1 Designing Pilot-Symbol-Assisted Modulation (PSAM) Schemes for Time- Varying Fading Channels Tharaka Lamahewa Research School of Information Sciences and Engineering ANU College of Engineering and Computer Science The Australian National University (ANU) Collaborators: Dr. Parastoo Sadeghi (ANU), Prof. Rod Kennedy (ANU), Prof. Predrag Rapajic (University of Greenwich, UK), Mr. Sean Zhou (ANU), Dr. Salman Durrani (ANU)

2 Outline Introduction Background and Motivation Contributions Optimum design of PSAM schemes in time- varying fading channels Optimum design under spatially correlated antenna arrays Conclusions

3 System Model y ` = g ` x ` + n ` FadingInput Noise Received Signal:

4 Adaptive Power Allocation (1) Aim: To save the overall transmitter power by adapting the power depending on the fading channel quality

5 Adaptive Power Allocation (2) Early studies: Perfect (Genie aided) channel state information (CSI) at Rx and possibly delayed CSI at Tx

6 Genie Aided Power Allocation Strategy Solution: Waterfill in time, with water level 0 (Goldsmith and Varaiya 1997) E d ( ½ ) = 1 ½ 0 ¡ 1 ½ ; ½ ¸ ½ 0 E d ( ½ ) = 0 ; ½ < ½ 0 ½ = j g j 2

7 Adaptive Power Allocation (3) Noise Variance: Later studies: A vicious genie provides imperfect CSI at Rx and (possibly delayed) imperfect CSI at Tx The channel is measured using a probe signal continuously Good Genie Bad Genie g ` = ^ g ` + ~ g ` ¾ 2 ~ g ( ` )

8 Power Allocation Strategy under Imperfect Channel State Information It is still a waterfilling strategy in time (with a more complex solution), with some water level 0 (Klein and Gallager 01, Yoo and Goldsmith 04 & 06) ^ ½ = j ^ g j 2 E d ( ^ ½ ) = f ( ^ ½ ; ½ 0 ; ¾ 2 ~ g ) ; ^ ½ ¸ ½ 0 E d ( ^ ½ ) = 0 ; ^ ½ < ½ 0

9 Our Model-Based Power Adaptation with Periodic Feedback There is no genie Channel estimate is based on pilot symbol transmission reduction in available resources possible tradeoffs. Channel estimates are fed back after some delay. Pilot SymbolsData Symbols Block Length = T

10 First Step Let us consider the case where there is no feedback, for comparison purposes, and optimize pilot transmission parameters

11 System Model without Feedback (1) Pilot symbols are transmitted periodically for channel estimation

12 System Model without Feedback (2) State equation: g ` = ®g ` ¡ 1 + w ` First-order AR or Gauss-Markov channel model: n ` » N C ( 0 ; N 0 ) g ` » N C ( 0 ; ¾ 2 g ) Observation equation: fading y ` = g ` x ` + n ` input noise small   fast channel large   slow channel 

13 Three Parameters for Optimization How often (pilot spacing)? How strong (pilot symbol power)? How many (pilot cluster size)? One 1 P 1 D 2 D 3 …... D T-1 D T ´ = 1 T ; 1. M. Dong, L. Tong, and B. M. Sadler, “Optimal insertion of pilot symbols for transmissions over time-varying flat fading channels,” IEEE Trans. Signal Processing, vol. 52, no. 5, pp. 1403–1418, May E p = ° ´ E ; E d = 1 ¡ ° 1 ¡ ´ E

14 Our Objective Function: A Capacity Lower Bound A closed-form expression for the information capacity of the time-varying Gauss-Markov fading channel in the presence of imperfect channel estimation is still unavailable. We utilize a lower bound to the channel capacity

15 Our Objective Function: A Capacity Lower Bound 1 1. M. Medard, “The effect upon channel capacity in wireless communications of perfect and imperfect knowledge of the channel,” IEEE Trans. Inform. Theory, vol. 46, no. 3, pp. 933–946, May Note the presence of signal power in the denominator C ( ^ g ) ¸ C LB ( ^ g ) = l og ³ 1 + j ^ g j 2 E d ¾ 2 ~ g E d + N 0 ´ Channel estimation: g = ^ g + ~ g estimate error y ` = ^ g ` x ` + ~ g ` x ` + n ` useful part additional noise effective instantaneous SNR

16 Average Capacity Lower Bound The lower bound is an average capacity per symbol and not per transmission block The lower bound, by itself, does not take into account models of channel time variations The lower bound, by itself, does not tell us how to optimize the two pilot parameters  and  C LB = E ^ g n l o g ³ 1 + j ^ g j 2 E d ¾ 2 ~ g E d + N 0 ´o expectations over channel estimate realizations

17 First Contribution (no Feedback Yet) Invoke the autoregressive (AR) or Gauss-Markov channel model and Kalman filtering principles to maximize the capacity lower bound Analyse optimum pilot scheme parameters pilot power γ and pilot spacing T to maximise the capacity lower bound for a wide range of SNR and normalized fading rates

18 Estimation errorincreases Kalman Filtering P 1 D 2 D 3 …... D T-1 D T M ( 1 ) = ¾ 2 ~ g ( 1 ) ^ g 1 ; ~ g 1 Estimate (true filtering) Predict (no pilots) ^ g ` = ® ` ¡ 1 ^ g 1 M ( ` ) = ¾ 2 ~ g ( ` ) = ¾ 2 g ¡ ® 2 ( ` ¡ 1 ) ( ¾ 2 g ¡ M ( 1 ))

19 Revisit the Capacity Lower Bound P 1 D 2 D 3 …... D T-1 D T SNRdecreases ^ g ` = ® ` ¡ 1 ^ g 1 ® < 1 ½ i ns t ( ` ) = j ^ g ` j 2 E d ¾ 2 ~ g ( ` ) E d + N 0 y ` = ^ g ` x ` + ~ g ` x ` + n ` signal noise

20 Revisit the Lower Bound (cnt’d) P 1 D 2 D 3 …... D T-1 D T Capacity per symboldecreases Instantaneous capacity per symbol: Instantaneous capacity per block: C LB ( ^ ½ 1 ) = 1 T T X ` = 2 C LB ( ` ; ^ ½ 1 ) C LB ( ` ; ½ i ns t ) = l og ¡ 1 + ½ i ns t ( ` ) ¢

21 LB versus Pilot Power and Spacing Average SNR = 10 dB ® = J 0 ¡ 2 ¼ f D T s ¢, f D T s = 0 : 1

22 Equal Power Comparison Pilot spacing  is optimized

23 Sensitivity of LB to Power Ratio 

24 Sensitivity of LB to Pilot Spacing 

25 Power Adaptation Based on Periodic Feedback

26 System Model with Feedback Switching period: T This model is still idealistic in the sense that the feedback link is noiseless.

27 A Closer Look at the Transmitter ^ ½ ` = j ^ g ` j 2 E d ( ^ ½ ` ) = f ( ^ ½ ` ; ½ 0 ; ¾ 2 ~ g ` ) ; E d ( ^ ½ ` ) = 0 ; Zero-delay power control: actual power allocation at time is performed based on the channel estimate ^ g ` = ® ` ¡ 1 ^ g 1 ` ^ g ` = ® ` ¡ 1 ^ g 1

28 Transmission Block Structure P 1 D 2 D 3 …... D d …. D T-1 D T Feedback Transmitted Fixed Power Strategy Adaptive Power Strategy

29 Power Distribution P 1 D 2 D 3 …... D d …. D T-1 D T Total Power:Adaptive Power per Symbol E d ( ^ ½ 1 ; ` ) = f ( ^ ½ ` ; ¾ 2 ~ g ` ; ¢¢¢ ) T X ` = d + 1 E ^ ½ 1 n E d ( ^ ½ 1 ; ` ) o = ( 1 ¡ ° ) ( T ¡ d ) T T ¡ 1 E E d = ( 1 ¡ ° ) ( d ¡ 1 ) £ T T ¡ 1 E E p = ° E £ T

30 Capacity Lower Bound per Block P 1 D 2 D 3 …... D d …. D T-1 D T Fixed Power CapacityAdaptive Power Capacity 1 T T X ` = d + 1 l og à 1 + ® 2 ( ` ¡ 1 ) ^ ½ 1 E d ( ^ ½ 1 ; ` ) ¾ 2 ~ g ` E d ( ^ ½ 1 ; ` ) + N 0 ! 1 T d X ` = 2 l og à 1 + ® 2 ( ` ¡ 1 ) ^ ½ 1 E d ¾ 2 ~ g ` E d + N 0 ! + C LB = No Information 0 +

31 Optimization Problem Maximize the total capacity per block subject to the power constraint in the adaptive transmission mode

32 Optimum Solution A generalized water-filling algorithm E ( ` ; ^ ½ 1 ) = ½ ¡ a c + b ºc ; ^ ½ 1 > º ® 2 ( ` ¡ 1 ) ; 0 ; o t h erw i se,

33 Slow Fading

34 Messages For the slow fading, even after a feedback delay of 8, we get good SNR improvement in low SNR In high SNR, even the delay-less feedback does not offer much compared to the case, where pilot power and spacing are optimized However, it continues to offer improvements at high SNR, compared the equal power allocation strategy.

35 Fast Fading

36 Messages Idealistically (no feedback delay), we would get a good SNR improvement at low SNR However, with a modest delay of d = 3, the performance rapidly drops No point in power adaptation for such a fast fading rate Note that, we have forced the transmitter that, in spite of feedback delay, perform power adaptation for at least one symbol

37 Optimal Block Length, Slow Fading

38 Message In the slow fading regime, the optimal block length without feedback is large enough to accommodate power adaptation

39 Optimal Block Length, Fast Fading

40 Message In the fast fading regime, the optimal block length without feedback is so small that it cannot accommodate any power adaptation Note that the optimal T* is less than half of what would have in a bandlimited fading channel according to the Nyquist rate (f D T = 0.1  T min =10)

41 Optimal Pilot Power Allocation

42 Message In the slow fading, the pilot power ratio in the without feedback scheme, is a reasonable choice for the feedback scheme too.

43 Multiple Antennas at the Receiver

44 A Capacity Lower Bound: SIMO Instantaneous capacity per symbol:

45 LB vs SNR: Spatially i.i.d. Channels Solid lines: using opt. parameters corresponding to SIMO bound Dashed lines: using opt. parameters corresponding to SISO bound

46 LB vs SNR: Spatially Correlated Channels Solid lines: using opt. parameters corresponding to SIMO bound Dashed lines: using opt. parameters corresponding to SISO bound

47 Conclusions We considered the achievable information rates in autoregressive fading channels with and without feedback under pilot-assisted channel estimation Under realistic delays, feedback can provide some gain at low SNR and in moderately slow fading channels (f D T s = 0.01) and even at high SNR (compared to equal power allocation case) In the relatively fast-fading case (f D T s = 0.1), use equal power allocation and just optimize the pilot spacing By optimally designing the training parameters for SISO systems, the same parameters can be used to achieve near optimum capacity in both spatially i.i.d. and correlated SIMO systems.

48 Current and Future Research What is the price of providing feedback in terms of information rate loss in the reverse channel? How much power should we allocate for feedback transmission? Requires proper modelling and formulation. So far, we have considered SISO and SIMO, extending the results to the MIMO Model-based comparison of PSAM and superimposed training in terms of achievable information rates