Spectral-null Codes Research Articles

Littlewood polynomials, spectral-null codes, and equipowerful partitions

Let $[n]$ denote $\{0,1, ... , n-1\}$. A polynomial $f(x) = \sum a_i x^i$ is a Littlewood polynomial (LP) of length $n$ if the $a_i$ are $\pm 1$ for $i \in [n]$, and $a_i = 0$ for $i \ge n$. Such an LP is said to have order $m$ if it is divisible by $(x-1)^m$. The problem of finding the set $L_m$ of lengths of LPs of order $m$ is equivalent to finding the lengths of spectral-null codes of order $m$, and to finding $n$ such that $[n]$ admits a partition into two subsets whose first $m$ moments are equal. Extending the techniques and results of Boyd and others, we completely determine $L_7$ and $L_8$ and prove that 192 is the smallest element of $L_9$. Our primary tools are the use of carefully targeted searches using integer linear programming (both to find LPs and to disprove their existence for specific $n$ and $m$), and an unexpected new concept (that arose out of observed symmetry properties of LPs) that we call regenerative pairs, which produce infinite arithmetic progressions in $L_m$. We prove that for $m \le$ 8, whenever there is an LP of length $n$ and order $m$, there is one of length $n$ and order $m$ that is symmetric (resp.~antisymmetric) if m is even (resp.~odd).

Computation of the Spectrum of dc2-Balanced Codes

We apply the central limit theorem for deriving approximations to the auto-correlation function and power density function (spectrum) of second-order spectral null (dc2-balanced) codes. We show that the auto-correlation function of dc2-balanced codes can be accurately approximated by a cubic function. We show that the difference between the approximated and exact spectrum is less than 0.03 dB for codeword length $n=256$ .

Estimated Spectra of Higher Order Spectral Null Codes

We present an estimate of the power density function (spectrum) of binary $K$ th-order spectral null codes. We work out the auto-correlation model in detail for second-and third-order spectral null codes. We compare the auto-correlation functions and spectra predicted by the model with those generated by full-set $K$ th-order spectral null block codes.

On Redundancy Reduction of Non-Recursive Second-Order Spectral-Null Codes

The code design problem of non-recursive second-Order Spectral Null (2-OSN) codes is to convert balanced information words into 2-OSN words employing the minimum possible redundancy. Let $k$ be the balanced information word length. If $k\in \,\,2 {{I}}{{N}}$ then the 2-OSN coding scheme has length $n=k+r$ , with 2-OSN redundancy $r \in \,\,2 {{I}}{{N}}$ and $n\in \,\,4 {{I}}{{N}}$ . Here, we use a scheme with $r=2 \log k+\Theta (\log \log k)$ . The challenge is to reduce redundancy even further for any given $k$ . The idea is to exploit the degree of freedom to select from more than one possible 2-OSN encoding of a given balanced information word. To reduce redundancy, empirical results suggest that extra information $\delta _{k}=0.5 \log k+\Theta (\log \log k)$ is obtained. Thus, the proposed approach would give a smaller redundancy $r^{\prime }=1.5 \log k+\Theta (\log \log k)$ less than $r=2 \log k+\Theta (\log \log k)$ .

New Distance Concept and Graph Theory Approach for Certain Coding Techniques Design and Analysis

AbstractA New graph distance concept introduced for certain coding techniques helped in their design and analysis as in the case of distance-preserving mappings and spectral shaping codes. A graph theoretic construction, mapping binary sequences to permutation sequences and inspired from thek-cube graph has reached the upper bound on the sum of the distances for certain values of the length of the permutation sequence. The new introduced distance concept in thek-cube graph helped better understanding and analyzing for the first time the concept of distance-reducing mappings. A combination of distance and the index-permutation graph concepts helped uncover and verify certain properties of spectral null codes, which were previously difficult to analyze.

Doppler tolerance, complementary code sets, and generalised Thue–Morse sequences

The authors generalise the construction of Doppler-tolerant Golay complementary waveforms by Pezeshki–Calderbank–Moran–Howard to complementary code sets having more than two codes, which they call Doppler-null codes. This is accomplished by exploiting number-theoretic results involving the sum-of-digits function and a generalisation to more than two symbols of the classical two-symbol Thue–Morse sequence. Two approaches are taken to establish higher-order nulls of the composite ambiguity function: one by rewriting it in terms of equal sums of powers (ESP) and the other by factoring it in product form to reveal a higher-order zero, analogous to spectral-null codes. They conclude by describing an application of minimal ESP sets to multiple-input–multiple-output radar.

Efficient Non-Recursive Design of Second-Order Spectral-Null Codes

A new efficient design of second-order spectral-null (2-OSN) codes is presented. The new codes are obtained by applying the technique used to design parallel decoding balanced (i.e., 1-OSN) codes to the random walk method introduced by some of the authors for designing 2-OSN codes. This gives new non-recursive efficient code designs, which are less redundant than the code designs found in the literature. In particular, if $k \in {\mathrm{I}}\mskip -2.5mu{\mathrm{I}}\mskip -7mu{\mathrm{N}} $ is the length of a 1-OSN code then the new 2-OSN coding scheme has length $n=k+r \in {\mathrm{I}}\mskip -2.5mu{\mathrm{I}}\mskip -7mu{\mathrm{N}} $ with an extra redundancy of $r\simeq 2\log _{2}k+(1/2)\log _{2}\log _{2} k-0.174$ check bits, with $k$ and $r$ even and $n$ multiple of 4. The whole coding process requires $O(k\log k)$ bit operations and $O(k)$ bit memory elements.

Reducing Code Length of Second-Order Spectral-Null Code

Tallini and Bose defined a random walk function on which an efficient encoding/decoding algorithm for second-order spectral-null (2-OSN) codes was based. Later, Yang enhanced Tallini-Bose code by using quasi 2-OSN codes. Both 2-OSN codes encode the information word into a balanced word and then append extra bits to make the length of this balanced word equal to 0 (or 2) modulo 4. The main innovation is to eliminate the need for appended bits, thereby increasing the code rate.

Generation of spectral nulls with permutation sequences

Abstract This paper describes the construction of classes of spectral null codes, which are obtained by shaping permutation sequences. The properties and the construction techniques of binary spectral null codes are used. The power spectral densities (PSD) of the proposed codes show nulls at the zero frequency and at submultiples of the frequencies of the transmitted symbols. The channel symbols or voltage levels mapped to each transmitted permutation symbol are chosen to satisfy the spectral null equation. Interesting results show the elimination of the DC component at the lower frequencies. The well-shaped power spectral densities of these codes may overcome some communications problem like zero frequency components. The proposed technique is generalized and the corresponding spectral null permutation sequences that generate spectral null codes are presented.

Efficient Encoding/Decoding for Second-Order Spectral-Null Codes by Reducing Random Walks

Second-order spectral-null (2-OSN) codes are, in general, constructed from concatenating any codewords in 2-OSN code. Most 2-OSN codes adopt the Tallini-Bose random walk method in encoding and decoding, which exchanges two adjacent bits each time in a binary vector. In this brief contribution, we give a new implementation of the Tallini-Bose random walk method for 2-OSN codes which is shown by experiment to be faster.

Moment Balancing Templates for Spectral Null Codes

The generalized moment balancing templates for spectral null codes are investigated in this paper. A new approach based on the insertion of a determined number of vectors at determinate indices of a spectral null code word, is found to balance the moment of the code word. Moment balanced code words preserve the spectral null properties and can guarantee the correction of one insertion or one deletion error. As compared to the universal template for an uncoded sequence, the redundancy is determined by the structure of the original spectral null code, however, it can be kept small. The new approach involves the applications of the optimized moment balancing template and nonbinary enumeration of the balancing moment values.

Efficient Encoding Algorithm for Second-Order Spectral-Null Codes Using Cyclic Bit Shift

Some efficient second-order spectral-null codes encoded an index of the random walk function recursively and ended with the short base second-order spectral-null codes. All of these codes used the Tallini-Bose random walk function that exchanges two consecutive bits. In this paper, we propose a new random walk function based on cyclic bit-shift, on which the redundancy can be improved. Moreover, the bit-shift can be implemented efficiently by either software or hardware.

Binary permutation sequences as subsets of Levenshtein codes, spectral null codes, run-length limited codes and constant weight codes

We investigate binary sequences which can be obtained by concatenating the columns of (0,1)-matrices derived from permutation sequences. We then prove that these binary sequences are subsets of a surprisingly diverse ensemble of codes, namely the Levenshtein codes, capable of correcting insertion/deletion errors; spectral null codes, with spectral nulls at certain frequencies; as well as being subsets of run-length limited codes, Nyquist null codes and constant weight codes.

Some new efficient second-order spectral-null codes with small lookup tables

Tallini and Bose introduced a recursive method to construct second-order spectral-null codes which result in short (information length 16) base optimal second-order spectral-null codes that can be easily implemented either with table lookup or enumerative coding. In this paper, the extended base suboptimal second-order spectral-null codes (17 information length 32) obtained from two short base codes by lookup table are used to reduce the overall code length.

Design of Efficient Second-Order Spectral-Null Codes

An efficient recursive method has been proposed for the encoding/decoding of second-order spectral-null codes, via concatenation by Tallini and Bose. However, this method requires the appending of one, two, or three extra bits to the information word, in order to make a balanced code, with the length being a multiple of 4; this introduces redundancy. Here, we introduce a new quasi-second-order spectral-null code with the length /spl equiv/ 2 (mod 4) and extend the recursive method of Tallini and Bose, to achieve a higher code rate.

On efficient high-order spectral-null codes over the m-ary alphabet

Let be the set of all words of length N over the m-polar alphabet Φ m ={−(m−1),−(m−3),…,+(m−3),+(m−1)}, having a q-th order spectral-null at zero frequency. Any subset of is a spectral-null code of length N and order q. This paper gives an equivalent formulation of in terms of codes over the m-ary alphabet ℤ m ={0, 1,…, m−1}, derives a recursive expression for the cardinality of , shows combinatorial properties of , gives new simple ways to obtain systematic m-ary q-th order spectral-null codes, and finally, presents new efficient recursive design methods to encode k m-ary information digits into second-order spectral-null codes over ℤ m of length N(k)=n(k)+N(⌈log m n(k)(n(k)−1)/2⌉+1), whose implementing algorithm requires T=O(mk log m k) m-ary digit operations and S=O(k) m-ary digit storing elements.

High-Order Spectral-Null Multimode Codes

A new class of codes with high-order spectral nulls at zero frequency is introduced. The codes, called high-order spectral-null multimode codes, are constructed through state-dependent encoding by use of the guided-scrambling coding technique, and result in efficiently coded sequences with good low-frequency characteristics.

A Performance Metric for Codes With a High-Order Spectral Null at Zero Frequency

Sum-variance is a well-known metric for assessing the performance of dc-free codes (first-order spectral-null codes), however, as we show in this paper, it is unsuitable for comparing the magnitude of spectral components of high-order spectral-null (HOSN) codes at low frequencies. In this paper, we introduce a new performance metric for evaluating the spectrum compression of arbitrarily HOSN codes around zero frequency; we call this metric the low-frequency spectrum weight (LFSW). We show that the asymptotic low-frequency spectral components of Kth-order spectral-null codes (K/spl ges/1) are exclusively determined by the order K and the LFSW, and that the LFSW equals the zero-frequency value in the spectrum of the corresponding sequence of Kth-order running digital sum values. We derive this result for symbol-by-symbol encoding, and then extend it to block HOSN codes. We then derive a closed-form expression for the LFSW of HOSN codes constructed through state-independent encoding. Closed-form expressions for LFSW of first-order zero-disparity codes and for the asymptotic LSFW of maxentropic dc-free sequences are also given.

On efficient high-order spectral-null codes

Let S (N,q) be the set of all words of length N over the bipolar alphabet (-1,+1), having a qth order spectral-null at zero frequency. Any subset of S (N,q) is a spectral-null code of length N and order q. This correspondence gives an equivalent formulation of S(N,q) in terms of codes over the binary alphabet (0,1), shows that S(N,2) is equivalent to a well-known class of single-error correcting and all unidirectional-error detecting (SEC-AUED) codes, derives an explicit expression for the redundancy of S(N,2), and presents new efficient recursive design methods for second-order spectral-null codes which are less redundant than the codes found in the literature.

Asymptotically exact bounds on the size of high-order spectral-null codes

The spectral-null code S(n, k) of kth order and length n is the union of n-tuples with /spl plusmn/1 components, having kth-order spectral-null at zero frequency. We determine the exact asymptotic in n behavior of the size of such codes. In particular, we prove that for n satisfying some divisibility conditions, log/sub 2/|S(n, k)|=n-k/sup 2//2log/sub 2/n+c/sub k/+o(1), where c/sub k/ is a constant depending only on k and o(1) tends to zero when n grows. This is an improvement on the earlier known bounds due to Roth, Siegel, and Vardy (see ibid., vol40, p.1826-40, 1994).