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6uC;,;/Xu&_ x$&7XXx/c81,'c P7PH{>q*"xxxxWWxxxWWkkxxxA.SSxSSJJSJSSSSSS8888JSSSSSSSSS.xJxJxJxJxJorJiJiJiJiJ8.8.8.8.{SxSxSxSxS{S{S{S{SxSxJ{xSxSxS{S`SxSxSxSrSrSrS{SiSiSiSiSxSxSxSxSxS{SS.SSSSz]SSuSiSiSi.i.{c{S{SxSxSxoSoSZAZSZSiSiS{S{S{S{S{SxxSlJ8SSS88/NxxxSSS8JDDSSSSSS;SSSS;8SVVS++SSffSSxSc]]8V;"xxSxWxxS唔S88xfxxxxxxxxxxd8SxS]SxoS8SxJS`xrxxxxxxxxxxPxxxxxxdofxGcxxxxxxxSxxxxxxxJxxxxJxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx8xxx8xxx8xxx8xxxxxxxxxxxxxxfi]f]oJiAlJ{JxJ8.uJo]]{JoSxJxf`SfSSiJxJofx]fffxi{8SxxxfJffffd88SSSSx{SSSxxxf8fSJ8dY Sa#4L p# !#March 1993`PT$5DOC: IEEE P802.1193/42Ѓ
Yԟ#Xw P7[AXP#
Submission'page #Xw P7[AXP#1 PREDICTION OF COVERAGE FOR DS MODULATION IN A
>
FREQUENCY SELECTIVE CHANNEL
1"Robert J. Achatz
U.S. Department of Commerce
National Telecommunications and Information Administration
F
Institute for Telecommunication Sciences
$325 Broadway
Boulder, Colorado 80303 USA
eTelephone: (303) 4973498
x#(Rev. 03/01/93
PERFORMANCE PREDICTION
The performance of a wireless modulation technique can be evaluated by its ability to transmit
data at a specified rate and quality of service (QOS) over a given percentage of a coverage area.
YQuality of service in this context refers to bit error rate (BER) or probability of a bit error (Pe). For
Yexample, wireless data can be transmitted at 2 Mbps with a maximum 1x106 Pe over 60 % of a room.
A summary of this performance prediction for a specified data rate can be displayed by a cumulative
Ydistribution function (CDF) of Pe versus percent of locations in the coverage area with that Pe or less.
A method for performing this prediction with measured impulse responses will now be described.
COVERAGE PREDICTION METHOD
If the data symbol period is less than the delay spread of the channel, the receiver sampling
Ywindow will have more than one data symbol within its limits. The received signal strength and Pe are
Ydependent on the values of the data symbols within the sampling window. Thus the Pe must be
averaged over all possible data sequences within the sampling window. All data sequences are assumed
to be equally likely since each symbol's value occurs with equal probability.
#0*$$Ԍ
The maximum and minimum indexes of the data symbols within the sampling window are first
determined. The maximum and minimum data indexes are computed by
d(min) = floor((tstds)/T)
d(max) = ceil(ts/T)
where ts is the sampling time in seconds
tds is the delay spread in seconds
T is the symbol period in seconds
ceil function rounds to higher integer
floor function rounds to lower integer
An array of all possible data sequences within these limits is then built. Data symbols can be either +1
or 1.
YsThe Pe for a BPSK modulator in Additive White Gaussian Noise (AWGN) without Intersymbol
Interference (ISI) is
Y-!# dddddddN :XP_e~=~Q sqrt {(2Eb/No)}x6X@87X@x6X@87X@x6X@87X@_P+e
_Q_EbI_No:_[99O_(i_2_/9_):߷$T$T$-T$T$!!S$$
where Eb is the signal energy per bit
No is the noise energy per Hertz
This expression can also be used for BPSK DS provided no wideband or narrowband jammers
are present.
0s#0*$$S$!!0Ԍ
Y]To determine the Pe with ISI and AWGN given the data sequence and impulse response
measurement, h(t), we must determine the demodulator's response to the "direct path" and the
remaining "multiple paths". The direct path component (DPC) is assumed to have zero delay and the
same phase as the sampling clock.
A# ]dddddddN X&stack {)_o ~=~0#_o~=~_s}
x6X@87X@x6X@87X@x6X@87X@39)__o~+ob+sH9&_90ߖ$T$T$T$T$!AS$$The remaining multiple path component's (MPC) delays and phases are referenced to the DPC's delay
and phase. Using the demodulators response, ro, from the appendix
a# dddddNddN :X%r_o~=~r~left ( )_s~+~mT~ right )~=x6X@87X@x6X@87X@x6X@87X@_r+o
_rq+s_mT:___e1l_):߷$T$T$T$T$!aS$$1# (ddddd>`ddN XSQRT E~Left( d_o_o~+ ~sum from {1=`inf} to inf~sum from{K=1}
to inf~d_1~~beta_k~cos (3_o ( )_K``)_s~)right ).`.`. x6X@87X@x6X@87X@x6X@87X@OQNhQjQiHNoHqHpI IEdoo+Kdm ko
K(sa3Z
))]+1++7+ff+
1-+1
cos}(().%..1$T$T$GT$T$!S$$# wddddd^ddN ,XBLeft (_s~~_K~)~{R'}_{cc'}~(t_,`n`) right )~+~{n'}`(+)x6X@87X@x6X@87X@x6X@87X@el<_ _++s+K_R+ccb_t_n
_nP_ddqY@ _
[_4_)_(_,6_)
_(_),ߩ$T$T$T$T$!S$$
Further simplifying
(# !ddddddddN X1r_o~=~SQRT E~ left ( DPC~+~MPC right )~+~{n'}~(t)x6X@87X@x6X@87X@x6X@87X@_r+oy_Ew_DPC_MPC_na
_t:_7__e y9
9OI9do9k _(
_)($T$T$`T$T$!S$$
where
# ;a'dddddPsddN X[{n'} (t)~=~ int from {tT} to t~n (t) SQRT {2 over
T}~cos~({3'} t~+~{'}) {c'} (t) dtx6X@87X@x6X@87X@x6X@87X@nGtt_+t+TntTxt(c]tMdtI+LIH
II()()d2wcos7
(
)()LH9TNSaR2W0
3
ߘ$T$T$T$T$!S$$y#0*$$a]S$WAS$8aS$^wS$!S$6$a'S$*Ԍ
For BPSK demodulation, an error occurs if a 1 is sent but a voltage less than 0 is detected or a 1
is sent and a voltage greater than or equal to 0 is detected. Since the data sequence and h(t) are known
in r(t), n(t) is the only random variable. Thus
# s
ddddd>ddN ,XLPe~=~1 over 2~Pr~Left ( SQRT E~ (`DPC~+~MPC`
)~+`N~<~0~line~1~\sent` right )x6X@87X@x6X@87X@x6X@87X@!PewPrXEDPCF MPC"N
sentiv 2WelXO182((
)<
0:1,ߩ$T$T$tT$T$!S$$!# sdddddddN CXQ+~~~1 over 2~Pr~ left ( ` SQRT E~ (DPC~+~MPC`)~+~N~ >=
~0~line~1~\sent` right )x6X@87X@x6X@87X@x6X@87X@!QVt
\2'elNO182(
)
01PrNEDPC& MPCDNsentC$T$T$ T$T$!!S$$where N is a noise voltage chosen from the noise probability density function (PDF).
If we assume n(t) to be zero mean AWGN we need only compute the variance of the demodulated noise
to determine its PDF.
JA# wdddddddN Xt%_{nn}^2~=~ ~left (~int from {sT} to S~n(s)~sqrt {2 over
T}~cos ({3'} s~+~{-'})~c(s)~ds~.~.~. right ) x6X@87X@x6X@87X@x6X@87X@%3
-b2(y)Ad29 cos
(+)s(c){.K..nnS+si+TnsATscs3dsY+I
I1h1Nj1j1ioNqqpLHITINSIaR20J$T$T$0T$T$!AS$$a# wIdddddddN 5X`~~Left (~~~~Int from {tT} to t~n (t)~sqrt {2 over T}~cos
({3'} t~+~{-'})~c (t)~dt right )x6X@87X@x6X@87X@x6X@87X@hNjjioNqqpLHTNSaR20tb+t+TntT{
t
ctdt+O
IKI"()d2cos: ()() 3-5߲$T$T$T$T$!aS$$# ;"dddddIsddN MXt%_{nn}^2~=~2 over T~~int from {sT} to s~~int from {tT} to
t~~ ~left (`n(s)~n`(t) right)~c (s) c (t) . . . x6X@87X@x6X@87X@x6X@87X@!%b2d2( ) ()
()u(e).U..nnTNs+s+Tt+t>+TZnJ s
nt
c
sctN++>207LLdkM$T$T$JT$T$!S$$j# z&dddddddN X?cos~( {3'}` s~+~{-'})~cos ( {3'}~ t~ +~ {-'})~dt~dsx6X@87X@x6X@87X@x6X@87X@8cos8(r8)B8cos8(8)J838-"83
8-z8Fzz
8`z)8sC 8t\8dt
8dsj$T$T$T$T$!S$$
#0*$$a
S$S$z!S$AIS$#"a"S$t&z&S$(Ԍbut
# sdddddddN !XS ~ left (~ n(s)~n~(t)~right )~=~~~~~stack {No/ 2} over
0~~{t=s} over \otherwisex6X@87X@x6X@87X@x6X@87X@0?dVk9 7n~snta No
ts_8 otherwise()()Q
/
2
80!ߞ$T$T$]T$T$!S$$=# dddddEddN X\and~ at~t~=~sx6X@87X@x6X@87X@x6X@87X@8and8at8t8s8=$T$T$T$T$!S$$
# sdddddddN X`cos`({3'}_s~+~{,'})~cos~({3'}~t~+~{,'})~=~1 over
2~(`cos~(2{3'}~+~2{,'})~+~1`)x6X@87X@x6X@87X@x6X@87X@!cos(H)cos()
1
82(Lcos(224)1b)3y,P3,3e,A
sq t2
ߓ$T$T$ T$T$!S$$!# Iddddd ddN Xc`(s)~c`(t)~=~1
x6X@87X@x6X@87X@x6X@87X@8c8s`8cf8t8(8)8(8)~818ߌ$T$T$T$T$!!S$$therefore
iA# sadddddddN X%_{nn}^2~=~{No} over 2x6X@87X@x6X@87X@x6X@87X@!%282nnNo
i$T$T$T$T$!AS$$Knowing the mean and variance of the AWGN we can write the noise PDF as
a# bddddd>4ddN XGf_n`()~=~{e ^{^{2}/2 (No/2)} over sqrt {2!`({No} over 2`)}}x6X@87X@x6X@87X@x6X@87X@ fnfeYNo]No ( )dd-2i/2 (/I2)2(82)p ! 2NNTSR
5$T$T$T$T$!aS$$# !ddddd8
ddN WX0~~~~~~~~~~=~{e^{^2/No} over sqrt {! No}}x6X@87X@x6X@87X@x6X@87X@)pGO?e]No8NoyG8!dd2
/W$T$T$cT$T$!S$$if
# &dddddddN )XHP_e~=~Pr~ left ( sqrt E~(DPC~+~MPC )~+`N~ >=
~0~line~1~\sent~ right )x6X@87X@x6X@87X@x6X@87X@_P+e
_Prc_E_DPC;_MPC_Na_sent:_|_k_s
__q
_ _R9e9lc99O3_( _)_0_1)ߦ$T$T$dT$T$!S$$j# )ddddd<ddN X>P_e~=~Pr~left (`N~>=~sqrt E~(DPC~+~MPC`)~line~1~\sent~right )x6X@87X@x6X@87X@x6X@87X@_P+e
_Pr_N_E_DPCy _MPC
_sent:_b___ o_R9e9l929Oq_(
_)_1j$T$T$"T$T$!S$$0*$$ S$S$S${IS$B!aS$7AbS$!a!S$'%&S$:))S$+Ԓ
# VddddduddN hXXP_e~=~int from {{sqrt E`(DPC ~+~MPC )}} to inf~e ^{}^{2/No}
/`sqrt{ ! No}~dx6X@87X@x6X@87X@x6X@87X@"Pey+E'+DPC+MPCs"edd$No
"Nof"d:"Q+dLry
N * O+(+)dd2dd/"/Td "!"h$T$T$]T$T$!S$$t# v
ddddd,ddN Xu~=~`/`sqrt {N_o}x6X@87X@x6X@87X@x6X@87X@_uI_N+o__L_/)IOt$T$T$T$T$!S$$u!# 1dddddddN Xdu~sqrt {N_o}~=~dx6X@87X@x6X@87X@x6X@87X@_du_NI+o_d)ZO_9_u$T$T$T$T$!!S$$then
A# IdddddI;ddN uXdP_e~=~~1 over sqrt !~~int from {{sqrt {E over N_o}~left (
DPC~+~MPC right )}} to inf~~e^{u^2} dux6X@87X@x6X@87X@x6X@87X@!PeRE/GNdd%oLDPCMPC e
uUduIvi
&OZLYPG~d~k,1dd?2&%!u$T$T$T$T$!AS$$a# (dddddr
`ddN CX?P_e~=~1 over 2~erfc~ left (sqrt {E over No}~(DPC~+~MPC) right
)x6X@87X@x6X@87X@x6X@87X@!PeWerfcE8NoDPC?MPCIo
27Nhji
No
q
pNNTSR
_1_827()C$T$T$JT$T$!aS$$# (zddddd
`ddN (X9P_e~=~Q~left ( sqrt {{2E} over {N_o}}~(DPC~+~MPC) right )x6X@87X@x6X@87X@x6X@87X@Pe
Q<E"_N+oDPCMPC:Nhjij
Noj
qj
pNNTSR
<2( )(ߥ$T$T$T$T$!S$$
In the event that excessive ISI causes a symbol error, an N which causes the voltage to cross zero again
Ycorrects the ISI error. In this case the Pe is
# (%dddddr
`ddN qXAP_e~=~1.0 ~Q left (~sqrt {{2E} over
{N_o}}~(DPC~+~MPC`)~right )x6X@87X@x6X@87X@x6X@87X@PeBQ<EZ_N+o2DPC
MPC:r
1.0J<2(@)Nhji
No
q
pNRNTRSRR
"q$T$T$T$T$!S$$
#0*$$qS$v
S$1S${!IS$AS$aazS$ %S$)Ԍ
Y]When the Pe is computed for all possible data sequences for a given impulse response measurement the
Yensemble of Pe are averaged and stored.
OTHER METHODS OF PERFORMANCE PREDICTION
Y. Chen [Chen, 1992] computed the average Pe for a DS transceiver in a frequency selective channel.
Y The Pe was averaged over all likely multipath component amplitudes, phases, and delays. Chen made
the assumption that delay spread did not exceed 2 symbol periods which reduced the number of possible
data sequences to 8 since only the current and two previous data symbols could appear in the sampling
window. Probability densities for the multipath component amplitudes, phases and delays were taken
from Saleh's [Saleh,1987] indoor model.
Y Chuang [Chuang,1987] generated Pe as a function of normalized rms delay spread using
Devasirvathem's [Devasirvathem, 1987] indoor power delay profile measurements. The study was
Ysinspired by Bello's GWSSUS channel research [Bello,1963] which predicted Pe as a function of the
power delay profile's rms delay spread to symbol period ratio, d. A large number of impulse responses
were stochastically generated from one of Devasirvatham's averaged power delay profiles. The
generated impulse responses are reasonable estimates of impulse responses that may exist in the
neighborhood of Devasirvatham's 4 foot measurement square. For each simulated impulse response,
YDPe was calculated. This Pe was then averaged over other impulse responses having the same d. A graph
Yof average Pe versus d was constructed and compared with Bello's prediction methods. The results
compared favorably for d < .2.
Winter [Winter,1985] predicted the outage of a receiver with maximal ratio combining antenna
diversity as a function of rms delay spread to symbol period, d. Computation of the BER and
r#0*$$Ԍ
probability of an outage was derived in terms of the squared sum of the antenna weighting function.
Outage was defined as the probability that a communication link cannot meet a specified BER
requirement.
Thoma [Thoma,1992] predicted the performance of pi/4 DQPSK modulation while moving.
Rappaport's [Rappaport,1990] channel model was used to simulate 1,125 complex impulse responses
over 1 meter. The error distribution was then calculated for a given data rate and velocity. At a
highway speed of 60 mph and a data rate of 1 Mbps the detection of 37,313 symbols can be simulated.
At a walking speed of 3.75 mph and a data rate of 1 Mbps the detection of 597,014 symbols can be
simulated. Performance was measured by outage probability where outage probability is defined as the
probability that the number of errors in a code block exceeds a threshold. The number of errors in a
code block below this threshold are assumed to be correctable with forward error correction and
therefore unimportant.
l%REFERENCES
[Bello, 1963], Bello, P.A., Nelin, B.D., "The Effect of Frequency Selective Fading on the Binary Error
YProbabilities of Incoherent and Differentially Coherent Matched Filter Receivers",#Xu&_ x$&7/XX# IEEE Trans
Yon Communications Systems#Xw P7[AXP#, June 1963
[Chen, 1992], K.C. Chen, "Performance Comparison Between Direct Sequence and Slow Frequency
YHopped Spread Sprectrum Transmission in Indoor Multipath Fading Channels",#Xu&_ x$&7/XX# doc: !!HIEEE T$T$MWP802.1192/80, July 1992#Xw P7[AXP#
[Chuang, 1987], J.C.I. Chuang, "The Effects of Time Delay Spread on Portable Radio Communications
Y6!Channels with Digital Modulation",#Xu&_ x$&7/XX# IEEE Journal on Selected Areas in Communications,#Xw P7[AXP# Vol
SAC5, No. 5, June 1987
#0*$$Ԍ
[Papazian, 1992a], Papazian, P.B., Achatz, R.J., "Wideband Propagation Measurements for Wireless
YIndoor Communication", IEE 802 Submission,#Xu&_ x$&7/XX# IEE P802.1192/83,#Xw P7[AXP# pp. 128
[Papazian, 1992b], Papazian, P.B., ety al., "Wideband Propagation Measurements for Wireless Infdoor
YaCommunication", #Xu&_ x$&7/XX#NTIA Report 93292,#Xw P7[AXP# January 1993
[Rappaport, 1990], Rappaport, T.S., Seidel, S.Y., Takamizawa, K., "Statistical Channel Impulse T$T$MResponse Model for Factory and Open Plan Building Radio Communication System Design", T$T$MY
#Xu&_ x$&7/XX#IEEE Trans. on Communications, Vol 39, No. 5#Xw P7[AXP#, May 1991
Y[Saleh, 1987], Saleh, A.A.M., "A Statistical Model for Indoor Multipath Propagation",#Xu&_ x$&7/XX# IEEE Journal
Yon Selected Areas in Communications, Vol SAC5, No. 2,#Xw P7[AXP# February 1987
[Thoma, 1992], Thoma, B., "Simulation of Bit Error Performance and Outage Probability of pi/4 T$T$MDQPSK in Frequency Selective Indoor Radio Channels Using A Measurement Based Channel T$T$MYlModel",#Xu&_ x$&7/XX# IEEE Trans. Globecomm 1992,#Xw P7[AXP# Dec 1992
[Winters, 1985], Winters, J.H., Yeh, Y.S., "On the Performance of Wideband Digital Radio T$T$MY)Transmission Within Buildings Using Diversity", #Xu&_ x$&7/XX#IEEE Globecom 1985 Proceedings,#Xw P7[AXP# 1985
! xP#^4L P: P#
Submission'Page #Xq4L P: [AXP#
# 0*$$Ԍ
&APPENDIX
This expression for the response of the demodulator assumes that delay spread is larger than symbol
period. No limit on delay spread to symbol period ratio is imposed. The sampling clock can have any
value between zero and the maximum delay spread.
WHERE
144
Transmitted Symbol Index
m44
Received Symbol Index
Yd144
1TH transmitted symbol
Y
k, k, )K
Amplitude, phase, and delay of kth. multipath component
Y)S, s44
Delay and phase of sample clock
Yy3o44
Carrier frequency
YbR'cc'(t)
Partial cross correlation function
YKt44
Sub chip offset
Y4nTc44
Integral chip offset
'(n)44
Code correlation
YTc44
Chip period
YNc44
Number of chips in PW word
c (t)44
PN code waveform
T44
PN word period
YNA, ND
Number of chips that agree/disagree
#
0*$$Ԍ
&APPENDIX
#
dddddddN Xr~()~+~mT~)~=~r_o~=x6X@87X@x6X@87X@x6X@87X@_r_mT_rW+o_(7_)Z_)___ߖ$T$T$_T$T$!S$$
L# xddddd9#ddN Xsqrt E~~sum from {1=inf} to inf~d_1~sum from K~ left (
`_K~cos`(3_o` ()_{k}~~)_s)~+`_s~~_k~)~{R'} _{cc'}~(t_e,~n
`right )x6X@87X@x6X@87X@x6X@87X@OIPIelETd+KK okssk`Rcct>e^n+1+%+u+1
!dd 3
)})|'cos(
(:
))F(,L$T$T$
T$T$!S$$
S# ;dddddusddN XW+~int from {tT} to t~n(t)~~sqrt {2 over T}~~cos~({3_o'}~
t~+~{'}`)~{c'}~(t)~ dtx6X@87X@x6X@87X@x6X@87X@2+ bOI.ILHlTNSaR202t+t+T*ntT o
t
c*trdt()d2cos()(): 3S$T$T$bT$T$!S$$where
|!# ddddddddN X{R'}_{CC'}~(t_,`)~=~left \{~(1~~{t_} OVER
{Tc})~{'}(n)~+~{t_} over {Tc}~{'}~(n+1)~\if~~T~<=
)_s~~()_K~~(m1)T)~