04/22/23
1
4. Mobile Radio Propagation
4.1 Small Scale Multipath Propagation
4.2 Impulse Response on a Multipath Channel
4.3 Small Scale Path Measurements
4.3 Frequency Domain Channel Sounding
4.4 Multipath Channel Parameters
4.5 Types of Small Scale Fading
4.6 Rayleigh & Ricean Distributions
4.7 Statistical Models for Multipath Fading Channels
4. mobile radio propagation
04/22/23
2
4. Mobile Radio Propagation – Small Scale Fading - Multipath
Fading: rapid fluctuations in amplitude over short time & distance
• antennas radiate electromagnetic (EM) waves over a wide angle
• EM waves reflect off of random objects
• transmitted signal arrives at a point from many different paths with different phases
• wide variation in phase & amplitude – depending on distribution• large-scale effects are negligible – not distance dependent
04/22/23
3
coherence bandwidth, Bc , is related to channel’s multipath structure• measure of maximum frequency difference between signals with strongly correlated amplitudes
Fading• multipath components (MPCs) – reflected EM wave components• if time delayof MPCs ≈ symbol duration then interference (fading) can result
d0
d1
d2 t0 ≈ t1 t2
04/22/23
4
4.1 Small Scale Multipath Propagation
important effects:(i) rapid changes in signal strength over small distances & time intervals
(ii) Doppler shifts on different multipaths random frequency modulation (iii) time dispersion (echoes) caused by multipath
Multipath in Urban Areas • NLOS path - mobile antenna height < surrounding structure’s height• even with LOS path multipath exists• received signal at any point consists of many plane waves with
- randomly distributed amplitudes phases, angles - components combine vectorially at receiver - causes distorted or faded signal reception
small scale MP propagation
04/22/23
5
Spatial Fading: all objects are stationary – only mobile moves• spatial variations appear as temporal variations at mobile receiver• constructive & destructive effects of multipath in multipath field• moving objects disturb the field – constructively or destructively
High Mobility: receiver passes through several fades in a short time
Low Mobility: extended time in a deep fade possible
Doppler shift: relative mobility between TX & RX proportional to direction & velocity
Characteristics of Fading
04/22/23
6
(i) multipath propagation and multipath components (MPCs) • presence of objects cause reflection & scattering• dissipates energy in amplitude, phase, time• multiple versions arrive at different times and spatial orientation• causes signal smearing - intersymbol interference (ISI)
(ii) relative speed between Tx-Rx results in Doppler shift – random frequency modulation
(iii) moving objects: induce time varying Doppler shift on MPCs• if speed of objects > mobile speed objects dominates fading • otherwise object speed can be ignored• impact also depends on size & shape of objects
4.1.1: Four Factors of Fading
04/22/23
7
• if signal bandwidth > multipath channel ‘bandwidth’- received signal will be distorted- signal strength won’t fade over a small area – insignificant small scale fading
• if signal bandwidth << physical channel bandwidth- rapidly changing signal amplitude - fading - but signal won’t be distorted
• signal bandwidth is proportional to 1/symbol duration (width)
(iv) bandwidth of transmitted signal
-0.4 -0.2 0 0.2 0.4time
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
edutilpma
2.5 3.0 4.0 4.5 5.0 5.5 6.0Frequency
-40
-30
-20
-10
0
PSD (d
B)
-0.4 -0.2 0 0.2 0.4time
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
edutilpma
2.5 3.0 4.0 4.5 5.0 5.5 6.0Frequency
-40
-30
-20
-10
0
PSD
(dB) 4.4GHz
0.23ns
04/22/23
8
4.1.2 Doppler shiftS = signal sourcev = velocityd = distance Y-X on mobiles path of movementt = d/vl = dcos1 = vtcos 1
if S is far away, assume 1 ≈ 2
=
cos22 tvl
(4.1)phase shift:
fd =
cos21 v
t
(4.2)
Doppler shift = relative frequency change during t
d YX
l
21
S
v
04/22/23
9
e.g. fc = 1850 MHz = c/fc = 0.162m v = 60mph = 28.62 m/s
a. mobile moving directly towards transmitter: = 0o cos = 1fd = v/ = 160Hz
f = fc + fd = 1850.00016MHz
b. mobile moving directly away from transmitter = 180o cos = -1
fd = -v/ = -160Hz
f = fc + fd = 1849.99984MHz
c. mobile moving to angle of signal’s arrival = 90o cos = 0 fd = 0
S
v
Sv
04/22/23
10
4.2 Impulse Response on a Multipath Channel
Channel Model used to predict & compare performance of(i) different communications systems(ii) different transmission bandwidths for a particular condition
Small Scale Signal Variations are related to RF channels impulse response
• channel impulse response is a wideband channel characterization• contains all information necessary to simulate/analyze any transmission• similar to concept of transfer function and input/ouput in control systems
h(t)x(t) y(t)
4.2 impulse response of MP channel
04/22/23
11
(1) Mobile RF Channel Models
(2) Multipath Channels and Delay Spread
(3) Channel’s Power Delay Profile
Organization of 4.2:
4.2.1 Bandwidth vs Received Power
(2) Continuous Wave (CW) signal
(1) wideband pulsed transmitted RF signal, x(t)
04/22/23
12
i. linear time-invariant (LTI) system
• MPCs have variable propagation delays - depending on receiver location• impulse response of LTI channel is a function of receiver position and does not depend on time
ii. linear filter with time-varying impulse response• time variation is due to receiver motion• at time t, many waves are arriving - filter’s nature is result of sum of all amplitudes & delays
(1) Mobile RF Channel Models
Channel Classification based only on h(t,) , noise is not considered
04/22/23
13
1.1 Assume time variation is strictly from receiver motion v = velocity d = fixed position between Tx & Rx
and y(d,t) =
t
dtdhx ),()( (4.4)
if channel is causal (no output until input is applied) then for t < 0 h(d,t) = 0
let y(d,t) = received signal at position d & time t h(d,t) = channel impulse responsex(t) = transmitted signal
then y(d,t) = x(t) h(d,t) =
dtdhx ),()( (4.3)
y(d,t)x(t) h(t,d)
04/22/23
14
1.2 Assume constant v position at any time t is given by
then y(vt,t) =
t
dtvthx ),()( (4.6)
y(t)x(t) h(vt,t)
since v is constant received signal y is just a function of t
y(t) =
t
dtvthx ),()( (4.7)
y(t) = x(t) h(vt,t) = x(t) h(d,t)
d = vt (4.5)
channel model: linear time-varying channel that changes with t & d
04/22/23
15
1.3 Assume v is constant over short intervals (distances) and let
x(t) = transmitted bandpass wave form
y(t) = received waveform
h(t,) = impulse response of time varying multipath RF channel• completely characterizes channel• is a function of variables t &
t represents time variation due to motion represents channel multipath delay for fixed t
(4.8)=
dthx ),()(
y(t) = x(t) h(t,)
y(t) is the convolution of input signal & channel impulse
04/22/23
16
Assume (reasonably) that multipath channel is • bandlimited• bandpass
h(t,) is equivalently described by its complex baseband impulse response hb(t,)
h(t, ) = Re{hb(t, ) exp(jwct) }
h(t, )=Re{hb(t, ) exp(jwct)} y(t)x(t)
i. Bandpass Channel Impulse Response Model
x(t) = channel input or transmitted signaly(t) = channel output or received signal
04/22/23
17
½ hb(t, ) r(t)c(t)
ii. Baseband Equivalent Channel Impulse Response Model
• use Complex Envelope to represent transmitted & received signals
r(t) = c(t) ½ hb(t, ) (4.9)
c(t) & r(t) are complex envelope representations of x(t) & y(t)
x(t) = Re{c(t)exp(j2fct)} (4.10)
y(t) = Re{r(t)exp(j2fct)} (4.11)
Complex Envelope properties cause ‘½ ‘ factor in 4.9
• needed to represent passband RF system at baseband
• low pass characterization removes high frequency variations from carrier – makes for easier signal analysis
04/22/23
18
overbar denotes - ensemble average for stochastic signal or - time average for deterministic signal
)(21)( 22 tctx
Average Power of bandpass signal, x(t) is given by (COU93)
04/22/23
19
(2) Multipath Channels and Delay SpreadRx receives a series of replicas of transmitted signal that are attenuated, time-delayed, phase-shifted
All MPCs received in ith bin are represented as one Resolvable MPC with delay i
0 3 2 1 4
• each bin has time delay width given by
= i+1 - i
Let = multipath delay axis of impulse response
• discretize into equal time-delay segments or excess delay bins
0 3 2 1 4
0 = 1st arriving signal if 0 = 0 = 1
• i = i· with N total equally spaced components
MP channel Delay Spread
04/22/23
20
• propagation delay from transmitter to receiver is often neglected by setting excess time delay of 1st MPC = 0
0 = 0
Excess Delay - denoted by i, is the relative delay of ith MPC relative to 1st arriving component
Maximum Excess Delay of the channel is given by MAX = N
Quantization of Delay Bins determines channel’s time delay resolution• model’s useful frequency span = maximum signal bandwidth that can be analyzed is given as
e.g. if = 1ns maximum signal bandwidth limited to 2GHz
model’s frequency span =
12
04/22/23
21
hb(t, )
0 1 2 3 4 5 6 7 8 9
t1
t0
t2
t3
Time-varying Discrete Impulse Response Model for multipath channel• different snapshots of hb(t, ) at times ti
t0 t1 t2 t3 t
04/22/23
22
hb(t, )
(t0)
(t1)
(t2)
t1
t2
t3
t0
t
0 1 2 3 4 … N-2 N-1
(t3)
Time-varying Discrete Impulse Response Model for multipath channel• time = t varies into the page• time delay bins are quantized to widths =
delayed multipath components
time
ampl
itude
04/22/23
23
hb(t,) = )()],()(2exp[),(1
0
tttfjt iiic
N
ii
(4.12)
Baseband Impulse Response of multipath channel:
N = total possible MPCs (bins)
() = unit impulse function – determines specific multipath bins which have components at time t & excess delay i
i(t) = excess delay of ith MPC at time t
i(t, ) = real amplitudes of ith MPC at time t
• if i(t,) = 0 excess delay bin has no MPC at time t & delay i
2fci(t)+i(t, ) = all mechanisms for phase shifts of one MPC in ith bin
• includes free-space propagation of ith MPC at time t & any additional phase shifts in channel
• often represented as one channel variable: i(t, ) = 2fci(t)+i(t, )
04/22/23
24
MPC model in a delay bin is affected by choice of and channel properties (geometry)
for example:
(i) with 1 MPC arriving in a delay bin - amplitude will not fade significantly over a small area
(ii) with 2 or more unresolveable MPCs arriving in a delay bin • they vectorially combine to yield instantaneous amplitude & phase of a single MPC model
• multipath amplitude in a bin will fade over local area
04/22/23
25
Assume channel impulse response is either time invariant or stationary over a small time-scale or distance interval
• hb(t,) in (4.12) can be simplified to :
hb(t,) = ii
N
ii j
)exp(1
0(4.13)
i = constant per excess delay bin
i = constant per excess delay bin
04/22/23
26
Pass Band Channel Modelsr(t) = s(t) h(t) + n(t)
• r(t) = received signal• s(t) = transmitted signal• h(t) = impulse response of channel• n(t) = AWGN
r(t) =
)()()( tndtsh
time invariant channel: i has constant amplitude
if h() =
L
lkk t
1
)( r(t) = )()(1
tntsL
lkk
time variant channel: i has time varying amplitude
if h() =
L
lkk tt
1
)()( r(t) = )()()(1
tntstL
lkk
04/22/23
27
(3) Channel’s Power Delay Profile (PDP) for small-scale channel model • determined by taking spatial average of |hb(t,)|2 over local area
• build ensemble PDP by taking several measurements of |hb(t,)|2
over a local area
• each ensemble represents possible small scale channel state
Probing pulse, p(t) is used to ‘sound’ the channel to measure hb(t,)
• p(t) approximates delta function
p(t) (t -) (4.14)
if p(t)’s time duration << impulse response of channel (Cox72,75)
• p(t)’s bandwidth >> channel bandwidth
• p(t) doesn’t need to be deconvolved from received signal, r(t) to determine relative multipath signal strengths
power delay profile
04/22/23
28
Received Power Delay Profile (PDP) in a local area given by:
P() k |hb(t,)|2 (4.15)
P() = single, time-invariant multipath power delay profile
• typically derived from many snapshots of |hb(t,)|2 averaged over local area (small scale)
• bar represents average over local area
• k = gain relating transmit power in p(t) to total power received in multipath delay profile
04/22/23
29
4.2.1 Bandwidth vs Received Power
(1) Wideband pulsed transmitted RF signal, x(t)
(2) Continuous Wave (CW) signal
• 2 extreme cases of small scale fading in the same channel – show variation in fading behavior for signals with different bandwidths
• impulse response of multipath channel is measured in a field using channel sounding techniques – aka probes
4.2.1 Bandwidth vs Received Power
04/22/23
30
(1) Consider wideband pulsed transmitted RF signal, x(t)
x(t) = Re{p(t)exp(j2fct)}
- p(t) = periodic baseband pulse train with very narrow pulse width Tbb
- MAX = maximum excess delay in channel
- TREP = pulse period, with TREP >> MAX wideband pulse p(t) will yield output hb(t,)
Tbb
TREP
Max
04/22/23
31
For all excess delay, i, of interest, assume p(t) is a square pulse with
for 0 t TbbbbMAX T/2 p(t) =
p(t) = 0 elsewhere
r(t) = )(21 1
0i
N
i
iji tpe
i
bb
bb
N
i
iji
TtrectT
e
2max
1
0r(t) =
(4.16)
Low Pass Channel Output, r(t) = hb(t) convolved with p(t)
• r(t) will closely approximate channel impulse response, hb(t)
04/22/23
32
e.g. excess delay: let MAX = 40ns and Tbb = 10ns
= 4 for 0 t Tbb,
then p(t) = 10/402
= 0 elsewhere
i
N
i
iji te
5rect21
0 =
i
bb
bb
N
i
iji
TtrectT
e
2max
1
0=
and r(t) = )(21 1
0i
N
i
iji tpe
04/22/23
33
Determine Received Power at time t0 measured power = |r(t0)|2
• |r(t0)|2 = energy received in time duration of multipath delay divided by MAX
• found by summing multipath power resolved in channel’s instantaneous multipath power delay profile, |hb(t,)|2
using 4.16 we have:
|r(t0)|2 = dttrtr )()(1 *
0max
max
(4.17)
|r(t0)|2 = dtetptpttN
i
jijij
N
j
ij
max
0
1
0
)(00
1
0max
)()()()(Re411
04/22/23
34
if all MPCs are resolved by probe p(t) then
1. the difference between any excess delay components is greater than the pulse width i j |j - i| > Tbb
(4.18)
= dtTtrectT
t kbbbb
N
kk
2
0
max1
00
2
max
max
)(1
dttptN
kkk
max
0
1
0
20
2
max
)()(Re411
2. |r(t0)|2 =
1
00
2 )(N
kk t=
and if i j
dtetptptt ijjijij
max
0
)(00 )()()()(
= 0
04/22/23
35
thus for wideband probe p(t) with narrow pulse width Tbb
if Tbb < delays between MPCs in the channel then from 4.18, total received power at any time is
• related to all powers in individual multipath components• scaled by
- ratio of p(t)’s width & amplitude - channel’s maximum observed excess delay
04/22/23
36
• E, [] denotes ensemble average in a local area over all possible values of i & i PWB = instantaneous received power from wideband pulse
• overbar denotes sample average over local measurement area- generally measured using multipath measurement equipment
E, [PWB]
1
0
21
0
2)(,
N
ii
N
i
ji
ie E (4.19)
Practically received power from MPCs is random process
• at any time t, each component has random amplitude & phase
• average small scale received power for wideband probe is found from (4.17) as
04/22/23
37
from 4.18 & 4.19: if transmitted signal is able to resolve multipaths then:
• small-scale received power = received power in each MPC• practically, amplitudes of individual MPCs don’t fluctuate widely in a local area
- received power of wideband p(t) does not fluctuate widely when receiver moves in a local area - reduced fading in a local area
next slide is continuous wave signal
04/22/23
38
(2) Continuous Wave (CW) signal
• denote complex envelope of CW signal by c(t) = 2 • CW signal transmitted in same multipath channel
hb(t,) = ii
N
ii j
)exp(1
0
• received signal’s instantaneous complex envelop given by phasor sum
r(t) =
1
0),(exp
N
iii tj (4.20)
• instantaneous received power given by:
|r(t)|2 = (4.21) 21
0),(exp
N
iii tj
04/22/23
39
channel changes as receiver moves in local area (distances )
• received signal strength varies from fluctuations in i & i
• i varies little over local area
• i varies greatly due to changes in propagation distance
• phase variations causes large fluctuations in r(t) as receiver moves
thus, since r(t) is phasor sum of individual MPCs• instantaneous phase of MPCs cause large fluctuations in r(t)• typifies small-scale fading for CW signals
04/22/23
40
average received power over a local area for CW signal is given by: E [PCW] =
21
0
)(,
N
i
ji
ie E (4.22)
E [PCW] 1011010110 ......
NN jN
jjN
jj eeeee (4.23)
)cos(2,
1
0
1
0
2ji
N
ijiij
N
i
N
ii r
(4.24)
(4.25)rij = E[i, j]
where rij = path amplitude correlation coefficient defined as:
overbar denotes average CW measurements at a mobile receiver in a local area
04/22/23
41
average received power of CW & wideband signals are equivalentIn a Small-scale Region, if 0)cos( ji and/or rij = 0 then
By comparison of 4.19 & 4.24 this situation can occur when either
(i) multipath phases are iid uniform over [0,2]• iid uniform distribution of is a reasonable assumption (iid = identically & independently distributed )
(ii) path amplitudes are uncorrelated• MPCs traverse differential path lengths that measure 102’s • MPCs are likely to arrive in random phases
if path phases are dependent on each other then path amplitudes are likely to correlated
- mechanisms affecting path phase will likely affect amplitude - highly unlikely at mobile RF frequencies
04/22/23
42
thus Received Local Ensemble Average Power of wideband & narrow band signals are equivalent in a small scale region
wideband transmission: transmitted signal bandwidth >> channel bandwidth
• multipath structure is completely resolved by received signal at any time• received power varies little – individual multipath amplitudes do not vary rapidly in a local area
narrow bandwidth: baseband signal duration > channel’s excess delay
• multipath is not resolved by received signal• large fluctuations (fading) occurs at receiver due to phase shifts of many unresolved MPCs
04/22/23
43
indoor RF channel measurements • assume measurement track = 5 (0.375m)• transmit CW signal and probing pulse signal
(i) CW transmitter fc = 4GHz = 0.075m (7.5cm)
- CW signal undergoes rapid fades with local mobility
(ii) wideband probing pulse Tbb = 10ns
- wideband measurements change little over measurement track
• Local Average Received Powers of Either are Nearly Equal !
indoor RF channel measurements
04/22/23
44
(2) Wideband Pulse, Tbb = 10ns
Rel
ativ
e R
ecei
ved
Pow
er0.
00
0.25
0.
50
0.75
1.
00
0 50 100 150 200 250 Excess Delay (ns)12 1
4 1
6
’s
(1/Tbb = 100MHz)
10dB
5 1.25ns (0.375m)
(1) Narrow Band CW Signal
=7.5cmfc= 4GHz
04/22/23
45
e.g. 4.2: discrete channel impulse response – used to model
Let MAX = max excess delay N
N = 64 multipath bins & = MAX/64
example 4.2
channel type distance MAX = MAX/N 2/1 urban channel 30km 100us 1.56us 1.28MHz2 micro-cell 1.2km 4us 62.5ns 32 MHz3 indoor 150m 500ns 7.81ns 256 MHz
(b) Maximum Bandwidth Accurately represented = 2/
(a) Width of Excess Delay Bin = MAX/N
04/22/23
46
e.g. 4.3 • mobile traveling at v = 10m/s• fc = 1GHz = c/fc = 0.3m• 2 multipath components received with properties given below
Mobile moves directly towards 1 & directly away from 2
a. find narrowband instantaneous power at intervals = 0.1s & 0.5s
b. find average narrowband power over interval 0.1s-0.5s
c. compare average narrowband & wideband received power over interval 0.1-0.5s
12v
component receive power receive power phase 1 -70dBm 100pW 0 02 -73dBm 50pW 1us 0
example 4.3
04/22/23
47
spatial interval = v · ttime interval spatial interval
0.1s 10m/s 0.1 =1m0.5s 10m/s 0.5 =
5m
21
0
),(exp
N
iii tj |r(t)|2 = (i) instantaneous power given by:
(ii) phase: mobile moves - phase of 2 components changes in opposite directions
i =
vtd 22
pWepWepW 291)50()100(200 |r(t)|2 =
at t = 0 phase of both components, i = 0 and instantaneous power given by:
04/22/23
48
at t = 0.1s
2 = 2(-10m/s)(0.1s)/0.3 = -20.94rad = -120o
1 = 2(10m/s)(0.1s)/0.3 = 20.94 rad 20.94 rad – (3·2 )rad = 2.09rad = 120o
the phase of each component given by 2vt/
pW
j
jj
jj
jpWjpW
781641087.8
1054.21054.8
10124.654.31066.85
866.05.01007.7866.05.01010
)120sin()120cos(50)120sin()120cos(100
26
266
266
266
2
instantaneous power, 2120120 )50()100(
jj epWepW |r(t)|2 =
|r(t)|2 = 78pW
04/22/23
49
1 = 2(10m/s)(t)/0.3
2 = 2(-10m/s)(t)/0.3
i =
vtd 22
phase
21
0
),(
N
i
tji
ie |r(t)|2 = instantaneous power
78.2pW-120o120o0.1
291pW-360o360o0.381.5pW-240o240o0.2
81.5pW-240o240o0.578.2pW-120o120o0.4
291pW0o0o0
instantaneous power |r(t)|2
21t
phase and instantaneous power over interval 0s - 0.5s
04/22/23
50
Average Narrowband Received Power
0.167 [2(291)pW + 2(81.5) + 2(72.2)] = 150.23pW
Average Wideband Received Power
E, [PWB]
1
0
21
0
2)(,
N
ii
N
i
ji
ieE
= 100pW + 50pW = 150pW
