- 1. Introduction
- Journal of the Optical Society of America
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Noise is then described entirely in terms of its variance, which is proportional to its energy. Code performance under this idealized assumption is often comparable to that achieved in practice, particularly for bandlimited channels such as voice-grade telephone lines Wilson et al.
The lower the noise energy, the more reliable the channel. Due to channel noise, the demodulated vector i? The conditional probability density function of is, given ak is where p is the dimensionality of the constellation. A hard-decision demodulator estimates ak based solely on Zk, then passes its estimate of ak to the decoder. In other words, the demodulator output is a point in the original constellation. From examination of Eq. TROTT correlated, using a nearest neighbor algorithm at the demodulator destroys information useful to the decoder.
Despite the increased complexity, virtually all modern coding schemes use soft-decision demodulation Forney et al. In the above discussion, we have implicitly assumed perfect carrier phase tracking and synchronization.
That is, both the modulator and demodulator have an absolute time base, and they agree on where each modulation interval starts. Specially designed rotationally invariant codes help solve this problem.
The analysis of rotationally invariant trellis codes using finite group homatons is addressed in Section VI. The Decoder The decoder is usually the most complicated part of a coding scheme, especially when combined with soft decision demodulation. Only the recent availability of inexpensive computing power has made powerful coding schemes, such as trellis coding, commercially feasible.
The structure of the decoder depends strongly on the type of code used, so we will not give a general description here. A trellis code maximum likelihood decoder is described in Section 1V. Sometimes, when the signal constellation and encoder have highly symmetric structures, a balance of hard- and soft-decision demodulation and decoding is used.
The result is slightly suboptimal performance with a large reduction in decoder complexity Wilson et al. Constellations Constellations are designed primarily for energy efficiency and error resistance. Then, the peak energy E, and averuge energy E, of a constellation C are where Pr u is the probability that the encoder selects point a at any time k. Using the common assumption that all points in C are used equally often, where ICI is the number of points in C. Peak energy determines the cost of the final amplifier stage in the transmitter.
In an effort to minimize peak energy, the points in a constellation are clustered around the origin. For purposes of comparison, constellations are always normalized to have equal average energy. Assume that successive encoder outputs are uncorrelated, so that harddecision demodulation is optimal. Assume also that channel noise is low, so that errors are rare. Then, mistaking a signal point for its nearest neighbor will be the most frequent type of decoding error see Eq. In this situation, the BER is minimized when the minimum distance between signal points is as large as possible.
For a given average energy, constellation dimensionality, and number of signal points, finding the optimum constellation is a problem in packing spheres around the origin Thompson, Perfect packing is neither possible nor necessarily desirable for several reasons. First, minimizing peak energy to reduce the cost of the transmitter and maximizing interpoint distance to minimize BER are conflicting goals.
Second, when sequential channel symbols are correlated, an ideally packed constellation is not always optimal in the sense of minimizing BER for a given average energy. Finally, and perhaps most importantly, the demodulator and decoder can be considerably simplified when the signal points fall on a regular lattice Forney et al. Generally, constellations with more points and dimensions require somewhat less energy per bit to achieve the same error resistance as a constellation with fewer points and dimensions.
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As the number of points and dimensions becomes large the asymptotic performance gains range from 1 to 2 dB Forney et al. Despite their suboptimality, the same constellations used for uncoded modulation are often used even when combined with coding. TROTT constellations give only small performance gains, increase decoder complexity, and behave unpredictably with certains types of non-Gaussian noise Ungerboeck, They are not used widely in practice.
The value w is called the carrier frequency, and the sin wt and cos wt axes are called the inphase and quadrature, respectively. Because all signal points have the same magnitude, PSK is commonly used in satellite communications where channels are subject to fading. The automatic gain control circuitry used to correct fading distorts any amplitude information in the signal. More elaborate schemes use signal points to specify discrete Fourier transform coefficients d t 0.
Viewed as a discrete-time system, the cardinality of the output set Y must be greater than the cardinality of the input set U, i. I U I where 0 denotes modulo two addition. After a single-bit error, any output block can still be uniquely identified. Hence, dfree for an uncoded system is the distance between the closest two signal points. It follows from Eqs. Equation 26 indicates that codes should have a high d,,,, for best performance. Catastrophic error propagation, as defined in Section III.
E, is an exception to Eq. Different methods are used to calculate dfreefor different types of codes Mulligan and Wilson, ; some sample calculations for trellis codes are given in Section 1V. Calculating free Euclidean distance is computationally difficult, but deriving the exact BER of a code is much harder. Hence, codes are usually designed to maximize dfreerather than to minimize BER, and their performance is verified using simulation. The ratio between the free Euclidean distance of a coded system and that of an uncoded system is called the coding gain.
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In addition, coding gain is a measure of relative energy efficiency only; the bandwidths of the compared systems must be separately considered. The Encoder Codes are designed from the standpoint of adding coding to an uncoded system. C , so the uncoded constellation must be at least doubled in size. Points can be added to a constellation in two ways.
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The first way is to concatenate adjacent channel symbols, as in Eq. This effectively increases the dimensionality of the constellation. The second way is to add points to the constellation directly, without increasing the number of dimensions. The former possibility increases the required bandwidth of the transmitted signal, which is unacceptable in a bandlimited environment. The latter choice requires a higher average energy to maintain interpoint distance, since there are more points per dimension.
However, the energy efficiency see Eq.
Codes that use the latter method of constellation expansion are termedalphabet redundant. All codes add redundancy. With alphabet redundant codes, the redundancy is absorbed by increasing the number of points in the constellation the channel alphabet , and not by transmitting more channel symbols Biglieri, Forward error correction FEC alphabet redundant codes are divided into two major types: those with memory, and those without.
Memoryless codes, or block codes, act as mapping functions. The input data is divided into sections, and each section is mapped into a larger output section. The inputs and outputs are generally, but not always, binary. A popular Reed-Solomon code, for example, has rate , Berlekamp et al.
Typically, a rate , code would not use a constellation with signal points. Instead, several small constellations with few dimensions would be used sequentially. This is effectively a large multidimensional constellation. Because the component constellations are identical and not optimized for consecutive use, the gains allowed by higher dimensions are usually not realized. Although block codes act as mapping functions, this description is useless when the block size is large.
For the rate , code, enumerating the mapping function requires a table with entries. This makes encoding difficult and decoding impossible. Instead, block codes are described using polynomials over a finite field.
follow url TROTT decoding to the simpler but far from trivial problem of finding roots of a finite polynomial Berlekamp, Thus, the encoder is just a discrete-time discrete-state dynamical system. The most general type of code with memory is the trellis code, whose design and structure are the subject of the next two sections. Section 1II.
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Trellis diagrams, the primary graphical tool for describing trellis codes, are explained in Section 1II. The distinction between finite and infinite memory systems and their relationship to trellis codes is addressed in Section 1II. The lack of a careful definition of finite and infinite memory systems has led to some confusion in the literature, which we attempt to resolve here.