Bit Error Rate Performance of TAPSK using Block Coded Modulation

Bit Error Rate Performance of TAPSK using Block Coded Modulation

Kirti Sharma, Suresh Sahadeorao Gawande Dept. of Electronics and Communication Engineering Bhabha Engineering Research Institute,

Bhopal-M.P

AbstractIn this paper, we calculate minimum non coherent distances of block-coded TAPSK (twisted amplitude and phase shift keying) and QAM (quadrature-amplitude modulation), both using hamming distance. According to the derived distances, non coherent block-coded TAPSK (NBC-TAPSK) and non coherent block-coded QAM (NBC-QAM) are presented. If we change the radius of NBC-TAPSK then it performs best among allnon coherent schemes and NBC-QAM performs worse due to its small minimum non coherent distance. However, if we consider block length is not short, NBC-QAM has the best error performance because the code words with small non coherent distances are rare.Here we also change the value of r and see the performance of BER and also see the effect of Rayleigh channel on BER.

codes , and , are also called components codes. the minimum non coherent Hamming distance of is defined by

, = min{,min, ,max}where d,min and

,maxdenote the minimum and maximum values of Hamming distance between any two code words corresponding to different data bits in .

II. NONCOHERENT BLOCKMODULATION USING LINEAR COMPONENT CODES

1. APSKFor TAPSK with labeling in Fig. 1, the bit in level

   decidesSymbol energy. The radiuses of the inner and outer

   Index TermsNon coherent detection, BCH Codes, channels Block coded modulation, multilevel coding.

   circles are denoted by

   r0 and r1 , respectively. The values of

   r1 and r0 (r0 1 r1)

   satisfy =2 when =0 has thesame

   1. INTRODUCTION

probability as =1, r2 r2 2 .With the proof given in

THE ADDITIVE white Gaussian noise (AWGN) channel 0 1

2

which introduces an unknown carrier phase rotation has been investigated in many works. This channel offers a useful

Appendix A, we have the following theoremDefine() by

()=

abstraction of the flat fading channel,when the effects of the phase rotation need to best studied independently of the

r2 r 2

1 0 d

2

(r2 (N d ) r r cos )2

• (r0r1d sin )

0 0 1

nc,a

amplitude variations. A simple model that is commonly used is one where the unknown carrier phase is constant over a

Block coded generalized-8TAPSK whose component codes are all linear, the minimum squared non coherent distance is

block of symbols, and independent from block to block.

This model is correct for frequency hopping systems. For this

2 mind 2

2

, d

, d

2

nc,b

2

nc,c

,where

d

nc

non coherent channel with large ,pilot symbols used for the carrier phase estimation with codes designed for coherent

d

2

nc,a

minf (d

a,min

), f (d

a,max

),

d

2

nc,b

r2 (N

(N d

ncH,b )

decoding perform well.However, for small , block

2

0

d)

nc

nc,a

nc,c

2r2d

ncH,c

For block-coded generalized-16TAPSK,

nc,a

codesdesigned for non coherent decoding outperform this training-based non coherent codes.The minimum non

by a similar derivation d 2 mind 2

0

, d

, d

2

nc,b

2

, d

nc,c

2

nc,d

0

coherent distances of codes are obtained by brute-force searching for all codeword-pairs.

Where, d 2 minf (d

a,min),

f (d

a,max

),

For the transmitted baseband codewordx =(1,2, , ),

the received baseband blocky = (1, 2, , ) is given by

2

d

nc,b

r2 (N

(N 2

2

2

d

ncH ,b

d 2

)2 ncH ,b ) ,

2

0

y = x exp{} + n .signal point in the signal constellation of 8PSK, is labeled by (, , ) where , , and {0, 1}. Let

2

d

0

nc,c

r2 (N

(N d

ncH,c

)2 d 2ncH,c ) and

(1, 1, 1), (2, 2, 2), , (, , ) bea block of

2

d

nc,d

2r2d

ncH,d

,for NBC-8TAPSK. Table I compares

transmitted signals. If ca = (1, 2, ,), cb = (1, 2, ,

) and cc = (1, 2, , ) are code words of binary block

NBC-8TAPSK with NBC-8PSK in terms of for =15, 31, 63, and 8. In this paper, only (15,11,1) code, (31,26, 1)

code and (63,57,1) BCH codes are used as componentcodes. The values of (,min,,min,,min).are shown in the column of code", and the values of which maximize the same rate and , NBC-8TAPSK always has larger ^2nc than NBC-8PSK.Figure 2 presents the results for =4. For the pilotoptimized 16QAM, the amplitude of the pilot signal is 1.225. NBC-16QAM has better BER than 16QAM(H) and 16QAM(L), but they all do not decrease exponentially.

1

0.5

Quadrature

0

-0.5

-1

Scatter plot of 16QAM with noise

-1 -0.5 0 0.5 1

In-Phase

From this diagram we can calculate minimum non coherent

nc min

distance d 2

.If we define d

min H

4 , then

d (d

/ 2 ) , d (d

/ 2 ) ,

0 min

min H 0

1,min

min H 1

d (d

/ 2 ) and d (d

/ 2 )

2,min

min H 2

3,min

min H 3

For blockcoded 16QAM whose component codes are all linear, the minimum squared non coherent distance

d

nc,a

nc

2 mind 2

2

, d

nc,b

2

, d

nc,c

2

, d

nc,d

(c) (d)

Fig. 1. Constellations with bit labeling for (a) 8PSK

(b) 8TAPSK ( = /4)(c) 16TAPSK ( =/8) (d) 16QAM

When =1, i.e. TAPSK becomes MPSK, we have ()=0and

()=( – ). Consequently, of block-coded MPSK is equal to (). Therefore, forBlockcoded MPSK, should be a binary block code withlarge . We proposed NBC-MPSK in [5] by setting da,max , dncH ,a N da,min such that

,= ,min at the priceof sacricing one data bit. But as increases, () also increases. For block-coded8TAPSK

Spectral efficiency	N=15		N=31		N=63
	Theo.	Simu.	Theo	Simu	Theo	simu
2.23	0.51	0.52	0.46	0.48	0.64	0.67
3.24	0.46	0.49	0.56	0.58	0.56	0.63
3.67	0.43	0.45	0.59	0.61	0.66	0.67
4.34	0.34	0.36	0.61	0.63	0.67	0.69

where is large enough, ()= (r r )2 N / 2 can be larger

Spectral efficiency	code	d 2 nc/p>
		N=15	N=31	N=63
4.34	8PSK	0.212	0.351	0.361
2.24	8TAPSK(H)	0.401	0.356	0.371
2.43	8TAPSK(L)	0.352	0.412	0.423
3.23	16QAM	0.450	0.453	0.464
2.56	16TAPSK	0.554	0.621	0.632

Table I compares NBC-8TAPSK with NBC-8PSK in terms of d 2 for =15, 31, 63,and 8.In this paper, only (15,11,1) code, (31,27,1) code and (63,57,1) BCH

nc

CODES are used as component codes. The values of

(,min,,min d,min)For the same rate and , NBC- 8TAPSK always has larger than NBC-8PSK. COMPARISON OF THEORETICAL BEST VALUES AND SIMULATION BEST VALUES OF FOR NBC-16TAPSK.

1 0

than (,min). If>1.61238, () is always larger than

(,min) for any value of ,min(= /2).In such case,

nc,a

since d 2

= (,min), ,mincould be anormal code with

large ,min and thus the one-bit loss isunnecessary.

B. 16QAM

The distance between the smallest-energy point and the origin in the 16QAM constellation is denoted by.

For NBC-16TAPSK, Table II compares the best values offor simulations with the theoretical best values of thatmaximize

. The values of (,min,,min,,min)are shown in the column of data rate" In the multistagedecoding, a decoding error in level probably causes errorpropagation, so slightly larger which results in better BER in level would have the

best overall BER. Let

Na and Nb denote the numbers of the

nearest-neighbor codewords for Ca and Cb

respectively,

shown in Table II also. We find that ifis less than or approximately equal to 1, the best forsimulations is close to

(slightly larger than) the best for.Butifis not small, the BER in level is increaseddue to the large number of the nearest-neighbor codewords,so the best for simulations is larger than the best for dnc .

NBC-16TAPSK is better than NBC16QAM at high SNRs which agrees with the minimum noncoherent distance analysis. For NBC-16QAM, the gap betweennoncoherent decoding and ideal coherent decoding is quite wide.

1. SIMULATION RESULTS AND DISCUSSIONS

   At high SNRs, the pilot-optimized 16QAM outperforms NBC-16QAM, and NBC-16TAPSK is the best among all noncoherent schemes. The results for =15 are shown in Fig. 3 in which the amplitude of the pilot signal is 1.673.We

   16TAPSK and NBC 16QAM areexplained in [7] is also compared .

   Figure 2 presents the results for =31. For the pilotoptimized 16QAM, the amplitude of the pilot signal is

   1.225. NBC-16QAM has better BER than 16QAM(L), but they all do not decrease exponentially because the average number of codewords with small noncoherent distances is little, but not little enough. For ideal coherent decoding,NBC- 16TAPSK is worse than NBC-16QAM. But for noncoherent decoding, NBC-16TAPSK is better than NBC16QAM at high SNRs which agrees with the minimum noncoherent distance analysis. For NBC-16QAM, the gap between noncoherent decoding and ideal coherent decoding is quite wide given as references.

   But here we take fixed minimum hamming distance, then find

   find that the average number of codewords with smallnoncoherent distances is too tiny to affect the curves

   2 , 2 , 2 , and then minimum required d 2

   min

   .for

   a b c

   above BER of 106 for all noncoherent 16QAM schemes.

   							16PSK
   							16TAPSK
   							16QAM
   							8TAPSK(L)
   							8TAPSK(H)

   -0.5

   10

   evaluating system performance, we compute BER versus E/Nb graph for the AWGN channel or Rayleigh channel. For encoding we use the BCH encoder, then transmitted the signals by thisencoding, at the receiver we use same type of decoder and see the error which place we have to correct.

   -0.6

   10

   								8PSK 16PSK 16 TAPSK 16QAM 8TAPSK(L) 8TAPSK(H) 16TAPSK

   -0.5

   BER

   10

   -0.7

   10

   -0.6

   10

   BER

   -0.8

   10

   1 1.5 2 2.5 3 3.5 4 4.5 5

   S/N

   -0.7

   10

   ( BER Vs S/Na t( r=o.35) )

   At the receivers, the channel-quantization decoding algorithmin [6, Sec. III] is used. This algorithm uses the estimate of from the family T={0,2/, ··· ,2(-

   -0.8

   10

   1 1.5 2 2.5 3 3.5 4 4.5 5

   S/N

   1)/}, =4for NBC-8TAPSK and NBC-16QAM,

   =8forNBC-8PSK and NBC-16TAPSK. In all simulations, we set =6,but if 'and care un-coded bits(,min=

   ,min=1), the labeling of bits and shouldbe Gray labeling of QPSK for theminimization of bit errorrate (BER). The labeling in Fig. 1(c) For NBCTAPSKand nonlinear NBC- TAPSK, we look for the value of that needs the lowest SNR at the BER of 106 by simulation results, and use it in simulations. In Fig. 2 and Fig. 3, we consider noncoherent

   blockcodes using sixteen signal points with data rate (4 – 4)/bits/symbol, including NBC-16TAPSK and NBC- 16QAMwhose (,min,,min, ,min,,min) is (2, 1, 1, 1), and the differentially-encoded16QAM scheme in [9] denoted by 16QAM(H). We modify the scheme in [9] by choosing the low energy codewordsinstead of the high- energy codewords, denoted by 16QAM(L),as suggested by [7]. The results of ideal coherent decoding for NBC-

   ( BER Vs S/Na t( r=o.40) )

   At high SNRs, the pilot-optimized 16QAM outperforms NBC-16QAM, and NBC-16TAPSK is the best among all noncoherent schemes. The results for =15 are shown in Fig. 3 in which the amplitude of the pilot signal is 1.673.We find that the average number of codewords with smallnoncoherent distances is too tiny to affect the curves above BER of 106 for all noncoherent 16QAM schemes.

   NBC16QAMoutperforms NBC-16TAPSK and the pilot- optimized16QAM, and its gap between noncoherent decoding and idealcoherent decoding is less than 1dB. Various non coherent block codes using eight or sixteen signal points with data rate (3 -3)/bits/symbol for =16 are compared in Fig. 4. NBC-16TAPSK and NBC-16QAM both use (,min,,min,,min,,min)=(8, 4, 1, 1),and NC-8TAPSK using () (denoted by NBC-8TAPSK(H) and NBC-8TAPSK(denoted by NBC-8TAPSK(L)) andboth

   use(,min,,min,,min) )=(1, 1, 1). NBC-8TAPSK using ()has almost the same BER as NBC-8TAPSK andthus is not shown in the gure-2. The used values of are1.94,

   1. and 1.6 for NBC 8TAPSK(H), NBC-8TAPSK(L)and NBC-8TAPSK, respectively. We find that NBC-8PSK isthe worst, and NBC-8TAPSK has better BER than NBC8TAPSK(L)and NBC-8TAPSK(H). At high SNRs, NBC16TAPSKoutperforms NBC-8TAPSK. This is reasonablesince its , 0.6277, is larger than ncof NBC- 8TAPSK,0.6030. After all, NBC-16QAM whose is only 0.1649 isthe best. It provides about 1.6dB gain over NBC-

      16TAPSK ata BERof106 .

      Quite different from NBC-MPSK and NBC-TAPSK, the average number of nearest neighbors of NBC-16QAM is very small. It is complicated to compute the average number of nearest neighbors of NBC-16QAM, so we take an example to illustrate this point as follows. Suppose that the transmitted has component codeword in level ca= 0. Consider another component codewordca.Help of scatter plot shown in above figure-4.then we compute BER for 16 QAM , For N=31, the minimum non coherent distance of energy constraint 16- MAPSK is larger than that of energy constraint 16-QAM. Therefore, it is reasonable that the performance of energy constraint 16-MAPSK is better than energy constraint 16- QAM.

      							8PSK 16PSK 16 TAPSK 16QAM 8TAPSK(L) 8TAPSK(H)

      -0.5

      10

      8 QAM

      -0.6

      BER

      10

      -0.7

      10

      -0.8

      10

      1 1.5 2 2.5 3 3.5 4 4.5 5

      S/N

2. CONCLUSION

In this paper, the minimum non-coherent distances of block- codedTAPSK and 16QAM using linear component codes arecalculated. The minimum non-coherent distance of block- codedQAM with more signal points can be derived similarly. Wend that the minimum noncoherent distance of block- codedMPSK derived is a special case of the derived minimumnoncoherent distance of block-coded TAPSK. According tothe derived distances, we propose NBC-TAPSK and NBC-QAM.The comparison of minimum noncoherent distancesshows the superiority of NBC-TAPSK over NBC- QAM athigh data rates. We compare various non-coherent block codesbased on the simulation results. By changing the value of radius in TAPSK we get optimum value of radius (r) in which TAPSK has better BER performance among all digital modulation techniques and QAM has worse error performance due to its smallminimum noncoherent distance.

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( BER Vs S/Na t( r=o.55) )

Quite different from NBC-MPSK and NBC-TAPSK, the average number of nearest neighbors of NBC-16QAM is very small. It is complicated to compute the average number of nearest neighbors of NBC-16QAM, so we take an example to illustrate this point as follows. Suppose that the transmitted codeword, denoted by x, has a component codeword in level ca= 0. Consider another component codewordcin leveland the Hamming distance between caand ca is dmin,Assume that = ,. For thiscase, we compute the number of nearest neighbors caused by Ca for NBC-16TAPSK and NBC-16QAM as follow.