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UWB Channels:Time-Reversal Signaling
NEWCOM, Dept. 1 Meeting Paris, 13 May 2005
Erdal ArıkanBilkent University
Ankara, Turkey
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Outline
• Time-reversal signaling• UWB channel model• Signaling and achievable rates for the UWB
channel – Fixed power– Time reversal– Water filling
• Simulation results• Conclusions
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Time Reversal Signaling
If channel is reversible, hRT(t) = hTR(t).
• R receives hTR(-t) hTR(t), which is
likely to be peaky.• C receives hTR(-t) hTC(t), which is
unlikely to be peaky if C is sufficiently far from R.• hXY(t) likely to have low coherence in time and space for
high delay-bandwidth product channels, such as the UWB channel.
1) R sends an impulse
3) T transmits hRT(-t)
T R
2) T receives hRT(t)
C
4) R receives hRT(-t) hTR(t)
5) R receives hRT(-t) hTC(t)
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Correlations of channel responses
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UWB Channel (FCC 2002)
• Frequency range: 3.1–10.6 GHz
• Radiated power : < -41.3 dBm/MHz
• Min. Bandwidth : 500 MHz
• Bandwidth > 20% of center frequency
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UWB Channel Indoor Emissions Limit
0.96 1.61
GPS Band
0.96 1.61
1.993.1 10.6
GPS Band
-41 dBm/MHz
7.5 GHz
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Maximum power emission:
–41.3 dBm/MHz 7.5 GHz = 0.56 mW.
UWB systems are not energy limited.
UWB Energy
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Spread or not?
• With fixed transmitter energy:Spreading the energy uniformly over a wide band deterioration of channel estimates collapse of achievable rates
(Médard-Gallager, Telatar-Tse, Subramanian- Hajek)
• In the UWB model, transmitter energy is allowed to increase as more bandwidth is used there is no collapse of achievable rates use all available bandwidth if possible.
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UWB Channel Model• The channel is modeled as a linear filter with
additive white Gaussian noise.
• The channel impulse response follows the Saleh-Valenzula model.
+h(t)x(t) y(t)
z(t)
s(t)
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Saleh-Valenzula Model for UWB
The channel impulse response is modeled as
– X lognormal shadowing gain– L number of clusters– Tl delay of cluster l– k index over rays within a cluster k,l excess delay of ray l in cluster k– Details in Report no. 02490r0P80215
(http://grouper.ieee.org/groups/802/15/pub/2002/Nov02)
lkl
L
l
K
klk TtXth ,
1 1,)(
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Model Characteristics
Parameter Value (CM1)Line of Sight YES
Range (m) 0-4
Coherence time (s) 200
Mean excess delay (nsec) 4.9
RMS delay (nsec) 5No. multipath components within 10 dB of peak component, NP10dB
13.3
No. paths capturing 85% of energy, NP(85%) 21.4
Channel energy mean (dB) -0.5Channel energy std (dB) 2.9
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Sample of a channel impulse response
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Frequency Domain Channel Model
An OFDM-like channel with subchannels
– Zi ~ CN(0,No) are independent noise – In each use of the vector channel, a new set of Ai are
chosen from a fixed distribution– K= W Ts
where W=RF bandwidth, Ts = signaling period– Input constraint: E[ Xi
2 ] Es for each i
– Assumption: Transmitter and receiver have perfect knowledge of the channel coefficients Ai
1,...,0 , KiZXAY iiii
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Perfect Channel Knowledge Assumption
• For the UWB channel, typical values are:– Coherence time Tc 100 – 200 s.
– Impulse response duration Td 50 – 100 ns
The receiver can estimate the channel impulse response with negligible overhead and feed it back to the transmitter.
• The signaling period should be chosen so as to satisfy Td << Ts<<Tc .
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Achievable Rates for the Given Channel Model
For any channel input X=(X0,..., XK-1 ) with a given covariance CX , the achievable rate is bounded by
where Y=(Y0,..., YK-1 ) is the channel output and A = diag(A0,..., AK-1 ).Equality holds iff X ~ CN(0,CX).
0
†
detln);(N
AACIEYXI X
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Fixed Power Allocation
Suppose each carrier is encoded independenly with Xk ~ CN(0,Es), k=0,...K-1. Then, the achievable rate is given by
This signaling scheme does not require the transmitter to know the channel transfer function.
kk
sFP A
NEEC 2
0
||1ln
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Water Filling SolutionWater-Filling
WF maximizes the achievable rate by optimum power allocation. In WF, the channel inputs Xk are independent Gaussian with optimal powers. The achievable rate by WF is given by
k ks
k
kWF
ANKE
NAEC
20
0
2
||
||ln
Here, total power is constrained not the power spectral density. Solution usually violates the UWB power constraint.
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Pulse Amplitude Modulation (PAM)
Samples of tranmitted signal:
pk = pulse samples, ck = data
1
0)(
r
kkmiki pcx
m samples
r pulses per signaling periodK = mr samples
index is mod K to simplify FD description
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PAM in Frequency Domain
• In frequency domain, PAM is given by
• Note that Ci is is periodic with period r.
mkijr
kki
KkijK
kki
iii
ecC
epK
P
PCX
/21
0
/21
0
1
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Time-Reversal: A form of PAM
In TR signaling, Xi=Ci Ai*, i.e. transmitted pulse is
the time-reversed channel impulse response. Then
• Here, C0 ,...,Cr-1 can be chosen independently, but the rest are determined by periodicity.
• In this study, we take C0 ,...,Cr-1 independent Gaussian with C0 ~ CN(0,i
2) subject to
.1,...,0 , 2 KiZACY iiii
1
0
1
0
22r
i
m
krkiis AKE
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Time Reversal Achievable Rates
The achievable rate by TR is given by
– m = # samples between successive pulses– r = # pulses per frame– Frame length K=mr– m=1 maximizes CTR, but also ISI
1
0
1
0
22
1
0
1
0
4
0
2
1ln
r
i
m
krkiis
r
i
m
krki
iTR
AKE
AN
EC
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TR with Fixed Power
• C0 ,...,Cr-1 are independent Gaussian with
• The achievable rate is then
1
01
0
2
1
0
4
0
1lnr
iK
kk
m
krki
sTR
A
A
NKEEC
.0~ 2
k
ksi AKE, CNC
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Simulation Results
IEEE Channel Model 1Bandwidth: 3.1-10.6 GHz8192 carriers
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Time Reversal + Water Filling
1
0
2
1
0
4
2
1
02
0
1
0 0
2
||
||
||ln
m
lmlk
m
lmlk
k
r
k ks
r
k
kTRWF
A
AD
DNKE
NDEC
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Simulation Results
IEEE Channel Model 1Bandwidth: 3.1-10.6 GHz8192 carriers
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Achievable Rates at Low SNR
• As SNR = Es/N0 0, WF power allocation becomes more frequency selective compared to FP and TR/FP.
• Under the assumption carrier gains are i.i.d. Ak ~ CN(0,1), it can be shown that
2
log
FP
TR
FP
WF
CC
KCC
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Achievable Rates at High SNR• At the SNR increases, FP allocation becomes
near optimal:
• TR deviates from optimal as the SNR increases:
where m is the number of samples between successive TR pulses.
1FP
WF
CC
mCC
TR
WF
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Power Allocation Against Channel OpaquenessA
lloca
ted
pow
er
Carrier no.
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Power Allocation: SNR = 10 dB
Es /N0= 10 dBPower constraint TR grossly violates
power constraint
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Power Allocation: SNR = 0 dB
Es /N0= 0 dBPower constraint TR violates
power constraint
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Power Allocation: SNR = -10 dB
Es /N0= -10 dB
TR & WF violate power constraint
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Power Allocation: SNR = -20 dB
WF violatespower constraint
Es /N0= -20 dB
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Conclusions• Fixed power allocation is the only power allocation
method consistent with the UWB specification.• WF may achieve significantly higher rates than FP but
they does so by violating the power spectral density constraint, especially at low SNR.
• The rate deficiency of TR/FP at low SNR can be fixed by TR/WF which combines TR with WF.
• At high SNR TR/WF and TR/FP have similar performance.
• TR should be used only at medium to low SNR and if possible in combination with WF.
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Other problems
• Multi-user power allocation:– Centralized algorithm with full knowledge of all
channels– Comparison of achievable rates
• Channel estimation problems