 Open Access
 Total Downloads : 123
 Authors : Kirti Sharma, Suresh Sahadeorao Gawande
 Paper ID : IJERTV3IS100516
 Volume & Issue : Volume 03, Issue 10 (October 2014)
 Published (First Online): 29102014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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,
BhopalM.P
AbstractIn this paper, we calculate minimum non coherent distances of blockcoded TAPSK (twisted amplitude and phase shift keying) and QAM (quadratureamplitude modulation), both using hamming distance. According to the derived distances, non coherent blockcoded TAPSK (NBCTAPSK) and non coherent blockcoded QAM (NBCQAM) are presented. If we change the radius of NBCTAPSK then it performs best among allnon coherent schemes and NBCQAM performs worse due to its small minimum non coherent distance. However, if we consider block length is not short, NBCQAM 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

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

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 generalized8TAPSK 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 blockcoded generalized16TAPSK,
nc,a
codesdesigned for non coherent decoding outperform this trainingbased 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 bruteforce searching for all codewordpairs.
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 NBC8TAPSK. Table I compares
transmitted signals. If ca = (1, 2, ,), cb = (1, 2, ,
) and cc = (1, 2, , ) are code words of binary block
NBC8TAPSK with NBC8PSK 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 , NBC8TAPSK always has larger ^2nc than NBC8PSK.Figure 2 presents the results for =4. For the pilotoptimized 16QAM, the amplitude of the pilot signal is 1.225. NBC16QAM 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
InPhase
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 blockcoded MPSK is equal to (). Therefore, forBlockcoded MPSK, should be a binary block code withlarge . We proposed NBCMPSK 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 blockcoded8TAPSK
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 NBC8TAPSK with NBC8PSK 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 NBC8PSK. COMPARISON OF THEORETICAL BEST VALUES AND SIMULATION BEST VALUES OF FOR NBC16TAPSK.
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 onebit loss isunnecessary.
B. 16QAM
The distance between the smallestenergy point and the origin in the 16QAM constellation is denoted by.
For NBC16TAPSK, 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
nearestneighbor 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 nearestneighbor codewords,so the best for simulations is larger than the best for dnc .
NBC16TAPSK is better than NBC16QAM at high SNRs which agrees with the minimum noncoherent distance analysis. For NBC16QAM, the gap betweennoncoherent decoding and ideal coherent decoding is quite wide.

SIMULATION RESULTS AND DISCUSSIONS
At high SNRs, the pilotoptimized 16QAM outperforms NBC16QAM, and NBC16TAPSK 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. NBC16QAM 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 NBC16QAM. But for noncoherent decoding, NBC16TAPSK is better than NBC16QAM at high SNRs which agrees with the minimum noncoherent distance analysis. For NBC16QAM, 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 channelquantization decoding algorithmin [6, Sec. III] is used. This algorithm uses the estimate of from the family T={0,2/, Â·Â·Â· ,2(
0.8
10
1 1.5 2 2.5 3 3.5 4 4.5 5
S/N
1)/}, =4for NBC8TAPSK and NBC16QAM,
=8forNBC8PSK and NBC16TAPSK. In all simulations, we set =6,but if 'and care uncoded 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 NBC16TAPSK and NBC 16QAMwhose (,min,,min, ,min,,min) is (2, 1, 1, 1), and the differentiallyencoded16QAM 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 pilotoptimized 16QAM outperforms NBC16QAM, and NBC16TAPSK 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 NBC16TAPSK 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. NBC16TAPSK and NBC16QAM both use (,min,,min,,min,,min)=(8, 4, 1, 1),and NC8TAPSK using () (denoted by NBC8TAPSK(H) and NBC8TAPSK(denoted by NBC8TAPSK(L)) andboth
use(,min,,min,,min) )=(1, 1, 1). NBC8TAPSK using ()has almost the same BER as NBC8TAPSK andthus is not shown in the gure2. The used values of are1.94,

and 1.6 for NBC 8TAPSK(H), NBC8TAPSK(L)and NBC8TAPSK, respectively. We find that NBC8PSK isthe worst, and NBC8TAPSK has better BER than NBC8TAPSK(L)and NBC8TAPSK(H). At high SNRs, NBC16TAPSKoutperforms NBC8TAPSK. This is reasonablesince its , 0.6277, is larger than ncof NBC 8TAPSK,0.6030. After all, NBC16QAM whose is only 0.1649 isthe best. It provides about 1.6dB gain over NBC
16TAPSK ata BERof106 .
Quite different from NBCMPSK and NBCTAPSK, the average number of nearest neighbors of NBC16QAM is very small. It is complicated to compute the average number of nearest neighbors of NBC16QAM, 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 figure4.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 16QAM. Therefore, it is reasonable that the performance of energy constraint 16MAPSK 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


CONCLUSION
In this paper, the minimum noncoherent distances of block codedTAPSK and 16QAM using linear component codes arecalculated. The minimum noncoherent 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 blockcoded TAPSK. According tothe derived distances, we propose NBCTAPSK and NBCQAM.The comparison of minimum noncoherent distancesshows the superiority of NBCTAPSK over NBC QAM athigh data rates. We compare various noncoherent 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 NBCMPSK and NBCTAPSK, the average number of nearest neighbors of NBC16QAM is very small. It is complicated to compute the average number of nearest neighbors of NBC16QAM, 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 NBC16TAPSK and NBC16QAM as follow.