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

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The Effect of Delay in CFO Tracking Loop

in Multi-Carrier Systems: Analysis

Presenter: Pin-Hsun Lin

Advisor: Prof. Tzi-Dar Chiueh

Date: Mar. 1

st

, 2004

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Outline

• Introduction

– Synchronization and tracking loop

– Number of delay in an OFDM CFO tracking loop and some alternatives of the loop to reduce the delay

• Analysis

– Find the loop difference equation

– Fundamental knowledge for further analysis – The transition matrix and it’s function

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Synchronization preliminaries

• Synchronization: estimation + compensation • All synchronization algorithms can be coarsely

divided into

– Feedforward type

fast but high complexity, usually used in acquisition *. – Feedback type

slow but low complexity, usually used in tracking.

(several complicated algorithms are achieved by iteration)

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Why there’s a tracking

“feedback loop” ?

• To make estimation more accurate, the data

(redundant) must be used as more sufficient as

possible.

• In CP-OFDM systems, there’re 2 redundancy can

be used:

– CP (time domain)

– Pilot (frequency domain)

• If we want to use pilots with time domain

compensation, a feedback loop is inevitable.

(Time domain compensation is intuitively the

easiest one

(not the only one)

but

it’s sensitive to ICI

and modulation type.

)

• If we want to use CP, there’s no feedback loop

but

it’s sensitive to timing error.

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Loop characteristics

• Performance indices of tracking loop:

– Convergence speed

– Steady state error variance – Steady state error distribution

• Several factors can affect the above:

– SNR

– Loop delay

– Constellation size

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The effect of loop delay and SNR to

steady state phase error distribution: an

example *

α =(K₁+K₂)/K₁

D is number of delay

* T. Koizumi and H. Miyakawa, “Statistical analyses of digital phase-locked loops with time delay,” IEEE Trans. Commun., July 1977, p. 731-p.735

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Number of Delay in the Loop and

Some Alternatives of the loop

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Alternatives of the tracking loop

BUF WLS NCO + -5_1 _{4_1} _{3_1} 2_1 FFT

Delay 1 Delay 1BR _{Delay 1}

Delay 1 1 CFO∧ arg{ } LF BUF WLS NCO + -4_1 _{3_1} _{3_1} 2_1 FFT Delay 1 _{Delay 1} Delay 1 1 CFO∧ arg{ } LF Original:

Remove the bit-reversal:

• Serial in/serial out deteriorates the convergence performance .

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Alternatives of the tracking loop

FFT _BUF WLS LF + -3_k 3_1 2_1 Delay 1 _{Delay 1} Delay 1 1 CFO∧ arg{ } Circular conv

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Alternatives of the tracking loop

FFT BUF WLS LF NCO + -3_N/2 2_N/2 1_N/2 Delay 1 _{Delay 1} Delay 1/2 1 CFO∧ arg{ }

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Find the

Loop Difference Equation

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Mapping the tracking loop as a

PLL problem

s_m,p FFT CFO Estimator LF AWGN _ej2πεp arg{ } LF Noise Delay 1 or 2 NCO NCO CFO Time domain CFO _{Tracking Loop}

Approximation of Time- domain CFO Tracking Loop

X_m,p

CFO

1/X_m,p

S_m,p

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The distribution of the Phase detector

output (1)

Sum of N-1 independent R.Vs given Φ

If can be approximated by Gaussian

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Test of the Gaussianity (1)

If

where then

The accuracy can be examined by Berry-Esseen theorem [6][9]:

In my case the upper bound is: where

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Test of the Gaussianity (2)

For good Gaussianity:

• FFT size is large • SNR is low

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The distribution of the Phase detector

output (2)

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The stochastic difference equation of

the loop

The difference equation of the loop filter:

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Fundamental Knowledge for

Further Analysis

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Problem formulation

t (sample) phase error φ (quantized by k-bit) acceptable phase error t=0 t=1 t=2 t=3 t=n t=n+1 1 2 3 2k 1 2 3 2k 1 2 3 2k 1 2 3 2k 1 2 3 2k 1 2 3 2k

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The Markov property and Markov

chain [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 transition probability and

transition matrix

[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

p1,4 1 2 3 4 1 2 3 4

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The existence of Markov property

Although ω depends on Φ, given:

3 m 2 m 1 m m 2 m 1 m m 1 m m

φ

,

φ

,

φ

,

φ

,

φ

,

φ

,

φ

-the p.d.f. is of Φ_mwithis known

No delay

1-symbol delay 2-symbol delay

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The Transition Matrix and

It’s Function

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Find the transition matrix

The number of terms ω depends only on the number of accumulator (only 2-term in 2nd order loop)

 combines these two terms as only one

(the resulting p.d.f. is the convolution of those of the 2 terms)

 Using the compound p.d.f. we can get the transition p.d.f.  integrating the transition p.d.f. to get the transition p.m.f.  Here’s the transition matrix

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The functions of the transition

matrix: steady state analysis

transition Matrix P

Stationary

Phase error variance

BER and

BER degradation Stationary

p.m.f.

Apply Eigenvalue decomposition

If P is nonnegative, Irreducible, aperiodic

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The functions of the transition

matrix: transient state analysis

transition matrix

Mean time to absorption (convergence)

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Complexity of higher order loop (1)

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 2nd order 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

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Complexity of higher order loop (2)

• Assume phase value is represented by k-bit, 2nd order loop, d-delay

 The transition matrix is 2k(d+2)x2k(d+2)

(actually the independent parameters are only 2k(d+2)(2k-1), as the following shows)

 The complexity is too large for computer to do matrix operation

 We must use some approximation

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Conclusion

• As the difference equation of the OFDM CFO tracking loop has Markov property, we can analyze it by

modeling it as a Markov chain (if finite word length effect is considered).

• After getting the transition matrix from the difference equation, we can find:

– Stationary p.m.f. – Stationary variance

– BER and it’s degradation – Mean time to converge

• For a higher order loop, the dimension of the transition matrix is too large for computer, it’s necessary to resort to some approximation of the transition matrix.

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