UWB Channels:Time-Reversal Signaling

NEWCOM, Dept. 1 Meeting Paris, 13 May 2005

Erdal ArıkanBilkent University

Ankara, Turkey

2

Outline

• Time-reversal signaling• UWB channel model• Signaling and achievable rates for the UWB

channel – Fixed power– Time reversal– Water filling

• Simulation results• Conclusions

3

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