UWB Channels: Time-Reversal Signaling NEWCOM, Dept. 1 Meeting Paris, 13 May 2005 Erdal Arıkan Bilkent 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, h RT (t) = h TR (t). R receives h TR (-t)  h TR (t), which is likely to be peaky. 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. 1) R sends an impulse 3) T transmits h RT (-t) T R 2) T receives h RT (t) C 4) R receives h RT (-t)  h TR (t) 5) R receives h RT (-t)  h TC (t)

4 Correlations of channel responses

5 UWB Channel (FCC 2002) Frequency range: 3.1–10.6 GHz Radiated power: < dBm/MHz Min. Bandwidth: 500 MHz Bandwidth > 20% of center frequency

6 UWB Channel Indoor Emissions Limit GPS Band GPS Band -41 dBm/MHz 7.5 GHz

7 Maximum power emission: –41.3 dBm/MHz  7.5 GHz = 0.56 mW.  UWB systems are not energy limited. UWB Energy

8 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.

9 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)

10 Saleh-Valenzula Model for UWB The channel impulse response is modeled as –X lognormal shadowing gain –L number of clusters –T l 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 r0P80215 (

11 Model Characteristics ParameterValue (CM1) Line of SightYES Range (m)0-4 Coherence time (  s) 200 Mean excess delay (nsec)4.9 RMS delay (nsec)5 No. multipath components within 10 dB of peak component, NP 10dB 13.3 No. paths capturing 85% of energy, NP(85%) 21.4 Channel energy mean (dB)-0.5 Channel energy std (dB)2.9

12 Sample of a channel impulse response

13 Frequency Domain Channel Model An OFDM-like channel with subchannels –Z i ~ CN(0,N o ) are independent noise –In each use of the vector channel, a new set of A i are chosen from a fixed distribution –K= W T s where W=RF bandwidth, T s = signaling period –Input constraint: E[ X i 2 ]  E s for each i –Assumption: Transmitter and receiver have perfect knowledge of the channel coefficients A i

14 Perfect Channel Knowledge Assumption For the UWB channel, typical values are: –Coherence time T c  100 – 200  s. –Impulse response duration T d  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 T d << T s <<T c.

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

16 Fixed Power Allocation Suppose each carrier is encoded independenly with X k ~ CN(0,E s ), 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.

17 Water Filling SolutionWater-Filling WF maximizes the achievable rate by optimum power allocation. In WF, the channel inputs X k 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.

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

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

20 Time-Reversal: A form of PAM In TR signaling, X i =C i A i *, i.e. transmitted pulse is the time-reversed channel impulse response. Then Here, C 0,...,C r-1 can be chosen independently, but the rest are determined by periodicity. In this study, we take C 0,...,C r-1 independent Gaussian with C 0 ~ CN(0,  i 2 ) subject to

21 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 C TR, but also ISI

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

23 Simulation Results IEEE Channel Model 1 Bandwidth: GHz 8192 carriers

24 Time Reversal + Water Filling

25 Simulation Results IEEE Channel Model 1 Bandwidth: GHz 8192 carriers

26 Achievable Rates at Low SNR As SNR = E s /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. A k ~ CN(0,1), it can be shown that

27 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.

28 Power Allocation Against Channel Opaqueness Allocated power Carrier no.

29 Power Allocation: SNR = 10 dB E s /N 0 = 10 dB Power constraint TR grossly violates power constraint

30 Power Allocation: SNR = 0 dB E s /N 0 = 0 dB Power constraint TR violates power constraint

31 Power Allocation: SNR = -10 dB E s /N 0 = -10 dB TR & WF violate power constraint

32 Power Allocation: SNR = -20 dB WF violates power constraint E s /N 0 = -20 dB

33 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.

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