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Statistical guarantees of performance for MIMO designs

Statistical guarantees of performance for MIMO designs. Jayanand Asok Kumar Shobha Vasudevan University of Illinois at Urbana Champaign. MIMO systems. Digital communication systems At the physical layer: Transmit and receive data bits Operation at high data rates is required

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Statistical guarantees of performance for MIMO designs

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  1. Statistical guarantees of performance for MIMO designs Jayanand Asok Kumar Shobha Vasudevan University of Illinois at Urbana Champaign

  2. MIMO systems • Digital communication systems • At the physical layer: Transmit and receive data bits • Operation at high data rates is required • Bit Error Rate(BER) is a performance metric • Bit error: Decoded bit does not match actual data bit • BER: Average probability of a occurrence of bit errors • Multiple Input Multiple Output (MIMO) systems • Complex wireless communication systems • Comprise several components implemented at the Register Transfer Level (RTL) • Components must meet area and power requirements as well MIMO RTL design is both time and resource-intensive

  3. Performance estimation • Traditionally, hardware is viewed as non-probabilistic • Data corruption at the receiver • Thermal noise, quantization etc. • Introduces randomness into the RTL designs • Performance metrics are probabilistic in nature • Simulation-based techniques • Reasonably accurate performance estimates • Time-consuming and incomplete • Need to estimate performance quickly and with a high degree of confidence We employ probabilistic model checking

  4. Sources of randomness • Analog to Digital Converter (ADC) • Discretizes signal in time (sampling) and value (quantization) • Imperfections in circuitry • Thermal fluctuations in current and voltage • Timing errors at the sampler • Lumped together and referred to as noise • Signal to Noise Ratio (SNR) • Provides a measure of the level of noise We assume an Additive White Gaussian Noise (AWGN) model

  5. Error modeling • External data corruption errors • Incorrect quantization level due to signal noise • Internal data corruption errors • Insufficient precision (number of bits) for data representation • Eg: Both values 0.95 and 0.55 represented by 0.75 • For a given q[n], bit decoding is non-probabilistic • Can map q[n] to bit errors by analyzing RTL code Area = probability Gaussian curve determined by SNR Value of q[n]

  6. Our methodology pCTL: probabilistic Computational Tree Logic • We use PRISM, a symbolic model checker Potentially leads to state-space explosion Best, worst and average case Sound techniques applicable to a large class of systems Rigorous analysis for performance estimation

  7. Case studies: MIMO systems • Case studies using components of MIMO systems I Error properties of a Viterbi decoder II Error properties of a MIMO detector IIIConvergence properties of a Viterbi decoder We assume Binary Phase Shift Keying (BPSK) signaling

  8. Case study IError properties of a Viterbi decoder

  9. Viterbi decoder basics • x[n] : Data bit at time step n • System has memorym=1 • q[n] : Received quantized sample • Insufficient to decode “most likely” transmitted bit • Viterbi decoder waits for L-1 time steps • L is traceback length • Heuristic choice: L > 5m (We choose L = 6) • Decodes bit using most likely “sequence”

  10. Viterbi algorithm • Pathmetrics pm0, pm1: Cost of paths • prev0, prev1: Most probable previous internal state • 1 time step = 1 clock cycle = 1 trellis stage • Decoding: • Trace back along path with least cost • Decoded bit ≠ actual data bit (Bit error!) • Latency of L-1 cycles pm0 Internal State 0 Decodes bit 0 prev0=1 Actual data bit pm0 < pm1 prev1=0 Internal State 1 pm1 Depend on q[n]

  11. DTMC modeling • DTMC model M: • State variables S (spans L trellis stages) • State μ: Unique assignment of values to S • State transition relation TP: S x S→[0,1] • State variable flag = 1 if decoded bit ≠ xL-1 0 otherwise Stage L-1 Stage L-2 Stage 0 pm0 prev0L-2 prev00 prev0L-1 prev01 xL-2 xL-1 x0 x1 prev1L-2 prev10 prev1L-1 prev11 pm1 Current stage Next clock cycle

  12. State transition relation TP • DTMC transition: State μ→ State μ’ • Probabilistic updates (pm0’, pm1’, x0’) = Γp(pm0, pm1, x0) • Probability computed based on x0and q0 (from SNR) • Non-probabilistic updates (prev00’, prev10’) = FS (pm0’, pm1’) Shift trellis to the right: (prev0i+1’, prev1i+1’, xi+1’) = (prev0i’, prev1i’, xi’), 0 ≤ i ≤ L-1 Traceback operation: flag’ = FE(prev0i’, prev1i’, xL-1) Above functions collectively define TP

  13. Property specification • We define a reward model on DTMC M • Assign a reward equal to value of flag in a state • P2(Average case): R=? [I = T] • Probability that an error occurs at the Tth step • In steady state, P2 corresponds to BER • P1 (Best case): P=? [G ≤ T (! flag)] • Probability that no error occurs in T steps • P3 (Worst case): P=? [F ≤ T (flag >1)] • Probability that more than 1 error occurs in T steps P1, P2 and P3 together can be used to make stronger claims about error-related performance of system

  14. Property-preserving reduction • For error properties • Actual value of decoded bit is not required (Data abstraction) • Replace prev0, prev1,x with c, w (data abstraction) (ci’, wi’) = Fabs (prev0i’, prev1i’, xi’), 0 ≤ i ≤ L-1 • c, w indicates “correctness” of traceback operation • Can discard variables storing past data bit values • Reduction in number of possible states Stage L-1 Stage L-2 Stage 0 pm0 cL-2 c0 cL-1 c1 xL-2 xL-1 x0 x1 wL-2 w0 wL-1 w1 pm1

  15. Reduced DTMC MR • Probabilistic function Γp is preserved (pm0’, pm1’, x0’) = Γp(pm0, pm1, x0) • Modified non-probabilistic updates (c0’, w0’) = Fcw(pm0’, pm1’) Shift trellis to the right: (ci+1’, wi+1’) = (ci’, wi’), 0 ≤ i ≤ L-1 Traceback operation: flag’ = FER(ci’, wi’, x0’) • flag still indicates correctness of decoded bit Reduction can be applied to systems with decoding latency

  16. Proof of correctness • Reduction is sound with respect to error properties • Proof involves showing two parts Part A) Value of flag is the same in both μ and μR Part B) P (μ→μ’) = P(μR → μR’) MR is a probabilistic bisimulation of M Fabs: Equivalence relation

  17. Probabilistic model checking • Verify properties P1, P2 and P3 on DTMC model MR • Probabilistic model checking • Explores all possible paths of length T High-confidence performance estimates can be obtained

  18. Case study IIError properties of a MIMO detector

  19. MIMO system: Channel • MIMO system: • NR receive antennas, NT transmit antennas y = Hx+ n y: Vector of NR received signals x: Vector of NT transmitted bits H: NRxNTChannel matrix (Rayleigh flat-fading) n: Noise vector (AWGN) H can be assigned complex values

  20. MIMO detector • Maximum Likelihood (ML) detection • Estimate most likely x, given y Estimate = arg min |y-Hs| where s is a possible value of x • Consider a 2x2 MIMO system with BPSK Split into real and imaginary parts: Estimate = arg min (|y1,R – h11,Rs1 – h12,Rs2| + |y1,I – h11,Is1 – h12,Is2| + |y2,R – h21,Rs1 – h22,Rs2| + |y2,I – h21,Is1 – h22,Is2|) = arg min (M1,R + M1,I + M2,R + M2,I) where s1, s2 are elements of s DTMC state variables

  21. Symmetry reduction State μ’ State μ • M1,R in stateμ = M2,R in stateμ’ • M1,RandM2,Rare symmetric with respect to MIN block • Receiver blocks are structurally symmetric • Well-known symmetry reduction techniques can be applied

  22. Case study IIIConvergence properties of a Viterbi decoder

  23. Convergence in a Viterbi decoder • Traceback paths converge • Decoded bit is independent of starting state • Typically, L > 5m is chosen • Convergence is a measure of confidence in the decoded bit • Convergent stage: prev0 = prev1 • Introduce the state variable count into M • count = count + 1 if stage is not convergent • If count > L, set flag =1 and reset count to 0. • Define a reward model on DTMC M using flag

  24. Convergence property • Property specification • C1: R=? [I = T] (similar to P2) • Probability that the decoded bit has non-converging traceback paths • Property-preserving reduction • Values from only the current trellis stage are required • State variables in MR : pm0, pm1, x0, flag and count • State transition relation: Γp is preserved from M Reduction techniques depend not just on the type of the system, but also on the property to be checked

  25. Results: Case study I • P2 matches BER from simulations • Reachability Iterations (RI) • PRISM explores all reachable states of DTMC • T > RI is sufficient for convergence of result • Faster than computing Steady State (SS) rewards RI = 263

  26. Results: Case study II • Model checking times • Time-bounded approach: < 0.5 seconds, for each T • SS reward computation: 53.27 seconds • Simulation-based computation • BER = 1.07x10-5 using 107 time steps • No bit errors found using 105 time steps! RI = 3

  27. Results: Case study III • MR has 61,000 states (Runtime: 120 seconds) • C1 stabilizes for L > 5 RI = 77

  28. Future work • Error diagnosis mechanism when a property fails • Characterization: Is corruption external or internal ? • In the absence of internal data corruption, a correct quantization level results in a correct decoded bit • Structural changes: (eg: Increase precision) • Remove internal data corruption • Analyze the effects of hardware faults • Hard and soft errors • Incorporate compositional reasoning to improve the scalability of our methodology

  29. Thank you

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