# The art of error correcting coding - Moreloz R.H.

ISBN 0471-49581-6

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Multistage decoding

In the first and second decoding stages, the decision variable is just the projection of the received signal sequence onto the X or Y axis, respectively. Figure 121 (c) shows a block

COMBINING CODES AND DIGITAL MODULATION

(8,1,8)

?'O O O O O O oy

(8,7,2)

186

THE ART OF ERROR CORRECTING CODING

2

3 4i . 1 • °

• •

5 '---4 k"'7

Figure 116 Overall trellis of an example MCM with 8-PSK modulation.

diagram of a multistage decoder for a three-level coded 8-PSK modulation with block partitioning. The decoders for the first and second stages operate independently on the inphase and quadrature component of the received signal sequences, fx and fy, respectively. Once decisions are made as to the estimates of the corresponding codewords, vi and $2, they are passed on to the third decoding stage.

Let ii = (v, \, vt2, ? ? ? ,Vin) ˆ Ci be the decoded codeword at the i-th stage, i = 1,2. Before the third-stage decoding, each two-dimensional coordinate (rXj,ryj) of the received signal r = (rx,fy) is projected onto a one dimensional coordinate r'XJ, 1 < j < n. The values r'XJ are the decision variables used by the decoder of C3. The projection depends on the decoded quadrant, which is indexed by the pair (vij,V2j), 1 < j < n, as shown in the table below.

Vij i>2j vr ? 1 xi

0 0 “n/2/2 (j*xi ^yi)

0 1 ->/2/2 {rxi -\~ryi)

1 1 y/2,/2 (vxi ^yi)

1 0 a/2/2 (txi H- tyi)

This is a scaled rotation of f by n/A, so that the rotated sequence f' = (r'rA. r',2, • • •, r'xn) can be decoded using a soft-decision procedure for component code C3. Note that, unlike Ungerboeck partitioning, the independence between the first and second levels in block partitioning results in no error propagation from the first decoding stage to the second.

For i = 1,2, - • •, 1/, let A$ denote the number of codewords in Ci of weight w. Assuming systematic encoding, a union bound on the bit error probability of the first decoding stage can be written as [MFLI, IFMLI]

where dp(i) = ^ [iA, + (w — ?') Ao]2. The probability of a bit error in the second decoding

COMBINING CODES AND DIGITAL MODULATION

187

2 -ary modulation

(a) Multilevel Encoder

(b) Multistage Decoder

Figure 117 Basic structures of an encoder and a decoder of multi-level coded modulation

systems.

stage upper is also bounded by (9.9) using the same arguments above.

The bound (9.9) can be compared with a similar one for the Ungerboeck’s partitioning (UG) strategy:

Hia) < E

l2REb

N0

wA\

(9.10)

From (9.9) and (9.10), it is observed that, while Ungerboeck’s partitioning increases exponentially the effect of nearest neighbor sequences, by a factor of 2W, the block partitioning has for dp(w) = wAj an error coefficient term, 2~w, that decreases exponentially with the distances of the first-level component code. As a result, for practical values of Eb/No, the block partitioning may yield, at the first stage, a real coding gain even greater than the asymptotic coding gain. This is a very desirable feature of a coded modulation with UEP.

For nonstandard partitioning (NS), the second level is generally designed to have a larger coding gain than the third level. Under this assumption, a good approximation is obtained by assuming that decoding decisions in the first and the second decoding stages are correct,

iE' ~

w=d3

12REb N0

wA\

(9.11)

188

THE ART OF ERROR CORRECTING CODING

(8,1,8)

?o O O O O O C)

b, =0:

b, = 1

Figure 118 Trellis and symbols used in metric computations in the first decoding stage.

bjb2=00: .

b,b2=01: .

b,b2=

10:

b, b,= 11:

1 Z I

OR ROTATE THE RECEIVED SIGNAL POINT

X blb2=

: 00 (0 deg)

bjb2=01 (-45 deg)

x 'x bjb2= 1 : (-90 deg) b,b2= 11 (-135 deg)

Figure 119 Trellis and symbols used in metric computations in the third decoding stage.

Example 102 Consider a three-level 8-PSK modulation for UEP with extended BCH (64,18, 22), (64,45,8) and (64,63, 2) codes as the first-, second- and third-level codes, respectively. This coding scheme has rate equal to 1.97 bits per symbol and can be compared with uncoded QPSK modulation, which has approximately the same rate (a difference of only 0.06 dB). Simulation results are shown in Figure 122. SI(n,k) and UB(n, k) denote simulations and upper bounds. An large coding gain of 8.5 dB is achieved at the BER of 10-5 for 18 most important bits (14.3%) encoded in the first level. In the second and third stages, the corresponding values of coding gain are 2.5 dB and -4.0 dB, respectively5.

5 Note that at this BER, the simulated coding gain at the first decoding stage is even greater than the asymptotic coding gain (8.1 dB), because of the reduced error coefficients.

COMBINING CODES AND DIGITAL MODULATION 189

Eb/No (dB)

Figure 120 Simulation results of a three-level coded 8-PSK modulation with Ungerboeck

mapping.

9.4 Bit-interleaved coded modulation (BICM)

In [CTB1, CTB2], the ultimate approach to pragmatic coded modulation is presented. The system consists of binary encoding followed by a pseudo-random bit interleaver. The output of the interleaver is grouped in blocks of u bits which are assigned, via a Gray mapping, to points in a 2"-ary modulation constellation. The capacity of this bit-interleaved coded modulation (BICM) scheme has been shown to be surprisingly close to the capacity of TCM, when Gray mapping is employed. Moreover, over flat Rayleigh fading channels, BICM outperforms a CM with symbol interleaving [CTB2], A block diagram of a BICM system is shown in Figure 123.

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