Turbo Trellis Coded Modulation with Iterative Decoding for Mobile Satellite Communications

Transcription

1 Turbo Trellis Coded Modulation ith Iterative Decoding for Mobile atellite Communications D. Divsalar and F. ollara Jet ropulsion Laboratory, California Institute of Technology 400 Oak Grove Dr., asadena, CA 909 hone: FAX: Abstract In this paper, analytical bounds on the performance of parallel concatenation of to codes, knon as turbo codes, and serial concatenation of to codes over fading channels are obtained. Based on this analysis, design criteria for the selection of component trellis codes for M modulation, and a suitable bit-by-bit iterative decoding structure are proposed. Examples are given for throughput of bits/sec/hz ith modulation. The parallel concatenation example uses to rate 4/5 -state convolutional codes ith to interleavers. The convolutional codes outputs are then mapped to to modulations. The serial concatenated code example uses an -state outer code ith rate 4/5 and a 4-state inner trellis code ith 5 inputs and outputs per trellis branch. Based on the above mentioned design criteria for fading channels, a method to obtain the structure of the trellis code ith maximum diversity is proposed. imulation results are given for AWG, and an independent Rayleigh fading channel ith perfect Channel tate Information (CI). I. Introduction Trellis coded modulation (TCM) proposed by Ungerboeck in 9 ] is no a ell-established technique in digital communications. ince its first appearance, TCM has generated a continuously groing interest, concerning its theoretical foundations as ell as its numerous applications, spanning high-rate digital transmission over voice circuits, digital microave radio relay links, and satellite communications. In essence, it is a technique to obtain significant coding gains (3-6 db) sacrificing neither data rate nor bandidth. Turbo codes represent a more recent development in the coding research field ], hich has raised large interest in the coding community. They are parallel concatenated convolutional codes (CCC) hose encoder is formed by to (or more) constituent systematic encoders joined through one or more interleavers. The input information bits feed the first encoder and, after having been scrambled by the interleaver, they enter the second encoder. A codeord of a parallel concatenated code consists of the input bits to the first encoder folloed by the parity check bits of both encoders. Analytical performance bounds for CCC ith uniform interleaver and maximum likelihood receiver ere obtained in 3], and 6] for AWG channel, and in 6] for Rayleigh fading channel ith binary modulation. The (suboptimal) iterative decoding structure 5] is modular, and consists of a set of concatenated decoding modules, one for each constituent code, connected through the same The research described in this paper as carried out at the Jet ropulsion Laboratory, California Institute of Technology, under contract ith the ational Aeronautics and pace Administration. interleavers used at the encoder side. Each decoder performs eighted soft decoding of the input sequence. Bit error probabilities as lo as 0 6 at E b / 0 = 0.6dB have been shon by simulation ] using codes ith rates as lo as /5. arallel concatenated convolutional codes yield very large coding gains (0- db) at the expense of a data rate reduction, or bandidth increase. In 4] e merged TCM and CCC in order to obtain large coding gains and high bandidth efficiency. In 4] and 3] e suggested merging TCM ith the recently discovered serial concatenated convolutional codes (CCC) ], adapting the concept of iterative decoding used in parallel concatenated codes. We refer to the concatenation of an outer convolutional code ith an inner TCM as serial concatenated TCM (CTCM). For parallel concatenated trellis coded modulation (CTCM), also addressed as turbo TCM, a first attempt employing the so-called pragmatic approach to TCM as described in 5]. Later, turbo codes ere embedded in multilevel codes ith multistage decoding 7]. Recently ], punctured versions of Ungerboeck codes ere used to construct turbo codes for - modulation. In 4] a different approach to construct CTCM as proposed. Results in 4] sho that the performance of the proposed codes is ithin db from the hannon limit at bit error probabilities of 0 7 over AWG channels. In this paper e used turbo trellis coded modulation and serial trellis coded modulation as discussed above, over fading channels for mobile satellite communications. For fading channels, e assume Rayleigh fading. Rician fading is actually a better model for mobile satellite communications since there is LO (line-of-sight), but hen an omni-directional antenna is used and LO is blocked by trees, poles, or buildings Rayleigh fading can be used as a orst-case scenario. II. Analytical Bounds on the erformance of codes over Channels Consider an (n, k) block code C ith code rate R c = k/n and minimum distance h m. An upper bound on the conditional bit-error probability of the block code C over fading channels, assuming coherent detection, maximum likelihood decoding, and perfect Channel tate Information (CI) can

2 be obtained in the form n k b (e ρ) k AC Q( Rc E b / 0 h=d min = h ρi ) () i= Input data C R c =p/q n To modulator and channel here E b / 0 is the signal-to-noise ratio per bit, and A C for the block code C represents the number of codeords of the block code ith output eight h associated ith an input sequence of eight. A C is the input output eight coefficient (IOWC). The Q function can be represented as 7] Q(x) = π π 0 e x sin θ dθ e x () To obtain the unconditional bit error rate, e have to average over the joint density function of fading samples. For simplicity assume independent Rayleigh fading samples. This assumption is valid if e use an interleaver after the encoder and a deinterleaver before the decoder. Thus the fading samples ρ i are independent identically distributed (i.i.d.) random variables ith Rayleigh density of the form f (ρ) = ρe ρ Using () and results in 0], by averaging the conditional bit error rate over fading e obtain π n k k AC sin θ ] h dθ b (e) π 0 h=d min sin = θ R c E b / o (3) We can further upper bound the above result and obtain 0] b (e) n k h=h m = k AC R c E b / o ] h (4) Extension of results to independent Rician fading is straightforard (see for example 0]). All these results apply to convolutional codes as ell, if e construct an equivalent block code from the convolutional code. Obviously results apply also to concatenated codes including parallel and serial concatenations. As soon as e obtain the input output eight coefficients A C for a particular code e can compute the performance. III. arallel Concatenated Convolutional Codes The structure of a parallel concatenated convolutional code (CCC) or turbo code is shon in Fig.. Figure refers to the case of to convolutional codes, code C ith rate Rc = p/q, and code C ith rate Rc = p/q, here the constituent code inputs are joined by an interleaver of length, generating a CCC, C, ith rate R c = R c R c. ote that Rc Rc is an integer multiple of p. The input block length k =, and the output codeord length n = n n as shon in Fig.. interleaver C R c =p/q Fig.. arallel Concatenated Convolutional Codes (CCC). A. Computation of input output eight coefficient (IOWC) for CCC (turbo codes) Uniform Interleaver. A crucial step in the analysis of concatenated codes and in particular CCC consists of replacing the actual interleaver that performs a permutation of the input bits ith an abstract interleaver called a uniform interleaver 3], defined as a probabilistic device that maps a given input ord of eight into all distinct ( ) permutations of it ith equal probability p = / ( ). An example for = 4, = is shon in Fig. 00 Uniform Interleaver n p = /6 00 Fig.. The action of a uniform interleaver of length 4 on sequences of eight Using the concept of uniform interleaver, i.e., averaging the b (e) over all possible interleavers, e can obtain for turbo codes With the knoledge of the A C for code C, and A C for code C, using the concept uniform interleaver, IOWC for CCC can be obtained as follos. The main property of the uniform interleaver is that it transforms an input block of eight at the input of the encoder C into all its distinct ( permutations. As a consequence, each input block of code C of eight, through the action of the uniform interleaver, enters the encoder C generating ( ) input-ords of code C. Thus, the number,h of codeords of the CCC ith output eights h,and h associated ith an input sequence of eight is given by,h = AC here,h is related to = A C ) ( as h,h : h h =h,h )

3 Bit Error Rate To Modulator Example. Consider a rate / CCC formed by to identical 4-state convolutional codes: Code C ith rate /3 and code C ith rate / (this is obtained by not sending the systematic bits of the rate /3 C convolutional code). The inputs of encoders are joined by a uniform interleaver of lengths = 50, 00 and 56. Both codes are systematic and recursive, and are shon in Fig. 3. Using the previously outlined analysis for CCC, e have obtained the bit-error probability bounds shon in Fig. 3. The performance is shon both for AWG and Rayleigh fading channels Capacity of AWG 0. db AW G Capacity of Rayleigh. db E b / o, db =50 =00 =56 6 π 7 = Input block size (bits) =50 =00 =56 Fig. 3. erformance of rate / CCC over AWG and Rayleigh Channels The reason for such a good performance of turbo codes is that the coefficients AC 9 0 decrease ith interleaver size. For large interleavers the maximum component of AC or equivalently AC,h, over all input eights and output eights h and h, is proportional to α M, ith corresponding output eights h (α M ), and h (α M ). If both convolutional codes are recursive (i.e., the output eight due to input eight one is very large) then α M. (This occurs for =.) Any other choice of encoders results in α M 0. When α M is negative e say that e have interleaving gain. The negative value of α M implies that the exponents of in the bit error rate expression are alays negative integers. Thus, for all h = h h, the coefficients of the exponents in h decrease ith, and e alays have an interleaving gain 9]. Define d i, f,ef f as the minimum eight of codeords of a recursive code C i, i =, generated by eight- input sequences. We call it the effective free Hamming distance of a recursive convolutional code. To maximize the interleaving gain, i.e., minimize α M corresponding to output eight h (α M ), and h (α M ) e should maximize the d i, f,ef f, i =,. The sum d, f,ef f d,f,ef f represents the effective free distance of the turbo code 9] ]. Thus, substituting the exponent α M into the expression for bit error rate approximated by keeping only the term of the summation in h, and h corresponding to h = h (α M ), and h = h (α M ), yields ] d,f,ef f d,f,ef f (5) lim b(e) B R c E b 0 here B is a constant independent of. IV. arallel Concatenated Trellis Coded Modulation The basic structure of parallel concatenated trellis coded modulation is shon in Fig. 4. b Data b π π Code Code not sent b not sent b Mapping Mapping ymbol MUX Fig. 4. Block Diagram of the Encoder for arallel Concatenated Trellis Coded Modulation. This structure uses to rate constituent convolutional b codes. The first most significant output bits of each convolutional code are only connected to the shift register of the TCM encoder and are not mapped to the modulation signals. The last b least significant output bits hoever are mapped to the modulation signals. This method requires at least to interleavers. The first interleaver permutes the b least significant input bits. This interleaver is connected to the b most significant bits of the second TCM encoder. The second interleaver permutes the b most significant input bits. This interleaver is then connected to the b least significant bits of the second TCM encoder. A. Design Criteria for CTCM over Rayleigh Channels To extend the asymptotic results e obtained for binary modulation to M-ary Modulation (e.g. M ), let x i represent the sequence of M-ary output (complex) symbols {x i, j } of trellis code i (i =, ). Complex symbols have unit average poer. Let x i represent another sequence of the output symbols {x i, j } for i =,. Then the above asymptotic result should be modified to b (e) B n η x,n x,n E R b c 4 0 n η x,n x,n E R b c 4 0 here, for i =,, η i is the set of all n i ith the smallest cardinality d i, f,ef f such that x i,ni x i,n i. Then d i, f,ef f represents the minimum (M-ary symbol) Hamming distance of b

4 Bit Error Rate trellis code i (i =, ) corresponding to input Hamming distance beteen binary input sequences that produce d i, f,ef f. The d i, f,ef f,i =, is also called the minimum diversity of trellis code i. We note that the asymptotic result on the bit error rate is inversely proportional to the product of the squared Euclidean distances along the error event paths hich result in d i, f,ef f i=,. Therefore the criterion for optimization of the component trellis codes is to maximize the minimum diversity of the code and then maximize the product of the squared Euclidean distances hich result in minimum diversity iteration=5 AWG iteration=5 B. bits/sec/hz CTCM ith for AWG and Channels The code e propose has b =, and employs modulation in connection ith to -state, rate 4/5 constituent codes. The selected code uses reordered mapping: If b, b, b 0 represents a binary label for natural mapping for, here b is the MB and b 0 is the LB, then the reordered mapping is given by b,(b b ), b 0. The effective Euclidean distance of this code is δ f,ef f = 5.7 (unit-norm constellation is assumed), using to interleavers. The structure of this code is shon in Fig. 5, and its BER for AWG and Rayleigh fading channels in Fig. 6. b4 b3 b b π π D D xd 0 D D xd 0 X Fig. 5. arallel Concatenated Trellis Coded Modulation,, bits/sec/hz. V. erially Concatenated Convolutional Codes The structure of a serially concatenated convolutional code (CCC) is shon in Fig. 7. Figure 7 refers to the case of to convolutional codes, the outer code C o ith rate Rc o = q/p, and the inner code C i ith rate Rc i = p/m, joined by an interleaver of length bits, generating an CCC C ith rate R c = k/n. ote that must be an integer multiple of p. The input block size is k = q/p and the output block size of CCC is n = m/p E b / o, db Fig. 6. BER erformance of arallel Concatenated Trellis Coded, bits/sec/hz. OUTER CODE RATE = q/p ITERLEAVER LEGTH = IER CODE RATE = p/m Fig. 7. erial Concatenated Convolutional Codes (CCC). A. Computation of input output eight coefficient (IOWC) for CCC A C Using the concept of uniform interleaver, i.e., averaging b (e) over all possible interleavers, e can obtain for serial concatenated codes. A C is the number of codeords of the CCC ith eight h associated ith an input ord of eight. A similar definition applies to input output eight coefficients (IOWC) of the outer code denoted by A C o,l and to IOWC of the inner code denoted by AC i l,h. With the knoledge of the A C o,l for the outer code, AC i l,h for the inner code, and using the concept of uniform interleaver, the IOWC A C for CCC can be obtained as follos. We recall that a uniform interleaver transforms a codeord of eight l at the output of the outer encoder into all its distinct ( l ) permutations. As a consequence, each codeord of the outer code C o of eight l, through the action of the uniform interleaver, enters the inner encoder generating ( l ) codeords of the inner code C i. Thus, the number A C of codeords of the CCC of eight h associated ith an input ord of eight is given by A C = l=0 A C o,l AC i l,h ( ) l Example. Consider a rate / CCC formed by a 4-state

5 4-Dimensional Modulation Bit Error Rate convolutional code C o ith rate / and an inner -state convolutional code C i ith rate / (this is obtained by not sending the systematic bits of the rate / C i convolutional code). The to codes are joined by a uniform interleaver. Input blocks of length = 50, 00 and 56 ere considered. The outer code is a nonrecursive code, the inner code is systematic and recursive, and the generators are shon in Fig.. Using the previously outlined analysis for CCC, e have obtained the bit-error probability bounds shon in Fig.. The performance as obtained both for AWG and Rayleigh fading channels. Comparing to Fig. 3, the performance of CCC is better than CCC both over AWG and fading channels k=input block size (bits) AW G 3 4 π 5 E b / o, db k=50 k=00 k= k=50 k=00 k=56 Fig.. erformance of rate / CCC over AWG and Rayleigh Channels For large interleavers the maximum component of AC over all input eights, and output eights h is proportional to α M ith corresponding output eights h(α M ).If the inner convolutional code is recursive (i.e., ith feedback) d o f then α M = here d o f is the free (minimum) distance of the outer convolutional code. The value of α M shos that the exponents of are alays negative integers. Thus, for all h, the coefficients of the exponents in h decrease ith, and e alays have an interleaving gain. Define d i f,ef f as the minimum eight of codeords of the inner code generated by eight- input sequences. We obtain a different eight h(α M ) for even and odd values of d o f. For even do f, the eight h(α M) associated to the highest exponent of is given by h(α M ) = do f di f,ef f ubstituting the exponent α M into the expression for bit error rate, approximated by only the term of the summation in h 9 0 corresponding to h = h(α M ), yields lim b(e) B even do f / R c E b 0 here B even is a constant independent of. For d o f odd, the value of h(α M ) is given by here h (3) m h(α M ) = (do f 3)d i f,ef f ] do f di f,ef f (6) h (3) m (7) is the minimum eight of sequences of the inner code generated by a eight-3 input sequence. Thus, substituting the exponent α M into the expression for bit error rate approximated by keeping only the term of the summation in h corresponding to h = h(α M ) yields lim b(e) B odd (do f )/ R c E b 0 here B odd is a constant independent of. ] (do f 3)di f,ef f h (3) m VI. erial Concatenated Trellis Coded Modulation The basic structure of serially concatenated trellis coded modulation is shon in Fig. 9. Data Outer π Inner Code Code b b b Mapping Fig. 9. Block Diagram of the Encoder for erial Concatenated Trellis Coded Modulation. We propose a novel method to design serial concatenated TCM for Rayleigh fading channels, hich achieves b bits/sec/hz, using a rate b/(b ) non-recursive binary convolutional encoder ith maximum free Hamming distance as outer code. We interleave the output of the outer code ith a random permutation. The interleaved data enters a rate (b)/(b) recursive convolutional inner encoder. The b output bits are mapped to to symbols belonging toa b level modulation (four dimensional modulation). In this ay, e are using b information bits for every to modulation symbol intervals, resulting in b bit/sec/hz transmission (hen ideal yquist pulse shaping is used) or, in other ords, b bits per modulation symbol. For the AWG channel the inner code and the mapping are jointly optimized based on maximizing the effective Euclidean distance of the inner TCM. The optimum -state inner trellis code is shon in Fig. 0. The effective Euclidean distance of this code is.76 (for unit norm constellation) and its minimum M-ary Hamming distance is. A. Design Criteria for CTCM over Rayleigh Channels To extend the asymptotic results obtained for binary modulation to to M-ary modulation (e.g., M), criteria simi- ()

6 b 4 b 3 b b b 0 D Fig. 0. Optimum -state inner trellis encoder for CTCM ith Modulation. lar to those discussed for parallel concatenated trellis coded modulation (CTCM) are no applied to serial concatenated trellis coded modulation (CTCM). The interleaving gain is still (do f )/, hoever no the minimum diversity is d o f di f,ef f for even d o f, and (do f 3)di f,ef f h (3) m for odd do f, here d i f,ef f represents the minimum (M-ary symbol) Hamming distance of the inner trellis code corresponding to input Hamming distance beteen binary input sequences to the trellis code that produce d i f,ef f. Therefore the criterion for optimizing the inner trellis code in CTCM is to maximize the minimum diversity of the code and then maximize the product of the squared Euclidean distances hich result in minimum diversity. For odd d o f, first e maximize di f,ef f, then among the codes ith maximum d i f,ef f, e maximize h(3) m, the minimum (M-ary symbol) Hamming distance of the inner trellis code corresponding to input Hamming distance 3 beteen binary input sequences to the trellis code that produce h (3) m. As is seen from the previous results, large d o f produces large interleaving gain and diversity. B. Design Method for Inner TCM The proposed design method is based on the folloing steps:. The ell knon set partitioning techniques for Rayleigh fading channels using multidimensional signal sets are used (see for example 0] and the references therein).. The input labels assignment is based on the codeords of the parity check code (b, b, ) and its set partitioning, to maximize the quantities described in the design criteria subsection. The assignment of codeords of the parity check code to the 4-dimensional signal points is not arbitrary. We ould like someho to relate the Hamming distance beteen input labels to the Euclidean distance beteen corresponding 4-dimensional signal points, under the constraint that the minimum Hamming distance beteen input labels for parallel transitions be equal to. To do so: Assign x x x 0 y y y 0 the b most significant bits of the input label to the first constellation ith b points by retaining only the b most significant bits of the Gray code mapping for the constellation. Use the same assignment for the b least significant bits of the input labels; the middle bit in the input label represents the overall parity check bit. 3. A sufficient condition to have very large output Euclidean and M-ary symbol Hamming distances for input sequences ith Hamming distance, is that all input labels to each state be distinct. 4. Assign pairs of input labels and 4-dimensional signal points to the edges of a trellis diagram based on the design criteria in subsection VI-A. To illustrate the design methodology e developed the folloing examples. C. Examples of the Design Methodology Example : et partitioning of and input labels assignment. Let the eight phases of be denoted by {0,,, 3, 4, 5, 6, 7}. Consider the signal set A 0 = (0, 0), (, 3), (, 6), (3, ), (4, 4), (5, 7), (6, ), (7, 5)]. Each element in the set has to components. The second component is 3 times the first one modulo. Also consider the signal set B 0 = (0, 0), (, 5), (, ), (3, 7), (4, 4), (5, ), (6, 6), (7, 3)]. Each element in the set has to components. The second component is 5 times the first one modulo. For these sets, the Hamming distance beteen elements in each set is, and the minimum of the product of square Euclidean distances is the largest possible. The folloing sets are constructed from A 0 and B 0 as: A = A 0 (0, ), A 4 = A 0 (0, 4), A 6 = A 0 (0, 6), A = B 0 (0, ), A 3 = B 0 (0, 3), A 5 = B 0 (0, 5), A 7 = B 0 (0, 7), here addition is component-ise modulo. Map the first and last bits of input labels to the signals as {00, 00, 0, 0,,, 0, 0} {0,,,3,4,5, 6,7}. The fifth bit for the input label is the parity check bit. Use an even parity check bit for signal sets A 0, A 4, A, A 5 and an odd parity check bit for signal sets A, A 6, A 3, A 7. This completes the input label assignments to signal sets. o the Hamming distance beteen input labels for each set A i i=0,,,...,7, is at least and the corresponding M- ary Hamming distance beteen signal elements in each set is. Consider a 4-state trellis code ith full transition. Assign A 0, A, A 4, A 6, to the first state, and A, A 3, A 5, A 7 to the second state, and permutations of these sets to the third and fourth states. This completes the input label and signal set assignments to the edges of the 4-state trellis. Therefore the minimum Hamming distance of the 4-state trellis code is. At this point to obtain a circuit that generates this trellis e need to use an output label. We used reordered mapping as it as discussed before to obtain the circuit for the encoder. The implementation of the 4-state inner trellis code is

7 Bit Error Rate shon in Fig.. The ROM maps 3 addresses in the range of 0 to 3 to a single output. The 3 binary outputs can be summarized in hex as 3A53ACC iteration=4 -state -state iteration=4 -state b 4 b 3 b b b 0 ROM D D X AWG -state iteration=4 4-state Fig.. 4-state inner trellis encoder for CTCM ith modulation for Rayleigh fading. VII. imulation of erial Concatenated Trellis Coded Modulation ith Iterative Decoding In this section the simulation results for serial concatenated TCM, ith over the Rayleigh fading channel are presented. For CTCM ith, the outer code is a rate 4/5, -state nonrecursive convolutional encoder ith d o f = 3, and the inner code is the 4-state TCM designed for in subsection VI-C. The bit error probability vs. bit signal-tonoise ratio E b / o for various numbers of iterations is shon in Fig.. The performance of the inner -state code is also shon in Fig.. This example demonstrates the poer and bandidth efficiency of CTCM, over a Rayleigh fading channel at lo BERs. References ] G. Ungerboeck, Channel coding ith multilevel phase signaling, IEEE Trans. Inf. Th., vol.it-5, pp.55-67, Jan. 9. ] C. Berrou, A. Glavieux, and. Thitimajshima, ear hannon Limit Error-Correcting Coding: Turbo Codes, roc. 993 IEEE International Conference on Communications, Geneva, itzerland, pp , May ]. Benedetto and G. Montorsi, Unveiling turbo codes: some results on parallel concatenated coding schemes, IEEE Trans. on Inf. Theory, March ]. Benedetto, D. Divsalar, G. Montorsi, and F. ollara, arallel Concatenated Trellis Coded Modulation, proceedings of ICC 96, June ]. LeGoff, A. Glavieux, and C. Berrou, Turbo Codes and High pectral Efficiency Modulation, roceedings of IEEE ICC 94, May -5, 994, e Orleans, LA. 6] D. Divsalar,. Dolinar, F. ollara, and R. J. McEliece, Transfer Function Bounds on the erformance of Turbo Codes, roceedings of IEEE Micom 95, ov. 5-,995, an Diego, CA. 7] L.U. Wachsmann, and J. Huber, oer and Bandidth Efficient Digital Communication Using Turbo Codes in Multilevel Codes, European Transactions on Telecommunications, vol. 6, o. 5, ept./oct. 995, pp E b / o, db Fig.. erformance of erial Concatenated Trellis Coded Modulation, -state outer, -state or 4-state inner, ith, bits/sec/hz ]. Robertson, and T. Woerz, ovel Coded modulation scheme employing turbo codes, Electronics Letters, 3st Aug. 995, Vol. 3, o.. 9]. Benedetto and G. Montorsi, Design of arallel Concatenated Convolutional Codes, IEEE Transactions on Communications, vol. 44, no. 5, pp , May ] D. Divsalar, and M.. imon, Design of Trellis Coded M for Channels: erformance Criteria, and et artitioning for Optimum Code Design, IEEE Trans. Commun., Vol. 36, o. 9, pp , ept. 9. ] D. Divsalar and F. ollara, On the Design of Turbo Codes, The Telecommunications and Data Acquisition rogress Report 4-3, July ept. 995, Jet ropulsion Laboratory, asadena, California, pp. 99-0, ov. 5, 995. ]. Benedetto, D. Divsalar, G. Montorsi, and F. ollara, erial Concatenation of Interleaved Codes: erformance Analysis, Design, and Iterative Decoding, The Telecommunications and Data Acquisition rogress Report 4-6, April June 996, Jet ropulsion Laboratory, asadena, CA, Aug. 5, 996. http : //edms. jpl.nasa.gov/tda/progress report/4 6/6D.pd f 3]. Benedetto, D. Divsalar, G. Montorsi, and F. ollara, erial Concatenated Trellis Coded Modulation ith Iterative Decoding: Design and erformance, ubmitted to IEEE Comm. Theory Mini Conference 97, (Globecom 97). 4]. Benedetto, D. Divsalar, G. Montorsi, and F. ollara, erial Concatenated Trellis Coded Modulation ith Iterative Decoding, IEEE IIT 97, June 9 - July 4, 97, Ulm, Germany. 5]. Benedetto, D. Divsalar, G. Montorsi, and F. ollara, oft-output Decoding Algorithms in Iterative Decoding of Turbo Codes, The Telecommunications and Data Acquisition rogress Report 4-4, Oct. Dec. 995, Jet ropulsion Laboratory, asadena, CA, pp. 63 7, Feb. 5, ] E.. Hall, and. G. Wilson, Design and Analysis of Turbo Codes on Rayleigh Channels, IEEE JAC special issue, submitted ept ] J. Craig, A ne, simple and exact result for calculating error probability for to-dimensional signal constellation, roceedings of IEEE Milcom 9,