Symbol Error Rate of Star QAM

Accurate SER Expressions for M-ary Dual Ring Star QAM in Fading Channels

Sourjya Dutta1 and Aniruddha Chandra2

1. Polaris Networks, Kolkata, India2. ECE Department, NIT, Durgapur, WB, India

December 28, 2012

2

Introduction

Our present work

• Derivation of analytical expressions for Symbol-Error-Rate for M-ary dual

Star QAM in channels corrupted by AWG noise with Rayleigh, Rician and

Nakagami-m fading.

• The expressions are computationally inexpensive.

• Monte Carlo simulations were performed to verify the expressions.

• The results eradicate the flaw in the paper by Barbounakis and Papadakis .

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

3

Outline

Introduction to Dual Ring Star QAM.

Motivation.

System Model.

SER Calculation.

Results.

Conclusion.

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

4

M-ary Star QAM Modulation

• It is a simple extension of phase modulation to a multi-amplitude phase modulation.

• It was first proposed by Cahn (1960).

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

5

Motivation

The Star QAM modulation scheme is of interest as :

1. It has better error performance for wireless fading channels.

2. Better suited for adaptive modulation.

3. Square and rectangular QAMs have high Peak to Average power ratio which is not so for star QAM

4. Simpler Encoder-Decoder structures.

SER of 16-ary Star and Square QAM in Rayleigh Fading Channel

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

6

System Model

• Modulator: {bk} →Input bit stream to the modulator.

ṡ(t) → Wave modulated using Star QAM Modulation.• Channel Description: n(t) →Gaussian Noise. α(t) → Amplitude variation due to fading. ṙ(t) = ṡ(t) α(t) + n(t)• Demodulator: ṙ(t) → received signal {ḃk} → Demodulated bit stream.

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

7

Monte Carlo Simulation

Flow Diagram showing the various steps of Monte Carlo simulation used for simulations.

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

8

SER Calculation

Following Craig’s Method we have derived the SER in AWGN channel for dual ring star QAM modulation as :

ɣ = Signal to Noise ratio;L = Ro/ RI

αk= rk/ RI and is called the scaling parameter

The SER in fading channel can be given as :

o

efadinge dPfP )()(

Where f(ɣ) is the fading distribution as a function of ɣ.

dL

MP

k k

kke

k

4

1 022

2

)(sin)1(2

)(sinexp

2

1

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

9

SER Calculation (contd…)

)(sin)1)(1(2

)(sin)(,

)(1exp

)(1

)exp(

2

1

22

2

4

1 0

k

kk

k

Riciane

KL

Mwhere

dKK

Pk

dL

MP

k k

kkRayleighe

k 14

1 022

2

)(sin)1(2

)(sin1

2

1

The expressions for SER in Rayleigh, Rician and Nalagami-m fading channels are

)(sin)1(2

)(sin)(,

)(2

1

22

2

4

1 0

k

kk

k

m

Nakagamie

L

Mwhere

dm

mP

k

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

Rayleigh Fading Channel►

Rician Fading Channel ►

K is the Rician Parameter.

Nakagami-m Fading Channel ►

m is the Nakagami shape factor.

10

Results

Analytical and Simulated SER values in Rician (K=10) fading channel

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

11

Results (contd…)

Analytical and Simulated SER values in Nakagami-m (m=2) channel

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

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Results (contd…)

Gray’s approximation which gives the Bit Error Rate as,

Pe,b= Pe/log2(M),

is not a satisfactory approximation for Star QAM Modulation.

The above estimation is somewhat practical for the SNR range 9dB to 14dB.

The formula gives errors above 30% for lower and higher SNR values.

For critical fading conditions the error shoots above 100%.

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

13

Conclusions

§ Expressions derived by Barbounakis and Papadakis are shown to be inaccurate.

§ Computationally efficient error rate expressions - contains summation of single definite integrals .

§ Derived expressions are validated through Monte Carlo simulation.

§ Expressions valid for M ≤ 128.

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

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References

[1] L. N. Binh, “Dual-ring 16-star QAM direct and coherent detection in 100 Gb/s optically amplified fiber transmission: simulation,” Optical and Quantum Electronics, vol. 40, no. 10, pp. 707–732, Aug. 2008.

[2] J. W. Craig, “A new, simple, and exact result for calculating the probability of error for two-dimensional signal constellations,” in Proc. IEEE Military Communications Conf. (MILCOM91), McLean, VA, USA, Oct.1991, pp. 571–575.

[3] X. Dong, N. C. Beaulieu, and P. H. Wittke, “Error probabilities of two dimensional M-ary signaling in fading,” IEEE Trans. Commun., vol. 47, no. 3, pp. 352–355, Mar. 1999.

[4] I. S. Barbounakis and A. M. Papadakis, “Closed-form SER expressions for star MQAM in frequency non-selective Rician and Nakagami-m channels,” International Journal of Electronics and Communication, vol. 59, no. 7, pp. 417–420, Nov. 2005.

[5] X. Lei, P. Fan, and Q. Chen, “Comment on “Closed-form SER expressions for star MQAM in frequency non-selective Rician and Nakagami-m channels”,” International Journal of Electronics and Communication, vol. 62, no. 9, pp. 715–716, Oct. 2008.

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29

Thank You

Please send comments on our work at

sourjya_dutta@polarisnetworks.net

S.Dutta, Symbol Error Rate of Dual Ring Star QAM in Fading Channels

CODIS 2012Dec. 28&29