A pragmatic approach to trellis-coded modulation - IEEE ...

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G. C. Clark, Jr. and J. B. Cain, Error-Correction Coding for Digital Com- munications, Plenum-Press, 1981. "QUALCOMM Announces Single-Chip K = 7 Viterbi ...
A Pragmatic Approach to Trellis-Coded Modulation Andrew J. Viterbi Jack K. Wolf Ephraim Zehavi Roberto Padovani

A T IS CURRENTLY WIDELY ACCEPTED THAT Forward Error Correcting (FEC) coding is a practical technique for increasing the transmission efficiency of virtually all digital communication channels. whatever their power or bandwidth limitations. Efficiency then applies to both the power and the bandwidth required to support a given information rate. or conversely to increase the information rate to be transmitted with a given power and bandwidth. This realization of the universal merits of FEC coding is relatively recent. When practical coding techniques were first proposed in the late sixties,’ it was generally believed that coding would benefit only power-limited channels where bandwidth was plentiful. Thus. the first applications were to space applications for which data rates, limited by low on-board transmitter power and large range losses, were much smaller than the available bandwidth. The first practical codes employed code rates of I / ? or even 1/3, which equated to less than 1 b/s/Hz with Quadrature Phase Shift Keying (QPSK). With the growth of digital satellite communication, transponder power grew but information rate requirements grew even faster, so that bandwidth limitations began to be felt. In response. code rates rose from 1/2 to 3/4 and even 7/8, but with QPSK modulation. even with code rates approaching 1, bandwidth efficiency is limited to less than 2 b/s/Hz. Incidentally. whereas for a given coding complexity a different code is optimum for each rate, a technique known as “puncturing” permits the use of a modified version of the basic rate 1/2 code for any rate (n l ) / n that is only moderately less efficient than the optimum code for that rate. This technique-first employed bq Linkabit Corporation but first published by Clark,

Cain, and Geist [ I ] for code rates 2/3 and 3/4 and later cxtended to all rates (n - I)/n by Yasuda et ul. [2]-deletes a fraction of the symbols generated by the rate 1/2 code, but utilizes the same decoder as the latter. with the deleted symbols replaced by erasures. This represents an earlier example of the pragmatic approach which, as will be further shown here, permits a single basic code to be used for both power-limited and bandwidth-limited channels. ~

With the growth of digital satellite comm unica tion, transponder power grew but information rate requirements grew even faster, so that bandwidth limitations began to be felt. T o attain bandwidth efficiencies in excess of 2 b/s/Hz, it is necessary to couple coding with symbols of a higher level than the four (phase) levels provided by QPSK.‘ Thus, if constant envelope (amplitude) signals are required (as for operation

~

This work was supported in part under National Science Foundation Grant ISI-8801254. ‘Throughout this paper, only convolutional and/or trellis codes are considered. Whereas some practical block codes have been used in communications systems, for any block code ofgiven complexity there \ irtuallq alw,ays is a convolutional code which exceeds i t in performance. e x c e p ~for those limited applications where messages are constrained to be very short and code length is thereby limited.

0 163-6804/89/0007-0011 $01.OO

1989 IEEE

‘The applications assumed throughout are to radio conimunication where two-dimensional signal constellations appl). Terminology for the constellations is more or less standard. Constant amplitude constellations where M signals are placed at equally spaced phase angles are called MPSK , For .\I= 8 and .21 = 16. .\Iis replaced bq the specific integer. while for M = 2 and .M = 4 . the more common B (for binar)) and Q (for quaternary) replaces M. The other commonly used constellation is the equally spaced grid of points. This is commonly called Q-\M (for quadrature amplitude modulation) or QASK (for quadrature amplitude shift keying). The one-dimensional version of equally spaced points on the line is simplq called ASK. Graphs ofthese carious constellations are given in [ 3 ] .Later in this paper. we precede this nomenclature by M or M2 to indicate the number of signals involved.

Jul) 1989 - IEEE Communications Magarinc

11

k Parallel Blts

Rb/s/lnput m e Convolutional 0 0

k-1

0

Parallel Bits

-

Encoder

Rate (k-l)/k

. . 0

MPSKor MASK Modulator

- -+ - -+

-

M=2

MPSK or MASK Coded SYmbols

Rsymbols/s

t

I Carrier

Fig. 1. Generic encoder/modulator .for trellis-coded tnodulatiorz.

over a nonlinear channel), the choice must be Multiple Phase Shift Keying (MPSK) with 8, 16, or even more levels. If the channels and power amplifier are linear. and non-constant envelope signals are acceptable, multiple constellations in two dimensions may be employed, such as 16, 64, or 256-point Quadrature Amplitude Shift Keying (QASK). In both cases, optimum trellis codes for a given complexity (constraint length or number of states) were found by Ungerboeck [3]. who provided the impetus for coding of bandwidth-limited channels through a technique called “set partitioning.” While he also showed that all such codes can be implemented using a binary convolutional encoder coupled with appropriate mapping of the binary code symbols onto the signal constellation, the resulting best codes for a given complexity (constraint length) were quite different from the classical binary convolutional codes used with BPSK and QPSK. Since the early seventies [4], for power-limited applications, the convolutional code of constraint length K = 7and rate 1/2, optimum in the sense of maximum free distance and minimum number of bit errors caused by remerging paths at the free distance, has become the de facto standard for coded digital communication. This was reinforced when punctured versions of this code also became the standard for rate 3/4 and 7/8 codes for moderately bandlimited channels. A major advantage of puncturing is that the same decoder can be employed, virtually unchanged other than in its branch metric generation, which inserts erasures for the punctured symbols. The remainder of this paper deals with methods for employing this very same K = 7, rate 1/2 convolutional code with signal phase constellations of 8-PSK and 16-PSK and quadrature amplitude constellations of 16-QASK, 64-QASK, and 256QASK to achieve, respectively, 2 and 3, and 2,4, and 6 b/s/Hz bandwidth efficiencies while providing power efficiency that in most cases is virtually equivalent to that of the best Ungerboeck codes for constraint length 7 or 64 states. This pragmatic approach to all coding applications permits the use of a single basic coder and decoder to achieve respectable coding (power) gains for bandwidth efficiencies from 1 b/s/Hz to 6 b/s/Hz. It may well become the universal coding standard.

A Comparative Summary of Ungerboeck Code Performance In his definitive paper on trellis-coded modulation, Ungerboeck [ 31 showed that with the generic encodermodulator of Figure 1 for any integer number of bits per second per Hertz, it was possible to achieve Asymptotic Coding Gains (‘4CGs)of as much as 6 dB in Eh/No within precisely the same signal spectral bandwidth, by doubling the signal con-

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July 1989 - IEEE Communications Magazine

stellation set from A4 = 2“‘ to .i4= 2k and employing a rate (k - I ) / k convolutional code. The signal selection or mapping process was based on a technique which he called set partitioning. Code searches were performed to obtain the rate (k - l)/k convolutional code with the highest ACG for a given signal configuration, of which three types were considered: MPSK, .4mplitude Shift Keying (ASK) in one dimension, and QASK in two dimensions. A number of extensions of this work have since appeared in the literature, both to formalize the theory in terms of lattices [ 5 ] [6] and to find good codes for new signal constellations [7-91. Probably the most significant of the latter was the extension to two-dimensional or more coding, which provided both improvements and more flexibility in accommodating other than integer data-rate-to-bandwidth ratios. The major drawbacks of most of this work were that: Each signal configuration seemed to require a different code, so that unlike the class of punctured binary codes, no single encoder-decoder could be used for a wide set of parameters (e.g., 2,3, and 4 b/s/Hz all with a single decoder). From a practical implementation perspective, the number oftrellis code states used in most applications has been very low (typically 4 and 8, occasionally 16. but rarely more). Performance was almost always measured by ACG, which is just a function of free Euclidean distance. This can give misleading results because not only may there be a large number of paths at this distance, but also the next higher Euclidean distance between unmerged paths may be a very small amount larger than the free distance (unlike the case for binary Hamming distances, which must be separated by integers). In this paper, we shall propose a remedy for each of these drawbacks. The next section treats the first and second of the above points, but first we note that the limitation of the third point has been removed by the work of Zehavi and Wolf [ 1 I], which showed that very tight union bounds on Bit Error Rates (BER), for all but very low Eb/No, can be obtained based on the classical generating function technique used for binary convolutional codes [ 121. To achieve this, the authors exploited the symmetry of the signal constellation to show that relative distances, rather than all distance pairs, sufficed, as they do for binary codes. We have employed this technique to produce complete curves of BER as a function of Eb/No for five classes of signal constellation^,^ which we shall require for comparisons in the next section-namely, the 8-PSK, 16-PSK,

3While we might have also considered codes for QASK configurations, the pragmatic approach to be presented treats QASK configurations by pairs of codes, one on each ASK dimension.

b '

'b

1.OE-02

1.OE-02

1.OE-03

1.OE-03

1.OE-04

1.OE-04 1.OE-05

1.OE-05

1.OE-06

1.OE-06

1.OE-07

1.OE-07

1.OE-08 1.OE-08

1

+

Q -.

u 1 6

P

~

States: U t

4 32

U

-LZ- 64

16

+ 32

* 64

Fig. 4. BER for best Ungerboeck codes ,for 4-.4SK.

Fig. 2. BER for best Ungerboeck codes .for 8-PSK.

'b

1.OE-02 1.OE-03

1.OE-03

1.OE-04

1.OE-04

1.OE-05

1.OE-05

1.OE-06 1.OE-07 1.OE-08 8

9

10

11

12

9

10

11

12

13

14

15

13

EJNo (dB)

U 16

t

32

* 64

Fig. 3. BER ,for best Ungerboeck codes .for 16-P.X

Fig. 5. BER .for best Ungerboeck codes for 8-.4SK

JuI) 1989 - IEEE Communications Magarinc

13

1.OE-03

1 .OE-04 1.OE-05 1.OE-06 -L%’.

.‘!

!

+&ASK

States:

u 1 6

4-4 -A- 32

!

!

+8

* 64

!

I

I

+

Fig. 6. BER for best Ungerboeck codes for 16-ASK.

and 4-ASK, %ASK, and 16-ASK configurations. These are given in Figures 2 through 6. In addition, Tables I and I1 give the ACG associated with each code. In each case, performance of the best binary convolutional code found by Ungerboeck [3] and others [ 131 is given and all possible code state values between 4 and 64 are included. It is worth noting, from Figures 2 through 6 and Tables I and 11, that ACG can often be optimistically misleading. For example, while the 64-state code for 8-PSK provides an ACG over QPSK of 5.0 dB, the coding gain at a BER of 1O r 5 is only 3.6 dB; worse still, for 16-PSK, the corresponding difference is 1.8 dB.

The Pragmatic View for MPSK A Sin le Coder-Decoder for All Values of M = 2

f

Figure 7 illustrates the pragmatic coded modulation system for M = 4, 8, and 16. Generalization to the kth power of 2 is immediate; k - 1 bits are input, with only the lowest order bit going to the binary convolutional encoder, while the remaining k - 2 bits select one of 2k-2sectors. The two bits output by the convolutional encoder select one of four phases according to the following Gray code:

00 01 11 10

---

0 rad. z / ~ ~ - rad. I 2n/Zk-’ rad. 3 ~ / 2 ~ -rad. ’

The remaining (uncoded) k - 2 bits select the sector lexicographically (i.e., the binary vector whose decimal equivalent is j (0 I j I 2k-2 - 1) selects the (i + 1)st ~ e c t o r ) . ~ The code chosen is the optimum rate 112, 64-state, binary convolutional code for which Very Large Scale Integration 4This mapping was first used by Clark and Cain [ 141 and also in [ 181 to demonstrate that the 4-state rate 112 code can achieve a 3 dB ACG as discussed below.

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(VLSI) implementations abound [ 15-1 71 (and which is also often converted, through puncturing, into a rate 314, 718, or generally (n - l ) / n code). The pragmatism refers to using the “industry standard,” which provides reasonably high coding gain, through a moderately high constraint length and a consequently large number of states, to generate trellis codes for a number of MPSK constellations. What remains to be shown is the degree of sub-optimality thus introduced relative to the best 64-state trellis code for each value of M considered. Before considering this comparison, we explore the maximum ACG achievable with the configuration of Figure 7. The trellis form generated by this encoderlmodulator is shown for M = 4,8, and 16 in Figures 8a, 8b, and 8c (vector X vanes over all 32 5-dimensional vectors). For M = 4(Figure 8a), ofcourse, it is the conventional trellis of a binary convolutional code with 64 states connected such that each state has two inputs and two outputs, each from two disjoint preceding and to two disjoint succeeding states. The trellis for M = 8 (Figure 8b) has twice as many branches, with basically the same connectivity but with all branches paired, since the input bit, which does not enter the rate 112 encoder (Figure 7-based on single rate 112 convolutional code: G,(D) = 1 + D2+ D3+ D4+ D5, G2(D) = 1 + D + D2 +D3 D6), produces two branches with identical connectivity, so that all branches of Figure 8a now become branch pairs with branch metrics identical except for opposite signs (i.e., the corresponding signal points are antipodal). Similarly, for M = 16 (Figure 8c), all pairs of states connected in Figure 8a by a single branch are now connected by four branches, each corresponding to one of four values of the two inputs, which do not enter the encoder. The squared Euclidean distance between nearest neighbors among these four “bundled” branches is proportional to sin2 (d4). Similarly, for any M = 2k, for which the modulator of Figure 7 is modified only by having k - 2 uncoded input bits, the bundle size increases to M / 4 = 2k-2 branches, which correspond to neighboring signal points separated by squared Euclidean distance proportional to sin2(4n/M). Now the ACG is defined, in the limit of asymptotically low error rates, to be the bit energy-to-noise Eh/ No required for coded operation with M = 2k signal points relative to that required for uncoded operation with M/2 = Zk-’ signal points (note that both cases provide the same information throughput, k - l bls1Hz). Bit errors can occur only because of an incorrect choice of path at any state. For M = 4, all pairs of paths are disjoint over at least one constraint length ( K = 7) branches, but for M 2 8, minimum length disjoint paths are only one branch long, because of the bundling of paths. Hence, the minimum Euclidean distance for an error event can be no greater than the single branch errors (although, under some circumstances, the 7-branch unmerged error events can accumulate less distance than the single branch error event-see the Appendix). Hence, while for an uncoded operation with M/2 signals, the BER Pbu is bounded by:

July 1989 - IEEE Communications Magazine

< 2 Q v ( 2 E s l N , 1 sin2(2rr/M) the coded BER Pbc with an M-signal constellation is lower bounded by considering only the minimum Euclidean distance among the bundled branches,

where, as we shall showj K may be less than unity. AsymptotiOnly the argument Of the error function Q into play. Hence, in terms of decibels, the ACG is:

0

0 0001

1001 0

0 1110

0110

___)

1010 1100

8-PSK Modulator

Select Half-Plane \

Input Information Bits

\

-

v

m

output Selector

c

100

4-

G,(D)

---+ 101 0 a-----

M-ary PSK Channel Symbol c

c

111

4-PSK Modulator

-

G,(D)

Fig. 7. Pragmatic encoder/modulator for QPSK, 8-PSK, and 16-PSK.

ACG

sin2(4n/M) 5

J O log,,[

1

1 (31

sin2(2rZlM)

= J O l o g l O [ 4cos2(2rr/M)] M 2 8 It is shown in the Appendix that the true ACG is lower bounded by the minimum of Equation (3) and a term that depends on an unmerged path which is at least one constraint length long; more precisely,

P , < - Q ( V 2 ( 2 E , / N O ) )+ 2 termsduetoerrorsover multi- branch unmerged paths

Table 1. Trellis-Coded MPSK Performance (dB)

4-PSK 8-PSK 16-PSK

...

M28

1 2 3

7.0

5.3

3.0*

3.2** 3.5

5.3

Best Ungerboeck 64-state code (rate 2/3) provides ACG = 5.0 dB. '* Best Ungerboeck 64-state code (rate 2/3) provides CG at 10 = 3.6 dB. *

with equality if the minimum is achieved by the first term. Thus, for A4 = 8 and 16, the bound given by Equation (3) becomes an equality, and for these cases:

ACG = 3 . 0 d B ,

M=8

ACC = 5 . 3 dB,

M = I6

-

15 )

Table I summarizes these results and compares them with the ACG values for the best Ungerboeck codes of the previous section. It also provides Coding Gains (CGs) at BER values of 10- 5. Note the apparent anomaly for M = 8, for which the CG at is 3.2 dB, while the ACG for this code is only 3.0 dB. This can be explained as follows: The BER for this code is upper bounded by:

Table II. Trellis-Coded QASK Performance (dB)

8-ASK + 64-QASK 16-ASK+ 256-QASK

6

"Ungerboeck codes slightly less (see Figure 10)

July 1989 - IEEE Communications Magazine

15

(a) QPSK

Figure 9 compares the best M = 8 and M = 16, 64-state codes as found by Ungerboeck with those generated by the pragmatic encoder of Figure 7. Our conclusion is that, for almost all purposes, the two sets of performance are equivalent, but the implementation of the latter is far simpler and more flexible. One more observation is important regarding MPSK modulation. In any phase modulation system, a stable phase reference is required for coherent demodulation at the receiver. In particular, in a PSK system with M signals, phase ambiguities of the form 2 d M , 4 d M , ..., (M - 1)(2dM) must be resolved. A method was found for resolving all phase ambiguities for our pragmatic approach to coding. This method is such that all integer multiples of 4n/M phase shifts are resolved using a form of differential encodingldecoding, while the remaining ambiguities are resolved by monitoring the growth of the state metrics in the convolutional decoder. The details of this scheme are beyond the scope of this paper.

The Pragmatic Approach for Linear MASK and M2-QASK Modulation (b)8-PSK

(c) 16-PSK Fig. 8. Trellis structure for pragmatic encoderhodulator.

The first term is the BPSK bit error probability, with energy doubled since two bits are sent per symbol, and multiplied by a factor of 112 because only half the input bits are involved in such potential (single branch) decision errors. The second term is negligible for M = 8, as shown in the Appendix and Equation (4). Hence, the CG over QPSK is 3 dB plus the effect of the multiplication factor of 112. At values of Eb/No for which Pb = the effect of the latter is an additional 0.2 dB, while as P, approaches 0, the effect disappears; hence the ACG is only 3.0 dB. On the other hand, we note from Table I that for the corresponding Ungerboeck code for M = 8 and 64-states, ACG = 5.0 dB while at Pb = CG = 3.6 dB, a difference of only 0.4 dB relative to the pragmatic code. Note also, from Table I and Figure 3, that for the M = 16,64-state code, ACG and CG are the same for both the Ungerboeck code and the pragmatic code. The explanation of this equality (see also the comparisons shown in Figure 9) for M = 16, as contrasted to the difference at M = 8, is that the optimal 64-state code at M = 16 is one for which the underlying binary convolutional code uses rate 112 and the pragmatic code performs essentially as well as that selected in Ungerboeck's search [3]. At M = 8, on the other hand, the best underlying 64-state binary convolutional code uses rate 213; thus, each state has four distinct input branches and four distinct output branches with no bundling. By limiting the pragmatic code to using a rate 112 binary convolutional code, sub-optimality is thus introduced.

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July 1989 - IEEE Communications Magazine

The simplest extension of the above technique to linear channels and power amplifiers is to use the classical twodimensional square grid of equally spaced points, and to code each dimension separately. Viewed as one-dimensional modulation, . this produces the same performance as the best Ungerboeck codes for MASK. Surprisingly, even compared to the more elaborate two-dimensional Ungerboeck codes, performance is roughly similar (as is also suggested in [ 131). The approach for MASK is to stretch the circular mapping into a mapping on the real line. The modulator is similar to that of Figure 7, but now the k - 2 uncoded symbols select one of 2k-2 = M / 4 line segments of length A, numbered lexicographically (i.e., from O..O to l...l) from left to right spanning a range of length M / 4 A centered about the origin. The two coded bits select one of 4 points within this segment, equally spaced at distance A/4 with 00,O 1, 1 1, 10 selecting the first, second, third, and fourth points, respectively, from left to right. This results in Mpoints equally spaced at A/4 extending from -(M/2 - 1) A/4 - A/8 to + (M/2 - I ) A/4 + A B . All distances are finally scaled according to the average energy to be transmitted. This mapping is the same as that for MPSK on the unit circle, with a conformal mapping of the circle onto the line segment of length A/4 (it is also that proposed by Clark and Cain [ 141). Hence, the minimum separation between bundled branches is A. For comparison with the uncoded case, use M / 2 = 2k-1 points, equally spaced at distance A'. If the average symbol energies for the coded and uncoded cases are made equal by normalizing both to unity, then, as shown in the Appendix,

M2- I

1 = (A/4?(-

12

)

(M/2)2- 1 = (A')21 1 12

whence

Consequently, based only on the single branch errors, the ACG is upper bounded by:

1 .E-03

required for QASK, one for each dimension, if the two dimensions are coded independently. On the other hand, a single encoder-decoder suffices if the encoder modulator alternates between selecting the I and Q dimensions on each alternate set of k - I input bits.

1 .E-04

Conclusions

1 .E42

1.E-05

1 .E-06

1 .E-07

1 .E-08

5:O

4:O

6:O

6.0

7.0 E&

9.0

11.0

10.0

+QPSK (Uncoded) +QPSK (R=1/2)

-8-PSK

016-PSK

-&16-PSK

-A-

12.0

(W

8-PSK Pragmatic

Ungerboeck Pragmatic

Ungerboeck

modulation).

which approaches 6 dB as Mbecomes large. Note the similarity between Equations (6) and (3) for large M. Including the effect of multi-branch unmerged paths, it is shown in the Appendix that for the K = 7 rate 112 code MS-4- 7, ACG L 1 0 1 0 g , o [ 4 ~ -I M2-1 ~

=

4

(7 1

25.2dn,

Appendix Bounds on Free Euclidean Distance and ACGfor Trellis-Coded Modulation Employing Rate 112 Code

Fig. 9. BER vs. Eb/N, for the pragmatic coder/modulator (PSK

4.5dn,

We have shown that with a very simple signal mapping, it is possible to utilize existing good binary convolutional encoderdecoders to implement trellis-coded modulators with performance gains (in required received energy-to-noise ratios) greater than 3 dB and in some cases approaching 6 dB, for moderately to highly bandlimited channels. The principal advantage of this approach is that a single rate 112 encoder-decoder, with only moderate modifications to handle parallel branch decoding, can provide this performance for a wide variety of trellis codes. Above all, considerable experience in VLSI implementation of this decoder type on a single integrated circuit [ 15171gives confidence that this relatively new set of applications is no more difficult to implement than the widely accepted applications that utilize binary convolutional codes.

M=X

5.4dB. M=16 (4) approach the Same Again for large M, Equations (7) limit. The of 4-ASK, &ASK, and 16-ASK are shown in Figure 10 and compared with the 64-state Ungerboeck code in each case. Performance is almost identical. CGs are summarized in Table 11. It is also shown in the Appendix that, if the free Hamming distance of the rate 112 binary convolutional code df2 12, then Equation (6) is satisfied as an equality and ACG = 6 dB. This, however, requires K 2 9 or 256 states [ 141. From the practical viewpoint of CG at small, but not vanishing, P, (such as the advantage of the higher constraint length and higher free Euclidean distance may be lost because of the large number of unmerged paths at that distance. For efficient use of bandwidth, both dimensions (I and Q) must be coded, each with its own MASK coded modulation, resulting in a coded M*-QASK constellation conveying 2(k - I ) b/symbol. All the above comparisons still hold, although CG is relative to an uncoded constellation conveying the same number of bits, but with one-quarter rather than one-half as many points. One consequence of the larger signal set size is the number of quantization bits to be provided in the branch metrics. This is compounded by the fact that unlike the situation for MPSK codes, the unmerged path at the free distance is a multibranch path and consequently the most probable error event is more sensitive to metric quantization. Also, two decoders are

MPSK Modulation As established by Equation (2), the squared Euclidean distance between nearest neighbors among bundled branches is proportional to sin2 ( 4 d M ) . This establishes the one-branch unmerged Euclidean distance. For the Euclidean distance between multi-branch unmerged paths, we start by noting that for any rate 112 binary convolutional code that, for the given constraint length, achieves maximum free Hamming distance, d , the first and last unmerged branch pairs differ in both symbols. Because of the mapping used, this guarantees that those two branches will each contribute squared Euclidean distance proportional to sin2 (21r/2~)= sin2 (ZdM), for A4 2 8. The remaining branches in the unmerged path therefore differ in dr - 4 binary symbols. At worst, each pair of non-identical b'ranches differ 'In just one symbol. Then there 4 branches each contributing the minimum are at most d Normalized Lqiared Euclidean Distance (NSED) of the mapping, sin2 (XlM). Consequently, the Euclidean distance for unmerged paths of length greater than one branch is lower bounded by:

For the best K = 7, rate 112 code, dr = IO. Combining the single branch minimum Euclidean distance with this lower bound yields: 4I1 2n Free NSED 2 Min Isin2( -1 2 sin2( -) M M

+ 6 sin2( M ) 1

ML8 Taking the ratio of this to the normalized Euclidean distance for uncoded performance sin2 (ZdM),as given in Equation (l), and taking ten times the logarithm to convert to ACG, results in:

July 1989 - IEEE Communications Magazine

17

'b

1.OE-03

I 9 0

I

9

5

.

I

I

I

1

I

I

I

1.OE-04 1.OE-05 1.OE-06 1.OE-07

1.OE-08 1.OE-09 1.OE-10 5

13

9

7

15

17

1

-4- Ungerboeck codes -0- Pragmatic codes Fig. 10. BER vs. Eb/N, for the pragmatic coderhodulator (ASK modulation).

Coded Auerugt?Norniulized Energy =

M28

Similarly, for the uncoded case of M / 2 points spaced A 'apart,

with equality if the minimum is achieved by the first term.

MASK Modulation The mapping for MASK is the same as for MPSK except that it is on the line rather than the circle. Letting the minimum Euclidean distance between equally spaced points be 414, the minimum NSED between bundled branches is A2 while that between multi-branch unmerged paths is:

NSED

2

A 2

2(-12

A

+ (df - 4 ) ( -41 2

1

(A'l2 with equality if the minimum in the numerator is A2. To complete the argument, it is necessary to normalize A and A' to yield equal average energies. For the coded case, = 2k points equally spaced at 414 and hence with there are values -t(A/8 + jA/4), j = 0,...M/2- 1; consequently,

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July 1989 - IEEE Communications Magazine

d -4

M2-4

Min[A2,A2/2+ (d, - 4)A2/161 I

12

Normalizing both energies to unity results in 48 192 A 2 = -, (A')2= M2- I M"4 and conseq~ently,~

ACG

If for the uncoded M / 2 point linear constellation the minimum normalized squared distance is A', it follows, from the previous argument, that

ACG 2 I O l o g l ~ , {

( M I 2)2 - I Uncoded Aueruge Normalized Energy = (A')2 -

2 10 log { (-

M2-1

).Min13,2

+11 4

with equality if 4 5 2 + (df- 4)/4, or dr 2 12. But for the K = 7 (64-state) code, df = IO, for which

M2-4

7

~ 2 - 1

8

A C G 2 IOlog 140-0-1

4.5, 25.2,

5.4,

M=4

M=8 M216

5Except for normalization, this same formula was derived by Clark and Cain [14].

Note the similarity to the corresponding result for MPSK.

References J. B. Cain, G. C. Clark, Jr., and J. M. Geist, 'Punctured Convolutional Codes of Rate In-l)/n and Simplified Maximum Likelihood Decoding," IEEE Trans. Info. Theory, vol. IT-25, pp. 97-100, Jan. 1979. Y. Yasuda, K. Kashusi, and Y. Hirata, 'High-Rate Punctured Convolutional Codes for Soft Decision Viterbi Decoding," IEEE Trans. on Commun., vol. COM-32, pp. 315-319, Mar. 1984. G. Ungerboeck, "Channel Coding with Multilevel/Phase Signals," /FEE Trans. on Info. Theory, vol. IT-28, pp. 55-67, Jan. 1982. J. A. Heller and I. M. Jacobs, "Viterbi Decoding for Satellite and Space Communication," IEEE Trans. on Comm. Tech.,vol. COM- 19, pp. 835848, Oct. 1971. G. D. Forney, Jr., R. G. Gallager, G. R. Lang, F. M. Longstaff, and S. U. Qureshi. "Efficient Modulation for Band-Limited Channels," IEEE Trans. onSelectedAreasinComm.,vol.SAC-2, pp. 632-647. Sept. 1984. A. R. Calderbank and N. J. A. Sloane, "New Trellis Codes Based on Lattices and Cosets," IEEE Trans. on Info. Theory, vol. IT-33, pp. 177195, Mar. 1987. A. R Calderbank and J. E. Mazo, "A New Description of Trellis Codes." /€€E Trans. on Info Theory, vol. IT-30, pp. 784-791, Nov. 1984. L. F. Wei, "Trellis-Coded Modulation with Multidimensional Constellations,"lEEETrans.onlnfo Theory,vol. IT-33, pp. 483-501, July 1987. A. R. Calderbank and N. J. A. Sloane. " A n Eight-Dimensional Trellis Code," IEEE Proceedings, vol. 74, pp. 757-759, May 1986. S. G. Wilson, H. A. Sleeper, P. J. Schottler, and M. T. Lyons, "Rate 3 / 4 Convolutional Coding of 16-PSK: Code Design and Performance Study," IEEE Trans. on Commun.. vol. COM-32. Dec. 1984. E. Zehavi and J. K. Wolf, "On the Error Performance of Trellis Codes," IEEE Trans. on Info. Theory, vol. IT-33, pp. 196-202, Mar. 1987. A. J Viterbi and J. K. Omura, Principles of Digital Communicationand Coding, NY: McGraw-Hill, 1979. G. Ungerboeck, "Trellis-Coded Modulation with Redundant Signal Sets," Pts. I and II, IEEECommunicationMag., vol. 25, pp. 5-21, Feb. 1987. G. C. Clark, Jr. and J. B. Cain, Error-Correction Coding for Digital Communications, Plenum-Press, 1981 "QUALCOMM Announces Single-Chip K = 7 Viterbi Decoder Device," IEEECommunication Mag., vol. 25, pp. 75-78, Apr. 1987. T. lshitani et al. "A Scarce-State-Transition Viterbi-Decoder for Bit Error Correction," IEEE Journal of Solid-state Circuits, vol. SC-22, pp 575-581, Aug. 1987. H Suzuki, M. Tajima, and M. Shinaja, 'Viterbi Decoder Chip Implemented with Sub-Parallel Architecture," IEEElntl. Symposium on lnfo. Theory. Kobe Japan, June 1988. J. Hagenauer and C. E. Sundberg. "On the Performance Evaluation of Trellis-Coded 8-PSK Systems with Carrier Phase Offset," Archiv Fue, Elektronik Und Uebertragungstecknick.Band 42, Heft 5, pp. 274-284, 1988

Biography Andrew J Viterbi, on July 1, 1985, became a founder and Vice Chairman and Chief Technical Officer of QUALCOMM. Inc , a company concentrating on mobile satellite communications for both commercial and military applications In 1968, he co-founded LlNKABlT Corp , which by 1980 had grown to become a sizable company The company grew at the same rate after being acquired by M/A-COM Inc , in 1980 Dr Viterbi was Executive Vice President of LlNKABlT from 1974 to 1982 In 1982, Dr Viterbi took over as President of M/A-COM LINKABIT, Inc From 1984 to 1985, he was appointed Chief Scientist and Senior Vice President of M/A-COM, Inc As a professor in the UCLA School of Engineering and Applied Science, from 1963 t o 1973, he did fundamental work in digital communication theory

and wrote t w o books on the subject. He has also received three major awards: the 1975 Christopher Columbus International Award (from the Italian National Research Council, sponsored by the City of Genoa); the 1980 Aerospace Communications Award, jointly with Dr. Irwin Jacobs (from AIAA); and the 1984 Alexander Graham Bell Medal (from IEEE, sponsored by AT&T) "for exceptional contributions t o the advancement of telecommunications." The practical development of these theoretical principles led t o the founding of LlNKABlT Corporation, together with Dr. Irwin Jacobs. In his first employment after graduating from MIT in 1957, he was a member of the project team at CIT Jet Propulsion Laboratory that designed and implemented the telemetry equipment on the first successful U.S. satellite, Explorer 1. In the early sixties, at the same laboratory, he was one of the first communication engineers t o recognize the potential of, and propose the digital transmission techniques for space and satellite telecommunication systems. Jack K. Wolf is a Chaired Professor in the Center for Magnetic Recording Research at the University of California at San Diego, La Jolla, CA. He received his B.S.E.E. degree from the University of Pennsylvania in 1956, and the M.S.E., M.A., and Ph.D. degrees from Princeton in 1957, 1958, and 1960, respectively. He was a member of the Electrical Engineering Department at New York University from 1963 t o 1965, and the Polytechnic Institute of Brooklyn from 1965 to 1973. He was Chairman of the Department of Electrical and Computer Engineering at the University of Massachusetts from 1973 to 1975, and was Professor there from 1973 t o 1984. Since 1985 he has been a Professor of Electrical Engineering and Computer Sciences and a member of the Center for Magnetic Recording Research at the University of California at San Diego. He is also a consultant t o Qualcomm, Inc., San Diego, California. During the 1968-69 academic year, he was a member of the Mathematics Research Center at Bell Laboratories. From 1971 to 1972, he was an NSF Senior Postdoctoral Fellow at the University of Hawaii. From 1979 to 1980, he was a Guggenheim Fellow at the University of California at San Diego and the Linkabit Corporation. His research interests are in information theory, coding theory, communications systems, and computer networks. Dr. Wolf is a Fellow of the IEEE. He was corecipient of the 1975 IEEE Information Theory Group Paper Award for the paper "Noiseless Coding for Correlated Information Sources' (coauthored with D. Slepian). He was Co-chairman of the 1969 IEEE International Symposium on Information Theory. He served on the Board of Governors of the IEEE Information Theory Group from 1970 to 1976 and from 1980 to 1986. Dr. Wolf was President of the IEEE Information Theory Group in 1974. He was International Chairman of Committee C of URSl from 1980 to 1983. Ephraim Zehavi received a B.S. degree in 1977 from the Technion-Israel Institute of Technology in Haifa Israel. He received an M.Sc. degree from Technton-Israel Institute in 1981 and a Ph.D. from the Department of Electrical and Computer Engineering, University of Massachusetts. Dr. Zehavi is currently a Lecturer in the Department of Electrical Engineering at the Technion-Israel Institute of Technology in Haifa, Israel. From 1985 to 1987 he was an engineer at QUALCOMM, Inc. He was a Research Assistant in the Department of Electrical and Computer Engineering at the University of Massachusetts from 1983 to 1985. Mr. Zehavi was a research and design engineer from 1977 t o 1983 at the Department of Communication, RAFAEL, Armament Development Authority, Haifa, Israel. Roberto Padovani is a Staff Engineer with Qualcomm, Inc., San Diego, CA. He received the Laurea degree from the University of Padova, Italy, and the M.S. and Ph.D. degrees from the University of Massachusetts in 1978, 1983, and 1985, respectively. Dr. Padovani was a research assistant in the Department of Electrical and Computer Engineering at the University of Massachusetts from 1982 to 1984. From 1984 to 1986 he was employed by M / A COM Linkabit where he was involved in the design and development of satellite communication systems and secure video systems. As a member of the Engineering Department of Qualcomm, Inc., Dr. Padovani has been involved in the development of a new CDMA modem for the mobile satellite channel, various satellite modems, and in the design of new-generation Viterbi decoders.

July 1989 - IEEE Communications Magazine

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