4. mobile radio propagation

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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

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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

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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

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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

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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

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(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

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• 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

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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

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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

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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

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(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)

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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

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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)

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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

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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

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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

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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

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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)

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(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

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• 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

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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

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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

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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, )

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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

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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

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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

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(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

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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

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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

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(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

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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)

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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

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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

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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

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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

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• 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

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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

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(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

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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

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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

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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

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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

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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

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(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

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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

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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

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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:

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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

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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

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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