Golay Complementary Sequences Research Articles

A construction of OFDM 16-QAM sequences having low peak powers

Using a realization of 16-QAM as set addition of two scaled versions of the 4-PSK constellation, we construct 16-QAM sequences having low peak-to-mean envelope power ratios (PMEPR) from 4-PSK Golay sequences. Various upper bounds on the peak envelope powers of these sequences are established. Transmission using these sequences provides twice as much rate as transmission using complementary Golay sequences with a maximum PMEPR of 3.6 (5.56 dB).

Generalised Rudin-Shapiro Constructions

A Golay Complementary Sequence (CS) has Peak-to-Average-Power-Ratio (PAPR) ≤ 2.0 for its one-dimensional continuous Discrete Fourier Transform (DFT) spectrum. Davis and Jedwab showed that all known length 2m CS, (GDJ CS), originate from certain quadratic cosets of Reed-Muller (1, m). These can be generated using the Rudin-Shapiro construction. This paper shows that GDJ CS have PAPR ≤ 2.0 under all unitary transforms whose rows are unimodular linear (Linear Unimodular Unitary Transforms (LUUTs)), including one- and multi-dimensional generalised DFTs. We also propose tensor cosets of GDJ sequences arising from Rudin-Shapiro extensions of near-complementary pairs, thereby generating many infinite sequence families with tight low PAPR bounds under LUUTs.

Generalized Reed-Muller codes and power control in OFDM modulation

Controlling the peak-to-mean envelope power ratio (PMEPR) of orthogonal frequency-division multiplexed (OFDM) transmissions is a notoriously difficult problem, though one which is of vital importance for the practical application of OFDM in low-cost applications. The utility of Golay complementary sequences in solving this problem has been recognized for some time. In this paper, a powerful theory linking Golay complementary sets of polyphase sequences and Reed-Muller codes is developed. Our main result shows that any second-order coset of a q-ary generalization of the first order Reed-Muller code can be partitioned into Golay complementary sets whose size depends only on a single parameter that is easily computed from a graph associated with the coset. As a first consequence, recent results of Davis and Jedwab (see Electron. Lett., vol.33, p.267-8, 1997) on Golay pairs, as well as earlier constructions of Golay (1949, 1951, 1961), Budisin (1990) and Sivaswamy (1978) are shown to arise as special cases of a unified theory for Golay complementary sets. As a second consequence, the main result directly yields bounds on the PMEPRs of codes formed from selected cosets of the generalized first order Reed-Muller code. These codes enjoy efficient encoding, good error-correcting capability, and tightly controlled PMEPR, and significantly extend the range of coding options for applications of OFDM using small numbers of carriers.

Performance of OFDM‐CDMA with Simple Peak Power Reduction

AbstractIn this paper we propose simple schemes which reduce the peak‐to‐average power ratio (PAPR) of the OFDM‐CDMA signals. The statistical behavior of the peak power in the OFDM‐CDMA signals is first examined by computer simulations, employing the two orthogonal sets of spreading sequences, namely, Walsh‐Hadamard sequences and Golay complementary sequences. It is shown that for small number of active users or uplink‐system, Golay complementary sequences yield much lower PAPR than Walsh‐Hadamard sequences, and for large number of active users, Walsh‐Hadamard sequences eventually becomes better. A simple scrambling scheme is introduced, which may result in considerable peak power reduction in the case of small number of the active users. For the downlink system, further reduction of the peak power can be achieved by a simple adaptive scheme based on the selection of the set of the spreading sequences.

Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes

We present a range of coding schemes for OFDM transmission using binary, quaternary, octary, and higher order modulation that give high code rates for moderate numbers of carriers. These schemes have tightly bounded peak-to-mean envelope power ratio (PMEPR) and simultaneously have good error correction capability. The key theoretical result is a previously unrecognized connection between Golay complementary sequences and second-order Reed-Muller codes over alphabets Z/sub 2/h. We obtain additional flexibility in trading off code rate, PMEPR, and error correction capability by partitioning the second-order Reed-Muller code into cosets such that codewords with large values of PMEPR are isolated. For all the proposed schemes we show that encoding is straightforward and give an efficient decoding algorithm involving multiple fast Hadamard transforms. Since the coding schemes are all based on the same formal generator matrix we can deal adaptively with varying channel constraints and evolving system requirements.

Efficient Golay correlator

A special set of orthogonal Golay complementary sequences is proposed for inclusion in the third generation cellular standard to be used as the set of synchronisation preambles for the random access channel. Besides excellent correlation properties, such sequences offer an efficient hardware implementation of the corresponding correlators based on the so-called efficient Golay correlator (EGG). This kind of correlator is derived here.

Golay sequences for identification of linear systems with weak nonlinear distortion

The design of bipolar excitation sequences for identifying linear time-invariant systems in the presence of weak nonlinear distortion is considered. A Volterra system model is used and system identification is carried out in the time domain by a crosscorrelation analysis. The measured system characteristic is the impulse response. A set of four pairs of Golay complementary sequences is presented which is very suitable for identification of linear systems suffering from weak second and third-order nonlinear distortion. Artefacts in the measured system characteristics caused by this kind of nonlinear distortion are effectively eliminated by averaging the four impulse responses obtained with the proposed four pairs of Golay complementary sequences. The key element in this study is higher-order autocorrelation properties of Golay complementary sequences. Experimental results obtained from a digital magnetic storage system are presented.

Dipulse-response measurement of a magnetic recording channel using Golay complementary sequences

We apply Golay complementary sequences for measuring the dipulse response and transfer functions of a digital magnetic-saturation-recording channel. Our experimental channel suffers from nonlinear distortion caused by a magnetoresistive (MR) sensor. The measured-channel characteristics are compared with the same characteristics obtained by the dipulse-response measurement technique by using feedback-shift register sequences. Based on a practical nonlinear Volterra model of a digital magnetic-saturation-recording channel, we analyze the artifacts caused by second- and third-order nonlinear distortion, which do-occur in the Golay dipulse responses and transfer functions. We show that with either of the considered dipulse-response measurement techniques, the artifacts caused by even-order nonlinear distortion, which occur in the measured-channel characteristics, can easily be removed. The measured-channel characteristics obtained show excellent performance for either of the considered excitation sequences.

Peak-to-mean power control and error correctionfor OFDM transmission using Golay sequences and Reed-Muller codes

A coding scheme for OFDM transmission is proposed, exploiting a previously unrecognised connection between pairs of Golay complementary sequences and second-order Reed-Muller codes. The scheme solves the notorious problem of power control in OFDM systems by maintaining a peak-to-mean envelope power ratio of at most 3 dB while allowing simple encoding and decoding at high code rates for binary, quaternary or higher-phase signalling together with good error correction.

Spreading sequences for multicarrier CDMA systems

The paper contains an analysis of the basic criteria for the selection of spreading sequences for the multicarrier CDMA (MC-CDMA) systems with spectrum spreading in the frequency domain. It is shown that the time-domain crosscorrelation function between the spreading sequences is not a proper interference measure for the asynchronous MC-CDMA users. Therefore, the spectral correlation function is introduced and, together with the crest factor and the dynamic range of the corresponding multicarrier waveforms, is used for the evaluation of MC-CDMA sequences. Some well-known classes of sequences, such as Walsh, Gold, orthogonal Gold, and Zadoff-Chu sequences, as well as Legandre and Golay complementary sequences, are evaluated with respect to the aforementioned basic criteria. It is also shown that the crest factors of the multicarrier spread spectrum waveforms based on the multilevel Huffman sequences are very close to or even lower than the crest factor of a single sine wave.

Synthesis of power efficient multitone signals with flat amplitude spectrum

It is shown that, generally, any binary or polyphase sequence belonging to a pair of complementary sequences can be used to construct the initial phases of tones forming a multitone signal with the crest factor less than or equal to 6 dB. The previously proposed Shapiro-Rudin sequences are representatives of a much larger family of binary Golay complementary sequences. Numerical investigations show that the values of the crest factors obtained by applying complementary sequences (binary or polyphase) are between 5 and 6 dB, regardless of the number of tones. >

Efficient pulse compressor for golay complementary sequences

The highly regular structure of binary Golay complementary sequences of length 2N makes possible a very simple correlator for these sequences. The algorithm called fast Golay correlation (FGC) performs the correlation of the input radar signal with the Golay sequence using only 2.log2(M) operations per input sample as opposed to M operations required by standard correlators. This algorithm is also valid for complex sequences of the same length. This fast correlator and the good autocorrelation properties of Golay sequences make them the ideal choice for high speed, long sequence pulse compression. Sequence agility on the pulse to pulse basis is also possible at the expense of some added complexity.

A new restriction on the lengths of golay complementary sequences

A new restriction on the lengths of golay complementary sequences

Three-dimensional imaging system based on Fourier transform synthetic aperture focusing technique

For planar scan surfaces, digitized ultrasonic RF-data can be adequately processed in terms of the Fourier transform synthetic aperture focusing technique algorithm, i.e. in terms of synthetic aperture pulse-echo backpropagation utilizing Fourier transforms only, to yield a quantitative three-dimensional image of defects residing in the homogeneous and isotropic bulk material. The implementation of this algorithm into an ultrasonic imaging system is described, which mainly comprises an array processor and high-resolution graphics to display the three-dimensional reconstruction volume as a walk-through along three orthogonal planes. To enhance the signal-to-noise ratio and the axial resolution of the system, controlled ultrasonic signals are transmitted as complementary Golay-sequences, cross-correlated with the received signals and deconvolved with similarly obtained reference signals.

Hadamard Matrices, Finite Sequences, and Polynomials Defined on the Unit Circle

If a $(\ast )$-type Hadamard matrix of order 2n (i.e. a pair (A, B) of $n \times n$ circulant (1,-1) matrices satisfying $AA\prime + BB\prime = 2nI$) exists and a pair of Golay complementary sequences (or equivalently, two-symbol $\delta$-code) of length m exists, then a $(\ast )$-type Hadamard matrix of order 2mn also exists. If a Williamson matrix of order 4n (i.e. a quadruple (W, X, Y, Z) of $n \times n$ symmetric circulant (1,-1) matrices satisfying ${W^2} + {X^2} + {Y^2} + {Z^2} = 4nI$) exists and a four-symbol $\delta$-code of length m exists, then a Goethals-Seidel matrix of order 4mn (i.e. a quadruple (A, B, C, D) of $mn \times mn$ circulant (1, -1) matrices satisfying $AA\prime + BB\prime + CC\prime + DD\prime = 4mnI$) also exists. Other related topics are also discussed.

Hadamard matrices, finite sequences, and polynomials defined on the unit circle

If a ( ∗ ) (\ast ) -type Hadamard matrix of order 2n (i.e. a pair (A, B) of n × n n \times n circulant (1,-1) matrices satisfying A A ′ + B B ′ = 2 n I AA\prime + BB\prime = 2nI ) exists and a pair of Golay complementary sequences (or equivalently, two-symbol δ \delta -code) of length m exists, then a ( ∗ ) (\ast ) -type Hadamard matrix of order 2mn also exists. If a Williamson matrix of order 4n (i.e. a quadruple (W, X, Y, Z) of n × n n \times n symmetric circulant (1,-1) matrices satisfying W 2 + X 2 + Y 2 + Z 2 = 4 n I {W^2} + {X^2} + {Y^2} + {Z^2} = 4nI ) exists and a four-symbol δ \delta -code of length m exists, then a Goethals-Seidel matrix of order 4mn (i.e. a quadruple (A, B, C, D) of m n × m n mn \times mn circulant (1, -1) matrices satisfying A A ′ + B B ′ + C C ′ + D D ′ = 4 m n I AA\prime + BB\prime + CC\prime + DD\prime = 4mnI ) also exists. Other related topics are also discussed.

There are no Golay complementary sequences of length 2.9 t

There are no Golay complementary sequences of length 2.9 t

There are no Golay complementary sequences of length 2.9 t

There are no Golay complementary sequences of length 2.9 t

Intersection preserving finite embedding theorems for partial quasigroups

is called a Golay complementary sequence of length k. For such a sequence to exist, k must be even and the sum of two squares. It is known no such sequences exist when k=18. Let k = 2 n . A necessary condition for a~, a 2 . . . . . a, to form the beginning of a complementary sequence is that there exist a partition of the set {1, 2 . . . . . n} into four sets Sa, $2, Dx and D2 such that for any 2n-th root of unity q:

Maximal binary matrices and sum of two squares

A maximal ( + 1 , − 1 ) ( + 1, - 1) -matrix of order 66 is constructed by a method of matching two finite sequences. This method also produced many new designs for maximal ( + 1 , − 1 ) ( + 1, - 1) -matrices of order 42 and new designs for a family of H-matrices of order 26.2 n {26.2^n} . A nonexistence proof for a ( ∗ ) (\ast ) -type H-matrix of order 36, consequently for Golay complementary sequences of length 18, is also given.