Coding vs. ARQ in Fading Channels: How reliable should the PHY be?

Transcription

1 Coding vs. ARQ in Fading Channels: How reliable should the PHY be? Peng Wu and Nihar Jindal University of Minnesota Department of Electrical & Computer Engineering

2 Introduction RC ayer ACK/NACK RC ayer Physical ayer TX R bits/symbol RX Physical ayer Two different and competing reliability mechanisms: Coding: Reducing transmitted rate R (bits/sym) reduces packet error probability ARQ: Retransmit packets not received correctly What is the optimal balance for reliable communication?

3 Main Assumptions Basic assumptions: Perfect CSI at receiver, transmitter only knows channel statistics (high mobility scenario) Packets retransmitted until successfully decoded Error free ACK/NACK Perfect error detection Errors independent across packet transmissions At PHY, either of HARQ and no HARQ can be used Important performance measure: long-term avg. successful rate X: number of PHY transmission attempts for each packet η η = R E[ X is also referred as goodput/successful throughput ] bits/symbol

4 PHY Reliability without PHY HARQ Number of packet TX s ~ Geometric (-) : E[# of packet TX' Successful throughput/goodput : s] = R ( ) bits/sym Error prob. increases in TX rate R A reliable PHY layer requires the sacrifice of TX rate Objective: Characterize * as a function of channel parameters (SNR, time/frequency selectivity) * = arg max R( ) Same issue arises for optimization of application layer fountain codes

5 Channel Model without PHY HARQ Fast fading: channel varies too rapidly to be tracked at TX Block-fading (order time/freq selectivity): y = SNR h x + h k : channel in k-th fading block, iid across blocks Channels are complex Gaussian (Rayleigh fading), known at RX SNR: average received signal-to-noise ratio t k t z t With infinite blocklength and strong channel code, successfully decode iff: log (+ h i SNR ) > R i = Errors due to bad channel realizations Packet error probability = mutual information outage probability = Ρ log (+ h i SNR ) < R i =

6 Gaussian Approximation Approximate RX mutual information by Gaussian: µ ( SNR ) - R = Ρ log (+ h i SNR ) < R Q i = σ ( SNR )/ ( µ ( SNR ), σ ( SNR ) / ) ~ N σ ( SNR R µ ( SNR ) Q ( ) Goodput approximation: ( κ ( ))( ) η = µ ( SNR ) Q - g κ: µ-normalized standard deviation κ = σ ( SNR) µ ( SNR) / κ decreases in both SNR and η g is strictly concave in ) Goodput (bits/symbol) =5 = SNR=0 db Exact Gaussian Approximation

7 Goodput Optimization η g - maximizing error probability: g satisfies ( SNR, ) g dq Q g g d is only determined by κ g ( ) ( ) ( ) = arg max Q = g ( κ ( ))( - ) = κ Basic analysis can be easily extended to MIMO and other fading distributions g monotonically increases in κ (decreases in SNR and )

8 Goodput vs. SNR 7 8 η (bits/symb bol) =0.00 =0.0 =0. optimal η (bits/symb bol) =0.00 =0.0 =0. optimal SNR (db) 0% near optimal for wide range of selectivity () SNR (db) Making PHY too reliable incurs significant penalty in open loop systems

9 Incremental Redundancy (IR) Outage if cannot decode after IR rounds-> triggers RC ARQ retransmission Frequency-flat channel during each IR round, i.i.d. fading across rounds If initial transmission rate is R bits/sym, # of IR rounds is smallest xˆ s.t. Xˆ ˆ x log ( + h i SNR ) > i = : random variable describing # of IR rounds k Ρ [ ] + > = Ρ ˆ = log ( h i SNR ) R k,.., - X k i = k = Error probability identical to expression without IR Relationship between R and unaffected by IR = Ρ log (+ h i SNR ) < R i = k Expectation of Xˆ is E[ Xˆ ] = + Ρ log (+ h i SNR ) < R k = i= R

10 Goodput Optimization with IR Successful throughput (i.e., goodput): E[ Xˆ R ]/( - ) = R E[ Xˆ ] ( ) = R ( - ) E[ Xˆ ] Optimal operating point: PHY Transmitted Rate E[Xˆ ] : an increasing function of R and Goodput without IR IR ( SNR, ) = arg max R ( ) E[ Xˆ Result: The optimizing error probability with IR is less than or equal to the optimizing error probability without IR: IR no IR ]

11 Goodput (IR) vs. Error Prob. 9 8 its/symbol) Goodput (bi SNR = 30 db Error Probability SNR = 0 db IR no IR If error prob. (or TX rate) too large, no early termination If error prob. (or TX rate) too small, always terminate in one IR round IR makes goodput less sensitive to the choice of error probability if parameters are carefully chosen

12 Conclusion and Future Work Don t make PHY reliable if it is difficult (e.g., lacking diversity) to do so, let ARQ achieve reliability instead Increase PHY reliability only when it is easy (e.g., plenty of diversity, closed loop system) to do so Same argument applies to rateless coding HARQ can achieve lower optimum PHY error probability Insensitivity of goodput to choice of reliability makes PHY optimization almost pointless Future work: The effect of HARQ feedback on the goodput/optimal reliabilty Taking RC delay into account