Variable-length balanced codes for quadrature phase shift keyed systems Xin Tu, Ivan J. Fair Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada E-mail:
[email protected] Published in The Journal of Engineering; Received on 15th September 2015; Accepted on 15th September 2015
Abstract: The authors outline an approach to construct capacity-approaching balanced quadrature phase shift keyed (QPSK) codes. These codes ensure an equal number of different symbol values and many symbol transitions in the encoded sequence in order to assist practical demodulators to accurately recover symbol values. Their codes are comprised of instantaneously decodable variable-length codewords that exhibit excellent performance with average code rates higher than previously reported fixed-length balanced QPSK codes.
1
Introduction
Balanced codes are widely used in binary communication systems to ensure the presence of many transitions and an equal number of logic 1’s and logic 0’s in the encoded bit stream. These characteristics enable practical demodulators to accurately recover bit-level synchronisation and establish the constant decision thresholds necessary for accurate symbol recovery [1]. Since balanced codes result in a null at DC in the spectrum of the encoded binary sequence, the terms balanced code and DC-free code are used interchangeably [2]. Owing to the advantages they offer to practical demodulators, the use of balanced codes has recently been suggested in bandpass systems such as those using quadrature phase shift keyed (QPSK) signalling [3, 4]. The balanced nature of a QPSK signal also results in a null at the carrier frequency of the bandpass spectrum, which corresponds to a null at DC in the spectrum of the equivalent complex baseband signal. This null allows for straightforward insertion and removal of a pilot tone without loss of informationcarrying signal power. Construction of fixed-length balanced QPSK codes using tablebased approaches [3] and guided scrambling [4] has recently been reported. In this paper, we outline the construction of variablelength DC-free QPSK codes based on the technique described in [5]. The codes we design demonstrate a tradeoff between code rate and suppression of low-frequency components. All variablelength codes reported in this paper have higher average code rates than those that have been reported for fixed-length codes satisfying the same DC-free constraint.
2
Balance constraint
A sequence is DC-free if and only if its running digital sum (RDS) is bounded [6], where RDS is the accumulation of signalling values from the beginning to any point in the sequence. We consider the QPSK signalling points {+1, +j, –1, –j} which result in a twodimensional RDS, and arbitrarily map, respectively, the symbol values {0, 1, 2, 3} to those points. Note that our results are independent of the one-to-one mapping of symbol values to signalling points. Our construction technique can be applied to the design of QPSK codes that satisfy various balance constraints. However, to demonstrate our approach we consider the specific case of restricting the RDS of the encoded sequence to the nine values depicted in Fig. 1. The capacity of sequences that satisfy this constraint is C = 1.5 bits of information per quaternary symbol [7]. It has been shown that this RDS constraint can be satisfied with fixed-length codes that map four binary source digits to three quaternary J Eng 2015 doi: 10.1049/joe.2015.0150
encoded symbols for a code rate of R = 4/3 and an efficiency of η = R/C = 88.9% [3]. 3
General approach
Following the technique outlined in [5], we propose constructing variable-length codewords by: (i) defining a minimal set of words that satisfy the constraint; (ii) constructing sets of instantaneously decodable codewords by concatenating words from this minimal set through partial extensions based on a tree structure; (iii) determining the optimal mapping of variable-length source words to each set of variable-length codewords using normalised geometric Huffman coding [8]; and (iv) evaluating the average code rate of each code constructed and selecting the code with the highest rate.
4
Minimal set
To design variable-length balanced QPSK codes, we first form minimal sets of variable-length words that start and end with RDS = 0. To ensure that the words in the minimal set are prefix free, it is sufficient that these words are unique and end the first time their RDS returns to 0. The codewords constructed through partial extensions are then guaranteed to be instantaneously decodable [5]. Specific to the balance constraint we consider here, we separate the nine permissible RDS values depicted in Fig. 1 into two subsets: the set of four values {+1, +j, –1, –j} that can be reached from RDS = 0 only after an odd number of symbols, and the other five values that can occur only after an even number of symbols. Since words in a minimal set both start and end with RDS = 0, it follows that they must have even length. We now determine the number of words of length l that start with RDS = 0 and end the first time their RDS returns to the value 0. In addition to having even length, we now show that the word length l is unbounded and that for each even l there exist 2l such words. Consider first the words of length l = 2. There are 22 such words: {02, 13, 20, 31}. Note that the word {02} describes the only way in which RDS = +1 after one symbol and that the RDS returns to 0 at the end of the length-2 word. Symmetrical reasoning applies to the other three length-2 words based on their 4-ary symmetry. Now consider the ways in which the last symbol of the word {02} can be replaced to form four words of length l = 4 that satisfy the conditions for inclusion in a minimal set. As noted above, the leading symbol of that word defines the only way in which RDS = +1 after one symbol. There are four ways in which
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Table 1 Parameters of best variable-length codes 4 1.347 0.898 1.338 0.892 8 11 153 1.543
lmax Rmax ηmax Rbest ηbest LC LS S W
Fig. 1 Permitted RDS values
the RDS can return to 0 for the first time after three additional symbols: these can be determined by replacing the last symbol {2} with each of the symbol sequences {123, 321, 132, 312}. Symmetrical arguments can be made for extension of the other three words of length l = 2. Since each of the 22 words of length two can be extended to 22 words of length four that return to RDS = 0 only at the end of the word, the result is the set of 2222 = 24 words of length l = 4 shown in Fig. 2. Note that the set of 24 words of length four includes the four possible ways in which RDS = +1 after three symbols; these four possibilities are described by the first three symbols of the four words that end with symbol {2}. Using the argument above, each of these words of length four can be extended to four words of length l = 6 by replacing their last symbol {2} with the four symbol sequences {123, 321, 132, 312}. By symmetry, each of the 24 words of length four in Fig. 2 can be extended to four words of length l = 6 with appropriate replacement of the last symbol resulting in 2422 = 26 words of length l = 6. Enumeration of longer words is straightforward. Each set of 2l−2 words of length l − 2 contains words that describe all possible ways of attaining an RDS from the set {+1, +j, −1, −j} prior to its last symbol. The last symbol can be replaced by one of four different sequences of three symbols that will result in RDS = 0 at the end of each length-l word. There are therefore 2l−222 = 2l words of length l that satisfy the properties required for inclusion in a minimal set. Clearly l is unbounded, and a complete minimal set would contain an infinite number of words. We instead use incomplete minimal sets with words limited in length to some finite lmax. Incomplete minimal sets result in some constraint-satisfying sequences never being used, and therefore restrict the code rate to values lower than capacity. The maximum possible code rate for codes constructed based on an incomplete minimal set is given by the logarithm of the largest real solution to the characteristic equation describing the number and length of words in the minimal set [2, 5]. Our incomplete minimal sets satisfying the balance constraint in Fig. 1 have the characteristic equation
5
6 1.440 0.960 1.430 0.954 10 14 665 2.312
8 1.473 0.982 1.467 0.978 12 17 2713 2.666
10 1.488 0.992 1.484 0.990 12 17 6816 2.778
Code construction
We propose constructing codes starting with several different incomplete minimal sets. Let N denote the number of words in a minimal set. The first level of the tree contains N leaves each labelled with a word from the minimal set. A partial extension is formed by extending a leaf with N branches each labelled with the label of the original leaf concatenated with a different word from the minimal set. The labels on the extended tree comprise a set of variable-length codewords. For each set of codeword lengths, we use normalised geometric Huffman coding [8] to evaluate the optimal corresponding source word lengths. The resulting average code rate is the ratio of the expected source word length to the expected codeword length. We select the code with the highest average code rate as the best code constructed based on that minimal set. For our particular DC-free constraint, we considered the minimal sets consisting of words with lmax = 4 up to lmax = 10. We restricted our search by constructing partial extensions to at most depth five and also limiting codeword length to at most 12. We assumed equiprobable and independent binary digits in the source stream. 6
Results
Parameters of our codes with the highest average code rate for each lmax are given in Table 1. This table reports values for the best code rate Rbest and its efficiency ηbest, along with values for LC and LS, the lengths of the longest codeword and longest source word in the code, respectively, and S, the number of words in the code. This table also reports values of low-frequency spectral weight W [9] which accurately describes suppression of low-frequency components in the power spectral density (PSD) of the equivalent complex baseband signal according to PSD = W v2 ,
v≪1
PSD [dB] = 10 log W + 20 log v,
v≪1
2lmax z−lmax + 2(lmax −2) z−(lmax −2) + . . . + 4z−2 − 1 = 0 The maximum values of code rate Rmax and efficiency ηmax for practical values of lmax are given in the first two rows of Table 1.
Fig. 2 Replacement of last symbol to construct words two symbols longer
Fig. 3 PSD of balanced QPSK codes
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J Eng 2015 doi: 10.1049/joe.2015.0150
Lower W reflects greater suppression of low-frequency components as demonstrated in Fig. 3. Spectra in this figure were generated by simulating the PSD of three million codewords. These spectra reveal a tradeoff of code rate and suppression of low frequencies. Note that the best fixed-rate code developed to date that satisfies the DC constraint in Fig. 1 has a lower code rate (R = 4/3 [3]) than any of the variable-length codes developed in this paper. It has worst spectral performance (W = 1.925) than the variable-rate code with lmax = 4 and provides only 1.6 dB additional suppression of low frequencies over our variable-length code that has a code rate within one per cent of capacity.
7
Conclusions
We have presented an approach to construct balanced QPSK codes with variable-length codewords. This technique involves the specification of an incomplete minimal set of words that satisfy the balance constraint, construction of codewords through partial extensions of this minimal set, evaluation of optimal source word lengths for each set of codewords, and selection of the code with the highest average code rate. The variable-length codes we construct have higher average code rates than fixed-length codes that satisfy the same constraint.
J Eng 2015 doi: 10.1049/joe.2015.0150
8
Acknowledgment
This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. 9
References
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