The Effect of Delay in CFO Tracking Loop in Multi-Carrier Systems: Simulation and Preliminary Analysis

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The Effect of Delay in CFO Tracking Loop

in Multi-Carrier Systems: Simulation and

Preliminary Analysis

Presenter: Pin-Hsun Lin

Advisor: Prof. Tzi-Dar Chiueh Date: Nov. 3rd, 2003

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Outline

• Model transformation

• Main factors affects the tracking performance with simulation

• Preliminary knowledge about Markov chain • Some preliminary analysis

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Motivation

• In the 802.11a project the time domain CFO tracking is said to be unstable since there’s a large delay (FFT block)

• Find out how the delay affects the burst

communication and how to solve the problem caused by the effect efficiently.

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Transformation from multi-carrier to

single carrier model

s_m,p FFT CFO Estimator LF AWGN _ej2πεp PD LF Noise Delay 1 or 2 ICI(Δε(m)) NCO NCO CFO Time domain CFO _{Tracking Loop}

Approximation of Time- domain CFO Tracking Loop

X_m,p N’_m,p φ(ε,m) CFO φ(ε,m)^ φ(Δε,m) Sample rate Symbol rate 1/X_m,p

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Main factors affect the tracking

accuracy

• Constellation size • SNR

• Number of delay in the tracking loop

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The effect of constellation size and SNR

Constellat ion (QAM) Angle (degre e) 16 16.87 64 7.69 256 3.7

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Selection of the lower bound of the

acquisition accuracy acquisition accuracy SNR π L 1 2

• The variance of the correlation type

estimation is lower bounded by: [3]

Correlation length=16 samples Correlation length=32 samples Correlation length=64 samples

L is the correlation length

L samples

CP L samples

correlate

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The delay effect to convergence:

simulation in the absence of noise

Parameter value

Constellation

size 64 QAM CFO Initial error 5

degree/sec Noise variance 0 ISI 0 FFT size (N) 64 Guard interval length (N_g) 16 2-symbol delay

No delay Number of symbol

Number of symbol Number of symbol

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The delay effect to convergence :

simulation model with noise

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The delay effect to convergence :

simulation result with noise

SNR

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Simulation results under different

constellation size, acquisition accuracy, and

delay

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Summary

• When a high speed communication system which requires short preambles (bad

acquisition) and large size of constellation

(low tolerance of constellation rotation) is

considered, a fast tracking mechanism is required.

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Fundamental knowledge for Analysis

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The Markov property and Markov

chain (1)[8]

• Using only transition probability and initial condition

can describe the whole process

• A discrete time Markov chain is a Markov process whose state space is finite or countable set and whose time

index is 0,1,2…

• In digital systems, finite word length makes the loop’s behavior as a finite state Markov chain.

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The Markov property and Markov

chain (2)[8]

• What kind of stochastic difference equation obey the Markov property? [7]

– Simple linear model

– Autoregressive (AR) model – Autoregressive (ARMA) model

• The 2nd order loop without delay generates

Markov-2 random sequence

• The 2nd order loop with K delay generates

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The transition probability and

transition matrix (1)

transition matrix (1) [8]_[8]

• A simple 1-step state transition diagram and transition matrix: p1,2 1 2 3 4 P2,3 p2,4 p3,3 p_4,4

Absorbing state Transient state

Starting state destination state

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The transition probability and

transition matrix (2) transition matrix (2) p12,24 1,2 p_23,33 p24,44 p4,4 p14,44 2,3 2,4 3,3 4,4 1,4 p12,23 p33,33 p44,44

•A 2-step state transition diagram and transition matrix:

1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 1,1 1,2 1,3 1,4 2,1 2,2 2,3 2,4 3,1 3,2 3,3 3,4 4,1 4,2 4,3 4,4 Previous state

This state This state

next state

42x42 transition matrix

• This kind of transition matrix can describe 1st order loop

with 1 delay or 2nd order loop without delay

• A 2nd order loop with 1 and 2 delay need 3 and 4-step

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Mean time to converge of the loop (1)

t (sample) phase error acceptable phase error t=0 t=1 t=2 t=3 t=n t=n+1 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Time to converge

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Mean time to converge of the loop (2)

The transition probability can be rearranged as:

Transient state

Absorption state

The mean time to converge can be proved as [8][8]:

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Some Preliminary Analysis

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The distribution of the Phase detector

output (1)

Sum of N-1 independent R.Vs given Φ

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The distribution of the Phase detector

output (2)

From [4], the pdf of the arg{.} term can be represented by:

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The stochastic difference equation of

the loop (1)

The difference equation of the loop filter:

Conversion to phase error difference equation:

which is an autoregressive moving average (ARMA) processes of order (4,2)

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The stochastic difference equation of

the loop (2)

The ARMA process can be transformed into Markov-2 [7] by:

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Conclusion

• Using the single carrier representation we can easily map the CFO tracking loop to the well known phase-locked loop (PLL).

• Based on the PLL model, we can find a high speed communication system requires a fast tracking

mechanism via simulation.

• Apply Liapounoff CLT to the sum of noise and ICI term, the output of PD can be derived as the Rician phasor. • Using Markov property of the stochastic difference

equation of the loop the mean time to converge can be derived.