Optimum Bit Detection of Quadrature Amplitude Modulation in Self Correlated Gaussian Noise Channel

Atif Bin Mansoor College of Aeronautical Engineering, National

University of Science and Technology, Pakistan Bmansoor@ieee.org ,

Awais Mansoor Center for Advanced Studies in Engineering , University

of Engineering and Technology, Taxila,

Pakistan

Syed Ismail Shah Iqra University, H-9, Islamabad, Pakistan

Syedismailshah@gmail.com

awais.mansoor@gmail.com

Abstract

This paper discusses the optimum bit detection for coherent QAM (Quadrature Amplitude Modulation) in the presence of colored Gaussian noise. We employ Gray-mapping for constellations, and derive the log-likelihood ratio (LLR) for each bit forming the symbol. Using LLR, we derive the average bit error probability for individual bits. We further investigate the equalization methods under colored noise to facilitate optimum detection.

1. Introduction

Optimum detection of symbols in digital modulation schemes has been a topic of great interest in digital communication circles. When referring a modulation scheme to be optimal we refer to its abilities to minimize the average probability of symbol error (Pe) but often in digital communications the engineers are interested in the probability of bit error (Pb) than Pe. In order to minimize Pb in the presence of channel noise, the most common and simplest approach is the use of Gray Codes, which guarantee that adjacent symbol errors result in no more than a single bit error. In practice additive noise is often Gaussian but its power spectral density (PSD) is not flat (white). The noise still remains independent of the input message signal but correlates with itself from time-to-time. Self-correlated noise can significantly alter the performance of receiver designed for white noise Gaussian (the extent of alteration is dealt with in detail in sections to follow), henceforth extending a need for the redrawing of decision regions and error probabilities. This paper discusses the optimality in bit detection for M-QAM modulation under non-flat Gaussian channel and compute probability of bit error for each modulation using loglikelihood method described in [3].

The rest of the paper is organized as follows:

Section-2 discusses the vector self correlated noise channel and its various properties. Section-3 derives various expressions to check the optimality of bit detection of 16-QAM (taken as representative of M-ary square QAM) using the log likelihood ratio. Section-4 generalizes the results for M-ary square QAM employing the derivations from Section-3. Section-5 discusses equalization techniques employed to whiten the channel effects. The paper is concluded in Section-6. 2. The vector self correlated noise channel

We consider 16-QAM square constellation as representatives of M-QAM constellations. After building the concept on optimality using the representative scheme, generalization for arbitrary size M-QAM constellations is presented. All constellation points in the analysis are assumed to be equally likely. Let x be the transmitted signal and n is the additive noise, then the received signal across the AGN (Additive Gaussian Noise) channel can be given as:-

y x n= +

where y is the received signal after going through additive Gaussian channel.

The autocorrelation matrix of the noise having non-flat power spectral density can be given using [1] as:-

[ ] 2* n nE nn R R σ= = Both nR and nR are positive definite matrices. This discrete additive Gaussian noise (AGN) channel can often be substituted in place for the continuous AGN channel. This

discrete noise vector can be “whitened”, transforming nR into an identity matrix.

The whitening filter “whitens” the non-flat colored noise and the analysis developed for AWGN channel can then be applied. Cholesky factorization determines the invertible whitening filter as:-

1/ 2 *1/ 2(1)n n nR R R=

where 1/ 2

nR is the lower triangle and *1/ 2

nR is the upper triangle matrix. A detailed review of the theory on whitening filters and their applications to AGN channels can be viewed in [1, 2]. 3. Optimality of 16-QAM square constellation

The constellation diagram for 16-QAM is shown in figure-1. Each of the 16-points is represented by bit vector (b1, b2, b3, b4) that is encoded using Gray mapping.

-4

-3

-2

-1

0

1

2

3

4

-4 -2 0 2 4

Figure-1: Original 16-QAM Constellation Diagram

Figure-2 shows the above 16-QAM constellation

when transformed under the colored noise correlation matrix Rn given by:-

112

1 12

nR

=

Figure-2 depicts that passing through self

correlated noise channel dmin has decreased and consequently error probability is increase. Due to the symmetry of the decision boundaries it is clear that the bit error probabilities for bit b1 and b3 are the same and so does the probabilities for b2 and b4. As a result it will be sufficient to derive expressions only for b1 and b2.

The Log Likelihood Ratio (LLR) for the first b1 denoted by LLR(b1), is defined as

11

1

Pr ( 1)( ) log( ) (2)

Pr ( 0)ob b

LLR bob b

==

=

Figure-2 suggested that the signal points s0, s1, s2,

s3, s4, s5, s6 and s7 contribute to the denominator of (2) while the rest contribute to the numerator. The denominator of

-6

-4

-2

0

2

4

6

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Figure-2: 16-QAM Constellation Diagram for signal

sets after passing through self correlated noise channel equation (2), with the help of Bayes’ rule and employing some algebraic manipulations can be written as:

0 1 2

1 3 4 5

6 7

( ) ( ) ( )1Pr ( 0) ( ) ( ) ( ) (3)

16 ( )( ) ( )

n n n

n n nz

n n

f z s f z s f z sob b f z s f z s f z s

f zf z s f z s

− + − + − = = + − + − + − + − + −

The individual terms for the constellation diagram shown in Figure-2 are calculated. For s0 = ( -3(1+1/ 2 )d, -3(1+1/ 2 )d)

2

0 2 2

22

2 2

1( ) exp2 2

( +3(1+1/ 2)d) ( +3(1+1/ 2)d)1 exp2 2

n

Q I

xf z s

z z

πσ σ

πσ σ

− = − + = −

Similarly, the calculations are made for the other six constellation points and for 1Pr ( 1)ob b = . Consequently we have both equations and can insert the results in LLR formula. The average BER can be found using the techniques of [3]. As stated earlier, the final

result shows enhancement of Probability of error due to reduction in dmin. In section-5, we employ equalization techniques to whiten the colored noise effects. This enables us to utilize the equations derived for AWGN in [2]. 4. Generalization to m-ary square QAM

Generalization of previous section’s results can be made to M-ary square QAM. The generalized correlation matrix R for an arbitrary colored noise can be written as:

1 cossin 1nR

θθ

= −

Any arbitrary constellation point ‘s’ will now map to ‘s`’ under the stated transformation. The pdf is now given by:

2

2 2

2 2

2 2

1( `) exp2 2

( ` ) ( ` )1 exp2 2

n

Q Q I I

xf z s

z s z s

πσ σ

πσ σ

− = − − + − = −

Where s`I and s`Q are the inphase and quadrature components of signal point s after passing through self correlated noisy channel. Now the log likelihood ratio (LLR) defined in (1), is recomputed for M-ary QAM in colored noise as:

1 1

1 1

`: ` ( cos , sin ), cos ( ), sin ( )

`: ` ( cos , sin ), cos ( ), sin ( )

Pr( )( ) log

Pr( )s s s s s X i s X j

ijs s s s s X i s X j

zLLR c

zθ θ θ θ

θ θ θ θ

= ∈ ∈

= ∈ ∈

=

∑∑

where cij is any arbitrary bit with i=j=k,…….,2k-1. 5. Equalization Techniques

In Additive Self-Correlated Gaussian Noise (ACGN) channel, the autocorrelation function of additive noise Rn(τ) need not equal to (No/2)δ(τ) .Using normalized noise autocorrelation function n (τ) = Rn (τ)/ (No/2), the power spectral density of un-normalized noise is then Sn(f) = (No/2) where is the Fourier Transform of n (τ).

We employed two approaches; one with whitening filter pushed back into transmitter and other reversible receiver with noise whitening filter as its part.

Figure-3: ACGN Channel Using Cholesky factorization the inverse of invertible whitening transformation is found:

1 0

1 2

−

Using this transformation the modified constellation for transmitter is found and is depicted in Figure-4. The choice of this constellation at transmitter results in an equivalent filtered AWGN channel and all results found in [2] can be employed.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-8 -6 -4 -2 0 2 4 6 8

Figure-4: Colored Noise compensated Constellation

The average bit error, using the techniques of [3] are obtained as

314

3 3

[ ( ) ( ) ( ) ( )

( ) ( )]

d dd db

d d

P Q Q Q Q

Q Q

η ησ σ σ σ

η ησ σ

− +

− +

= + + +

+ −

where 2( ) 1/ 2 exp( / 2)x

Q x u duα

π= −∫ and

2 1 2 2( / ) cosh ((1/ 2) exp(4 / ) 3)d dη σ σ−= +

5.1 Adaptive filter receiver

Correlation matrix for colored noise in channel is mostly unknown. To cater for varying channel behavior, we considered an adaptive technique to train the receiver. A predefined sequence of symbols is utilized for estimation of correlation matrix for colored noise channel. The simulation results showed the exact calculation of the correlation matrix. This matrix is subsequently used for whitening by receiver. 6. Conclusion

In this paper, we studied the optimality of bit detection for M-QAM signal set under the additive colored noise environment. For M-QAM signal constellation, we employed 16-QAM signal set as representative of the general results, and derived the log likelihood ratio (LLR) for each bit forming the symbol under the colored noise environment. We used three different approaches for this purpose. Approaches utilizing whitening filter at receiver and transmitter provide practical means to calculate the Pb for colored noise channel utilizing existing methods for AWGN channels. Varying decision boundary approach appears to give extended mathematical insight and provide means to calculate Pb independent of AWGN techniques. This, however, shows an increment in the probability of error due to reduction in dmin. 7. References [1] J. Cioffi, Digital Communication-Signal Processing and Detection, Preprint, available at www.ece.stanford.edu/ee356. [2] M. K. Simon and R. Annavajjala, “On the Optimality of Bit Detection of Certain Digital Modulation,” IEEE Trans.Commun. vol. 53, No.2, February 2005. [3] M. S. Raju, A. Ramesh, and A. Chockalingam, “BER Analysis of QAM with Transmit Diversity in Raleigh Fading Channels” in Proc. IEEE GLOBECOM’03, San Francisco, CA, Dec. 2003, pp. 641-645. [4] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1995.