multiple tone modulation for digital communication channels with arbitrary intersymbol interference (ISI) and additive Gauss- ian noise. Multiple tone modulation ...
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Transactions Papers Discrete Multiple Tone Modulation with Coset Coding for the Spectrally Shaped Channel Antonio Ruiz, John M. Cioffi, Senior Member, IEEE, and Sanjay Kasturia
Abstract- In this paper we develop a discrete approach to (DFT) to this data transmission system, together with a time multiple tone modulation for digital communication channels window to reduce the ISI. Peled and Ruiz [ 5 ] extended this with arbitrary intersymbol interference (ISI) and additive Gauss- DFT technique and introduced a “cyclic extension” procedure ian noise. Multiple tone modulation is achieved through the concatenation of a finite block length modulator based on discrete for elimination of IS1 and applied this technique together Fourier transform (DIT) code vectors, and well known high with a frequency-domain equalizer to the voiceband changain coset or trellis codes. Symbol blocks from an inverse D I T nel. Hirosaki [6] has also explored similar techniques for (IDIT) are cyclically extended to generate ISI-free channel- multiplexed QAM using the DFT, including [7] a very high output symbols that decompose the channel into a group of orthogonal and independent parallel subchannels. The design of rate groupband data modem. Feig et al. [8]-[lo] have also the energy distribution and coded information allocation over the explored discrete multiple tone techniques using the DFT for subchannels is optimized for the finite block length case and for application to the linearized magnetic storage channel. Areas the coset code concatenation, leading to an implementable coding of recent activity include application to the voiceband channel, system with optimized performance for the channel with ISI. as in a recent product introduction by Telebita [ll], multitone Asymptotic performance of this system is derived, and examples of asymptotic and finite block length coding gain performance techniques [12], [13], and investigation of application to the for several channels is evaluated at different values of bits per high-data-rate digital subscriber loop [141- [ 161. Also, there is sample. Using sufficiently long blocks for a particular IS1 pattern, concurrent work by the coathors of this paper in [17] where it can be shown that the implementable techniques presented here the discrete multiple tones of the IDFT are replaced by the achieve the cut-off rate for any channel with IS1 in the presence channel eigenvectors. of additive Gaussian noise. In addition, this discrete multiple tone One difficulty in the application of the multitone modulation technique is linear in both the modulation and the demodulation, and is free from the effects of error propagation that often afflict methods has been the practical synthesis of the systems with systems employing bandwidth-optimized decision feedback plus finite complexity. In this paper, we investigate a discrete coset codes. multiple tone approach through the IDFT, which uses a finite number of subchannels (analogous to the “tones” in multitone) and a finite block length. We expand on the recently I. INTRODUCTION introduced discrete multiple tone modulation with coset coding ultiple tone modulation has long been considered a using the IDFT (henceforth called “DFT codes”) of [18] candidate for modulation on channels with severe in- to allocate energy and coded bits according to the spectral tersymbol interference (ISI). Multiple tone techniques can be characteristics of the channel. The features that distinguish traced to Holsinger [l], and later Chang [2] and Saltzberg [3]. DFT codes from previous work are the concatenation of the These so-called “multitone” or “multichannel” transmission trellis codes (or coset codes) with the discrete multiple tone systems attempt to subdivide ,the channel with IS1 into a modulation through the channel independent vectors of the bank of orthogonal frequency-indexed subchannels. Later, IDFT, the optimal allocation of energy and coded information Weinstein and Ebert [4] applied the discrete Fourier transform according to the spectral characteristics of the channel, and the computational complexity savings achieved in using fast Paper approved by the Editor for Coding Theory and Applications of the Fourier transform algorithms for the DFT. IEEE Communications Society. Manuscript received April 19, 1988; revised August 23, 1989. This work was supported in part by an IBM Faculty Trellis coded modulation, originally developed by UngerDevelopment Award, by NSF under grant number MIP 86-57266, and by the Stanford Joint Services Electronics Program under Contract DAAG 29- boeck [19]-[21] and subsequently analyzed and formulated 85-K-0048. in terms of coset codes [22], [23] is modified for applicaA. Ruiz is with IBM T. J. Watson Research Center, Hawthorne, NY 10532. tion to the channel with IS1 in this paper. The new codes J. M. Cioffi and S. Kasturia are with the Department of Electrical Engineerhave significant coding gain improvements with respect to ing, Information Systems Laboratory, Stanford, CA 94305. IEEE Log Number 9200330. previously developed codes for partial response channels or
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RUlZ et al.: CODING FOR THE SPECTRALLY SHAPED CHANNtL
codes that also shape the transmitted spectruin [24]-[26]. The DFT codes can also significantly improve on the recently introduced vector coding methods of [17] by increasing the computationally feasible block length for a given coding gain or equivalently, by reducing the computation required to achieve a particular gain. In Section 11, we present general expressions for DFT code symbol sequences and average spectra. In Section 111, we discuss the so-called “continuous approximation” relation between average energy for a signal set from a cubic lattice contained in a hypersphere and the number of points in that signal set. Coset codes for two types of channels are discussed, namely, for the flat “Nyquist” channel in Section IV and for frequency-specified channels (or channels with ISI) in Section V. General design procedures and asymptotic bounds are also described for both types of channels. The computational complexity issues related to the computation of the DFT are discussed in Section VI, in particular, the number of required real additions and multiplications. And finally, we conclude the paper in Section VII. As with all coding methods that do not increase bandwidth, the coding gain is a good way to evaluate coding performance. Before we provide the basis for our coding gain evaluations, we discuss the benchmark coset codes used in our coding technique. A . Benchmark Coset Codes in Four and Eight Dimensions
In all our code design for four and eight dimensions we use good known trellis codes [19] -[21] or coset codes [23] of code depth L,L = 2 and four or more stdtes, using signal sets in Z4 and Z8. These benchmark coset codes achieve a coded Euclidean distance
where is the “free distance” factor over the minimum distance in the received signal set. This means that the codes increase effective detection distance by a factor @ over the minimum distance in the signal sets received. For example, coding gains of 4.77, 4.52, and 5.27 dB can be realized for 2, 4, and 8 dimensional codes, respectively, using 16 states [21]. More states can be used, if desired, to reduce the nearest neighbor counts, B. Coding Gain and the Matched Filter Bound
For the evaluation of the coset codes in this paper, we use the definition of coding gain given by
where Pcoded and Puncoded are the average coded and uncoded powers, and dcoded and dullLodedare the resulting coded free distance and the uncoded minimum distance respectively. Also, all uncoded systems in this paper are assumed to use i bits per dimension or per sample coming from representative PAM ( 2 2 1) signal sets with 22 components located at &1 for i = 1, or & l through f(22 - 1) for all other values of i, with minimum distance drnin = 2 .
+
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We use the matched filter bound (MFB) as the reference for all coding gain performance evaluations, as it represents an easily computed and well-known performance reference. Using the above representative signal sets per sample in an uncoded system, the minimum distance dInin at the input to the channel is given by dIniIl= 2 , and for a normalized channel, the MFB results in d~x,,,,= 4. Clearly, not all channels can achieve the MFB through symbol by symbol detection, as the best performance for a given channel is at most that obtained from observing all possible output sequences and finding the smallest received output Euclidean distance between two samples. MFB performance for the flat “Nyquist” channel and all of the frequency-specified normalized channels that we use gives d~llill = 4. On the other hand, the best symbol by symbol detection (with Viterbi detection) is only equal to the MFB for the flat “Nyquist” channel and some partial response channels, including ( l / f i ) ( l - D ) , ( l / f i >(1 - D’), and (1/2)(1 - D)(1 D)’,for all number of bits per sample. All our designs assume the same sampling rates for coded and uncoded systems, and thus the coded and uncoded average energies per sample, per dimension, or per symbol are equivalent measures of average power. Besides, we use the same uncoded and coded average energies, henceforth we refer to average energy instead of average power. With the above assumptions and using the fact that all the known coset codes (or benchmark coset codes) that we use result in dzoded = Gd;,,, then the coding gain expression above simplifies to
+
Coding Gain = 10 log (Gd:,,/4.0)
(2)
where do,, is the minimum distance of the received signal set in the coded system. 11. A SYSTEM FOR FREQUENCY-DESIGNED SYMBOLS OR SYMBOLSFOR DIT CODES
This section describes the symbol construction method used to obtain sequences to be transmitted on the channel. This method takes advantage of the properties of the discrete Fourier transform to generate sequences with specified frequency content at the frequency bins of interest determined by the right combination of sampling rate and transform size. However, for channels with intersymbol interference (ISI), finite length symbols have to be specially mapped to preserve symbol integrity because of the smearing effects of the intersymbol interference. A cyclic extension procedure is described, which eliminates intersymbol interference. A . Modulation wid Demodulation through the IDFTIDFT
The method of modulation uses the inverse discrete Fourier transform (IDFT). We show in Fig. 1 that a block of encoded and mapped data (from signal sets) is transformed by the IDFT in the transmitter. At the receiver, we perform demodulation through the inverse operation, the discrete Fourier transform (DFT).The DFT output is then decoded from signal sets to obtain the received information. We use normalized versions
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DFT PaTallel to
dt) Fig. 1. System configuration.
of the N point DFT and IDFT as follows:
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to the IDFT has Hermitian symmetry and is constrained as follows:
N-1
XN-k
(0 5 n
5N
- 1 ) . (4)
In using the IDFT/DFT we take advantage of the properties of the transformation [27]: linearity, circular shift, symmetry, and circular convolution. We also have the properties that the transform matrix [as defined in (3) and (4)] consists of orthonormal eigenvectors, and that the Fourier transform of AWGN is AWGN. Two types of IDFT output can be designed which result in complex or real sequences depending on the construction of the IDFT input, taking advantage of the symmetry properties of the IDFT. The sections below describe both types of outputs. I ) General Expression for Complex Output Symbol Sequence: To better understand the structure of the symbol sequence to be transmitted on the channel, we derive a general expression for the multidimensional IDFT output symbol sequence of length N . Using a set of complex inputs { X } , we usually obtain a set of complex outputs {x}. We rewrite (4) and obtain values X ( k ) , for k = 0 to N - 1, from two-dimensional signal sets (or two dimensional subset components of larger dimensional . signal set c k carries Lk signal sets) CO through C N - ~Each levels (or equivalently log, Lk bits) with coordinates ( u k l , b k l )
5 L, and L
n LI, is the total number
N-1
=
k=O
of levels per block. Thus, if we let x ( k ) = (Lkl resulting IDFT output samples are of the form
.(n) =
o < < $!
(for N even)
with X ( O ) , and X ( N / 2 ) both real quantities, or
1 N-l ~ ( n=)X(k)e’v k=O
where 1 5 1
=X i
+ j b k l , the
XN-k = X i
0
(O)h,(l)h(2) 0 0 0 0 h(0) h(1) h ( 2 ) 0 h ( O ) h(1) h(2) 0 0 0 0 0 h(0) h(1) h ( 2 ) 0 h ( 0 ) h(1) 0 h(2) 0 0 0 h(O) h(1) h(2) 0
Equivalently, there exists a cyclic convolution between -c and h and the following DFT transform pair holds:
(h(71)) @3& H ( H ( k ) } X
(12)
where X is the DFT of U, and with knowledge of N ( k ) at the frequency bins ti we could undo the interbin interference coming from H ( k ) inside each symbol. However, as we shall see in Section V, we design codes according to the spectral characteristics of the channel so by proper design of the signal
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sets, which are mapped to the input of the IDFT, the interbin interference need not be undone. We note that the price we pay for eliminating IS1 through the cyclic extension is extra energy and extra uncoded bits. The cyclic extension means that an additional ( M - 1)E, units of average energy (where E; = (22i- 1 ) / 3 is the average energy per sample), and ( M - l)i uncoded bits are carried by the cyclically extended symbol in N‘ = N M - 1 samples. At the receiver, only N samples are processed by the DFT and M - 1 samples are discarded. The resulting SNR loss because of the cyclic extension is given by
with LYAa proportionality constant dependent on A. Similarly, for the cubic lattice ZA, where dJ = d, the average energy as a function of the number of levels L and the minimum distance d is given by
+
SNRI,,, = lO10g
N+M-1 N ’
111. HYPERSPHERICAL SIGNAL SETS IN THE CUBIC AND
RECTANGULAR LATTICESAND THE “CONTINUOUS APPROXIMATION"^ In the DFT codes, we use anywhere from one code of multiple dimensions to several codes of smaller dimension. All examples deal with one or more sets of four or eightdimensional codes. For example, for the flat “Nyquist” channel we design codes on the Z” lattice, and for four and eight dimensions, we use signal sets on the Z4 and Z 8 lattices respectively. For all channels, we find the lowest energy multidimensional signal sets to obtain the maximum coding gain performance for a given block length. We investigate in this section the approximate average energy relations for these multidimensional signal sets which are contained in a hypersphere. If we let A be the dimensionality of the code, the optimal signal sets in ZA are those contained within the lowest radii shells for a given number of levels. For example, the first shell in Z” contains 2” levels, all with the same average energy EL = A0.25d2. The second shell contains 8, 64, and 2048 additional levels for the Z 2 , Z4, and Z8 lattices, respectively. For other shells in the same lattices, their levels and corresponding average energies EL can be tabulated as in [31]. In [31], we also consider signal sets in the rectangular lattice RA with minimum distances d j in each of the dimensions (1 5 j 5 A). In is important to have an approximation to the average energy for any signal sets in A dimensions of the types used for DFT codes. In Appendix A, we derive the average energy for the signal set on the rectangular lattice RA contained in a hypersphere, with minimum distances d3 (1 5 j 5 A ) and L points or levels to obtain
‘The term “continous approximation” is due to Forney [36].
+
(with r($ 1) = $! for A even). These expressions are for a reasonably large number of lattice points contained in a A-dimensional hypersphere. For the cases required in this paper, the d j ’ s are either all equal (i.e., from the cubic lattice Z”), or equal in pairs (where each equal pair comes from the same frequency bin). We use this result in our derivations of DFT codes for the flat “Nyquist” channel and frequency-specified channels. The multiplicative constants (1” for the cases of A = 2 , 4, and 8 are 0.1592, 0.3001, and 0.5636, respectively. Equation (15) is a good approximation with average energy errors of about one percent or less for levels approximately greater than 43, 450, and 47000 for 2, 4, and 8 dimensions, respectively.
Iv.
DFT CODES WITH SPECTRAL NULLSFOR THE FLAT “NYQUIST”CHANNEL
In this section we design coset codes with spectral nulls using the IDFT by zeroing corresponding frequency bins. As a function of dimensionality, one dimension is sacrificed for each of the nulls at D.C. and the Nyquist frequency. For a block length N that is even, and with both D.C. and the Nyquist frequency zeroed, we have N - 2 dimensions remaining. Similarly, for a block length N that is odd, and with D.C. zeroed, we have N - 1 dimensions remaining. Other spectral nulls or notches at other frequency bins (as in pilot tone implantation applications) require sacrificing two dimensions per null. Clearly, the implications of spectral nulls in these DFT codes is that as one or more dimensions are sacrificed, information bits and energy normally carried by the sacrificed frequency bins must now be carried by the other dimensions or frequency bins. A . Four and Eight-Dimensional Codes with Notches
at D.C. andlor the Nyquist Frequency Depending on the spectral constraint, we have codes of two types: those with a notch at D.C. only, and those with notches at both D.C. and the Nyquist frequency. Using signal sets in Z4 and Z8 contained in a hypersphere as in [31], [33], [34], we consider each of these types of notch codes. Fig. 2 shows the average spectrum of two 4D notch codes for the flat “Nyquist” channel, reflecting the two types of codes discussed in this section. For codes with a notch at D.C., we use a block length N which is odd since we do not explicitly need the presence of the Nyquist frequency. On the other hand, for codes with notches at D.C. and the Nyquist frequency, we use a block length which is even since we need the explicit presence of
RUIZ et ai.: CODING FOR THE SPECTRALLY SHAPED CHANNEL
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-. - -.- _- - ~~ _- .......................................
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’
’
..................
’
1 Bit/Dimension
~
......... .
2 Bits/Dimension 3 Bits/Dimension
. 4~ Bits/Dimension ~
N=10. m = 2 1=2. Notch ot D C and Nyqust Freq ( 7 = 2 ) N=9. m=2 #=2. Notch ot 0 C (7=1)
J O 1
0 2
03
04
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1
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01
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80
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0
Block Lenqth
Normalized Frequency
Fig. 2. Average spectrum for 4-D notch codes for the flat “Nyquist” channel.
Fig. 3. Coding gains for 4-D DFT codes on the flat “Nyquist” channel with a notch at D.C.
,
the Nyquist frequency component to put zero energy in it. If we let A be the dimensionality of the code (i.e., A = 4 or S), and r according to the type of notches desired, the block length N is given by the expression
N=Am+r
,
1
(16)
B.- zN . + m = m(Ai+ I) + ~ i
where Ei = (2” - 1)/ 3 is the average energy per dimension (or per sample). To design the codes and perform the calculations for coding gain, it is better to specify coded bits and average energy per code, that is Bk and Ek, respectively, as follows:
Bi Bk = m
(2
= 1;2 , 3 , . . . )
1 Bit/Dimension
......... 2 Bits/Dimension ...~ 4 Bits/Dimension
(17)
+
-
~
3 Biis/Dimension
where i is the number of bits per dimension (or equivalently, per sample) in the uncoded system. We have also assumed rate n / ( n 1) encoders for each code which results in the number of coded bits greater than the number of uncoded bits by the quantity m, the number of codes per block. The total average energy per block E,, is given by
+ 1) + Tmi
-.
.....................................................
where m is the number of 4-D or 8-D codes present in the block. Thus, the total number of coded bits to be transmitted per block is given by
= (Ai
I I ~7 (______..__.___._-_l.-.l
I
(IC = 1, . . . . m ) (19)
NE, EI, = m
,
N
0
Fig. 4.
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,
1
120
160
Coding gains for 8-D DFT codes on the flat “Nyquist” channel with a notch at D.C.
channel with a notch at D.C. and the Nyquist frequency. These coding gains are plotted for values of i = 1, 2, 3, and 4 bits per dimension as a function of the block length. Coding gains are computed by distributing energy and coded bits evenly over all remaining dimensions, obtaining d:ut for the various block lengths, and using (2) with a value of G = 4. We note that in all the 4-D and 8-D notch codes for the flat “Nyquist” channel, for a reasonably large block length, the coding gain increases as the number of bits per dimension increases. This effect occurs since for the flat “Nyquist” channel with notches, energy per coded bits is better allocated in the signal sets in Z4 and Z8 for the larger number of bits per dimension. However, an opposite effect occurs for frequency-specified channels as we will see in Section V.
B. Asymptotic Coding Gain Bounds for DFT Codes on the Flat “Nyquist” Channel
Figs. 3 and 4 plot coding gains for 4-D an 8-D codes for flat “Nyquist” channel with a notch at D.C. Fig. 5 and Fig. 6 plot coding gains for 4-D and 8-D codes for the flat “Nyquist”
In this section, we investigate the asymptotic performance of the codes for the flat “Nyquist” channel with notches at D.C. and/or the Nyquist frequency. Analyzing expressions for the number of coded bits per code Bk and average energy per code EI, as in (19) and (20) as m goes to a very large value,
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TABLE I
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Code 4-D 4-D 4-D 4-D 8-D 8-D 8-D 8-D
I
1 Bit/Dimension Bits/Dimension 3 Bits/Dimension 4 Bits/Dimension
.......... 2
80
1 2 3 4 1
2 3 4
Eo0 4 20 84 340 8 40 168 680
B, 5 9 13 17 9 17 25 33
3.01 4.72 4.90 4.95 4.26 5.70 5.93 5.98
*
Block Length Fig. 5. Coding gains for 4-D codes on the flat “Nyquist” channel with notches at D.C.and the Nyquist Frequency.
40
Bits per sample (i)
Asymptotic Coding Gain2 Bound, Gain,
120
160
For 1 and 2 bits per dimension, the coding gain bounds were computed using tables for average energy versus number of levels (as in [31]) because the number of levels per code is low and (15) is a poor approximation in that case.
Using the above asymptotic bound expression, we summarue coding gain bounds as a function of the number of bits per dimension for coset codes on the flat “Nyquist” channel with a value of G = 4 and signal sets on the cubic lattice inside a hypersphere as in Table I. The above coding gain bounds show that as the number of bits per dimension i grows and the block length N increases, the coding gains obtained are bounded approximately by the coding gains of the benchmark codes on a flat “Nyquist” channel plus any “shaping” gains obtained from using optimal signal sets in Z4 and Z 8 ,respectively. These shaping gains are substantial and add about 0.45 to 0.73 dB to the overall gain.
Block Length Fig. 6. Coding gains for 8-D codes on the flat “Nyquist” channel with frequency. notches at D.C. and the Nyquist .. .
they become
B,=Ai+l
(21)
and
E, = A(i(22i
- l)),
(22)
respectively. using the continuous approximation as in (15), we obtain approximate expressions for and subsequently for coding gain. using (151,(22) and L = 2B” with B, as in (211, and solving for 8 ,we solve for the asymptotic coding gain (using (2)) as a function of i and A to obtain
(23)
C. Comparison of DFT Codes and Other Coset Codes with spectral ~ ~for the l ~l~~ l “Nyquist” ~ Channel Another recently developed successful technique for generating coset codes with spectral nulls for the flat “Nyquist” channel is bounding running digital sums (henceforth called RDS codes) [24]. The RDS codes achieve spectral nulls through bounding the running digital sum of the samples of the symbol sequence. For example, to achieve a null at D.C. the RDS is maintained at or near zero, and to achieve a null at the Nyquist frequency, the RDS of the alternately negated samples is maintained at or near zero. The RDS code symbols use signal sets specially designed to effectively achieve the bound on running digital sums, together with known Coset codes to obtain coding gains. w e refer the reader to [241 for further details on RDS codes and turn to a comparison. We also note that in our codes the Dec. and the Nyquist frequency bin is the running digital sum, and is approximately bounded by 0.5N. We compare the coding gain performance of the DFT codes for the flat “Nyquist” channel with the RDS codes. To effect a fair comparison we use gains with respect to the matched filter bound as discussed previously, and we use coset codes with similar number of states. The comparisons are summarized in the Tables I1 and 111. From these tables we conclude that, in most cases, DFT codes for the flat “Nyquist” channel outperform RDS codes by up to 1 dB of coding gain for 4-D and 8-D codes with nulls at
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TABLE I1 Coding Gain for 4D DFT Code (Null at D.C.)
Coding Gain for 4D RDS Code (null at D.C.)
3.01 dB 4.50 dB Coding Gain for 4-D DFT Code (Null at D.C. & Nyquist) 3.00 dB 4.20 dB
2.04 dB 3.34 dB3 Coding Gain for 4-D RDS Code (null at D.C. & Nyquist 2.04 dB 3.34 dB3
Bits/Dimension 1 2 Bits/Dimension 1 2
3This coding gain was computed using the uncoded signal set: ( l , l , l , l ) , (3,1,1,1), (3,3,1,1), (3,3,3,1) and a subset of (3,3,3,3), with Eavg= 20, instead of the uncoded signal set: ( l , l , l , l ) , (3,1,1,1), (3,3,1,1), and a subset of {(3,3,3,1) U (5,1,1,1)}, with E,, = 19.5. TABLE 111 Coding Gain for 8D DFT Code (Null at D.C.)
Coding Gain for 8D RDS Code (null at D.C.)
4.23 dB 5.50 dB Coding Gain for 8-D DFT Code (Null at D.C. & Nyquist) 4.23 dB 5.45 dB
4.64 dB 4.52 dB Coding Gain for 8-D RDS Code (null at D.C. & Nyquist) 4.64 dB 4.52 dB
Bits/Dimension 1 2 Bits/Dimension
1 2
D.C. and/or the Nyquist frequency. Moreover, the DFT codes have average spectra which are predictably flat as computed from (11) and illustrated in Fig. 2. Regarding computational complexity, the main issue between the RDS codes and the DFT codes is that the first has to keep track of the signal set selection through RDS calculation at the transmitter, while on the other hand, the second carries out an IDFT at the transmitter and a DFT at the receiver. Although we use fast algorithms for the DFT’s (as briefly discussed in Section VI) the operational complexity of these codes is higher, but still computationally within easy reach of most digital signal processors used in communication systems today. Furthermore, we can use the optimal Viterbi detector without increasing the number of states. The main disadvantage of our new codes is that the transmitted levels do not fall on 2”.This can lead to a significant increase in the complexity of timing recovery and other system functions indirectly related to the number of transmitted time-domain levels.
v.
DFT
CODES FOR FREQUENCY-SPECIFIEDCHANNELS OR
CHANNELS WITH IS1 We discuss here coset codes specifically designed for channels with arbitrarily prescribed magnitude characteristics. These channels may contain characteristic nulls at D.C. and/or the Nyquist frequency. Furthermore, these frequency-specified channels have a spectral shape that is not flat and can be expressed with a finite length polynomial
where the coefficients h, are real numbers so the channel h; = 1 This normalcharacteristic is normalized or, ization has no effect upon the relative coding gains of the compared coding methods. Since our design method uses the spectral characteristics of the channel IH(e3”) as we will see
1,
below, energy and information (bits or levels) are distributed according to H ( e j ” ) The design of 4-D and 8-D coset codes for the frequencyspecified channels is based on optimal signal sets in Z4 and Zs, respectively at the receiver. However, as explained in more detail in the next section, the minimum output distance clout, which in turn determines the minimum coded free distance of the code and thus the coding gain performance, is directly related to the transmitted minimum distance d, and channel spectral characteristic ( H ( j )1 on each frequency bin. Furthermore, we show that this relation results because of the parallel channel decomposition relation obtained from the “cyclic convolution” property of the DFT.
1
1.
A. Coset Code Design Specifications for Frequency-Specified Channels
We develop here the prerequisites for designing coset codes for frequency-specified channels. We assert the following specifications: Number of coded signal levels per block L. Spectral magnitude frequency characteristic I H ( j )I = ( e J d > f = f J = 3 f * , N of the channel where j = 0.1, . . . , N - 1 for a block length N . Equal output minimum distance squared d&t on all codes at the receiver. Total average energy per block Eavgconstrained at the input to the channel. Ek is the desired average energy level specification for each code where E&is obtained from the total average energy budget for the block. Besides, Ek is shown to be optimally equal for all codes for equal output minimum distance However, Bk, the number of coded bits per code (or equivalently Lk = 2Bfi the number of coded levels per block) varies with the code location on the frequency bins.
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To start, we use a relation between average energy per frequency bin between input and output of the channel as follows:
(25)
Using (14) and (27) for 2,4, and 8 dimensions, we obtain the following approximations for average energy in the code Ek as a function of the number of levels in the code Lk, the output distance squared d;ut, and the channel spectral characteristics IH;,kl:
1 This simple relation for average energies holds true, in this E ( ~ - ~0.15927 ) d;,,Lk (28) particular case, because we use intersymbol-interference-free IHl,kl (henceforth called ISI-free) symbols as detailed in Section IIE ( 4 - D ) E 0.3001 d;,,L:I2 (29) B, and therefore, there exists a cyclic convolution relation IHl,kllH2,k1 between the input and the output of the channel as in (12). Since it is a DFT relation, this only holds true for the integer arguments. Thus, there exists a relation between { Y }and { X } where IH;,kl is the spectral characteristic of the “parallel changiven by nel” at the jth frequency bin in the kth code. Or generically Y ( j )= H ( j ) X ( j ) for 0 5 j 5 N - 1. (26) for two, four, and eight dimensions.
We call this relation the “parallel channel” interpretation of the DFT codes. We comment also, that this relation can be applied to symbols that are not ISI-free; however, it holds strongly only for long block lengths where N >> M . Codes are designed for equal minimum distance dout at the receiver. This implies that 2
dout
where A is the dimensionality of the code (A = 2, 4, or 8), and K k , A is a constant that depends on the code specifications and the dimensionality of the code as follows:
= ‘:ut; = d?utk
for all frequency bins j (0 5 j 5 N - 1) and codes k ( l 5 IC 5 m). With the same output minimum distance there is one common unique minimum Euclidean distance at the receiver, which for equally likely symbols determines the coded free distance parameter used to evaluate the coding gain performance of the DFT codes. Hence, using (25), (14), and (15) for each frequency bin j , we obtain
Before describing the coset code design procedure, we first prove the equal energy per coset code specification. B. Energy Allocation for DFT Codes
Because of the frequency-design, the codes we obtain consider the shape of the channel H (@”) to distribute information (bits) according to the channel characteristic with average energy per code Ek as a given input parameter to the design equations. We must now consider how to partition the average total energy per block E,, into the various E k ’ S . The allocation of average energy and information according to channel characteristic I H ( j )I and signal-to-noise ratio (SNR) is reminiscent of the “water filling theorem” [29] of communications, commonly used for computing the capacity of bandlimited channels that are not flat. We recall, however, that our interest is coset codes with the least amount of total average energy per block, designed for optimal (i.e., the largest possible) minimum output distance dout for all codes. In this way, we obtain the maximum free distance dfree on the coset codes as seen by the receiver. Here, we describe such a method which pertains to the frequency-specified case of interest in this section and is similar to a procedure independently developed by Aslanis for vector codes in [17].
I
I
with U A as in (14). Equation (31) is a general expression that applies not only to DFT codes but to other multidimensional codes as well. The general case is one where multiple parallel channels with many dimensions are partitioned into multiple codes of dimensionality A. For the DFT codes, every pair of dimensions is associated with one frequency bin (other than D.C. and the Nyquist frequency) and one magnitude spectral transfer function value IH ( j )I. With (31) we can now derive the optimal energy allocation for equal output minimum distance dout and derive, along the way, the required partial coded levels (or equivalently, the number of transmitted coded bits) for each code. We do this for a block with m codes as follows: we wish to minimize (31) subject to the constraint that the total number of coded levels is equal to the product of partial coded levels m
L = ~ L , , n=l
or equivalently ni
In L =
In L,.
(33)
n=l
To minimize Ek, we proceed by forming the Lagrangian [28] as follows:
As usual in Lagrangian problems we must satisfy the following set of m equations:
dC
-= 0
aLk
IC = 1,2, ‘ . . ,m.
RUIZ et al.: CODING FOR THE SPECTRALLY SHAPED CHANNEL
1021
As developed in more detail in [31], we obtain L2/A
k
P A = ____ Kk,Ad:,t 2
(34)
and
2
L,a = LA
(fi
(35)
n=1
Kk,a
to obtain ISI-free symbols. For modulation and demodulation through the IDFT and DFT, respectively, we are interested in the frequency characteristic IH(j)l at frequencies f j = j f , / N where N is the block length and 0 5 j 5 N / 2 for N even, or 0 5 ,j 5 ( ( N - l ) / 2 )for N odd. The codes obtained here are
the energy is better used in other frequency bins with much higher IH(j)l value. Therefore, the number of 4-D or 8-D codes used in a given block length is bounded from above, contrary to the case in the flat “Nyquist” channel with spectral nulls where all other dimensions could be used. We use the symbol length (with cyclic extension) N’ and the IDFTDFT size N as the main parameters that characterize the DFT
For real symbol sequences, we characterize two types of channels among the frequency-specified channels, namely, type I with one frequency null at D.C. and type I1 with two frequency nulls, one at D.C. and another at the Nyquist frequency. Type I channels use an IDFTDFT size N that is odd and can use as many as N - 1 dimensions to carry information. On the other hand, type I1 channels use an IDFTDFT size N that is even and can use as many as N - 2 dimensions to carry information. The block length N’ and the size of the IDFT/DFT used N are related to the length of h(n)the impulse response of the channel M by the expression of IC, and thus the optimal allocation of energy for minimum N’ = N + M - 1 where the additional M-1 value comes from total average energy using the best possible signal sets in ZA the cyclic extension. Thus, the total number of information (A = 2, 4, and 8) and for equal output minimum distance bits carried per block equals N’i where i is the number of dout on all codes k (1 5 IC 5 m) requires equal average energy bits per dimension. However, the number of coded bits per block B , given by B = N’i m (and thus the number of distribution on each code. coded levels L = 2 B ) varies according to m, the number of 4-D or 8-D codes used in a block. For the 4-D and 8-D codes, since the number of dimensions used has to be a multiple of 4 and 8, respectively, the number of codes m is limited (37) by
and substituting into (31) we obtain (36) which is independent
+
We also note that equal average energy distribution applies if (14) and (15) represent a good approximation to average energy for a given number of levels in a signal set. This is usually the case for a reasonably large number of levels. Besides, this holds for all values of m and correspondingly, at each frequency bin, each of the two dimensions (in-phase and quadrature) has the same average energy. C. Design Procedure for Coset Codes for Frequency-Specified Channels
We consider here 4-D and 8-D coset codes for channels whose frequency characteristic is accurately modeled by (24), and thus a cyclic extension of length M - 1 can be used
where 7 varies according to the number of nulls, A is the dimensionality of the code, and y is the fraction of frequency bins used (i.e., equivalently, fraction of total bandwidth, fraction of parallel channels, or fraction of available dimensions that are used). Also, codes with notches at other frequency bins can be easily designed by nullifying the desired frequency bin with zero energy, carrying the energy and information bits from the two dimensions elsewhere, and excluding such frequency bin from the design procedures that follow. Using (35), (32), and L = 2(N+M-1)i+m, we derive the expression for the optimal partial coded levels LI, [31],
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IEtk IKANSAClIONS ON LOMMUNICAIIONS, VOL 40, NO 6, JUNk I Y Y ?
(39)
results obtained for the flat "Nyquist" channel with notches. where there is no coding loss component from the channel. With expressions fur Lk, d:ut, and the coding gain, we are ready to find optimal 4-D and 8-D codes for the channel with ISI. Assuming ideal conditions as explained before, the procedure is as follows. 1) Use a normalized channel characteristic
CY=;'
Using (36), (37), (18), and L = 2(N+M-1)i+m, we derive an expression for the output minimum distance squared d:,t to obtain [31]
Once the output minimum distance is available, the coding gain about the MFB is readily available from (2) as follows: Gain = 10 log(Gd~,,/4.0)
(41
On closer examination, the expressions for Lk, d:,t, and coding gain contain a dependence on the frequency response coming from the parallel orthogonal channels (because of modulation through the IDFT) where the codes are located. In the coding gain expression, we can identify two components. The first component contains the coding gain from the coset codes used, the shaping gain, and the SNR loss, and the second component contains the coding (gain) loss from the frequency-specified channel shape. As we shall see in the examples, the coding loss usually increases dramatically as the number of bits per dimension increases, and thus, best results are usually obtained for a low number of bits per dimension. This is completely opposite of the
f? JfL-&
2
IH(ejnnf)Idf = lhj12 = 1.0. 2) Select the desired IDFT/DFT length N you wish to work with. This value determines the added complexity of the system because of the IDFT/DFT computation as investigated in Section VI. The overall coded symbol length after cyclic extension is given by N' = N + M - 1 where M is the length of { h ( , n ) }the impulse response of the channel. 3) Once N and N' are determined, calculate the total N E , where E, is the average encrgy average eiiergy per sample as a function of the number of bits per sample or per dimension. (We must note that average energy per symbol is Esyr,,bul 1 N'E,, which is not all available for coding because some of this average energy is spent in the redundancy of the cyclic extension). The number of coded levels per block L is calculated from L = 2(N=M-1)i+m. 4) Using (41), we calculate the coding gain for several values of rn to choose the one with the best coding gain for the given block length N . To place the codes on the frequency bins, the coding gain is maximized if the codes are placed on the larger I H j , k 1's. Starting with the frequency characteristics I H ( e j 2 " f ) at the frequencies j 3 = j f s / N (0 5 j 5 N / 2 for N even or 0 5 j 5 ( ( N - l ) / 2 ) for N odd), we sort the IH(j)l's and then distribute codes from the maximum IH(j)l and so on from the next maximum, until all codes have been allocated I H j , k I 's. 5) With this value of rri and the corresponding IH3,kl'S where the codes are located, we can calculate the partial coded levels for each code Lk. using (39). The resulting partial coded levels are optimal if Lk 2 2A for each A-dimensional code [311. 6) To further characterize the code, we calculate d:ut using (40). With this value of d:ut, and using (27) we calculate the input minimum distances squared d; for optimal multidimensional signal sets at the transmitter (i.e., the ones that-for each code-lie in a hypersphere of average energy E'k ;= E,,Jin). For 4-D codes, there are two frequency bins (for the lcth code) IH3,kl and IHj+l,kl with corresponding input minimum distances squared d; and d;+,. Similarly for the 8-D codes, there are four frequency bins (for the kth code) lHj,kl, IHj+l,kl, I H j + 2 , k l , and IHj+3,kl with corresponding minimum distances squared d ; , d;+, , d;+2, and d;+3. A procedure for optimal multidimensional signal set characterization and implementation methods are covered in [31], [33], [34]. With these procedures, it is possible to fully characterize the resulting signal sets given the parameters Lk, d 1 , k through d + , k , and Ek.
I
D.Exumples oj UFI
code3 - Coding Gums
Following the design procedures outlined in the previous
RUIZ et al.: CODING FOR THE SPECTRALLY SHAPED CHANNEL
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0
I
N
~
H(3)=[-
I
12,-.13.-
I
I
I
I
I
16,- 18,-.22,-26.-.12..68,46,26,07,-04,-
I 10.- 121
1 Bit/Dimension
m
.........
LL
2 Bits/Dimension 3 Bits/Dimension 4 Bits/Dimension
= N
0,
0
c ._ c3 0 m
.I 0 U 0 0
_ _ - - - .-
,,,,-N I
I
1
,-
I
I
I
Fig. 9.
I
I
I
200
Coding gains for 4-D codes on the H ( D ) = 0.53 - 0 . 5 3 0 0.40D2 0.53D3 - 0.13D4 channel.
+
n ~
LL
I ? N ?
C
Uc
I
150
I
m
c3 0
I
Block Length
Fig. 7. Average spectra for frequency-specified channel examples.
l
I
100
Normolized Frequency
e
I
50
1 2 3 4
Bit/Dimenson Bits/D menslon Blts/D m e n s f o n Bits/Dimenslon
N
c 0
-
(3
-
C in
U 0
~
0 0
0
1 2 3 4
Bit/Dimension B ts/Dimension Bits/Dimens on Bits/Dimens on
80
40
120
U 0 0 0
160
Block Length
50
1 00
150
200
250
300
Block Length
Fig. 8. Coding gains for 4-D codes on the H ( 0 ) = 0.71 - 0.71D2 channel.
Fig 10 Coding gains for 4-D codes on the H ( D ) = [- 12. - 13. - 16, - 18, - 2 2 , - 26. - 12. 68, 46, 26, 07, - 04, - 10 - 121 channel
section, we investigate DFT codes for three examples of frequency-specified channels as follows: 1) H ( 0 ) = 0.71 - 0.71D2 2 ) H ( 0 ) = 0.53 - 0 . 5 3 0 - 0.40D2 0.53D3 - 0.13D4 3) H ( D ) = -0.12 - 0 . 1 3 0 - 0.16D2 - 0.18D3 - 0.22D4 - 0.26D5 - 0.12D6 0.68D7 f 0 . 4 6 0 ’ 0.260’ 0.07O1O - 0 . 0 4 0 ~-~0 . 1 0 0-~0~. 1 2 0 ~ ~ . These channels have been normalized and their average spectra are shown in Fig. 7. Coding gains for 4-D codes and 8-D codes (at a value of G = 4) for various block lengths and number of bits per dimension i = 1, 2, 3, and 4 are plotted in Figs. 8-13. All of the above channels are characterized by one or two frequency nulls and a smooth frequency characteristic. Block Length A typical result is that the coding gains are higher for lower Fig. 11. Coding gains for 8-D codes on the H ( 0 ) = 0.71 - 0.71D2 number of bits per dimension. This is opposite to the gains for channel. DFT codes for the flat “Nyquist” channel where the gains are higher for larger number of bits per dimension. This is because, otherwise be using a higher value of y (i.e., more codes on for lower bits per dimension, the code uses a lower fraction of more frequency bins or available dimensions) in locations of available frequency bins y,and since this lower fraction takes the channel where the transfer function H (e j Z T f ) has less advantage of the best part of the frequency characteristic of the gain. channel IH (e J 2 * f ) then the coding gain is better than would Fig. 14 shows the average spectra of two 4-D code trans-
+
+ +
+
I
1,
I
1024
IEEE TRANSACTIONS ON COMMUNICATIONS,VOL. 40, NO. 6, JUNE 1992
U7
1 Bit/Dimension ......... 2 Bits/Dimension - - 3 Bits/Dimension 4 Bits/Dimension
m LL I
.E
[
I
I
I
Block Length = 18, Block Length = 20,
I
I
I
I
I
=0.67. m=2. i=l. for H(D)=[.53.-.53.-.40..53,-.13] =0.75,m=3. i=l. for H(D)=[.71.0.-.71]
,.-__. _.-_._ ................. ,,-._. ,....,_,-._. .-.:
N}
U
0
Block Length
t
al+
I
I
I
I
I
I
0.2
0.1
0.3
0.4
0.5
Normalized Frequency
I
I
I
F I
negligible. Also, the coded level constraint (33) becomes in the limit the asymptotic coded level constraint [31]
s
logz(W)df =
(42)
%+1
-1 --
Bit/Oimension 2 Bits/Dimension 3 Bits/Dimension 4 Bits/Dimension
Block Length
Fig. 13. Coding gains for 8-D codes on the H(D)= [-.12, -.I6 -.18, -.22, -.26,-.12, .68, .46, .26,.07, -.04,-.lo, -.12] channel.
where the region f c F m is the asymptotic bandwidth corresponding to y, the fraction of available frequency bins used by the codes. The asymptotic y is labeled y*,and is the optimal y for any block length and for a given number of bits per dimension. Using' (39), we can derive the asymptotic optimal level allocation to obtain [31]
mitted sequence examples for the H ( D ) = 0.71 - 0.71D2 and H ( D ) = 0.53 - 0 . 5 3 0 - 0.40D2 0.53D3 - 0.13D4 channels. More detailed coding results for specific block lengths with information on partial coded levels, minimum distances, and coding gains are contained in Appendix B for the first of these two channels.
+
E. Asymptotic Performance of Coset Codes for Frequency-Specified Channels
Just as we investigated asymptotic bounds for coset codes for the flat "Nyquist" channel, we investigate here the bounds that result when the block length grows very large for the various number of bits per dimension for coset codes on frequency-specified channels. This analysis provides further insight into the behavior of DFT codes for the frequencyspecified channels. First, as the block length grows very large then N' G N since A4
