Vector Permutation Modulation - IEEE Xplore
IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 8, AUGUST 2005
673
Vector Permutation Modulation Danilo Silva and Weiler A. Finamore, Member, IEEE
Abstract— This article introduces a new class of codes for transmission over a Gaussian channel. The new scheme, named vector permutation modulation (VPM), is an extension of Slepian’s (scalar) permutation modulation (SPM) to the case where codeword components are L-dimensional vectors. A method is presented for maximum-likelihood detection in additive white Gaussian noise and fading channels. Since SPM is a special case of VPM, this method can also be used for SPM detection in fading channels. We solve the problem of finding the reference codeword which maximizes code rate for a given average energy. It is shown that VPM performs significantly better than its scalar counterpart. Index Terms— Permutation modulation, permutation codes, multidimensional signaling.
I. I NTRODUCTION
P
ERMUTATION codes for transmission over a Gaussian channel have been introduced in [1]. With this definition, codewords are obtained by permuting the scalar components of a reference codeword in all possible ways. The overall transmission system is known as permutation modulation [1]– [5]. The same permutation codes can also be used for source quantization. In this context, it is shown in [6] that the performance of (scalar) permutation codes is closely related to that of optimum entropy-constrained scalar quantizers. Recently, the concept of permutation codes was extended to the case where codeword components are L-dimensional vectors instead of scalars [7]. These new codes, called vector permutation codes, have a rate-distortion performance tied to that of entropy-constrained vector quantizers, being superior thus to scalar permutation codes and optimum scalar quantizers [8]. In this paper, we introduce vector permutation modulation, a transmission scheme which employs vector permutation codes for the purpose of modulation. It is shown that also in this context vector permutation codes have better performance than their scalar counterpart. One possible explanation for this behavior is that vector permutation modulation is, in a sense, analogous to nonuniform multidimensional signaling [9]. This article is organized as follows. Section II reviews the geometric model of a communication system. In Section III, vector permutation modulation is introduced together with a maximum-likelihood detection algorithm for both additive white Gaussian noise and fading channels. An important problem is to find the count of each component in a codeword Manuscript received October 18, 2004. The associate editor coordinating the review of this letter and approving it for publication was Prof. Chi-Chao Chao. This work was partially supported by the Brazilian National Council for Scientific and Technological Development (CNPq). The authors are with the CETUC–Center for Telecommunications Studies, Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil (e-mail: {danilo, weiler}@cetuc.puc-rio.br). Digital Object Identifier 10.1109/LCOMM.2005.08026.
such that the code rate is maximized. This problem is solved in Section IV, wherein we also compare the performances of scalar and vector permutation codes. Finally, Section V provides some concluding remarks. II. G EOMETRIC M ODEL OF A C OMMUNICATIONS S YSTEM Transmission of signals over a bandlimited channel disturbed by additive white Gaussian noise (AWGN) can be modeled in the N -dimensional Euclidean space RN [10]. At each interval of duration T , one out of M equally likely messages is chosen for transmission. Each message is associated to a signal vector, or codeword, xi . The set of possible codewords C = {xi }M i=1 is called the constellation or codebook. When message i is transmitted, the received signal vector is given by y = xi + z, where z is a Gaussian random vector of i.i.d. components with zero mean and variance N0 /2. The maximum-likelihood receiver for this system asserts that message ˆı was the transmitted message, where ˆı = arg min y − xi 2
(1)
i∈{1,...,M }
and · denotes Euclidean vector norm. When transmission takes place over a fading channel, the received signal vector is commonly given by y = Hxi + z, where H = diag(h1 , . . . , hN ) is the diagonal channel fading matrix, and H can change independently from one codeword to another [4]. We say that the channel is slow fading if h1 = · · · = hN = h, otherwise it is fast fading. Assuming that the receiver has knowledge of the channel state, the maximum-likelihood detection rule yields ˆı = arg min y − Hxi 2 .
(2)
i∈{1,...,M }
Some parameters typically used for comparison of codes are the information rate1 1 log M (bits/dimension) (3) R N the average energy per codeword per dimension2 E
M 1 xi 2 N M i=1
(4)
and the minimum Euclidean distance between codewords3 dmin
min xi − xj
i,j, i=j
(5)
which we consider as our reliability measure. 1 If we assume that the channel is limited to a bandwidth W , then it is mathematically possible to assign at most N = 2T W dimensions to the vector space. Letting Rb be the transmission bit rate, we find that R is proportional to the spectral efficiency Rb /W = T 1W log M bit/s/Hz. 2 Since signals are used with equal probability, E is related to the average 1 energy per bit Eb = N E/ log M = R E. 3 At moderate to high signal-to-noise ratios, d min can be used for obtaining a reasonable approximation to the codeword error probability of a system [9].
c 2005 IEEE 1089-7798/05$20.00
674
IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 8, AUGUST 2005
III. V ECTOR P ERMUTATION M ODULATION Let {µ1 , . . . , µK } be a set of distinct L-dimensional real vectors and let {m1 , . . . , mK } be a sequence of positive integers such that K n= mk . (6) k=1
Consider the reference codeword of length n composed of mk repetitions of each symbol µk in ascending order of index4 : x1 = (µ1 , . . . , µ1 , µ2 , . . . , µ2 , . . . , µK , . . . , µK )T . m1
m2
(7)
mK
A vector permutation code is the set C of codewords obtained by taking all possible permutations of the components of x1 . The total number of such codewords is M = K
n!
k=1
mk !
IV. V ECTOR P ERMUTATION M ODULATION D ESIGN AND P ERFORMANCE
.
In the special case L = 1, this code reduces to that originally defined by Slepian [1], which will be referred here as a scalar permutation code. Transmission schemes employing scalar and vector permutation codes are called, respectively, scalar permutation modulation (SPM) and vector permutation modulation (VPM). Note that a VPM codeword has N = nL dimensions overall. The information rate of this scheme is thus 1 log M. (8) R= nL Since every codeword has the same norm, the average energy per codeword per dimension is K
E=
where Hk = diag(h(k−1)L+1 , . . . , hkL ). Therefore, complexity remains unchanged. More efficient methods exist for maximum-likelihood detection in the special case of L = 1. A simple algorithm based on a sorting operation can be applied to the reception of SPM over AWGN and slow fading channels, with complexity O(n log n) [1], [4]. In fast fading channels with K = 2, a similar algorithm exists whose complexity is also O(n log n) [4]. When K > 2, however, the only method currently proposed for fast fading channels is based on a trellis construction and K K appears to have complexity O( k=1 mk ) = O( k=1 pk n) = O(nK ) [5]. Therefore, since SPM is a special case of VPM, the vector assignment approach described above can be used √ to reduce this complexity to O(n2 n log n).
1 1 x1 2 = mk µk 2 . nL nL
(9)
k=1
Furthermore, the minimum distance between codewords is √ 2 µk − µ (10) dmin = min k,, k=
since the two closest codewords will differ only by an exchange of two symbols. Maximum-likelihood detection of VPM in an AWGN channel is performed by searching for the permutation of x1 = T (x11 , . . . , x1n ) which is closest in Euclidean distance to the T received codeword y = (y1 , . . . , yn ) . If we let δkj = yk − x1j 2
(11)
be the cost of assigning vector yk to vector x1j , then the problem becomes that of finding n the permutation π of {1, . . . , n} for which the total cost k=1 δk,π(k) is minimum. We have thus an assignment problem, which can be solved √ with complexity O(n2 n log n) by the CSA algorithm [11]. The same method can be used when reception of VPM is performed in a fading channel. To take into account the fading coefficients in (2), a slight modification has to be made on the computation of the costs: δkj = yk − Hk x1j 2
(12)
When alphabet-dimension L, codeword-length n and alphabet-size K are given, a VPM is completely characterized K by its alphabet {µk }K k=1 and composition vector {mk }k=1 . If we assume that the minimum distance between codewords dmin is fixed, then our goal is to design the alphabet and the composition vector such that the resulting codes have a maximum rate R for a given energy E, or, equivalently, a minimum energy E for a given rate R. Vector permutation modulation design problem is similar to that of designing nonequiprobable multidimensional constellations [9]. The difference, in case of VPM, is that the probabilities of the symbols µk are restricted to values pk such that pk n = mk is an integer. It is shown in the Appendix that the optimum composition vector (i.e., the composition vector which maximizes rate for a fixed energy) is given by expressions similar to that of a Maxwell–Boltzmann distribution. These expressions are a generalization of the results in [3]. Following [9], we apply the optimum composition vector to alphabets based on the densest known lattices in L dimensions. In one, two and three dimensions, the alphabets are based on, respectively, the integer lattice Z, the hexagonal lattice A2 and the face-centered cubic lattice D3 . Performance of VPM’s with such alphabets5 is shown in Fig. 1, where it can be seen that performance improve with both increasing codeword-length n and increasing alphabetdimension L. Fig. 1 also shows that the asymptotical performance of one VPM can be attained by another VPM with finite n and larger L. The asymptotical performance of a VPM is obtained by making n → ∞ in the expressions for rate, energy and optimum composition vector, which then become6 K
R=− E=
bold underlined notation is used here to denote a vector of vectors.
1 L
k=1 K
pk µk 2
(13)
(14)
k=1
SPM, alphabets based on Z are proven to be optimum [2]. that these expressions are identical to those obtained in the design of nonequiprobable multidimensional constellations [9]. 5 For
6 Note
4 The
1 pk log pk L
SILVA and FINAMORE: VECTOR PERMUTATION MODULATION
675
−8
−8.5
−8.5
−9.5
2 E/dmin (dB)
E/d2min (dB)
−9
L =1 L =2 L =3
L = 1, n = 200 L = 1, n = 500 L = 1, n = ∞ L = 2, n = 100 L = 2, n = 250 L = 2, n = 500 L = 2, n = ∞ L = 3, n = 333 L = 3, n = ∞
−9
L=1
−10
−9.5
−10.5
−10
L=2
L=3 −11 0.9
0.92
0.94
0.96
0.98
1
1.02
1.04
1.06
1.08
1.1
R (bits/dimension)
−10.5 1 10
2
3
10
10
4
10
N = nL
Fig. 1. Performance of VPM’s with L = 1, 2, 3. Curves for n = ∞ correspond to the asymptotical performance of each code.
Fig. 2. Performance of VPM’s with L = 1, 2, 3 for a rate R = 1 as a function of the overall number of dimensions N = nL.
2
2−λµk p k = K , k = 1, . . . , K. −λµk 2 k=1 2
(15)
Fig. 2 shows, for R = 1, the performance of VPM’s with L = 1, 2, 3 as a function of the overall number of dimensions N = nL, along with the asymptotical performance of VPM for each L. It can be observed that, from a certain value of nL, the performance of an SPM or VPM is surpassed by that of another VPM with greater L. V. C ONCLUSIONS In this paper, we introduce vector permutation modulation, a new transmission scheme which offers significant performance gains over traditional scalar permutation modulation. A method is presented for maximum-likelihood detection of VPM in AWGN and fading channels. Considering the criterion which minimizes the average transmitted energy for a given bit rate, the optimum composition vector is shown to be given by the rounded values of a Maxwell–Boltzmann distribution. A PPENDIX K
Given L, K and {µk }k=1 , we want to find the integers K {mk }k=1 that minimize −R and E under a restriction in n. By using (6), (9) and n mk K 1 1 log M = R= log − log (16) nL nL =1
k=1 =1
we form the Lagrangian f (m1 , · · · , mK ) =
mk K k=1 =1
log +λ
K k=1
mk µk 2 +ν
K
mk
k=1
(17) where λ > 0. The first sum in (16) is constant and is thus omitted in (17). When mk is increased, f (·) achieves a minimum when the following inequalities are satisfied: f (m1 , · · · , mk +1, · · · , mK )−f (m1 , · · · , mk , · · · , mK ) > 0 f (m1 , · · · , mk −1, · · · , mK )−f (m1 , · · · , mk , · · · , mK ) ≥ 0
Substituting (17), we have log(mk + 1) + λµk 2 + ν > 0
(18)
log(mk ) + λµk 2 + ν ≤ 0
(19)
Combining (18) and (19), the optimum values for mk arise:
2 2 mk = 2−λµk −ν = β · 2−λµk , k = 1, . . . , K (20) where x denotes the integral part of x. Here, β > 0 is adjusted to satisfy restriction in n, while λ is adjusted so that a desired E or R is achieved. R EFERENCES [1] D. Slepian, “Permutation modulation,” in Proc. IEEE, vol. 53, pp. 228– 236, Mar. 1965. [2] E. M. Biglieri and M. Elia, “Optimum permutation modulation codes and their asymptotic performance,” IEEE Trans. Inform. Theory, vol. 22, pp. 751–753, Nov. 1976. [3] I. Ingemarsson, “Optimized permutation modulation,” IEEE Trans. Inform. Theory, vol. 36, pp. 1098–1100, Sept. 1990. [4] A. Nordio and E. Viterbo, “Permutation modulation for fading channels,” in Proc. Int. Conf. Telecommunications, vol. 2, Feb. 2003, pp. 1177–1183. [5] E. Viterbo, “Trellis decoding of permutation modulations,” in Proc. IEEE Int. Symp. Information Theory, June 2003, p. 393. [6] T. Berger, “Optimum quantizers and permutation codes,” IEEE Trans. Inform. Theory, vol. 18, pp. 759–765, Nov. 1972. [7] W. A. Finamore, S. V. B. Bruno, and D. Silva, “Vector permutation encoding for the uniform sources,” in Proc. IEEE Data Compression Conference, Mar. 2004, p. 539. [8] D. Silva and W. A. Finamore, “Vector permutation code design algorithm,” in Proc. Int. Symp. Information Theory and its Applications, Oct. 2004, pp. 753-757. [9] F. R. Kschischang and S. Pasupathy, “Optimal nonuniform signaling for Gaussian channels,” IEEE Trans. Inform. Theory, vol. 39, pp. 913–929, May 1993. [10] J. Wozencraft and I. M. Jacobs, Principles of Communication Engineering. New York: Wiley, 1965. [11] A. V. Goldberg and R. Kennedy, “An efficient cost scaling algorithm for the assignment problem,” Mathematical Programming, vol. 71, pp. 153–178, Dec. 1995.
673
Vector Permutation Modulation Danilo Silva and Weiler A. Finamore, Member, IEEE
Abstract— This article introduces a new class of codes for transmission over a Gaussian channel. The new scheme, named vector permutation modulation (VPM), is an extension of Slepian’s (scalar) permutation modulation (SPM) to the case where codeword components are L-dimensional vectors. A method is presented for maximum-likelihood detection in additive white Gaussian noise and fading channels. Since SPM is a special case of VPM, this method can also be used for SPM detection in fading channels. We solve the problem of finding the reference codeword which maximizes code rate for a given average energy. It is shown that VPM performs significantly better than its scalar counterpart. Index Terms— Permutation modulation, permutation codes, multidimensional signaling.
I. I NTRODUCTION
P
ERMUTATION codes for transmission over a Gaussian channel have been introduced in [1]. With this definition, codewords are obtained by permuting the scalar components of a reference codeword in all possible ways. The overall transmission system is known as permutation modulation [1]– [5]. The same permutation codes can also be used for source quantization. In this context, it is shown in [6] that the performance of (scalar) permutation codes is closely related to that of optimum entropy-constrained scalar quantizers. Recently, the concept of permutation codes was extended to the case where codeword components are L-dimensional vectors instead of scalars [7]. These new codes, called vector permutation codes, have a rate-distortion performance tied to that of entropy-constrained vector quantizers, being superior thus to scalar permutation codes and optimum scalar quantizers [8]. In this paper, we introduce vector permutation modulation, a transmission scheme which employs vector permutation codes for the purpose of modulation. It is shown that also in this context vector permutation codes have better performance than their scalar counterpart. One possible explanation for this behavior is that vector permutation modulation is, in a sense, analogous to nonuniform multidimensional signaling [9]. This article is organized as follows. Section II reviews the geometric model of a communication system. In Section III, vector permutation modulation is introduced together with a maximum-likelihood detection algorithm for both additive white Gaussian noise and fading channels. An important problem is to find the count of each component in a codeword Manuscript received October 18, 2004. The associate editor coordinating the review of this letter and approving it for publication was Prof. Chi-Chao Chao. This work was partially supported by the Brazilian National Council for Scientific and Technological Development (CNPq). The authors are with the CETUC–Center for Telecommunications Studies, Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil (e-mail: {danilo, weiler}@cetuc.puc-rio.br). Digital Object Identifier 10.1109/LCOMM.2005.08026.
such that the code rate is maximized. This problem is solved in Section IV, wherein we also compare the performances of scalar and vector permutation codes. Finally, Section V provides some concluding remarks. II. G EOMETRIC M ODEL OF A C OMMUNICATIONS S YSTEM Transmission of signals over a bandlimited channel disturbed by additive white Gaussian noise (AWGN) can be modeled in the N -dimensional Euclidean space RN [10]. At each interval of duration T , one out of M equally likely messages is chosen for transmission. Each message is associated to a signal vector, or codeword, xi . The set of possible codewords C = {xi }M i=1 is called the constellation or codebook. When message i is transmitted, the received signal vector is given by y = xi + z, where z is a Gaussian random vector of i.i.d. components with zero mean and variance N0 /2. The maximum-likelihood receiver for this system asserts that message ˆı was the transmitted message, where ˆı = arg min y − xi 2
(1)
i∈{1,...,M }
and · denotes Euclidean vector norm. When transmission takes place over a fading channel, the received signal vector is commonly given by y = Hxi + z, where H = diag(h1 , . . . , hN ) is the diagonal channel fading matrix, and H can change independently from one codeword to another [4]. We say that the channel is slow fading if h1 = · · · = hN = h, otherwise it is fast fading. Assuming that the receiver has knowledge of the channel state, the maximum-likelihood detection rule yields ˆı = arg min y − Hxi 2 .
(2)
i∈{1,...,M }
Some parameters typically used for comparison of codes are the information rate1 1 log M (bits/dimension) (3) R N the average energy per codeword per dimension2 E
M 1 xi 2 N M i=1
(4)
and the minimum Euclidean distance between codewords3 dmin
min xi − xj
i,j, i=j
(5)
which we consider as our reliability measure. 1 If we assume that the channel is limited to a bandwidth W , then it is mathematically possible to assign at most N = 2T W dimensions to the vector space. Letting Rb be the transmission bit rate, we find that R is proportional to the spectral efficiency Rb /W = T 1W log M bit/s/Hz. 2 Since signals are used with equal probability, E is related to the average 1 energy per bit Eb = N E/ log M = R E. 3 At moderate to high signal-to-noise ratios, d min can be used for obtaining a reasonable approximation to the codeword error probability of a system [9].
c 2005 IEEE 1089-7798/05$20.00
674
IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 8, AUGUST 2005
III. V ECTOR P ERMUTATION M ODULATION Let {µ1 , . . . , µK } be a set of distinct L-dimensional real vectors and let {m1 , . . . , mK } be a sequence of positive integers such that K n= mk . (6) k=1
Consider the reference codeword of length n composed of mk repetitions of each symbol µk in ascending order of index4 : x1 = (µ1 , . . . , µ1 , µ2 , . . . , µ2 , . . . , µK , . . . , µK )T . m1
m2
(7)
mK
A vector permutation code is the set C of codewords obtained by taking all possible permutations of the components of x1 . The total number of such codewords is M = K
n!
k=1
mk !
IV. V ECTOR P ERMUTATION M ODULATION D ESIGN AND P ERFORMANCE
.
In the special case L = 1, this code reduces to that originally defined by Slepian [1], which will be referred here as a scalar permutation code. Transmission schemes employing scalar and vector permutation codes are called, respectively, scalar permutation modulation (SPM) and vector permutation modulation (VPM). Note that a VPM codeword has N = nL dimensions overall. The information rate of this scheme is thus 1 log M. (8) R= nL Since every codeword has the same norm, the average energy per codeword per dimension is K
E=
where Hk = diag(h(k−1)L+1 , . . . , hkL ). Therefore, complexity remains unchanged. More efficient methods exist for maximum-likelihood detection in the special case of L = 1. A simple algorithm based on a sorting operation can be applied to the reception of SPM over AWGN and slow fading channels, with complexity O(n log n) [1], [4]. In fast fading channels with K = 2, a similar algorithm exists whose complexity is also O(n log n) [4]. When K > 2, however, the only method currently proposed for fast fading channels is based on a trellis construction and K K appears to have complexity O( k=1 mk ) = O( k=1 pk n) = O(nK ) [5]. Therefore, since SPM is a special case of VPM, the vector assignment approach described above can be used √ to reduce this complexity to O(n2 n log n).
1 1 x1 2 = mk µk 2 . nL nL
(9)
k=1
Furthermore, the minimum distance between codewords is √ 2 µk − µ (10) dmin = min k,, k=
since the two closest codewords will differ only by an exchange of two symbols. Maximum-likelihood detection of VPM in an AWGN channel is performed by searching for the permutation of x1 = T (x11 , . . . , x1n ) which is closest in Euclidean distance to the T received codeword y = (y1 , . . . , yn ) . If we let δkj = yk − x1j 2
(11)
be the cost of assigning vector yk to vector x1j , then the problem becomes that of finding n the permutation π of {1, . . . , n} for which the total cost k=1 δk,π(k) is minimum. We have thus an assignment problem, which can be solved √ with complexity O(n2 n log n) by the CSA algorithm [11]. The same method can be used when reception of VPM is performed in a fading channel. To take into account the fading coefficients in (2), a slight modification has to be made on the computation of the costs: δkj = yk − Hk x1j 2
(12)
When alphabet-dimension L, codeword-length n and alphabet-size K are given, a VPM is completely characterized K by its alphabet {µk }K k=1 and composition vector {mk }k=1 . If we assume that the minimum distance between codewords dmin is fixed, then our goal is to design the alphabet and the composition vector such that the resulting codes have a maximum rate R for a given energy E, or, equivalently, a minimum energy E for a given rate R. Vector permutation modulation design problem is similar to that of designing nonequiprobable multidimensional constellations [9]. The difference, in case of VPM, is that the probabilities of the symbols µk are restricted to values pk such that pk n = mk is an integer. It is shown in the Appendix that the optimum composition vector (i.e., the composition vector which maximizes rate for a fixed energy) is given by expressions similar to that of a Maxwell–Boltzmann distribution. These expressions are a generalization of the results in [3]. Following [9], we apply the optimum composition vector to alphabets based on the densest known lattices in L dimensions. In one, two and three dimensions, the alphabets are based on, respectively, the integer lattice Z, the hexagonal lattice A2 and the face-centered cubic lattice D3 . Performance of VPM’s with such alphabets5 is shown in Fig. 1, where it can be seen that performance improve with both increasing codeword-length n and increasing alphabetdimension L. Fig. 1 also shows that the asymptotical performance of one VPM can be attained by another VPM with finite n and larger L. The asymptotical performance of a VPM is obtained by making n → ∞ in the expressions for rate, energy and optimum composition vector, which then become6 K
R=− E=
bold underlined notation is used here to denote a vector of vectors.
1 L
k=1 K
pk µk 2
(13)
(14)
k=1
SPM, alphabets based on Z are proven to be optimum [2]. that these expressions are identical to those obtained in the design of nonequiprobable multidimensional constellations [9]. 5 For
6 Note
4 The
1 pk log pk L
SILVA and FINAMORE: VECTOR PERMUTATION MODULATION
675
−8
−8.5
−8.5
−9.5
2 E/dmin (dB)
E/d2min (dB)
−9
L =1 L =2 L =3
L = 1, n = 200 L = 1, n = 500 L = 1, n = ∞ L = 2, n = 100 L = 2, n = 250 L = 2, n = 500 L = 2, n = ∞ L = 3, n = 333 L = 3, n = ∞
−9
L=1
−10
−9.5
−10.5
−10
L=2
L=3 −11 0.9
0.92
0.94
0.96
0.98
1
1.02
1.04
1.06
1.08
1.1
R (bits/dimension)
−10.5 1 10
2
3
10
10
4
10
N = nL
Fig. 1. Performance of VPM’s with L = 1, 2, 3. Curves for n = ∞ correspond to the asymptotical performance of each code.
Fig. 2. Performance of VPM’s with L = 1, 2, 3 for a rate R = 1 as a function of the overall number of dimensions N = nL.
2
2−λµk p k = K , k = 1, . . . , K. −λµk 2 k=1 2
(15)
Fig. 2 shows, for R = 1, the performance of VPM’s with L = 1, 2, 3 as a function of the overall number of dimensions N = nL, along with the asymptotical performance of VPM for each L. It can be observed that, from a certain value of nL, the performance of an SPM or VPM is surpassed by that of another VPM with greater L. V. C ONCLUSIONS In this paper, we introduce vector permutation modulation, a new transmission scheme which offers significant performance gains over traditional scalar permutation modulation. A method is presented for maximum-likelihood detection of VPM in AWGN and fading channels. Considering the criterion which minimizes the average transmitted energy for a given bit rate, the optimum composition vector is shown to be given by the rounded values of a Maxwell–Boltzmann distribution. A PPENDIX K
Given L, K and {µk }k=1 , we want to find the integers K {mk }k=1 that minimize −R and E under a restriction in n. By using (6), (9) and n mk K 1 1 log M = R= log − log (16) nL nL =1
k=1 =1
we form the Lagrangian f (m1 , · · · , mK ) =
mk K k=1 =1
log +λ
K k=1
mk µk 2 +ν
K
mk
k=1
(17) where λ > 0. The first sum in (16) is constant and is thus omitted in (17). When mk is increased, f (·) achieves a minimum when the following inequalities are satisfied: f (m1 , · · · , mk +1, · · · , mK )−f (m1 , · · · , mk , · · · , mK ) > 0 f (m1 , · · · , mk −1, · · · , mK )−f (m1 , · · · , mk , · · · , mK ) ≥ 0
Substituting (17), we have log(mk + 1) + λµk 2 + ν > 0
(18)
log(mk ) + λµk 2 + ν ≤ 0
(19)
Combining (18) and (19), the optimum values for mk arise:
2 2 mk = 2−λµk −ν = β · 2−λµk , k = 1, . . . , K (20) where x denotes the integral part of x. Here, β > 0 is adjusted to satisfy restriction in n, while λ is adjusted so that a desired E or R is achieved. R EFERENCES [1] D. Slepian, “Permutation modulation,” in Proc. IEEE, vol. 53, pp. 228– 236, Mar. 1965. [2] E. M. Biglieri and M. Elia, “Optimum permutation modulation codes and their asymptotic performance,” IEEE Trans. Inform. Theory, vol. 22, pp. 751–753, Nov. 1976. [3] I. Ingemarsson, “Optimized permutation modulation,” IEEE Trans. Inform. Theory, vol. 36, pp. 1098–1100, Sept. 1990. [4] A. Nordio and E. Viterbo, “Permutation modulation for fading channels,” in Proc. Int. Conf. Telecommunications, vol. 2, Feb. 2003, pp. 1177–1183. [5] E. Viterbo, “Trellis decoding of permutation modulations,” in Proc. IEEE Int. Symp. Information Theory, June 2003, p. 393. [6] T. Berger, “Optimum quantizers and permutation codes,” IEEE Trans. Inform. Theory, vol. 18, pp. 759–765, Nov. 1972. [7] W. A. Finamore, S. V. B. Bruno, and D. Silva, “Vector permutation encoding for the uniform sources,” in Proc. IEEE Data Compression Conference, Mar. 2004, p. 539. [8] D. Silva and W. A. Finamore, “Vector permutation code design algorithm,” in Proc. Int. Symp. Information Theory and its Applications, Oct. 2004, pp. 753-757. [9] F. R. Kschischang and S. Pasupathy, “Optimal nonuniform signaling for Gaussian channels,” IEEE Trans. Inform. Theory, vol. 39, pp. 913–929, May 1993. [10] J. Wozencraft and I. M. Jacobs, Principles of Communication Engineering. New York: Wiley, 1965. [11] A. V. Goldberg and R. Kennedy, “An efficient cost scaling algorithm for the assignment problem,” Mathematical Programming, vol. 71, pp. 153–178, Dec. 1995.
