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

ISBN 0471-49581-6

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Several signal sets used in digital communication systems are shown in Figure 97. The I and Q axis represent orthogonal signals that are used in transmission.1 In digital communication systems, generally

I = cos(27r/cf), Q = sin(27r/cf), (9.1)

where / stands for in-phase and Q for quadrature. If a point in the IQ-plane has coordinates (x, y), then the transmitted signal is

s(t) = Rcos(2wfct + <f>), (k — 1 )T <t< kT, (9.2)

where R = y/x2 + y2, 4> = tan~1(y/x), and T is the symbol duration. In other systems

(e.g., storage), orthogonal pulses may be used, such as those illustrated in Figure 98. The set of signal symbols in the two-dimensional IQ-plane is called a constellation, and symbols are called signal points.

From the viewpoint of digital signal processing, modulation is a mapping. That is, the process of assigning an ^-dimensional binary vector b to a signal point (x(b),y(b)) in the

1 For simplicity of exposition, let I = /(i) and Q = Q(t).

COMBINING CODES AND DIGITAL MODULATION

171

1 bit/symbol

.Q

2 bits/symbol

*Q

3 bits/symbol

*Q

BPSK

QPSK

8PSK

4 bits/symbol

*Q

5 bits/symbol

iQ

6 bits/symbol

lQ

16-QAM 32-QAM 64-QAM

Figure 97 Examples of M-PSK and M-QAM signal constellations.

constellation. In previous chapters, only BPSK was considered, in which case v = 1. For v > 1, there are many possible assignments of bits to signal points. That is, many ways to label the signal points. Figure 99 shows an example of a QPSK constellation (v = 2) with two different (non-equivalent) mappings.

Moving from binary modulation to 2*'-ary modulation has the advantage that the number of bits per symbol is increased by a factor of v, thus increasing the spectral efficiency of the system. On the other hand, the required average energy of the signal increases (QAM), or the distance between modulation symbols decreases (PSK). In practice, transmitted power is limited to a maximum value. This implies that the signal points become closer to each other. Recall that the probability of error in an AWGN channel between two signal points separated by an Euclidean distance equal to De is [Hay, Pro]

where Q(x) is given by (1.2). As a result, a higher probability of error is experienced at the receiver end. In this sense, the function of error correcting coding is to reduce the probability of error Pr(e) and to improve the quality of the system.

172

THE ART OF ERROR CORRECTING CODING

1= Sj(t)

0 Ts '

Figure 98 An example of two orthogonal pulses.

10 ' ‘ 00

• •

• •

11 01

11 . 00

• •

• •

10 01

(a) GRAY MAP (b) NATURAL MAP

Figure 99 Two different mappings of a QPSK constellation.

9.1.2 Coded modulation

In 1974, Massey introduced the key concept of treating coding and modulation as a joint signal processing entity [Mas3], see Figure 100. That is, the coordinated design of error correcting coding and modulation schemes.

(Massey, 1974) Combine coding and modulation

Figure 100 The idea of joint coding and modulation.

Two fundamental questions on combining coding and modulation arise:

1. How to construct the bits-to-symbols mapping?

2. How to assign coded bit sequences to coded symbol sequences?

COMBINING CODES AND DIGITAL MODULATION

173

Two basic approaches were proposed in the 1970s to design coded modulation systems:

1. Trellis Coded Modulation (TCM) [Ungl]

Apply a natural mapping of bits to signals, through set partitioning. Given an underlying finite-state machine, assign symbol sequences to trellis paths. Perform Viterbi decoding at the receiver.

2. Multilevel Coded Modulation (MCM) [IH]

Apply a mapping of codewords to bit positions, through a binary partition. For 2"-ary modulation, use v error correcting codes, one per label bit. Perform multistage decoding at the receiver.

In both TCM and MCM, the basic idea is to expand the constellation in order to obtain the redundancy needed for error correcting coding, and then to use coding to increase the minimum Euclidean distance between sequences of modulated signals.

9.1.3 Distance considerations

To illustrate how error correcting codes and digital modulation can be combined, and the consequent increase in the minimum distance between signal sequences from an expanded signal constellation, consider the following.

Example 95 A block coded QPSK modulation scheme is shown in Figure 101. Codewords of the extended Hamming (8,4,4) code are divided in four pairs of symbols and mapped to QPSK signal points with Gray mapping. A nice feature of Gray labeling of QPSK points is that the squared Euclidean distance between points is equal to twice the Hamming distance between their labels. As a result, a block coded modulation scheme with p = 1 bits/symbol and a minimum squared Euclidean distance, or MSED, Dr2rlin = 8 is obtained. An uncoded system with the same spectral efficiency is BPSK, which has an MSED D\nc = 4. Consequently, the asymptotic coding gain of this scheme is

Over an AWGN channel, this coded QPSK modulation requires half the power of uncoded BPSK modulation, to achieve the same probability of error P(e). Figure 102 shows simulation results of this scheme.

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