Introduction

Intelsat has adopted a new standard, which employs trellis-coded modulation (TCM), using 8-PSK, with mandatory Reed-Solomon (R-S) (219, 201) outer coding over GF(2⁸). This standard has twice the bandwidth efficiency (at almost 2 bps/ Hz) than the IESS-308 standard it replaces (at almost 1 bps/Hz). The standard states, "Since 8-PSK TCM uses practically the same satellite power and is twice as bandwidth efficient, its usage will permit more efficient use of orbital spectrum". This standard, reviewed in the Background section, does not use the optimum 64-state Ungerboeck TCM for the chosen 8-PSK modulation. Instead the standard chooses a "pragmatic" TCM (PTCM) scheme, based on the methodology of Viterbi and two patents. The use of PTCM is justified in Viterbi as follows:

1. There exists a widely used "industry standard" constraint-length 7 (64-state), rate ½ convolutional code that is optimum for BPSK and QPSK.
2. While the use of this convolutional code in PTCM results in a 2dB clear sky loss relative to the optimum 64-state, rate 2/3 Ungerboeck TCM at very low BERs, there is only a 0.4dB loss at a BER of 10^-5. The mandatory R-S outer code further reduces this BER to an acceptable level.

One patent title reflects the chief benefit of PTCM for 8-PSK: reduction of the trace-back memory and computation complexity (per decoded bit) associated with the PTCM decoder relative to a decoder for the optimum 64-state Ungerboeck TCM decoder. The patent also describes a metric-setting method, reviewed in the Branch-Metric Computation section, that requires a conversion from in-phase and quadrature data to phase (this requires a divide, table look-up, and other four-quadrant logic to be provided external to the "industry-standard" Viterbi decoder). The Efficient Branch-Metric Calculations section describes a simple metric-setting procedure, suitable for DSP software implementation, yielding the desired performance using only multiplies and saturation logic.

The second patent describes the phase-ambiguity-resolution circuit you need if the PTCM scheme is used by itself (in other words, without Reed-Solomon outer coding). The use of this circuit effects the branch-metric computation at low-to-moderate values of E_b/N₀. You can minimize the multiplication of errors caused by the ambiguity-resolution circuit by erasing (in other words, setting to 0) some branch metrics when the received signal is close to intermediate significant-bit (ISB) transitions.

This article continues with a review of two proposals for PTCM schemes at 2.5 bps/Hz using 8-PSK. Also in this section is the performance of simple branch-metric computations for the more promising scheme. We conclude with a description of a procedure that uses the R-S outer code to resolve phase ambiguity.

Background

Figure 1 shows the PTCM phase-ambiguity-resolving encoder proposed in the two patents.

Figure 1:  Phase Ambiguity Encoder for rate 2/3 PTCM using 8-PSK

The 8-ary symbol formed by ENCC[2:0] is mapped to the 8-PSK constellation (Figure 2).

Figure 2:  Symbol mapping to 8-PSK constellation

Figure 3 shows a block diagram of the PTCM decoder. We are principally concerned with a description and simplification of the first module in Figure 3 (the computation of the branch metrics to the Viterbi Decoder). The decoder's performance enhancement with respect to uncoded QPSK may be intuitively understood using the following description. The distance between closest signal pairs for uncoded BPSK is a factor of √2 times that of uncoded QPSK with equal signal power. However, the latter's signal power may be increased by 3dB while maintaining the same energy per bit as the former; thus, uncoded BPSK and QPSK have identical performance. Assume the Viterbi decoding/ re-encoding in Figure 3 produces 2 least-significant bits (of the 3 bits) that are error-free. Therefore, the distance between antipodal signals is greater (by a factor of √2) than the minimum distance for uncoded QPSK; thus, the asymptotic coding gain (ACG) is 3dB (when compared to uncoded QPSK, also 2bps/Hz).

Figure 3:  PTCM decoder block diagram

For more exact estimates of coding gain at various E_b/N₀'s (neglecting-error-multiplication by the phase ambiguity resolution decoder), it is typical to compare the performance of a rate (N-1)/N PTCM scheme using M=2^N signals with the performance of an equivalent bandwidth uncoded system using M/2 signals. For uncoded operation, the BER, P_bu is bounded by:

while the coded BER, P_bc, with an M-signal constellation, is lower bounded by:

where K≤1. For rate 2/3 PTCM using 8-PSK, using a standard 64-state convolutional code, this reduces to:

The following description will help to explain these results. The minimum distance path is only one branch long, due to the two parallel transitions from a state, X, at stage n to stage n+1 (Figure 4). The single-branch error probability is merely the BPSK bit-error probability, with energy doubled since two bits are sent per symbol, and multiplied by a factor of 0.5 because only one out of two input bits is involved in such single-branch decision errors. This is a lower bound, since errors from multi-branch paths must also be considered. At higher E_b/N₀ for the standard 64-state code, the multi-branch errors may be neglected (in comparison to single-branch errors).

Figure 4:  Trellis for PTCM

To achieve this performance, the branch metrics must be set according to the Euclidean distance between the received signal and the four closest transmitted points in the signal constellation.

For gray-coded QPSK used in IESS-308, with the constellation points being e^jkπ/2, k=0,1,3 and 2, four (soft-decision) branch metrics computed from the incoming in-phase and quadrature matched filter outputs, I and Q, are simply I, Q, -I, and -Q corresponding to symbols 00, 01, 11, and 10. These matched filter values, when negated, may be considered relative squared Euclidean distances.

Figure 5:  Branch metrics calculated using the Euclidean distances to four nearest neighbors

For IESS-310, without loss of generality, we consider received in-phase and quadrature components as shown in Figure 5 and the squared Euclidean distances between the received signal and the four closest transmitted points in the 8-PSK constellation. The squared distances with respect to constellation points on a circle of radius R are:

R (which, assuming no fading, is constant) may be estimated using an automatic gain control circuit (AGC). If the effects of varying P (=I²+Q²+R²) are ignored, the branch metrics may be taken to be the last terms in the right-hand side of the equations, the correlation of the received signal with the closest four transmitted signals. In order to avoid determining which four signals (of the eight) are closest to the received signal, you can use the absolute values of the correlation of the received signal with only those vectors depicted in Figure 4. You can compute the four correlations using two additions, two multiplications by a constant, and four comparisons.

There are two difficulties with directly using the previously computed correlations in a Viterbi decoder:

1. When P>>2R2 or P<<2R2 (due to noise), the correlations (the relative distances) are given undue weight in the Viterbi decoder
2. ISB decoding (ISB hard-decision boundaries are shown by the dashed lines in Figure 2) errors cause error multiplication in the phase-ambiguity resolution-circuit.

To resolve the first difficulty, there are upper and lower limits on the correlations. To resolve the second difficulty, erasures (0s) take the place of correlations with respect to the furthest two (of the four closest) transmit signals when the received signal is close to these ISB decision boundaries.

In current practice, the I and Q matched-filter outputs are first converted to an angle (using a division and a four-quadrant arctangent table look-up) and then the four metrics are set according to a table. Note that these metrics have two periods in (0,2π), due to taking the absolute values of the correlations).

Motivated by the periodicity of the correlations described above, more efficient metric calculations are:

These surrogate squared distances may be computed using three multiplies. These are then limited symmetrically with respect to 0 (this involves an additional eight comparisons and, in the worst case, four substitutions). You can also choose the limit value such that the relative distances are, for all practical purposes, the same as the relative distances previously shown (after limiting). For simplicity, modification of metrics for received signals close to ISB transitions are omitted.

The performance of this metric setting procedure, using 11-bit quantized I and Q values, a trace-back memory of 38 states, and empirically optimized metric saturation, are shown in Figure 6. The performance approaches the theoretical lower bound at high E_b/N₀ and is comparable to a commercially available PTCM decoder. Details of the Viterbi decoder used for this implementation are provided in Singh and Jayasimha.

Figure 6:  Rate 2/3, 8-PSK PTCM performance using simplified branch metrics (red line) compared to uncoded QPSK (blue line). The green line is the bound given in the Background section.

A rate 5/6 code for 8-PSK, using a "industry-standard" rate 1/2, 64-state convolutional encoder punctured to rate 3/4, is described in Wolf and Zehavi and is shown in Figure 7. The normalized square Euclidean distance for this code is 1.465 (the punctured code has a free Hamming distance of 5). Thus, though this PTCM provides 2.5 bps/Hz as compared to 2 bps/Hz for uncoded QPSK, it still provides an ACG of 1.66dB.

Figure 7:  Rate 5/6 PTCM using a standard rate 1/2 code punctured to rate 3/4. The odd and even sets of tri-bits are mapped to 8-PSK constellation according to Figure 2

You may use a phase-ambiguity-resolution circuit, based on similar ideas to those shown in Figure 2. However, McCallister et al point out that, when mandatory R-S outer coding is used, it may be used to resolve phase ambiguity. This avoids error multiplication due to the phase-ambiguity-resolution circuit. The PTCM scheme in McCallister et al does not use a punctured 64-state code; instead, the pair of bits produced by the convolutional code are time interleaved on odd and even baud rates (Figure 8).

Figure 8:  Rate 5/6 PTCM using an unpunctured rate code. The odd and even sets of tri-bits are mapped to 8-PSK constellation in lexicographic order (rather than the gray-coded order of Figure 2)

For this rate, 5/6 PTCM using 8-PSK, using a 64-state convolutional code, the BER at high SNRs is:

The minimum-distance path is only one branch long, because of the two parallel transitions from a state, X, at stage n to stage n+1 (Figure 4). The single-branch error probability is merely the QPSK bit-error probability, with energy multiplied by 1.25 since 2.5 bits are sent per symbol as compared to 2 bits in QPSK, and multiplied by a factor of 0.8 because four of five input bits are involved in such single-branch decision errors. At higher E_b/N₀, for the 64-state code, the multi-branch errors may be neglected (in comparison to single-branch errors). Thus, the ACG is 10log₁₀(1.25)=0.97dB, which is worse than the TCM obtained using a rate 1/2 code punctured to rate 3/4 by 0.69dB.

However, McCallister et al states that ACG is not the sole criterion used in selecting a TCM scheme; instead, the coding gain at the operating range of BERs should be considered. As the BER of the scheme in Wolf and Zehavi is ultimately limited by the rate 3/4 punctured code, it exhibits a sharper "knee" than the scheme of McCallister et al. Thus, in a range of BERs (typically between 10^-3 and 10^-5 where R-S outer coding further reduces BERs to make them acceptable), the scheme in McCallister et al performs better than the scheme of Wolf and Zehavi. Figure 9 shows the performance of the scheme of McCallister et al.

Figure 9:  Rate 5/6, 8-PSK PTCM performance compared to uncoded QPSK. The approximate performance derived in this section is shown as a green line. The circles show the performance with a (225,205) Reed-Solomon outer code over GF(2⁸)

As suggested in McCallister et al, phase ambiguity (as well as symbol-pair synchronization in the PTCM of Figure 8) may be resolved using the R-S outer codes with periodically inserted unique words (UWs) (avoiding error multiplication in phase ambiguity resolution circuits). For example, IESS prescribed mandatory R-S (219, 201) outer coding over GF(2⁸) with periodically inserted unique words but, curiously, includes a testing requirement as follows:

"Due to the steepness of the BER versus E_b/N₀ response curve when using the Reed-Solomon outer coding, an inordinately long period of time is necessary to detect a sufficient number of errors to determine the BER performance with a reasonable degree of confidence at even moderate E_b/N₀ values. Assuming that the Reed-Solomon outer codec is functioning, determining the BER performance of the TCM codec without Reed-Solomon outer-coding would enable users to quickly determine whether or not the modem is functioning correctly".

Evidently, any scheme that uses the R-S synchronization pattern to resolve phase ambiguities cannot cater to testing without R-S outer coding. Furthermore, the scheme of McCallister et al avoids the error multiplication caused by the ambiguity-resolution circuit in Wolf. As seen in the Efficient Branch-Metric Calculations section, setting of branch metrics in the scheme of Viterbi due to an ambiguity-resolution circuit is also made more complex. However, the absence of a phase-ambiguity-resolution circuit may allow the inner code, for some repetitive data patterns, to indicate node synchronization, but the outer code to fail to synchronize. The following procedure ensures synchronization with all data patterns, assuming that inner and concatenated codes are not synchronized and timer=0 initially:

if (inner code synchronized)
if (outer code errors in s-bit UW< r)
concatenated code is synchronized
else {
set inner code is not synchronized;
increment inner code phase reference by 2π/M (mod 2π)
if (phase==0) change symbol pair alignment;
}
else
if (timer++==timeout) {
timer=0;
increment inner code phase reference by 2π/M (mod 2π)
if (phase==0) change symbol pair alignment;
}

The inner code usually correlates the re-encoded decoded sequence and the (suitably delayed) hard-decision decoded received symbols in order to determine phase synchronization. The expected outer-code synchronization time using this method, calculated using the methods described in Jayasimha and Kumar, is not significantly different than the outer-code synchronization time when the inner code incorporates a phase-ambiguity-resolution circuit such as the one in Wolf.

The performance of a simplified metric-setting procedure for a 2 and 2.5 bps/Hz PTCM decoder, suitable for the trade-off criteria used in selecting a TCM scheme, are:

• E_b/N₀ at operating BER versus decoder memory/complexity
• Performance loss associated with phase-ambiguity-resolution methods versus synchronization time
• bps/Hz versus sensitivity to phase error/spectral re-growth DSP implementation, is described and shown to be comparable to that provided by a commercially available PTCM decoder chip. We also extend this procedure to one 2.5 bps/Hz PTCM proposal. A method for phase ambiguity resolution and/or symbol set alignment using the R-S outer-code unique words, that is data pattern insensitive, is also described.