Codes over Tori
740
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 2, MARCH 1997
Codes over Tori Klaus Huber, Member, IEEE
Abstract—In this correspondence it is shown how the elements of finite fields when mapped on the surface of a torus can be used for block coding over a three-dimensional signal space. The block codes are useful for data transmission using the recently defined Mannheim distance. Index Terms—Block codes, tori, Gaussian integers, Mannheim distance, coded modulation.
I. INTRODUCTION For digital data transmission very often signals over a two-, three-, or higher dimensional signal space are selected for bandwidth efficiency (see e.g., [1] for the most common signal constellations; for a standard reference of orthogonal functions for communications engineering see [5]). Most high-performance communication systems use convolutional codes at least as inner codes. This is mainly due to the fact that there exist efficient soft-decision decoding algorithms which allow maximum-likelihood decoders. For block codes most good algebraic decoders employ the Hamming distance, which is not very efficient for codes over two- or three-dimensional signal spaces. Recently, the Mannheim distance was introduced to make QAMlike signal constellations more susceptible for algebraic decoding methods (see [3]). For codes over hexagonal signal constellations a similar metric was introduced in [4]. In this correspondence we show that the Mannheim distance is useful for block codes over tori. The resulting signal constellations can be seen as a sort of three-dimensional phase-shift keying. II. CODES
OVER
Fig. 1. GF (97) represented as Gaussian integers modulo 9 + i4.
TORI
Let p be a prime of the form p 1 mod 4. Then, from a basic theorem of number theory we know that p can be represented in an essentially unique way as sum of two squares
p = a2 + b2 :
(1)
In this correspondence, we use the convention a > b > 0. In [3] it was shown how the elements of GF (p) for p 1 mod 4 or of GF (p2 ) for p 3 mod 4 can be mapped on a subset of the Gaussian integers using the complex modulo function given below. The Gaussian integers are those complex numbers whose real and imaginary parts are integers. The Mannheim distance between two Gaussian integers and with respect to the Gaussian integer a + ib is the sum of the absolute values of real and imaginary part of the difference ( 0 ) mod a + ib. The modulo reduction, before summing the absolute values of real and imaginary part, is the difference between the well-known Manhattan distance and the Mannheim distance, for further details see [3]. (For simplicity, we only consider the case p 1 mod 4, leaving the straightforward modifications of the case GF (p2 ) for p 3 mod 4 to the reader.) To obtain the representation of the field GF (p) = f0; 1; 1 1 1 ; p 0 1g as Gaussian integers, we take the elements of GF (p), interprete them
Fig. 2. GF (257) represented as Gaussian integers modulo 16 + i.
as integers k = 0; 1; 1 1 1 ; p 0 1, and use the complex modulo function
(k) = k 0 k(a 0 ib) (a + ib) p to get the corresponding Gaussian integers ([1] denotes rounding to the closest Gaussian integer). For instance, the element 67 of GF (97) is represented as 67 mod 9 + i4 = 67
Manuscript received May 16, 1995; revised October 7, 1996. The author is with Deutsche Telekom AG, Technologiezentrum, P.O. Box 10 00 03, 64276 Darmstadt, Germany. Publisher Item Identifier S 0018-9448(97)00779-7.
0
67(9
0 i4)
97
1
(9 + i4) = 1 + i 3:
Figs. 1 and 2 display the fields GF (97) and GF (257) when represented as Gaussian integers modulo 9 + i4 and 16 + i, respectively.
0018–9448/97$10.00 1997 IEEE
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 2, MARCH 1997
Fig. 3. Cut through torus having radii
R
741
and r .
The elements of GF (p) = f0; 1; 1 1 1 ; p 0 1g can be placed on surface points Tj of a torus using the mapping Tj = tx(j ) ; ty(j ) ; tz(j ) = (cos
(j )(R
0r cos '(j )); sin
(j )(R
0 r cos '(j )); r sin '(j ))
Fig. 4.
T (x; a; b)
Fig. 5.
T (x; b;
for
= 257;
p
a
= 16;
b
= 1;
R
= 2; and
r
= 1.
(2) where tx ; ty ; tz are Cartesian coordinates, and R and r are the radii of a torus as shown by the cut in Fig. 3. To get a better comprehension of the mapping of (2) assume that the points of Fig. 1 are placed on a rubber plane. Now wrap the portion of the plane where the points are located around a torus such that 5 is neighbor to 6, 26 neighbor to 27, etc. For the remainder of the correspondence we define (x) and '(x) as follows: '(x) =
2a
1
(3)
1
(4)
x p 2b x: (x) = p
The signal points Tj are on the line T (x; a; b), where T (x; a; b) = (cos (x)(R
0 r cos '(x)); (x)(R 0 r cos '(x)); r sin '(x))
sin
(5)
with x in the real interval [0; p]. For p = 257 = 162 + 12 ; a = 16; b = 1; R = 2, and r = 1 the line T (x; a; b) is displayed in Fig. 4. The figure makes clear the geometric significance of the integers a and b: a gives the number of rounds around the ring of radius r and b gives the number of rounds around the circle of radius R. The curve T (x; b; 0a) is displayed in Fig. 5 for the parameters of Fig. 4. At the intersection points of T (x; a; b) and T (x; b; 0a) we get the points Tj ; j = 0; 1; 1 1 1 ; p 0 1. To see this take the integer i which fullfills a + ib 0 mod p, i.e., i is a square root of 01 modulo p. Then we have Tj = T (x = j ; a; b)
and Tij mod p = T (x = j ; b; 0a):
This means that going from Tk to Tk+1 corresponds to moving along the line T (x; a; b) (see Fig. 4), whereas going from Tk to Tk+i corresponds to moving along the line T (x; b; 0a) (see Fig. 5). Now let us determine the ratio of R=r such that the Euclidean distance dE (T0 ; T61 ) between T0 and T61 equals the Euclidean distance dE (T0 ; T6i ) between T0 and T6i . A simple derivation leads to R = 2r;
for dE (T0 ; T61 ) = dE (T0 ; T6i ):
v= =
2 (T0 ; T1+i ) dE 2 dE (T0 ; T1 )
3 + cos
2 (a+b)
p
3 + cos
(6)
2 (
)
2
a)
for
p
= 257;
a
= 16;
b
= 1;
R
= 2; and
r
= 1.
Determining the radius r such that dE (T0 ; T61 ) = dE (T0 ; T6i ) = yields r=
6
04
2a
cos p
2b
+ cos p
+ 2 cos
2a
p
1 cos
2b
: (7)
p
For p not too small, a simple computation shows that r is wellapproximated by r
pp=(2):
(8)
The value is quite close to the minimum Euclidean distance dE between any two points Tj ; Tj (see Table I). In general, however, is not exactly equal to dE . This is due to the fact, that the torus line starts at ' = 0, therefore depending on p, there can be points Tk and Tk61 (or Tk and Tk6i ) which are slightly closer together than . Another parameter of interest is the ratio v of the following squared Euclidean distances (see the bottom of this page). It is not difficult to show that limp!1 v = 2. For p small, v is also displayed in Table I. As we are considering the use of tori for signal transmission we are interested in the average signal energy Es of a transmitted point. We get p01 1 2 2 Es = 1 (R + r 0 2Rr cos '(j )): p j =0
1 cos ap0b 0 2 b p 1 cos p 0 2
2a
0
cos
2 (a+b) 2a
cos( p
p
+ cos
+ cos
2b
p
0b)
2 (a
p
:
742
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 2, MARCH 1997
In general, the required energy to transmit a bit of information is smaller than for codes over Gaussian integers, see [3], or codes over Eisenstein–Jacobi integers, see [4] (i.e., for signal points lying on a hexagonal grid which has a higher density than QAM signal constellations). The reason for the better performance lies of course in the fact that we now have signal constellations which are in three dimensions instead of two. More precisely, from the value of Es for Gaussian integers given by (p2 0 1)=6p (see the Appendix) and (7), (9) with R = 2r, we immediately get the improvement (see the bottom of this page) which for p not too small and = 1 is well-approximated by
TABLE I PARAMETERS p; a; b; ; r; v; Es ; and Eb for R = 2r AND EUCLIDEAN MINIMUM DISTANCE dE = 1
10 log10
2
1 p p0 1 10 log
2
2
2
15
10
2
2
15
1:2 dB:
Compared on a bit-per-dimension basis, the toroidal signal set does not improve on usual QAM signal constellations. The toroidal signal constellation should rather be seen as a sort of three-dimensional PSK scheme, since it retains some of the advantages of PSK while making it possible to use more signal points per dimension. Namely, the energy per bit for the toroidal signal set is obtained using (8) which leads to
pp
5
2
2
= log2 (p)
p= log (p) 2
whereas for the M -PSK signal constellation with distance 1 between adjacent points the corresponding value is 2
M 2
= log2 (M )
p = = log (p) 4 3
2
where for a fair comparison we have set M = p2=3 . The most notable advantage of the toroidal signal set compared to the QAM signal set is the much smaller dynamic range of the amplitudes. Remark 1: A reviewer noted that there is still room for improvement if more efficient curves are used, since there are big open areas inside the torus. Indeed, this is true and we briefly mention the most straightforward change which preserves the “Mannheim neighborhood.” Namely, we modify the torus such that the cut (see Fig. 3) consists of ellipses. The toruspoints are then given by Tj = tx(j ) ; ty(j ) ; tz(j )
0 r cos '(j )); (j )(R 0 r cos '(j )); r
(j )(R
= (cos
From '(j ) = 2aj=p we have
sin
cos(2aj=p) =
cos(2j=p)
which gives p zero, as is seen by considering the sum over all real parts of exp( 01 1 2j=p). Hence we obtain 2
Es = R + r
2
(9)
from which we get the energy per bit Eb = Es = log2 p. In Table I the values of p; a; b; ; r; v; Es ; and Eb are listed for R = 2r and dE = 1. From (7) it is clear that the signal energy is smallest if a and b are as close together as possible. For p 1 mod 4 this is the case if a = b + 1. In Table I we can see, for example, that the energy for p = 41 is smaller than the energy for p = 37.
10 log10
p2
0 1 =5r
6p 2
10 log10
(p
2
1
1
2
sin '(j )):
Setting R = 2 1 r1 and using standard numerical techniques, the value of r2 can be found such that the normalized Euclidean distance 2 dE =Eb is as great as possible. In Table II the results of such computations are displayed. Significant improvements over the data of Table I occur if a and b are close together. These improvements have been obtained by permitting dE (T0 ; T1 )2 6= dE (T0 ; Ti )2 . In the general case, the condition dE (T0 ; T1 )2 = dE (T0 ; Ti )2 leads to 2R(R
0 2r ) = 1
2
r2
0r 1 2 1
cos
2a
p
+ cos
2b
p
(10)
which, of course, contains the solution R = 2r; r1 = r2 = r. Using (10) as constraint and finding the minimum of Es = R2 +(r12 +r22 )=2,
=
0 1)(6 0 4(cos(2a=p) + cos(2b=p)) + 2 cos(2a=p) 1 cos(2b=p)) 30 2 p
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743
higher number of signal points per dimension. The main parts are the mapping of the elements of finite fields GF (p) onto the surface of a torus, and the usage of the Mannheim metric. The Mannheim metric is quite well suited for codes over tori, as neighbor points have Mannheim distance 1. Errors with a given Mannheim weight however can be corrected with block codes having fewer check symbols than codes able to correct the same number of Hamming distance errors. Since in data communication errors tend to occur with higher probability closer to the signal points sent, such Mannheim metric codes outperform block codes using the Hamming metric. Note also that the Mannheim metric codes are not limited to correct small Euclidean-distance errors. From [3] we know, that the maximal Mannheim weight which an error at a position can have is given by p a 0 1 which has the order p. Codes for the Mannheim metric, notably icyclic codes, have been constructed in [3].
TABLE II PARAMETERS p; a; b; R = 2r1 ; OPTIMIZED r2 ; Es ; and Eb FOR EUCLIDEAN MINIMUM DISTANCE dE = 1
APPENDIX In this appendix we determine the average signal energy for the Gaussian integers modulo the Gaussian prime = a + ib, where p = 1 3 with p 1 mod 4. The star denotes complex conjugation. We therefore determine the sum
S
p01
=
N (k mod )
k=0
where the norm N ( ) of a Gaussian integer is defined by N ( ) =
1 3 . The average energy then equals S=p. To compute A+iB mod one can proceed as follows. Let x + iy be A + iB mod , then 3 (A + iB )(a 0 ib) mod —i.e., modulo p—equals f + ig where
aA 0 bB = F 1 p + f aB 0 bA = G 1 p + g:
Hence
A + iB mod
=
x + iy
=
f + ig : a 0 ib
(12)
Equation (12) can already be found in [2]. Hence setting k mod xk + iyk with xk + iyk = (fk + igk )=(a 0 ib) and
=
a 1 k = Fk 1 p + fk 0b 1 k = Gk 1 p + gk
we get i.e.,
Es
=
S
R2 + r12 + (2R2 0 4Rr1 )=(cos(2a=p) + cos(2b=p))
=
r2
=
and
R
R1
2(cos(2a=p) + cos(2b=p)) cos(2a=p) + cos(2b=p)
0 3:
(11)
This solution however (apart from the fact that it does not exist for 2 all possible p) leads to values of dE =Eb which are not better than the solution R = 2r; r1 = r2 = r to which r1 and r2 of (11) converge for p ! 1. III. CONCLUSION In this correspondence it has been shown how codes over tori can be used for signal transmission. The toroidal signal constellations can be regarded as a sort of three–dimensional phase shift keying which makes it possible to retain some of the advantages of PSK with a
=
k=0
N (fk + igk ) N (a + ib) k=0
=
cos(2a=p) + cos(2b=p)
p01
x2k + yk2
k=0 p01
with respect to r1 leads to
r1
p01 =
=
N (xk + iyk )
1
p01
fk2 + gk2 :
p k=0
Now if k runs from 0 to p 0 1 then ak mod p which equals fk runs through all residues modulo p. The same is true for gk . Clearly, to get the residues k mod of smallest norm, we select fk and gk from the interval 0(p 0 1)=2 1 1 1 (p 0 1)=2. Thus p01 k=0
fk2
p01
=
k=0
gk2
01)=2
(p
=2
k2 ;
k=1
) S = 4p
p01
k2 :
k=0
The latter sum is well known. We thus get
S
= =
1 p 4
01)
(p
2
p2 0 1
as was to be proved.
6
01) + 1
(p
2
(p
0 1 + 1)
6
) S=p = p 60p 1 2
(13)
744
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 2, MARCH 1997
ACKNOWLEDGMENT The author wishes to thank the reviewers and K. H. Hofmann for helpful comments on the manuscript. REFERENCES [1] R. E. Blahut, Digital Transmission of Information. Reading, MA: Addison-Wesley, 1990. [2] C. F. Gauss, “Theorie der biquadratischen Reste, Zweite Abhandlung,” Commentationes soc. reg. Gotting.recentiores, Vol. VII. Gottengae 1832, also contained in C. F. Gauss, Untersuchungen u¨ ber H¨ohere Arithmetik (German translation of the Latin Disquisitiones Arithmeticae by H. Maser, 1889), 2nd reprint. New York: Chelsea, 1981. [3] K. Huber, “Codes over Gaussian integers,” IEEE Trans. Inform. Theory, vol. 40, no. 1, pp. 207–216, Jan. 1994. [4] K. Huber, “Codes over Eisenstein–Jacobi integers,” in Finite Fields: Theory, Applications and Algorithms, Contemporary Math. (Las Vegas, 1993), vol 168, pp. 165–179 (American Math. Soc., Providence, RI). [5] Wozencraft and Jacobs, Principles of Communication Engineering. New York: Wiley, 1965.
Novel Analytical Performance Bounds for Symbol-by-Symbol Decoding of Digital-Data Impaired by ISI and AWGN E. Baccarelli, R. Cusani, Member, IEEE, and S. Galli, Student Member, IEEE Abstract— New analytical upper bounds are derived for the performance of symbol-by-symbol maximum-likelihood (ML) Abend– Fritchman-type detectors for multi-amplitude/phase-modulated digital sequences transmitted over time-dispersive (generally) time-variant linear channels introducing finite Intersymbol Interference (ISI) and additive white Gaussian noise (AWGN). The proposed bounds are obtained starting from a novel form of the standard Bhattacharyya bound and take explicitly into account the allowed decoding delay. Both deterministic and random multipath slow-fading Rayleigh transmission ISI channels are considered. Simulation results obtained for some test channels of practical interest show the tightness of the derived analytical bounds. Index Terms—Symbol-by-symbol detection, Bhattacharyya-like bound, time-dispersive channels, multipath random channels, Forney’s bound, Abend–Fritchman detection algorithm.
I. INTRODUCTION Time dispersion introduced by the channel constitutes one of the most significant performance degradation sources for wideband, high-speed digital data-communication systems, such as codedivision multiple access mobile satellite systems, or time-division multiple access terrestrial mobile systems [1, ch. 8]. When the impairment of the received signal due to the resulting intersymbol interference (ISI) is severe (as in the case of transmission channels with spectral nulls), nonlinear detection techniques are generally requested in order to mitigate noise-enhancement phenomena. Extensive research has been devoted in the past to derive analytical Manuscript received June 21, 1995; revised June 28, 1996. The authors are with the INFO-COM Department, University of Rome “La Sapienza,” 00184 Rome, Italy. Publisher Item Identifier S 0018-9448(97)00805-5.
performance bounds for the usual maximum-likelihood sequence estimators (MLSE) (see, in particular, the fundamental work of G. D. Forney [4] and, among the others, the contributions in [2], [3, ch. 4], [6], [7] and, more recently, [10]). On the other hand, to the best of our knowledge no results concerning analytical performance bounds for symbol-by-symbol (SBS) Abend–Fritchman-like detectors [1, Sec. 6.6], [11], [14], [15], minimizing the decoding symbol-errorprobability (SEP) are available. Indeed, as also explicitly pointed out in [1, p. 605], analytical performance evaluation for the above mentioned decoders has been considered a difficult task, so that only fragmentary results obtained by means of Monte Carlo simulations have been reported until now [1, ch. 6]. The aim of this correspondence is to fill the mentioned gap, giving some analytical upper bounds for the performance evaluation of finite decision-delay SBS-ML detectors for unquantized soft-decision decoding of multi-amplitude/phase digitally modulated data sequences transmitted over time-dispersive, finite-memory, linear waveform channels impaired by AWGN. The case of data-communication systems with quantized-decision decoding has been separately analyzed in [9]. Indeed, there are several reasons to support the theoretical and practical interest in such kind of analytical bounds. First, they supply theoretical insight about the relationships between receiver performance, channel behavior, and modulation techniques, thus allowing evaluation of the potential benefits arising from the utilization of various digital modulation formats [16]. Secondly, analytical performance bounds allow a more meaningful comparison between different types of ISI channels than can be carried out on the basis of simpler single-parameter performance indices as, for example, signal-to-noise ratio, or capacity [13]. Thirdly, the computation of analytical bounds provides a practical way for evaluating the receiver performance without resorting to time-consuming simulations. Finally, a renewed interest in the performance of SBS detectors has recently arisen in the literature with reference to the development of low-complexity iterative turbo-code decoding algorithms [5]. The presented upper bounds for the decoding SEP are derived starting from a novel form of the Bhattacharyya bound for the case of deterministic (i.e., nonrandom) ISI channels (see Section III). The followed approach presents some analogies with that developed by Biglieri in [2] for the computation of the cutoff rate of channels with memory; however, in [2] sequence decoders are assumed instead of SBS decoders (with finite values of the decision delay), as in the present work. The proposed bounds depend on a set of suitably defined distance parameters whose values essentially dominate the limit of ultimate detector performance. The obtained analytical results are also extended to the case of random multipath slow-fading Rayleigh noisy channels, under the assumption of a perfect channelstate information (CSI) at the receiver site (see Section IV). As far as the tightness of the presented bounds is concerned, it is well known that for SEP’s of practical interest (generally, below 1003 ) the simple Bhattacharyya-like bounds are as tight as more complex upper bounds such as the Chernoff and Gallager bounds (see [16, sec. V] and [3, Secs. 2.3 and 2.4]). The simulation results of Figs. 2–8 discussed in Sections III–IV directly support this conclusion. II. THE DISCRETE-TIME DATA-TRANSMISSION MODEL We refer to the discrete-time complex baseband data-transmission systems of Fig. 1, equivalent to the cascade of a digital linear
0018–9448/97$10.00 1997 IEEE
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 2, MARCH 1997
Codes over Tori Klaus Huber, Member, IEEE
Abstract—In this correspondence it is shown how the elements of finite fields when mapped on the surface of a torus can be used for block coding over a three-dimensional signal space. The block codes are useful for data transmission using the recently defined Mannheim distance. Index Terms—Block codes, tori, Gaussian integers, Mannheim distance, coded modulation.
I. INTRODUCTION For digital data transmission very often signals over a two-, three-, or higher dimensional signal space are selected for bandwidth efficiency (see e.g., [1] for the most common signal constellations; for a standard reference of orthogonal functions for communications engineering see [5]). Most high-performance communication systems use convolutional codes at least as inner codes. This is mainly due to the fact that there exist efficient soft-decision decoding algorithms which allow maximum-likelihood decoders. For block codes most good algebraic decoders employ the Hamming distance, which is not very efficient for codes over two- or three-dimensional signal spaces. Recently, the Mannheim distance was introduced to make QAMlike signal constellations more susceptible for algebraic decoding methods (see [3]). For codes over hexagonal signal constellations a similar metric was introduced in [4]. In this correspondence we show that the Mannheim distance is useful for block codes over tori. The resulting signal constellations can be seen as a sort of three-dimensional phase-shift keying. II. CODES
OVER
Fig. 1. GF (97) represented as Gaussian integers modulo 9 + i4.
TORI
Let p be a prime of the form p 1 mod 4. Then, from a basic theorem of number theory we know that p can be represented in an essentially unique way as sum of two squares
p = a2 + b2 :
(1)
In this correspondence, we use the convention a > b > 0. In [3] it was shown how the elements of GF (p) for p 1 mod 4 or of GF (p2 ) for p 3 mod 4 can be mapped on a subset of the Gaussian integers using the complex modulo function given below. The Gaussian integers are those complex numbers whose real and imaginary parts are integers. The Mannheim distance between two Gaussian integers and with respect to the Gaussian integer a + ib is the sum of the absolute values of real and imaginary part of the difference ( 0 ) mod a + ib. The modulo reduction, before summing the absolute values of real and imaginary part, is the difference between the well-known Manhattan distance and the Mannheim distance, for further details see [3]. (For simplicity, we only consider the case p 1 mod 4, leaving the straightforward modifications of the case GF (p2 ) for p 3 mod 4 to the reader.) To obtain the representation of the field GF (p) = f0; 1; 1 1 1 ; p 0 1g as Gaussian integers, we take the elements of GF (p), interprete them
Fig. 2. GF (257) represented as Gaussian integers modulo 16 + i.
as integers k = 0; 1; 1 1 1 ; p 0 1, and use the complex modulo function
(k) = k 0 k(a 0 ib) (a + ib) p to get the corresponding Gaussian integers ([1] denotes rounding to the closest Gaussian integer). For instance, the element 67 of GF (97) is represented as 67 mod 9 + i4 = 67
Manuscript received May 16, 1995; revised October 7, 1996. The author is with Deutsche Telekom AG, Technologiezentrum, P.O. Box 10 00 03, 64276 Darmstadt, Germany. Publisher Item Identifier S 0018-9448(97)00779-7.
0
67(9
0 i4)
97
1
(9 + i4) = 1 + i 3:
Figs. 1 and 2 display the fields GF (97) and GF (257) when represented as Gaussian integers modulo 9 + i4 and 16 + i, respectively.
0018–9448/97$10.00 1997 IEEE
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 2, MARCH 1997
Fig. 3. Cut through torus having radii
R
741
and r .
The elements of GF (p) = f0; 1; 1 1 1 ; p 0 1g can be placed on surface points Tj of a torus using the mapping Tj = tx(j ) ; ty(j ) ; tz(j ) = (cos
(j )(R
0r cos '(j )); sin
(j )(R
0 r cos '(j )); r sin '(j ))
Fig. 4.
T (x; a; b)
Fig. 5.
T (x; b;
for
= 257;
p
a
= 16;
b
= 1;
R
= 2; and
r
= 1.
(2) where tx ; ty ; tz are Cartesian coordinates, and R and r are the radii of a torus as shown by the cut in Fig. 3. To get a better comprehension of the mapping of (2) assume that the points of Fig. 1 are placed on a rubber plane. Now wrap the portion of the plane where the points are located around a torus such that 5 is neighbor to 6, 26 neighbor to 27, etc. For the remainder of the correspondence we define (x) and '(x) as follows: '(x) =
2a
1
(3)
1
(4)
x p 2b x: (x) = p
The signal points Tj are on the line T (x; a; b), where T (x; a; b) = (cos (x)(R
0 r cos '(x)); (x)(R 0 r cos '(x)); r sin '(x))
sin
(5)
with x in the real interval [0; p]. For p = 257 = 162 + 12 ; a = 16; b = 1; R = 2, and r = 1 the line T (x; a; b) is displayed in Fig. 4. The figure makes clear the geometric significance of the integers a and b: a gives the number of rounds around the ring of radius r and b gives the number of rounds around the circle of radius R. The curve T (x; b; 0a) is displayed in Fig. 5 for the parameters of Fig. 4. At the intersection points of T (x; a; b) and T (x; b; 0a) we get the points Tj ; j = 0; 1; 1 1 1 ; p 0 1. To see this take the integer i which fullfills a + ib 0 mod p, i.e., i is a square root of 01 modulo p. Then we have Tj = T (x = j ; a; b)
and Tij mod p = T (x = j ; b; 0a):
This means that going from Tk to Tk+1 corresponds to moving along the line T (x; a; b) (see Fig. 4), whereas going from Tk to Tk+i corresponds to moving along the line T (x; b; 0a) (see Fig. 5). Now let us determine the ratio of R=r such that the Euclidean distance dE (T0 ; T61 ) between T0 and T61 equals the Euclidean distance dE (T0 ; T6i ) between T0 and T6i . A simple derivation leads to R = 2r;
for dE (T0 ; T61 ) = dE (T0 ; T6i ):
v= =
2 (T0 ; T1+i ) dE 2 dE (T0 ; T1 )
3 + cos
2 (a+b)
p
3 + cos
(6)
2 (
)
2
a)
for
p
= 257;
a
= 16;
b
= 1;
R
= 2; and
r
= 1.
Determining the radius r such that dE (T0 ; T61 ) = dE (T0 ; T6i ) = yields r=
6
04
2a
cos p
2b
+ cos p
+ 2 cos
2a
p
1 cos
2b
: (7)
p
For p not too small, a simple computation shows that r is wellapproximated by r
pp=(2):
(8)
The value is quite close to the minimum Euclidean distance dE between any two points Tj ; Tj (see Table I). In general, however, is not exactly equal to dE . This is due to the fact, that the torus line starts at ' = 0, therefore depending on p, there can be points Tk and Tk61 (or Tk and Tk6i ) which are slightly closer together than . Another parameter of interest is the ratio v of the following squared Euclidean distances (see the bottom of this page). It is not difficult to show that limp!1 v = 2. For p small, v is also displayed in Table I. As we are considering the use of tori for signal transmission we are interested in the average signal energy Es of a transmitted point. We get p01 1 2 2 Es = 1 (R + r 0 2Rr cos '(j )): p j =0
1 cos ap0b 0 2 b p 1 cos p 0 2
2a
0
cos
2 (a+b) 2a
cos( p
p
+ cos
+ cos
2b
p
0b)
2 (a
p
:
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 2, MARCH 1997
In general, the required energy to transmit a bit of information is smaller than for codes over Gaussian integers, see [3], or codes over Eisenstein–Jacobi integers, see [4] (i.e., for signal points lying on a hexagonal grid which has a higher density than QAM signal constellations). The reason for the better performance lies of course in the fact that we now have signal constellations which are in three dimensions instead of two. More precisely, from the value of Es for Gaussian integers given by (p2 0 1)=6p (see the Appendix) and (7), (9) with R = 2r, we immediately get the improvement (see the bottom of this page) which for p not too small and = 1 is well-approximated by
TABLE I PARAMETERS p; a; b; ; r; v; Es ; and Eb for R = 2r AND EUCLIDEAN MINIMUM DISTANCE dE = 1
10 log10
2
1 p p0 1 10 log
2
2
2
15
10
2
2
15
1:2 dB:
Compared on a bit-per-dimension basis, the toroidal signal set does not improve on usual QAM signal constellations. The toroidal signal constellation should rather be seen as a sort of three-dimensional PSK scheme, since it retains some of the advantages of PSK while making it possible to use more signal points per dimension. Namely, the energy per bit for the toroidal signal set is obtained using (8) which leads to
pp
5
2
2
= log2 (p)
p= log (p) 2
whereas for the M -PSK signal constellation with distance 1 between adjacent points the corresponding value is 2
M 2
= log2 (M )
p = = log (p) 4 3
2
where for a fair comparison we have set M = p2=3 . The most notable advantage of the toroidal signal set compared to the QAM signal set is the much smaller dynamic range of the amplitudes. Remark 1: A reviewer noted that there is still room for improvement if more efficient curves are used, since there are big open areas inside the torus. Indeed, this is true and we briefly mention the most straightforward change which preserves the “Mannheim neighborhood.” Namely, we modify the torus such that the cut (see Fig. 3) consists of ellipses. The toruspoints are then given by Tj = tx(j ) ; ty(j ) ; tz(j )
0 r cos '(j )); (j )(R 0 r cos '(j )); r
(j )(R
= (cos
From '(j ) = 2aj=p we have
sin
cos(2aj=p) =
cos(2j=p)
which gives p zero, as is seen by considering the sum over all real parts of exp( 01 1 2j=p). Hence we obtain 2
Es = R + r
2
(9)
from which we get the energy per bit Eb = Es = log2 p. In Table I the values of p; a; b; ; r; v; Es ; and Eb are listed for R = 2r and dE = 1. From (7) it is clear that the signal energy is smallest if a and b are as close together as possible. For p 1 mod 4 this is the case if a = b + 1. In Table I we can see, for example, that the energy for p = 41 is smaller than the energy for p = 37.
10 log10
p2
0 1 =5r
6p 2
10 log10
(p
2
1
1
2
sin '(j )):
Setting R = 2 1 r1 and using standard numerical techniques, the value of r2 can be found such that the normalized Euclidean distance 2 dE =Eb is as great as possible. In Table II the results of such computations are displayed. Significant improvements over the data of Table I occur if a and b are close together. These improvements have been obtained by permitting dE (T0 ; T1 )2 6= dE (T0 ; Ti )2 . In the general case, the condition dE (T0 ; T1 )2 = dE (T0 ; Ti )2 leads to 2R(R
0 2r ) = 1
2
r2
0r 1 2 1
cos
2a
p
+ cos
2b
p
(10)
which, of course, contains the solution R = 2r; r1 = r2 = r. Using (10) as constraint and finding the minimum of Es = R2 +(r12 +r22 )=2,
=
0 1)(6 0 4(cos(2a=p) + cos(2b=p)) + 2 cos(2a=p) 1 cos(2b=p)) 30 2 p
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higher number of signal points per dimension. The main parts are the mapping of the elements of finite fields GF (p) onto the surface of a torus, and the usage of the Mannheim metric. The Mannheim metric is quite well suited for codes over tori, as neighbor points have Mannheim distance 1. Errors with a given Mannheim weight however can be corrected with block codes having fewer check symbols than codes able to correct the same number of Hamming distance errors. Since in data communication errors tend to occur with higher probability closer to the signal points sent, such Mannheim metric codes outperform block codes using the Hamming metric. Note also that the Mannheim metric codes are not limited to correct small Euclidean-distance errors. From [3] we know, that the maximal Mannheim weight which an error at a position can have is given by p a 0 1 which has the order p. Codes for the Mannheim metric, notably icyclic codes, have been constructed in [3].
TABLE II PARAMETERS p; a; b; R = 2r1 ; OPTIMIZED r2 ; Es ; and Eb FOR EUCLIDEAN MINIMUM DISTANCE dE = 1
APPENDIX In this appendix we determine the average signal energy for the Gaussian integers modulo the Gaussian prime = a + ib, where p = 1 3 with p 1 mod 4. The star denotes complex conjugation. We therefore determine the sum
S
p01
=
N (k mod )
k=0
where the norm N ( ) of a Gaussian integer is defined by N ( ) =
1 3 . The average energy then equals S=p. To compute A+iB mod one can proceed as follows. Let x + iy be A + iB mod , then 3 (A + iB )(a 0 ib) mod —i.e., modulo p—equals f + ig where
aA 0 bB = F 1 p + f aB 0 bA = G 1 p + g:
Hence
A + iB mod
=
x + iy
=
f + ig : a 0 ib
(12)
Equation (12) can already be found in [2]. Hence setting k mod xk + iyk with xk + iyk = (fk + igk )=(a 0 ib) and
=
a 1 k = Fk 1 p + fk 0b 1 k = Gk 1 p + gk
we get i.e.,
Es
=
S
R2 + r12 + (2R2 0 4Rr1 )=(cos(2a=p) + cos(2b=p))
=
r2
=
and
R
R1
2(cos(2a=p) + cos(2b=p)) cos(2a=p) + cos(2b=p)
0 3:
(11)
This solution however (apart from the fact that it does not exist for 2 all possible p) leads to values of dE =Eb which are not better than the solution R = 2r; r1 = r2 = r to which r1 and r2 of (11) converge for p ! 1. III. CONCLUSION In this correspondence it has been shown how codes over tori can be used for signal transmission. The toroidal signal constellations can be regarded as a sort of three–dimensional phase shift keying which makes it possible to retain some of the advantages of PSK with a
=
k=0
N (fk + igk ) N (a + ib) k=0
=
cos(2a=p) + cos(2b=p)
p01
x2k + yk2
k=0 p01
with respect to r1 leads to
r1
p01 =
=
N (xk + iyk )
1
p01
fk2 + gk2 :
p k=0
Now if k runs from 0 to p 0 1 then ak mod p which equals fk runs through all residues modulo p. The same is true for gk . Clearly, to get the residues k mod of smallest norm, we select fk and gk from the interval 0(p 0 1)=2 1 1 1 (p 0 1)=2. Thus p01 k=0
fk2
p01
=
k=0
gk2
01)=2
(p
=2
k2 ;
k=1
) S = 4p
p01
k2 :
k=0
The latter sum is well known. We thus get
S
= =
1 p 4
01)
(p
2
p2 0 1
as was to be proved.
6
01) + 1
(p
2
(p
0 1 + 1)
6
) S=p = p 60p 1 2
(13)
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ACKNOWLEDGMENT The author wishes to thank the reviewers and K. H. Hofmann for helpful comments on the manuscript. REFERENCES [1] R. E. Blahut, Digital Transmission of Information. Reading, MA: Addison-Wesley, 1990. [2] C. F. Gauss, “Theorie der biquadratischen Reste, Zweite Abhandlung,” Commentationes soc. reg. Gotting.recentiores, Vol. VII. Gottengae 1832, also contained in C. F. Gauss, Untersuchungen u¨ ber H¨ohere Arithmetik (German translation of the Latin Disquisitiones Arithmeticae by H. Maser, 1889), 2nd reprint. New York: Chelsea, 1981. [3] K. Huber, “Codes over Gaussian integers,” IEEE Trans. Inform. Theory, vol. 40, no. 1, pp. 207–216, Jan. 1994. [4] K. Huber, “Codes over Eisenstein–Jacobi integers,” in Finite Fields: Theory, Applications and Algorithms, Contemporary Math. (Las Vegas, 1993), vol 168, pp. 165–179 (American Math. Soc., Providence, RI). [5] Wozencraft and Jacobs, Principles of Communication Engineering. New York: Wiley, 1965.
Novel Analytical Performance Bounds for Symbol-by-Symbol Decoding of Digital-Data Impaired by ISI and AWGN E. Baccarelli, R. Cusani, Member, IEEE, and S. Galli, Student Member, IEEE Abstract— New analytical upper bounds are derived for the performance of symbol-by-symbol maximum-likelihood (ML) Abend– Fritchman-type detectors for multi-amplitude/phase-modulated digital sequences transmitted over time-dispersive (generally) time-variant linear channels introducing finite Intersymbol Interference (ISI) and additive white Gaussian noise (AWGN). The proposed bounds are obtained starting from a novel form of the standard Bhattacharyya bound and take explicitly into account the allowed decoding delay. Both deterministic and random multipath slow-fading Rayleigh transmission ISI channels are considered. Simulation results obtained for some test channels of practical interest show the tightness of the derived analytical bounds. Index Terms—Symbol-by-symbol detection, Bhattacharyya-like bound, time-dispersive channels, multipath random channels, Forney’s bound, Abend–Fritchman detection algorithm.
I. INTRODUCTION Time dispersion introduced by the channel constitutes one of the most significant performance degradation sources for wideband, high-speed digital data-communication systems, such as codedivision multiple access mobile satellite systems, or time-division multiple access terrestrial mobile systems [1, ch. 8]. When the impairment of the received signal due to the resulting intersymbol interference (ISI) is severe (as in the case of transmission channels with spectral nulls), nonlinear detection techniques are generally requested in order to mitigate noise-enhancement phenomena. Extensive research has been devoted in the past to derive analytical Manuscript received June 21, 1995; revised June 28, 1996. The authors are with the INFO-COM Department, University of Rome “La Sapienza,” 00184 Rome, Italy. Publisher Item Identifier S 0018-9448(97)00805-5.
performance bounds for the usual maximum-likelihood sequence estimators (MLSE) (see, in particular, the fundamental work of G. D. Forney [4] and, among the others, the contributions in [2], [3, ch. 4], [6], [7] and, more recently, [10]). On the other hand, to the best of our knowledge no results concerning analytical performance bounds for symbol-by-symbol (SBS) Abend–Fritchman-like detectors [1, Sec. 6.6], [11], [14], [15], minimizing the decoding symbol-errorprobability (SEP) are available. Indeed, as also explicitly pointed out in [1, p. 605], analytical performance evaluation for the above mentioned decoders has been considered a difficult task, so that only fragmentary results obtained by means of Monte Carlo simulations have been reported until now [1, ch. 6]. The aim of this correspondence is to fill the mentioned gap, giving some analytical upper bounds for the performance evaluation of finite decision-delay SBS-ML detectors for unquantized soft-decision decoding of multi-amplitude/phase digitally modulated data sequences transmitted over time-dispersive, finite-memory, linear waveform channels impaired by AWGN. The case of data-communication systems with quantized-decision decoding has been separately analyzed in [9]. Indeed, there are several reasons to support the theoretical and practical interest in such kind of analytical bounds. First, they supply theoretical insight about the relationships between receiver performance, channel behavior, and modulation techniques, thus allowing evaluation of the potential benefits arising from the utilization of various digital modulation formats [16]. Secondly, analytical performance bounds allow a more meaningful comparison between different types of ISI channels than can be carried out on the basis of simpler single-parameter performance indices as, for example, signal-to-noise ratio, or capacity [13]. Thirdly, the computation of analytical bounds provides a practical way for evaluating the receiver performance without resorting to time-consuming simulations. Finally, a renewed interest in the performance of SBS detectors has recently arisen in the literature with reference to the development of low-complexity iterative turbo-code decoding algorithms [5]. The presented upper bounds for the decoding SEP are derived starting from a novel form of the Bhattacharyya bound for the case of deterministic (i.e., nonrandom) ISI channels (see Section III). The followed approach presents some analogies with that developed by Biglieri in [2] for the computation of the cutoff rate of channels with memory; however, in [2] sequence decoders are assumed instead of SBS decoders (with finite values of the decision delay), as in the present work. The proposed bounds depend on a set of suitably defined distance parameters whose values essentially dominate the limit of ultimate detector performance. The obtained analytical results are also extended to the case of random multipath slow-fading Rayleigh noisy channels, under the assumption of a perfect channelstate information (CSI) at the receiver site (see Section IV). As far as the tightness of the presented bounds is concerned, it is well known that for SEP’s of practical interest (generally, below 1003 ) the simple Bhattacharyya-like bounds are as tight as more complex upper bounds such as the Chernoff and Gallager bounds (see [16, sec. V] and [3, Secs. 2.3 and 2.4]). The simulation results of Figs. 2–8 discussed in Sections III–IV directly support this conclusion. II. THE DISCRETE-TIME DATA-TRANSMISSION MODEL We refer to the discrete-time complex baseband data-transmission systems of Fig. 1, equivalent to the cascade of a digital linear
0018–9448/97$10.00 1997 IEEE
