UWB Channels TimeReversal Signaling NEWCOM Dept 1 Meeting

UWB Channels: Time-Reversal Signaling NEWCOM, Dept. 1 Meeting Paris, 13 May 2005 Erdal Arıkan Bilkent University Ankara, Turkey

Outline • Time-reversal signaling • UWB channel model • Signaling and achievable rates for the UWB channel – Fixed power – Time reversal – Water filling • Simulation results • Conclusions 2

Time Reversal Signaling 2) T receives h. RT(t) T 3) T transmits h. RT(-t) If channel is reversible, h. RT(t) = h. TR(t). • R receives h. TR(-t) h. TR(t), which is likely to be peaky. 1) R sends an impulse R 4) R receives h. RT(-t) h. TR(t) C 5) R receives h. RT(-t) h. TC(t) • C receives h. TR(-t) h. TC(t), which is unlikely to be peaky if C is sufficiently far from R. • h. XY(t) likely to have low coherence in time and space for high delay-bandwidth product channels, such as the UWB channel. 3

Correlations of channel responses 4

UWB Channel (FCC 2002) • Frequency range : 3. 1– 10. 6 GHz • Radiated power : < -41. 3 d. Bm/MHz • Min. Bandwidth : 500 MHz • Bandwidth > 20% of center frequency 5

UWB Channel Indoor Emissions Limit -41 d. Bm/MHz 7. 5 GHz GPS Band 3. 1 10. 6 1. 99 1. 61 0. 96 GPS Band 0. 96 1. 61 6

UWB Energy Maximum power emission: – 41. 3 d. Bm/MHz 7. 5 GHz = 0. 56 m. W. UWB systems are not energy limited. 7

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. 8

UWB Channel Model • The channel is modeled as a linear filter with additive white Gaussian noise. z(t) x(t) y(t) s(t) h(t) + • The channel impulse response follows the Saleh. Valenzula model. 9

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. 02490 r 0 P 80215 (http: //grouper. ieee. org/groups/802/15/pub/2002/Nov 02) 10

Model Characteristics Parameter Line of Sight Value (CM 1) YES Range (m) 0 -4 Coherence time ( s) 200 Mean excess delay (nsec) 4. 9 RMS delay (nsec) 5 No. multipath components within 10 d. B of peak component, NP 10 d. B 13. 3 No. paths capturing 85% of energy, NP(85%) 21. 4 Channel energy mean (d. B) -0. 5 Channel energy std (d. B) 2. 9 11

Sample of a channel impulse response 12

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 13

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. 14

Achievable Rates for the Given Channel Model For any channel input X=(X 0, . . . , XK-1 ) with a given covariance CX , the achievable rate is bounded by where Y=(Y 0, . . . , YK-1 ) is the channel output and A = diag(A 0, . . . , AK-1 ). Equality holds iff X ~ CN(0, CX). 15

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. 16

Water-Filling Solution 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 Here, total power is constrained not the power spectral density. Solution usually violates the UWB power constraint. 17

Pulse Amplitude Modulation (PAM) Samples of tranmitted signal: index is mod K to simplify FD description pk = pulse samples, ck = data m samples r pulses per signaling period K = mr samples 18

PAM in Frequency Domain • In frequency domain, PAM is given by • Note that Ci is is periodic with period r. 19

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, C 0 , . . . , Cr-1 can be chosen independently, but the rest are determined by periodicity. • In this study, we take C 0 , . . . , Cr-1 independent Gaussian with C 0 ~ CN(0, i 2) subject to 20

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 21

TR with Fixed Power • C 0 , . . . , Cr-1 are independent Gaussian with • The achievable rate is then 22

Simulation Results IEEE Channel Model 1 Bandwidth: 3. 1 -10. 6 GHz 8192 carriers 23

Time Reversal + Water Filling 24

Simulation Results IEEE Channel Model 1 Bandwidth: 3. 1 -10. 6 GHz 8192 carriers 25

Achievable Rates at Low SNR • As SNR = Es/N 0 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 26

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. 27

Allocated power Power Allocation Against Channel Opaqueness Carrier no. 28

Power Allocation: SNR = 10 d. B Es /N 0= 10 d. B Power constraint TR grossly violates power constraint 29

Power Allocation: SNR = 0 d. B Es /N 0= 0 d. B Power constraint TR violates power constraint 30

Power Allocation: SNR = -10 d. B Es /N 0= -10 d. B TR & WF violate power constraint 31

Power Allocation: SNR = -20 d. B Es /N 0= -20 d. B WF violates power constraint 32

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. 33

Other problems • Multi-user power allocation: – Centralized algorithm with full knowledge of all channels – Comparison of achievable rates • Channel estimation problems 34