Chapter 5-Small Scale Fading and Multipath
Chapter 5 Small-Scale Fading and Multipath
School of Information Science and Engineering, SDU Outline l Small-Scale Multipath Propagation l Impulse Response Model of a Multipath Channel l Small-Scale Multipath Measurements l Parameters of Mobile Multipath Channels l Types of Small-Scale Fading l Rayleigh and RiceanDistributions l Statistical Models for Multipath Fading Channels Small Scale Fading l Describes rapid fluctuations of the amplitude, phase of multipath delays of a radio signal over short period of time or travel distance l Caused by interference between two or more versions of the transmitted signal which arrive at the receiver at slightly different times. l These waves are called multipath waves and combine at the receiver antenna to give a resultant signal which can vary widely in amplitude and phase. Small Scale Multipath Propagation
l Effects of multipath l Rapid changes in the signal strength § Over small travel distances, or § Over small time intervals l Random frequency modulation due to varying Doppler shifts on different multiples signals l Time dispersion (echoes) caused by multipath propagation delays l Multipath occurs because of l Reflections l Scattering Multipath l At a receiver point l Radio waves generated from the same transmitted signal may come l from different directions l with different propagation delays l with (possibly) different amplitudes (random) l with (possibly) different phases (random) l with different angles of arrival (random). l These multipath components combine vectoriallyat the receiver antenna and cause the total signal § to fade § to distort Multipath Components
Radio Signals Arriving from different directions to receiver
Component 1
Component 2
Component N
Receiver may be stationary or mobile. Mobility l Other Objects in the radio channels may be mobile or stationary l If other objects are stationary l Motion is only due to mobile l Fading is purely a spatial phenomenon (occurs only when the mobile receiver moves) l The spatial variations as the mobile moves will be perceived as temporal variations § Dt= Dd/v l Fading may cause disruptions in the communication Factors Influencing Small Scale Fading
l Multipath propagation l Presence of reflecting objects and scattererscause multiple versions of the signal to arrive at the receiver § With different amplitudes and time delays § Causes the total signal at receiver to fade or distort l Speed of mobile l Cause Doppler shift at each multipath component l Causes random frequency modulation l Speed of surrounding objects l Causes time-varying Doppler shift on the multipath components Factors Influencing Small Scale Fading l Transmission bandwidth of the channel l The transmitted radio signal bandwidth and bandwidth of the multipath channel affect the received signal properties: § If amplitude fluctuates or not § If the signal is distorted or not Doppler Effect l Whe a transmitter or receiver is moving, the frequency of the received signal changes, i.e. İt is different than the frequency of transmissin. This is called Doppler Effect. l The change in frequency is called Doppler Shift. l It depends on l The relative velocity of the receiver with respect to transmitter l The frequenct (or wavelenth) of transmission l The direction of traveling with respect to the direction of the arriving signal. Doppler Shift – Transmitter is moving
The frequency of the signal The frequency of the signal that is received behind the that is received in front of the transmitter will be smaller transmitter will be bigger Doppler Shift –Recever is moving S
d = XY Ll = SX - SY = d cosq Ll = vLt cosq
The phase change in the received signal: Ll 2pvLt DF = 2p= cosq ll
Dl Doppler shift (The apparent change in frequency) : 1 DF v XYq f = = cosq dd 2pDt l v A mobile receiver is traveling from point X to point Y Doppler Shift l The Dopper shift is positive l If the mobile is moving toward the direction of arrival of the wave. l The Doppler shift is negative l If the mobile is moving away from the direction of arrival of the wave. Impulse Response Model of a Multipath Channel l The wireless channel charcteristics can be expressed by impulse response function l The channel is time varying channel when the receiver is moving. l Lets assume first that time variation due strictly to the receiver motion (t = d/v) l Since at any distance d = vt, the received power will be combination of different incoming signals, the channel charactesitics or the impulse response funcion depends on the distance d between trandmitter and receiver. Multipath Channel Modeling
Impulse Response Model of a Multipath Wireless Channel Impulse Response Model of a Multipath Channel l The wireless channel characteristics can be expressed by impulse response function l The channel is time varying channel when the receiver is moving. l Lets assume first that time variation due strictly to the receiver motion (t = d/v) l Since at any distance d = vt, the received power will ve combination of different incoming signals, the channel charactesitics or the impulse response funcion depends on the distance d between trandmitter and receiver Impulse Response Model of a Multipath Channel
d = vt v
d A receiver is moving along the ground at some constant velocity v. The multipath components that are received at the receiver will have different propagation delays depending on d: distance between transmitter and receiver. Hence the channel impulse response depends on d.
Lets x(t) represents the transmitter signal y(d,t) represents the received signal at position d. h(d,t) represents the channelimpulse response which is dependent on d (hence time-varying d=vt). Multipath Channel Model
Building Multipath Channel
2nd MC
Base 1st MC Mobile 2 Station
B u il di ng B u ild 1st MC in 4th MC g Multipath Channel 2nd MC B Mobile 1 uild 3rd MC ing (Multipath Component) Impulse Response Model of a Multipath Channel
Wireless Multipath Channel x(t) y(t) h(d,t)
The channel is linear time-varying channel, where the channel characteristics changes with distance (hence time, t = d/v)
¥ y(d,t) = x(t) Ä h(d,t) = ò x(t )h(d,t -t )dt -¥ For a causal system, h(d,t) = 0 for t < 0; hence t y(d,t) = ò x(t )h(d,t -t )dt -¥ Impulse Response Model
d= vt assume v is constant over time t y(vt,t)=-òx(t)h(vt,)tdtt -¥ t y(t)=òx(t)h(vt,t-tt)d=x(t)Äh(vt,t)=Äx(t)h(dt,) -¥
We assume v is constant over short time. x(t): transmitted waveform y(t): received waveform h(t,t): impulse response of the channel. Depends on d (and therefore t=d/v) and also to the multiple delay for the channel for a fixed value of t. t is the multipath delay of the channel for a fixed value of t.
¥ y(t) = ò x(t )h(t,t )dt = x(t) Ä h(t,t ) -¥ ...Continue with Multipath Channel Impulse Response Model Impulse Response Model
x(t) y(t) jwct h(t,t ) = Re{hb (t,t )e } jw t x(t) = Re{c(t)e jwct } y(t) = Re{r(t)e c } Bandpass Channel Impulse Response Model y(t) = x(t) Ä h(t,t )
c(t) 1 r(t) hb (t,t ) 2 1 r(t) = c(t) Ä h (t,t ) 2 b
Baseband Equivalent Channel Impulse Response Model Impulse Response Model
1 r(t) = c(t) Ä h(t,t) 2 b
j2pfct x(t) = Re{}c(t)e wc= 2pfc y(t) = Re{}r(t)ej2pfct
c(t) is the complex envelope representation of the transmittedsignal r(t) is the complex envelope representation of the received signal
hb(t,t) is the complex baseband impulse response Discrete-time Impulse Response Model of Multipath Channel
Amplitude of Multipath Component There are N multipath components (0..N-1)
to= 0 t1= Dt Excess Delay ti= (i)Dt Bin tN-1= (N-1)Dt
t (excess delay) Dt tN-1 t0 t2 ti Excess delay: relative delay of the ithmultipath componentascompared to the first arriving component th ti : Excesssdelay of i multipath component, NDt: Maximum excess delay Multipath Components arriving to a Receiver
Ignore the fact that multipath components arrive with differentangles, and assume that they arriving with the same angle in 3D.
1 2NN-2 N-1 th Component
......
t t0=0 t1 tN-3 tN-2 tN-1 (relative delay of multipath Comnponent)
Each component will have different Amplitude (ai) and Phase (θi) Baseband impulse response of the Channel
N -1 j(2pfct i (t)+fi (t,t )) hb (t,t ) = å ai (t,t )e d (t -t i (t)) i=0
ai (t,t ) : the real amplitude of the ith multipath component at time t.
t i (t) : excess delay of the ith multipath component at time t.
2pfct i (t) +fi (t,t ) : Phase term that represents phase shift due to
free space propagatio n of the ith component. Simply represent it with : q (t,T ) i d (·) : unit impulse function. Discrete-Time Impulse Response Model for aMultipath Channel
hb(t,t)
t
t3 t(t3)
t2 t(t2)
t1 t(t1) t0 t t(t0) o t1 t2 t3 t4 t5 t6 tN-2 tN-1 Time-Invariance Assumption
If the channel impulse response is assumed to be time-invariant over small-scale time or distance interval, then the channel impulse response can be simplified as:
N -1 jqi hb (t ) = å aie d (t -t i ) i=0
When measuring or predicting hb(t), a probing pulse p(t) which approximates the unit impulse function is used at the transmitter. That is:
p(t) » d (t -t ) This is called sounding the channel to determine impulse response. Complex Baseband Impulse Response
Baseband impulse response hb(t) is a complex number and therefore has a magnitude (amplitude) ai and a phase θi.
jqi hb(t) = aie hb(t)
ai hb(t) = ai(cosqi+jsinqi) qi
|hb(t)| = ai
you can think of it also as a vector that starts at origin. Amplitudes and Phases of Multipath Components
1st Arriving Multipath Component (Say 0th Component)
q0=0 (phase)
fc 2a0
Two components emerge from the same source at the same time. They belong to the same transmitter signal. But they travel different paths. They arrive at the
same receiver with time difference equal to ti.
fc 2ai
th i Multipath Component qi=2pfcti t 0 ti qi is expressed in radians Components arriving at the same time
What happens if two or more multipath components are with the same access delaybin(arrive at the same time)? Thenthe received signal is the vectorial addition of twomultipath signals. R Example: Lets assume two signals S1 and S2 arrive at
a3 S1 the same time at the receiver: a1 S2 q3 a2 q2 q1 jq1 jq 2 S1 = a1e S2 = a2e
R = S1 + S2
jq1 jq 2 jq 3 = a1e + a2e = a3e
R is the combined receiver signal. Components arriving at the same time
The amplitude and phase of the combined signal (R) depends on the amplitudes and phases of the two components.
Depending on the valuesof the phases of the components, the combined affect may weaken or strengthen the amplitude of the combined signal.
It is possible that the two signalsmay totally cancel each other depending on their relative phaseson their amplitudes. Example 1 – Addition of Two Signals
MC: Multipath Component 3 cos(x+pi/16) st 1 MC cos(x+pi) cos(x+pi/16)+cos(x+pi) 2st MC 2
Combined 1 Signal 0
a1/a2=1 -1
q1=p/16 -2 q2 =p
-3 -10 -5 0 5 10 Example 2 – Addition of Two Signals
3 cos(x+pi/16) st 1 MC 3*cos(x+pi) cos(x+pi/16)+3*cos(x+pi) 2st MC 2
Combined 1 Signal 0
a1/a2=1/3 -1
q1=p/16 -2 q2 =p
-3 -10 -5 0 5 10 Power Delay Profile
For small-scale fading, the power delay profile of the channel is found by
taking the spatial average of 2 over a local area (small-scale area). hb (t;t )
If p(t) has a time duration much smaller than the impulse response of the multipath channel, the received power delay profile in a local area is given by:
2 P(t ) » k hb (t;t )
2 The bar represents the average over the local area of hb (t;t )
Gain k relates the transmitter power in the probing pulse p(t) to the total received power in a multipath delay profile. Example power delay profile
Taken from DimitriosMavrakisHomepage:http://www.ee.surrey.ac.uk/Personal/D.Mavrakis/ Relationship between Bandwidth and Receiver Power l What happens when two different signals with different bandwidths are sent through the channel? l What is the receiver power characteristics for both signals? l We mean the bandwith of the baseband signal l The bandwidth of the baseband is signal is inversely related with its symbol rate.
One symbol Bandwidth of Baseband Signals Highbandwidth (Wideband) Signal
Lowbandwidth (Narrowband) Signal
Continuous Wave (CW) Signal t A pulsed probing signal (wideband) Tbb
Transmitter p(t) x(t): transmitted signal
TREP
TREP >> t max (t max : maximum measured excess delay)
j 2pfct x(t) = Re{ p(t)e } = p(t) cos(2pfct) x(t) Multipath y(t) p(t) Multipath r(t) Wireless Channel Wireless Channel
Bandpass signals Baseband signals Received Power of Wideband Sİgnals
p(t) Multipath r(t) Wireless Channel
The output r(t) will approximate the channel impulse response since p(t) approximates unit impulses.
N -1 1 jqi r(t) = å aie × p(t -t i ) 2 i=0 Assume the multipath components have random amplitudes and phases at time t.
N -1 2 N -1 é jqi ù 2 Ea,q [PWB ] = Ea,q êå aie ú = å ai = E[PWB ] ë i=0 û i=0 Received Power of Wideband Sİgnals
This shows that if all the multipath components of a transmittedsignal is resolved at the receiver then: The average small scale received power is simply the sum of received powers in each multipath component.
In practice, the amplitudes of individual multipath components do not fluctuate widely in a local area (for distance in the order of wavelength or fraction of wavelength).
This means the average received power of a wideband signal do not fluctuate significantly when the receiver is moving in a local area. Received Power of Narrowband Sİgnals
A CW Signal x(t): transmitted signal Transmitter
c(t)
Assume now A CW signal transmitted into the same channel.
Let comlex envelope will be: c(t) = 2 N -1 The instantaneous complex envelope jqi (t,t ) r(t) = å aie of the received signal will be: i=0
2 N -1 2 The instantaneous power will be: jqi (t,t ) r(t) = åaie i=0 Received Power of Narrowband Sİgnals
Over a local area (over small distance – wavelengths), the amplitude a multipath component may not change signicantly, but the phase may change a lot.
For example: -if receiver moves l meters then phase change is 2p. In this case the component may add up posively to the total sum S.
-if receiver moves l/4 meters then phase change is p/2 (90 degrees) . In this case the component may add up negatively to the totalsum S, hence the instantaneous receiver power.
Therefore for a CW (continues wave, narrowband) signal, the small movements may cause large fluctuations on the instantenous receiver power, which typifies small scale fading for CW signals. Wideband versus Narrowband Baseband Signals
However, the average received power for a CW signal over a localarea is equivalent to the average received power for a wideband signal on the local area. This occurs because the phases of multipath components at different locations over the small-scale region are independently distributed (IID uniform) over [0,2p].
In summary: 1.Received power for CW signals undergoes rapid fades over small distances 2.Received power for wideband signals changes very little of smalldistances. 3.However, the local area average of both signals are nearly identical. Small-Scale Multipath Measurements l Several Methods l Direct RF Pulse System l Spread Spectrum Sliding Correlator Channel Sounding l Frequency Domain Channel Sounding l These techniques are also called channel sounding techniques Direct RF Pulse SysTxtem
fc
Pulse Generator
RF Link
Rx
Digital BPF Detector Oscilloscope Parameters of Mobile Multipath Channels l Time Dispersion Parameters l Grossly quantifies the multipath channel l Determined from Power Delay Profile l Parameters include § Mean Access Delay § RMS Delay Spread § Excess Delay Spread (X dB) l Coherence Bandwidth l Doppler Spread and Coherence Time Measuring PDPs l Power Delay Profiles l Are measured by channel sounding techniques l Plots of relative received power as a function of excess delay l They are found by averaging intantenous power delay measurements over a local area § Local area: no greater than 6m outdoor § Local area: no greater than 2m indoor § Samples taken at l/4 meters approximately § For 450MHz – 6 GHz frequency range. Timer Dispersion Parameters Determined from a power delay profile.
2 akt k P(t k )(t k ) Mean excess delay( ): å å t k k t = 2 = å ak å P(t k ) k k
2 Rms delay spread (st): 2 st = t - (t ) 2 2 2 å akt k å P(t k )(t k ) 2 k k t = 2 = å ak å P(t k ) k k Timer Dispersion Parameters
Maximum Excess Delay (X dB):
Defined as the time delay value after which the multipath energy falls to X dB below the maximum multipath energy (not necesarilybelonging to the first arriving component).
It is also called excess delay spread. RMS Delay Spread PDP Outdoor PDP Indoor Noise Threshold l The values of time dispersion parameters also depend on the noise threshold (the level of power below which the signal is considered as noise). l If noise threshold is set too low, then the noise will be processed as multipath and thus causing the parameters to be higher. Coherence Bandwidth (BC)
l Range of frequencies over which the channel can be considered flat (i.e. channel passes all spectral components with equal gain and linear phase). § It is a definition that depends on RMS Delay Spread.
l Two sinusoids with frequency separation greater than Bc are affected quite differently by the channel.
f1
Receiver
f2 Multipath Channel Frequency Separation: |f1-f2| Coherence Bandwidth
Frequency correlation between two sinusoids: 0 <= Cr1, r2 <= 1.
If we define Coherence Bandwidth (BC) as the range of frequencies over which the frequency correlation is above 0.9, then
1 B = s is rmsdelay spread. C 50s
If we define Coherence Bandwidth as the range of frequencies over which the frequency correlation is above 0.5, then 1 B = C 5s This is called 50% coherence bandwidth. Coherence Bandwidth l Example: l For a multipath channel, s is given as 1.37ms. l The 50% coherence bandwidth is given as: 1/5s = 146kHz. § This means that, for a good transmission from a transmitter to a receiver, the range of transmission frequency (channel bandwidth) should not exceed 146kHz, so that all frequencies in this band experience the same channel characteristics. § Equalizers are neededin order to use transmission frequencies that are separated larger than this value. § This coherence bandwidth is enough for anAMPS channel (30kHz band needed for a channel), but is not enough for a GSM channel (200kHz needed per channel). Coherence Time l Delay spread and Coherence bandwidth describe the time dispersive nature of the channel in a local area. l They don’t offer information about the time varying nature of the channel caused by relative motion of transmitter and receiver. l Doppler Spread and Coherence time are parameters which describethe time varying nature of the channel in a small-scale region. Doppler Spread l Measure of spectral broadening caused by motion l We know how to compute Doppler shift: fd l Doppler spread, BD, is defined as the maximum Doppler shift: fm = v/l l If the baseband signal bandwidth is much
greater than BD then effect of Doppler spread is negligible at the receiver. Coherence Time Coherence time is the time duration over which the channel impulse response is essentially invariant.
If the symbol period of the baseband signal (reciprocal of the baseband signal bandwidth) is greater the coherence time, than the signal will distort, since channel will change during the transmission of the signal .
T S Coherence time (TC) is defined as:
TC 1 TC » fm
f2 f1
Dt=t2 -t1 t1 t2 Coherence Time
Coherence time is also defined as: 9 0.423 TC » 2 = 16pf m f m
Coherence time definition implies that two signalsarriving with a time
separation greater than TC are affected differently by the channel. Types of Small-scale Fading Small-scale Fading (Based on Multipath Tİme Delay Spread)
Flat Fading Frequency Selective Fading
1. BW Signal < BW of Channel 1.BW Signal > Bw of Channel 2. Delay Spread < Symbol Period 2.Delay Spread > Symbol Period
Small-scale Fading (Based on Doppler Spread)
Fast Fading Slow Fading
1.Low Doppler Spread 1.High Doppler Spread 2.Coherence Time > Symbol Period 2.Coherence Time < Symbol Period 3.Channel variations smaller than baseband 3.Channel variations faster than baseband signal variations signal variations Flat Fading l Occurs when the amplitude of the received signal changes with time l For example according to Rayleigh Distribution l Occurs when symbol period of the transmitted signal is much larger than the Delay Spread of the channel § Bandwidth of the applied signal is narrow. l May cause deep fades. § Increase the transmit power to combat this situation. Flat Fading s(t) r(t) h(t,t)
t << TS
0 TS 0 t 0TS+t
Occurs when: BC: Coherence bandwidth B << B S C BS: Signal bandwidth and TS: Symbol period T >> s S t st: Delay Spread Frequency Selective Fading l Occurs when channel multipath delay spread is greater than the symbol period. l Symbols face time dispersion l Channel induces Intersymbol Interference (ISI) l Bandwidth of the signal s(t) is wider than the channel impulse response. Frequency Selective Fading s(t) r(t) h(t,t)
t >> TS
0 T T +t S 0 t 0 TS S
Causes distortion of the received baseband signal
Causes Inter-Symbol Interference (ISI) Occurs when: As a rule of thumb: T < s BS >BC S t and
TS < st Fast Fading l Due to Doppler Spread l Rate of change of the channel characteristics is larger than the Rate of change of the transmitted signal l The channel changes during a symbol period. l The channel changes because of receiver motion. l Coherence time of the channel is smaller than the symbol period of the transmitter signal
Occurs when: BS: Bandwidth of the signal B < B S D BD: Doppler Spread and TS: Symbol Period T >T S C TC: Coherence Bandwidth Slow Fading l Due to Doppler Spread l Rate of change of the channel characteristics is much smaller than the Rate of change of the transmitted signal
Occurs when: BS: Bandwidth of the signal B >> B S D BD: Doppler Spread and TS: Symbol Period T << T S C TC: Coherence Bandwidth Different Types of Fading
TS
Flat Fast Flat Slow Fading Fading
Symbol Period of Transmitting Signal
st Frequency Selective Frequency Selective Slow Fading Fast Fading
TC TS Transmitted Symbol Period
With Respect To SYMBOL PERIOD Different Types of Fading
BS Frequency Selective Frequency Selective Fast Fading Slow Fading Transmitted
Baseband BC Signal Bandwidth
Flat Fast Flat Slow Fading Fading
BD BS Transmitted Baseband Signal Bandwidth
With Respect To BASEBAND SIGNAL BANDWIDTH Fading Distributions l Describes how the received signal amplitude changes with time. l Remember that the received signal is combination of multiple signals arriving from different directions, phases and amplitudes. l With the received signal we mean the baseband signal, namely the envelope of the received signal (i.e. r(t)). l Its is a statistical characterization of the multipath fading. l Two distributions § Rayleigh Fading § RiceanFading Rayleigh and Ricean Distributions l Describes the received signal envelope distribution for channels, where all the components are non-LOS: l i.e. there is no line-of–sight (LOS) component. l Describes the received signal envelope distribution for channels where one of the multipath components is LOS component. l i.e. there is one LOS component. Rayleigh Fading Rayleigh
Rayleigh distribution has the probability density function (PDF)given by:
æ r 2 ö ì ç - ÷ r ç 2 ÷ ï eè 2s ø (0 £ r £ ¥) p(r) = ís 2 ï î0 (r < 0)
s2 is the time average power of the received signal before envelope detection. s is the rmsvalue of the received voltage signal before envelope detection
Remember: 2 (see end of slides 5) P (average power) µ Vrms Rayleigh The probability that the envelope of the received signal does not exceed a specified value of R is given by the CDF:
R R2 - P(R) = P (r £ R) = p(r)dr = 1- e 2s 2 r ò 0 ¥ p r = E[r] = rp(r)dr = s = 1.2533s mean ò 0 2 1 rmedian r = 1.177s found by solving = p(r)dr median ò 2 0
rrms = 2s Rayleigh PDF 0.7 0.6065/s 0.6 mean = 1.2533s median = 1.177s 0.5 variance = 0.4292s2
0.4
0.3
0.2
0.1
0 0 s1 2s2 3s3 4s4 5s5 RiceanDistribution l When there is a stationary (non-fading) LOS signal present, then the envelope distribution is Ricean. l The Riceandistribution degenerates to Rayleigh when the dominant component fades away. Level Crossing Rate (LCR)
Threshold(R)
LCR is defined as the expected rate at which the Rayleigh fading envelope, normalized to the local rmssignal level, crosses a specified threshold level R in a positive going direction. It is given by:
-r 2 N R = 2p fm re
where
r = R / rrms (specfied envelope value normalized to rms)
N R : crossings per second Average Fade Duration Defined as the average period of time for which the received signal is belowa specified level R.
For Rayleigh distributed fading signal, it is given by:
1 1 2 t = Pr[r £ R] = (1- e-r ) N R N R 2 er -1 R t = , r = rfm 2p rrms Statistical Models for Multipath Fading Channels l Clarke's Model for Flat Fading l Two-ray Rayleigh Fading Model l Salehand Valenzuela Indoor Statistical Model l SIRCIM and SMRCIM Indoor and Outdoor Statistical Models Fading Model – Gilbert-Elliot Model Fade Period Signal Amplitude
Threshold
Time t
Good Bad (Non-fade) (Fade) Gilbert-Elliot Model 1/AFD
Good Bad (Non-fade) (Fade) 1/ANFD
The channel is modeled as a Two-State Markov Chain. Each state duration is memory-less and exponentially distributed.
The rate going from Good to Bad state is: 1/AFD (AFD: AvgFade Duration) The rate going from Bad to Good state is: 1/ANFD (ANFD: AvgNon-Fade Duration)