discontinuities Rician Fading •there is a LOS or dominant path, producing fewer deep fades occurs in small cells
Chapter 5 –Small-scale multipath propagation 2 Dr. Sheng-Chou Lin
Wireless Communication Multipath Fading: Duration and Frequency
AT LEAST 18 dB C/I 10 0 -10 THRESHOLD FOR dB ESTIMATING n and t
time or wavelength
for a Rayleigh signal 95% of the amplitude is above -10 dB V = vehicle speed, = carrier wavelength n = average number of level crossings at 10 dB below average power t = average fade duration at 10 dB below RMS power n = 0.75V crossings/second and t = 0.132 seconds V e.g. at 850 mHz and 15 miles per hour, n = 16 crossings/second, and t = 6 msec
Chapter 5 –Small-scale multipath propagation 3 Dr. Sheng-Chou Lin
Page 2 Wireless Communication
Multi-path Propagation Effects
Multi-path Propagation Signal levels vary as user moves Slow variations come from blockage and shadowing by large objects such as hills and buildings Rapid Fading comes as signals received from many paths drift into and out of phase •phase cancellation occurs, causing rapid fades that are occasionally deep Rayleigh Fading
•the fades are roughly /2 apart: A 7 inches apart at 800 MHz. 10-15 dB 3 inches apart at 1900 MHz •called Rayleigh fading, after the statistical model that describes it t
Chapter 5 –Small-scale multipath propagation 4 Dr. Sheng-Chou Lin
Wireless Communication
Selective Diversity
. maximum amplitude A ...... path 1 ...... path 2
t
Use a diversity scheme to take advantage of uncorrelated fading Use the dominant instantaneous amplitude This eliminates most of the deep nulls
Chapter 5 –Small-scale multipath propagation 5 Dr. Sheng-Chou Lin
Page 3 Wireless Communication Space Diversity A Method for Combating Rayleigh Fading
D Fortunately, Rayleigh fades are very short and last a small percentage of the time Two antennas separated by several wavelengths will not generally experience fades at the same time “Space Diversity”can be obtained by using two receiving antennas Signal received by Antenna 1 and switching instant-by-instant to whichever is best
Signal received by Required separation D for good Antenna 2 decorrelation is 10-20(BS) •12-24 ft. @ 800 MHz.
Combined Signal •5-10 ft. @ 1900 MHz. Required separation D is (MS)
Chapter 5 –Small-scale multipath propagation 6 Dr. Sheng-Chou Lin
Wireless Communication
Space Diversity D Application Limitations
Space Diversity can be applied only on the receiving end of a link. Transmitting on two antennas would: •fail to produce diversity, since the two signals combine to produce only one value of signal level at a given Signal received point -- no diversity results. by Antenna 1 •produce objectionable nulls in the radiation at some angles Signal received Therefore, space diversity is by Antenna 2 applied only on the “uplink”, i.e., reverse path Combined •there isn’t room for two sufficiently Signal separated antennas on a mobile or handheld
Chapter 5 –Small-scale multipath propagation 7 Dr. Sheng-Chou Lin
Page 4 Wireless Communication
Doppler Shift
Doppler spreading increases the signal bandwidth •fd : moving toward, –moving away •fd = cos() (v/ ) Example: fc 1850 MHz, 60mile/hour (mph) •= c / fc = 3 × 108 / 1850 × 106 = 0.162 m •v = 60 mph = 26.82 m/s
•The mobile is moving toward the transmitter, fd = 26.82 / 0.162 = 1850.0 Hz
•The mobile is moving away the transmitter, fd = - 1850.0 Hz •fd = 0, as = 90 cos() = 0 v
D
Chapter 5 –Small-scale multipath propagation 8 Dr. Sheng-Chou Lin
Wireless Communication Impulse Response Model of a multipath Channel The small-scale variations of a mobile radio signal ; assumptions •The velocity may be assumed constant over a short time (or distance) interval. •The multipath channel is a band-limited band-pass channel •The high frequency variations caused by carrier is removed. (baseband). N-1
hb (t, ) = ai ( t, ) exp [ j2fc i ( t ) + i (t, )] (-i ( t )] I = 0 i (t, ) ai ( t, ) : real amp. and excess delay of A time-varying discrete-time impulse response for a multipath radio channel Ith multipath component at time t.
i (t, ) = 2fc I ( t ) + i (t, ) : the phase shift due to free space propagations of the Ith multipath component + additional phase shifts
i (t, ) : lumps together all the mechanisms for phase shifts of a single multipath component within the ith excess delay bin. Chapter 5 –Small-scale multipath propagation 9 Dr. Sheng-Chou Lin
Page 5 Wireless Communication
Impulse Reponses Model (time invariant)
If the channel impulse response is assumed to be time invariant or is at WSS over a small-scale time or distance interval, then N-1
hb () = ai exp [ ji ] (- i ) i = 0 x + i y 2 Power delay profile: the spatial of | hb (t, ) | over a local area. 2 P ( t ; ) = k | hb (t, ) | base band ,K relates the transmitted power • average over a local area to provide a single time -invariant multipath power make several local area measurement in different location P ()
Measurement : hb ( ) can be predicted by a probing pulse p( t ) p( t ) ( t - ) x2(t) = 1/2c(t)2
Chapter 5 –Small-scale multipath propagation 10 Dr. Sheng-Chou Lin
Wireless Communication
Bandwidth and Received power
Wideband signal : a very narrow pulse, p( t ), does not fluctuate when a receiver is moved about a local area The received power varies very little Narrowband signal : the CW signal strength will vary at a rate governed by
the fluctuations of ai and i large signal fluctuations (fading) occur
• ai varies little over local area Narrowband • I varies greatly due to changes in propagation distance • When the path amplitudes are uncorrelated, multipath phases Wideband are I.I.d over [ 0 , 2] • multipath is not resolved • fading due to the phase shifts of the many unresolved multipath components
– Ex: Tb = 10ns wideband signal and CW signal, fc = 4GHz
Chapter 5 –Small-scale multipath propagation 11 Dr. Sheng-Chou Lin
Page 6 Wireless Communication Bandpass and Baseband channel impulse response Mobile radio channel as a function of time and space. • Channel impulse response = h(d, t), x(t) = transmitted signal • the received signal y(d,t) = x(t) h(d,t), d: position of the receiver • d = vt, v : assumed constant over a short time interval. • h(d, t) h( t, ) , t : time due to motion, : multipath delay for a fixed t. Bandpass channel Complex baseband impulse response
Bandpass channel factor of 1/2 are due to the properties of the complex envelope y(t) 1 f -f f c c f
Complex Baseband 2 ~y(t) f f ~ ~ ~ ~ ~ ~ y(t) = 1/2 x(t ) h(t ), x(t ) = c(t ), h(t ) = hb(t ), y(t ) = r(t ) Chapter 5 –Small-scale multipath propagation 12 Dr. Sheng-Chou Lin
Wireless Communication
Complex Envelop of Bandpass System
1 x(t) htReh tej2fct , h th tjh t b b R I f
Immediate complex -fc fc 1 j2fct 2 ht hb te c.c term 2 c(t) 1 f xt ctej2fct c.c 2 1 ytxththxt d h(t) f -fc f 2 c 1 h (t) h c t de j2fct c.c b b 4 Cos(2fct )+ j sin(2fct ) f 0 2 1 r(t)= ½ (c(t) h (t) h ct e j2fcdej2fct c.c b 4 b f 1 j2f t 1 c 1 ytRe hbtcte rt hbtct y(t) 2 2 f Chapter 5 –Small-scale multipath propagation 13 Dr. Sheng-Chou Lin
Page 7 Wireless Communication
Channel Baseband Complex Envelope
Baseand impulse of a multipath channel N-1 Initial phase
hb ( t, ) = ai ( t, ) exp [ j2fc i ( t ) + i (t, ) ] (- i ( t )) I = 0 i (t, ) Time invariant baseand impulse h(t) Re[hb N-1
hb ( ) = ai exp [ ji ] (- i ) i = 0 x + i y
• Power delay profile : the spatial 2 average of hb (t, ) over a local area ~ d ~ t. • To provide a single time-invariant multipath power delay profile P () d(t) • Maximum bandwidth that this model can accurately represent is equal to 1 / 2 t Chapter 5 –Small-scale multipath propagation 14 Dr. Sheng-Chou Lin
Wireless Communication
Wideband Signal in Mutipath Channel
Tbb A pulsed, transmitted RF signal 2 a1 x( t ) = Re { p( t ) exp [ j2fc( t ) ], T p t 2 T REP • max bb a repetitive baseband pulse a 2 train with very narrow pulse width Tbb and period TREP Wideband signal
• TREP >> max , max : maximum excess delay • Low-pass channel output r(t) hb (t) Instantaneous power delay profile N-1 a1 Measure at t ~ d r (t ) = ai exp [ ji ] p(t - i ) 0 0 i = 0 a2 a3 To determine the received power at ai some time t Complex Base Band 0 • The measured power if the multipath 1 2 3 i t components are resolved Resolved N-1 average N-1 d d2 0 2 2 2 d PWB = |r (t0 ) | = ak (t0 ) Ea, [PWB ]= ak 1 k= 0 k= 0 Sum of powers received in each multipath bins received f power does not fluctuate with d ~ ai , Chapter 5 –Small-scale multipath propagation 15 Dr. Sheng-Chou Lin
Page 8 Wireless Communication
Narrow Signal in Mutipath Channel
Tbb A CW, transmitted RF signal a1
x( t ) = Re { p( t ) exp [ j2fc( t ) ], • The complex envelope is given by c(t) = 2 a2 • Instantaneous complex envelope of the received signal N-1 r (t ) = a exp [ j ( t, ) ] i i fading i = 0 Instantaneous envelope a1 + a2 Instantaneous power (Complex base band )
N-1 Measured at t0 ~ d0 2 2 Pcw = |r (t0 ) | = | ai exp [ ji ( t, ) ] | i = 0 N-1 N-1 N 2 Ea,[Pcw ] = ai + 2 r i, j cos(i - j ) i = 0 i = 0 j i d
• r i, j = Ea [ai aj ]: path amp. Correlation coefficient d0 d1 • Ea,[Pcw ]= Ea, [PWB ] as r i, j and/or cos(i - j ) =0
This can occur i are i.i.d over [0,2] or path amplitudes are uncorrelated f Chapter 5 –Small-scale multipath propagation 16 Dr. Sheng-Chou Lin
Wireless Communication
An Example (SMRCIM)
This technique of quantizing the delay bins determines the time delay resolution of the channel model •Maximum bandwidth that the SMRCIM model (Simulation of Mobile Radio Channel Impulse-response Models) can accurately represent is equal to 1 / 2 (useful frequency span of the model) Example: A discrete channel impulse response model, If number of multipath bins is 64, •urban radio cahannel with excess delays up to 100 s.
–= N / N = 100/ 64 =1.5625 s –1 / 2 = 1/ (2(1.5625 s)) = 0.32 MHz DELAY SPREAD FUNCTION •microcellular channels with excess delays < 4 s.
–= N / N = 4/ 64 =62.5 ns –1 / 2 = 1/ (2(62.5s)) = 8 MHz •indoor channels with excess delays < 500ns -9 –= N / N = 500 10 / 64 =7.8125 ns –1 / 2 = 1/ (2(7.8125 ns)) = 64 MHz N = Chapter 5 –Small-scale multipath propagation 17 Dr. Sheng-Chou Lin
Page 9 Wireless Communication
An Example (Narrow band v.s.Wideband)
A mobile traveling at a velocity of 10 m/s, two multipath components, fc = 1000MHz, The first path with 0 = 0 and power = - 70dBm, Second path with 1 = 1s and power = -73dBm. Mobile moves directly towards the first path and away from the second.
•0 = 0, 1 = 0, = c / f = 0.3m second •P0 = -70dBm = 100pW, P1 = -73dBm = 50pW complex •at t =0, the narrow instantaneous power = r(t)2 first =100pW exp(0)+ 50pW exp(0)2 = 291 pW
• at t = 0.1s, 0 = 2d / = 2vt / = 210 0.1 / 0.3 = 20.94 rad = 2.09 rad= 120 – 1 = -120 , since mobile moves away from the second component. – r(t)2 = 100pW exp(j120)+ 50pW exp(-j120)2 = 79.3 pW t = 0 2 • at t = 0.3s, 0 = 360= 0, 1 = -360= 0= r(t)= 291 pW • at t = 0.4s, r(t)2 = 79.3 pW, at t = 0.5s, r(t)2 = 79.3 pW. • The average narrowband received power = (2)(291)+(4)(79.3)/6 = 149 pW t = 0.1 • The average wideband received power = P0 + P1 = 100+50 = 150 pw – PW,B PN,B, The wideband signal power remains constant over the same interval
Chapter 5 –Small-scale multipath propagation 18 Dr. Sheng-Chou Lin
Wireless Communication
Delay Profile
Measured multipath power delay profiles •900 MHz cellular in San Francisco •Inside a grocery store at 4GHz
Chapter 5 –Small-scale multipath propagation 19 Dr. Sheng-Chou Lin
Page 10 Wireless Communication
Time Dispersion Parameters
Power delay profile •Mean excess delay •RMS delay spread •Excess delay spread
Mean excess delay
2 ak k P( k ) k k k = a 2 k P( k ) k k RMS deplay spread a 2 2 2 k k P( k ) k 2 k 2 2 k = - ( ) Where = 2 ak P( k ) k k • In outdoor mobile: RMS ~ s • In indoor mobile: RMS ~ ns
Chapter 5 –Small-scale multipath propagation 20 Dr. Sheng-Chou Lin
Wireless Communication
An Example
Maximum excess delay ( xdB ): •time delay during which multipath energy falls to X dB below the maximum.
•I.e. x - 0 , where 0 is the first arriving signal, x is the maximum delay at which a multipath component X dB of the strongest arriving component (which does not necessarily arrive at 0 ) 2 Threshold level: , , depend on the choice of noise threshold 2 •noise threshold , , , Pr() 0dB
Example: -10dB 1 = 4.38 sec Bc = -20dB 5 -30dB 2 = 21.07 sec = 146 kHz = 1.37 sec 0 1 2 5 ( s)
Chapter 5 –Small-scale multipath propagation 21 Dr. Sheng-Chou Lin
Page 11 Wireless Communication
Typical measured RMS delay spread
• Outdoor mobile channel : RMS is on the order of s • Indoor radio channel : RMS is on the order of ns
Chapter 5 –Small-scale multipath propagation 22 Dr. Sheng-Chou Lin
Wireless Communication
Coherence Bandwidth
Relation derived from RMS delay spread
•BW Bc , the channel can be considered as “flat” •Flat channel: a channel which passes all spectral components with equal gain and linear phase •Two frequency components have a strong potential for amplitude correlation over the range of frequencies.
Relation between Frequency correlation function and Bc 1 •correlation function > 0.9 B c CR > 0.5 50 1 CR > 0.9 •correlation function > 0.5 Bc 5
Ex: = 1.37 sec, Bc 1/ 5= 146 kHz •AMP BW = 30k no equalizer required. •GSM 200 k equalizer required
Chapter 5 –Small-scale multipath propagation 23 Dr. Sheng-Chou Lin
Page 12 Wireless Communication
Signal BW v.s. Coherent Bandwidth
t f T BW > Bc Freq. Selective channel Narrowband Channel 1 Bc BW 1/T
t f BW B Flat channel Wideband Channel c
Chapter 5 –Small-scale multipath propagation 24 Dr. Sheng-Chou Lin
Wireless Communication
Flat and Frequency-selective Fading
2-ray multipath channel (point-point)
i
Flat fading Frequency-selective fading
Chapter 5 –Small-scale multipath propagation 25 Dr. Sheng-Chou Lin
Page 13 Wireless Communication Channel Delay Spread, A Phenomenological Model
DELAY SPREAD FUNCTION | H(f) |
TRANSMITTER RECEIVER CHANNEL TRANSFER FUNCTION
• The delay spread of a channel d is the RMS value of the channel impulse response (delay spread function) • In a mobile environment, the delay spread function is constantly changing (i.e., |h(f)| is a nonlinear time-varying filter) • The channel transfer function |h(f)| has a lowpass characteristic with multiple delays (time dispersion) • The delay spread represents the time it takes most of the energy from the transmitter to propagate (at c = 3e+8 m/s) to the receiver • can be considered the group delay of the channel model |h(f)| • For in-building propagation, = 0.1 s; for urban propagation = 3s
Chapter 5 –Small-scale multipath propagation 26 Dr. Sheng-Chou Lin
Wireless Communication Coherence Bandwidth, Frequency Diversity Gain and Delay Spread
The channel coherence bandwidth BC can be computed from the delay spread d of a channel: 1 B = C 2
If a signal has a bandwidth b greater than BC , then the signal has frequency components that fade independently. the signal has a frequency diversity gain, G G = 1+B , B : Bandwidth of signal
Signals with bandwidths greater than BC are more resistant to channel fading effects
EXAMPLES:
•Compute the coherence bandwidth of a channel with = 3s (Bc = 53 khz) •Show there is no frequency diversity gain for amps.(AMPS = 30 khz < 53 khz) •Compute the frequency diversity gain for CDMA. ( g = 1 + 1.25 x 3 = 4.75) Chapter 5 –Small-scale multipath propagation 27 Dr. Sheng-Chou Lin
Page 14 Wireless Communication
INTERSYMBOL INTERFERENCE (ISI) AND DELAY SPREAD
To avoid isi in the standstill (nonfading) case, the maximum data rate RB is related to the delay spread d of the channel 1 R = B
To avoid ISI in mobile environments (fading case), the maximum data rate R is given by: B 1 R = B 2
EXAMPLE: • You are interested in buying a wireless modem from a vendor for indoor data transmission at rates less than 300 kbits/sec. • the vendor insists that you buy modems equipped with equalizers which doubles the price.
• is this necessary? no. assume a fading case with = 0.5s, then RB = 318 kbits/sec
Chapter 5 –Small-scale multipath propagation 28 Dr. Sheng-Chou Lin
Wireless Communication
Doppler Spread
To describe time varying nature of the channel in a small-scale region.
•Doppler spread BD : a measure of the spectral boarding channel caused by the time rate of change of the channel.
•Doppler spectrum : components in the range fc-fd to fc-fd.
•Effect of Doppler spread are negligible, as BWsignal BD t T Coherence time TC is the time domain c dual of Doppler •To characterize time varying nature
•Tc 1/ fm under Rayleigh fading
•Ts Tc channel will change during the transmission of the baseband message distortion •Time correlation function > 0.5,
Tc 9 / 16fm , fm: the max. Doppler shift Chapter 5 –Small-scale multipath propagation 29 Dr. Sheng-Chou Lin
Page 15 Wireless Communication
A thumb rule
A popular rule of thumb for modern digital communications is
9 0.423 T = = C 2 16 fm fm
• Tc 1/ fm suggests a time duration during Rayleigh fading
• Tc 9 / 16fm is often too restrictive
• Definition of coherence time : two signals arriving with a time separation > Tc are affected differently by the channel.
Example: A vehicle, speed = 60 mile/per hour, fc = 900 MHz
• Tc = 9 / 16fm = 2.22ms
• If a digital transmission is used, max. symbol rate Rc = 1/ Tc = 454bps. –Distortion could result from multipath time delay spread = • Using the practical rule, Tc = 0.423/fm 6.77ms , max. symbol rate Rc = 1/ Tc = 150bps
Chapter 5 –Small-scale multipath propagation 30 Dr. Sheng-Chou Lin
Wireless Communication
An Example
Small-scale propagation measurements •Determine the proper spatial sampling interval •consecutive samples are highly correlated in time
•fc = 1900 MHz and v = 50m/s.
•For correlation, the sampling time is Tc /2. Use the smallest Tc for conservative design.
–Tc 9 / 16fm = 9/16 v = 565 s Tc /2 = 282.5 s •How many samples is required over 10m travel distance.
–Spatial sampling interval: x = vTc /2 = 50 565 s /2 = 1.41 cm
–Required samples = Nx = 10 / x =708 samples •How long would it take to make these measurements –d / v = 10m/50 = 0.2 seccond
•The Doppler spread BD = fm = vfc / c = 316.66 Hz
Chapter 5 –Small-scale multipath propagation 31 Dr. Sheng-Chou Lin
Page 16 Wireless Communication
Types of small-scale fading
Depending on the relation Small-Scale fading between the signal and (based on multipath time delay spread) channel parameters, different transmitted signals will undergo Flat Fading Frequency selective Fading different types of fading 1. BW of signal < BW of channel 1. BW of signal > BW of channel 2. Delay spread < Symbol period 2. Delay spread > Symbol period
•Signal parameters: Small-Scale fading Bandwidth, symbol period (based on Doppler spread) •Channel parameters: RMS delay spread, Doppler spread Fast Fading Slow Fading 1. High Doppler spread 1. Low Doppler spread 2. Coherence time < Symbol period 2. Coherence time > Symbol period 3. Channel variations faster than 3. Channel variations slower than baseband signal variations baseband signal variations
Chapter 5 –Small-scale multipath propagation 32 Dr. Sheng-Chou Lin
Wireless Communication Types of fading
l T
o s b
f m o y Flat Slow Flat Fast d S
o Fading Fading i g
Type of fading experienced r n i e t t P i
by a signal as a function of l m o s
b
n Frequency Selective Frequency Selective m a y r Slow Fading Fast Fading •Signal parameters: S T Ts Tc –Symbol period (Ts ) Transmitted Symbol Period –Baseband signal B
s
bandwidth ( B ) d s n a h t b d e •Channel parameters: i
s Frequency Selective Frequency Selective w a d b
Fast Fading Slow Fading n d –RMS delay spread () a e b t
t l B i c Coherent BW ( B ) a m c n s g
i Flat Fast Flat Slow n s –Doppler spread ( B ) a D r Fading Fading Coherent Time ( T ) T c Bs BD Transmitted baseband signal bandwidth Chapter 5 –Small-scale multipath propagation 33 Dr. Sheng-Chou Lin
Page 17 Wireless Communication
Rayleigh Distribution
To describe statistical time varying nature of the received envelope • A flat fading signal • An individual multipath component • The envelope of the two quadrature Gaussian noise
x + i y r = x2 + y2
Zero-mean Gaussian dist. with 2
r r 2 exp ( ) , 0 r P(r) = 2 22 0 , r 0 Rayleigh fading beams : rms before envelope 2 : time-average power before x, y envelope t
Chapter 5 –Small-scale multipath propagation 34 Dr. Sheng-Chou Lin
Wireless Communication
Rayleigh Distribution Parameters
Cumulative distribution function (CDF)
R2 P( R ) = P ( r R) = 1- exp ( ) r 22
Mean value of Rayleigh distribution rmean = E [ r ] = r p( r ) dr = / 2 = 1.2533 0 Variance of Rayleigh distribution
E [ r 2 ] = E [ x 2 ] + E [ y 2 ] = 2 2 1.2533
2 2 2 2 2 r = E [ r ] –(E [ r ]) = 2 - (/2) = 0.4292 rms of the envelope = square root of the mean square = E [ r 2 ] = 2
Chapter 5 –Small-scale multipath propagation 35 Dr. Sheng-Chou Lin
Page 18 Wireless Communication
Ricean Fading Distribution
There is a dominant stationary (nonfading) signal component
•line-of-sight ( LOS ) Rayleigh •small-scale fading envelope distribution is Ricean •Ricean Rayleigh as LOS fades away
r ( r 2 +A2 ) Ar exp ( ) I0 ( 2 ) ,A 0, 0 r P(r) = 2 22 , r 0 0
A : peak amplitude of LOS Random multipath LOS I0 ( ) : Modified Bessel function of the first kind of zero-order K = A2 / ( 2 2 ) : describe Ricean distribution K (dB) = 10 log [ A2 / ( 2 2 ) ] dB
• A 0, K dB, Ricean Rayleigh
Chapter 5 –Small-scale multipath propagation 36 Dr. Sheng-Chou Lin
Wireless Communication
Clarke’s Model for Flat Fading
Assumptions •A fixed transmitter with a vertically polarized antenna •The field on the mobile antenna comprises of N azimuthal plane waves with –arbitrary carrier phases z –arbitrary azimuthal angles of arrival vertically polarized y –each wave having equal average amplitude in absence of a direct LOS experience similar attenuation over small-scale distances Vertically polarized plane waves at BS x N Azimuthal plane Ez = E0 Cn cos ( 2fct + n ) n =1
• Doppler shift is very small Tc( t ) and Ts( t ): Gaussian • The phase angles uniformly distributed on [0, 2] Random processes 2 2 Ez = Tc( t ) cos ( 2fct ) + Ts( t ) sin( 2fct ) r ( t ) = Tc ( t ) + Ts ( t ) 2 2 2 2 2 Rayleigh distribution = p (r) Tc = Tc = Ez = E0 / 2 = Chapter 5 –Small-scale multipath propagation 37 Dr. Sheng-Chou Lin
Page 19 Wireless Communication Spectral-Shape with Doppler-spread
Spectral analysis for Clark’s model •Total received power z vertically polarized
2 y
Pr = AG() p() d 0 G() = Antenna Azimuthal gain pattern x A : average received power w.r.t an Azimuthal plane isotropic antenna y p() = incoming power of the angle
•instantaneous freq. Of the received signal (CW, freq.= fc) component arriving at an angle v f () = f = cos() + f = f cos() + f df = d-sinf c m c m
fm : maximum Doppler shift, an even function f () = f (-) •The received power with frequency S( f ) df = A [ G() p() + G(-) p(-)] d Chapter 5 –Small-scale multipath propagation 38 Dr. Sheng-Chou Lin
Wireless Communication Doppler power spread
0º 180º Doppler power spectrum (unmodulated CW carrier) -1 = cos [ ( f-fc )/ fm ]
2 sin = 1-[( f-fc )/ fm ] •For the case, vertical /4 Antenna G() =1.5 and p() = 1/2over [0, 2] 1.5 SEz( f ) = f 2 m 1-[( f-fc )/ fm ] A Baseband power spectral density 1 K 2 SbbEz( f ) = 1- ( f / 2fm ] 8fm K[ ] : complete elliptical integral of the first kind • not intuitive
Chapter 5 –Small-scale multipath propagation 39 Dr. Sheng-Chou Lin
Page 20 Wireless Communication
Two-Rayleigh Fading Model
To consider multipath time delay spread as well as fading
hb ( t ) = 1 exp [ j1 ] ( t ) + 2 exp [ j2 ] ( t - )
•1 , 2 : independent and Rayleigh distributed •1 , 2 : independent and uniformly distributed over distributed •: time delay between two rays •To create a wide range of frequency selective fading effects by varying
1st Rayleigh fading neam 2nd Rayleigh fading neam Rayleigh fading beam
Chapter 5 –Small-scale multipath propagation 40 Dr. Sheng-Chou Lin
Wireless Communication
Simulation of Flat Fading Model
Quadrature Amplitude modulation Quadrature Amplitude modulation • RF Doppler filter Two indep. Gaussian low-pass noise • Baseband Doppler filter for in-phase and quadrature fading Spectral filter Accurate time domain of Doppler fading IFFT at the last stage
Construct negative components of the noise source
Chapter 5 –Small-scale multipath propagation 41 Dr. Sheng-Chou Lin
Page 21 Wireless Communication
Simulation of frequency-selective Fading
To produce both flat and frequency-selective fading effects • Several Rayleigh fading simulators • Variable gains • Time delays
Rayleigh Ricean •Add a single freq. Component dominant in amplitude within Doppler fading spectrum
Chapter 5 –Small-scale multipath propagation 42 Dr. Sheng-Chou Lin
Wireless Communication
Small-scale multipath measurements
Measurements • Direct RF pulse system • Spread spectrum sliding correlator – Narrow BW wideband • Frequency Domain – FFT, IFFT
Chapter 5 –Small-scale multipath propagation 43 Dr. Sheng-Chou Lin
Page 22 Wireless Communication
Digital Modulation under flat fading
Performance in slow, flat fading channels in AWGN
•Binary modulation •Rayleigh fading •To average the error probability in AWGN over the possible ranges of signal strength due to fading
2 • = [ Eb/ No] is the average value of signal-to- noise ratio, has a Rayleigh distribution •Mean SNR is significantly larger than that required Fading v.s. nonfading when operating over a nonfading channel (~20- 50dB)
Chapter 5 –Small-scale multipath propagation 44 Dr. Sheng-Chou Lin
Wireless Communication Level Crossing Rate (LCR)
Level Crossing Rate (LCR): the rate at which the Rayleigh fading envelope, normalized to RMS level (2 ), crosses a specified level.
•NR : the number of level crossings per second ( r = R ) 2 NR rpR,rdr 2fm e 10 0 0 -10 R fm : the maximum Doppler frequency r: time derivative of r(t) dB pR,r: joint density function ofr andr at r = R 1 sec.
= R / Rrms : the value of the specified NR R normalized to RMS amplitude of • Few crossings at both high and low levels fading envelope • Maximum at = 2 Example: For a Rayleigh fading signal, = 1, maximum Doppler freq. fm= 20Hz, fc = 900 MHz. -1 •NR = 2(20) 1e = 18.44 crossings per second
• f d,max = v / v = 20 (1/3) = 6.66 ms = 24 km/hr Chapter 5 –Small-scale multipath propagation 45 Dr. Sheng-Chou Lin
Page 23 Wireless Communication
Average fade duration
Average fade duration: average period of time for which the received signal a specified R. For a Rayleigh fading signal
2 1 e - 1 10 = Pr [ r R ] = N f 2 0 R m R 2 • Pr [ r R ] = 1- exp( ) : probability that the received signal less than R = the fading time in one second.
• Helps determine the most likely NR number of signaling bits that may be 1 sec. lost during a fade Pr [ r R ] = fading time in 1 sec.
Example: Threshold level = 0.707, Doppler freq. = fm = 20Hz, Binary digital modulation with bit duration of 50 bps, bit error occurs for 0.1 2 • The average fade duration = (e 0.707 -1) / (0.707)20 2= 18.3 ms • bit period = 1/50 = 20ms the signal undergoes than fast Rayleigh fading
• for =0.1, = 0.002 s = 20ms one bit will be lost during a fade, NR = 4.96 the total number of bits in error is 5/sec. BER = 5/50 = 0.1
Chapter 5 –Small-scale multipath propagation 46 Dr. Sheng-Chou Lin
Wireless Communication
Lesson 5 Complete
Chapter 5 –Small-scale multipath propagation 47 Dr. Sheng-Chou Lin
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