Maximum-length Shift-register Sequences Research Articles

New algebraic studies of pattern attributes in maximum-length shift-register sequences

In order to facilitate the ubiquitous heterogeneous and medium-independent communication networks, transmitter identification is crucial especially as the number of transmitters becomes tremendous in reality. Since the birth of spread-spectrum communications, pseudo-random identification sequences have been widely adopted due to their preferable nearly Dirac-Delta auto-correlation property, which would lead to the advantages in synchronization and multi-user interference mitigation. Maximum-length shift-register sequences (a.k.a. m-sequences) are pseudo-random sequences often adopted for multi-access communications, but categorization or grouping of the patterns (bit substrings) contained in m-sequences has not been investigated to the best of our knowledge. Thus, this paper is dedicated to the new study on the categorization of m-sequences, which would give rise to many applications involving assignment, addressing, and management of identification or spreading sequences. We define new essential parameters, namely single-pattern-searching parameters, and design a new parallel algorithm to spot such inherent parameters associated with each selected underlying pattern. Our proposed new approach will facilitate a highly computationally-efficient solution to find the pattern-attributed (pattern-contained) m-sequences without actually generating any m-sequence for subsequence-matching between an m-sequence and the underlying pattern(s). We further utilize these single-pattern-searching parameters to establish the new analysis of pattern capacity and quality by characterizing multiple pattern attributes within m-sequences. The first goal of this work is to find single-pattern-attributed m-sequences recursively until the final subset of m-sequences is acquired so that each m-sequence contains all underlying patterns. The second goal of this work is to find the shortest sequences containing specified patterns. To achieve this goal, we first construct the shortest binary sequences subject to the underlying patterns and it can be accomplished by solving the generalized traveling salesman problem (GTSP). Then, according to the constructed shortest binary sequences, the number of pattern-contained m-sequences can be determined thereby. Memory- and computational-complexity analyses are also presented to demonstrate that our proposed new scheme is much more computationally efficient than the conventional subsequence-matching method for searching and counting the pattern-attributed m-sequences. On the other hand, our proposed new scheme leads to the same memory-complexity as the conventional method for registering the spotted pattern-attributed m-sequences. Finally, three new metrics for assessing the underlying patterns are proposed. They are attributability, discriminability, and ambiguity. The associated numerical evaluation is also provided in this paper.

Spectrally-Constrained Sequences: Bounds and Constructions

We investigate spectrally-constrained sequences (SCSs), which are applicable to the communication and radar systems operating over non-contiguous carriers or frequency slots. Typical examples of such systems are overlay cognitive radio or cognitive radar networks. First, we derive the periodic- and aperiodic-correlation lower bounds for single-channel SCSs and multi-channel SCSs by convex optimization in the frequency domain. Each of these bounds reduces to a Welch bound when the number of forbidden carriers is set to zero. We then propose systematic constructions of optimal unimodular single-channel SCSs with the aid of cyclic difference sets and the theory of maximal-length shift register sequences.

Analysis of Optimal Sounding Signal Design in OFDM Systems

We focus on a sounding signal design for single-input single-output orthogonal frequency division multiplexing (SISO-OFDM) systems. We show that the frequency spectrum of an optimum sounding signal has a constant magnitude across the frequency band for the cases with or without Doppler effects. Simulation results show that the designed optimum sounding signal outperforms random sounding signals and that the performance of a maximal-length shift register sequence is indistinguishable from that of the optimum sounding signal.

Spatio-temporal dynamics of impulse responses to figure motion in optic flow neurons.

White noise techniques have been used widely to investigate sensory systems in both vertebrates and invertebrates. White noise stimuli are powerful in their ability to rapidly generate data that help the experimenter decipher the spatio-temporal dynamics of neural and behavioral responses. One type of white noise stimuli, maximal length shift register sequences (m-sequences), have recently become particularly popular for extracting response kernels in insect motion vision. We here use such m-sequences to extract the impulse responses to figure motion in hoverfly lobula plate tangential cells (LPTCs). Figure motion is behaviorally important and many visually guided animals orient towards salient features in the surround. We show that LPTCs respond robustly to figure motion in the receptive field. The impulse response is scaled down in amplitude when the figure size is reduced, but its time course remains unaltered. However, a low contrast stimulus generates a slower response with a significantly longer time-to-peak and half-width. Impulse responses in females have a slower time-to-peak than males, but are otherwise similar. Finally we show that the shapes of the impulse response to a figure and a widefield stimulus are very similar, suggesting that the figure response could be coded by the same input as the widefield response.

Method and software for using m-sequences to characterize parallel components of higher-order visual tracking behavior in Drosophila.

A moving visual figure may contain first-order signals defined by variation in mean luminance, as well as second-order signals defined by constant mean luminance and variation in luminance envelope, or higher-order signals that cannot be estimated by taking higher moments of the luminance distribution. Separating these properties of a moving figure to experimentally probe the visual subsystems that encode them is technically challenging and has resulted in debated mechanisms of visual object detection by flies. Our prior work took a white noise systems identification approach using a commercially available electronic display system to characterize the spatial variation in the temporal dynamics of two distinct subsystems for first- and higher-order components of visual figure tracking. The method relied on the use of single pixel displacements of two visual stimuli according to two binary maximum length shift register sequences (m-sequences) and cross-correlation of each m-sequence with time-varying flight steering measurements. The resultant spatio-temporal action fields represent temporal impulse responses parameterized by the azimuthal location of the visual figure, one STAF for first-order and another for higher-order components of compound stimuli. Here we review m-sequence and reverse correlation procedures, then describe our application in detail, provide Matlab code, validate the STAFs, and demonstrate the utility and robustness of STAFs by predicting the results of other published experimental procedures. This method has demonstrated how two relatively modest innovations on classical white noise analysis—the inclusion of space as a way to organize response kernels and the use of linear decoupling to measure the response to two channels of visual information simultaneously—could substantially improve our basic understanding of visual processing in the fly.

Divisibility of polynomials over finite fields and combinatorial applications

Consider a maximum-length shift-register sequence generated by a primitive polynomial f over a finite field. The set of its subintervals is a linear code whose dual code is formed by all polynomials divisible by f. Since the minimum weight of dual codes is directly related to the strength of the corresponding orthogonal arrays, we can produce orthogonal arrays by studying divisibility of polynomials. Munemasa (Finite Fields Appl 4(3):252---260, 1998) uses trinomials over $${\mathbb{F}_2}$$ to construct orthogonal arrays of guaranteed strength 2 (and almost strength 3). That result was extended by Dewar et al. (Des Codes Cryptogr 45:1---17, 2007) to construct orthogonal arrays of guaranteed strength 3 by considering divisibility of trinomials by pentanomials over $${\mathbb{F}_2}$$ . Here we first simplify the requirement in Munemasa's approach that the characteristic polynomial of the sequence must be primitive: we show that the method applies even to the much broader class of polynomials with no repeated roots. Then we give characterizations of divisibility for binomials and trinomials over $${\mathbb{F}_3}$$ . Some of our results apply to any finite field $${\mathbb{F}_q}$$ with q elements.

Performance of DS/SS System Using Pseudo-Ternary M-Sequences

In this paper, newly-found properties of the pseudo-ternary maximum-length shift register sequences (pseudo-ternary M-sequences) are described. In particular, the balance properties, the run-length distribution, the cross-correlation properties, and the decimation relationships are shown. The pseudo-ternary M-sequence is obtained by subtracting the one-chip shifted version from the {+1, -1}-valued M-sequence. Moreover, in this paper, performances of the direct sequence spread spectrum (DS/SS) system using the pseudo-ternary M-sequence are analyzed. In the performance evaluation, tracking error performance (jitter) and bit error rate (BER) performance that takes the jitter into account in DS/SS system with a pseudo-ternary M-sequence non-coherent DLL are evaluated. Using the pseudo-ternary M-sequence instead of the conventional M-sequences can improve the tracking error performance about 2.8[dB]. Moreover, BER of the DS/SS system using the pseudo-ternary M-sequence is superior about 0.8[dB]to that using the {+1, -1}-valued M-sequence.

Distributive Pilot Arrangement Based on Modified M-sequences for OFDM Intercarrier Interference Estimation

The intercarrier interference (ICI) matrix for the orthogonal frequency division multiplexing systems usually has a fairly large dimension and changes over time for mobile communications. In this paper, we design the modified maximum-length shift-register sequences and propose a new distributive allocation scheme together with a linear interpolation for both pilot-aided and non-pilot-aided ICI vector estimates; this new method can combat the realistic non-circulant ICI matrix problem, where almost all other existing schemes could not perform effectively. Monte Carlo experiments show the significant advantages of our new ICI matrix estimator especially when the signal-to-noise ratio is larger than 15 dB

A note on chaotic unimodal maps and applications

Based on the word-lift technique of symbolic dynamics of one-dimensional unimodal maps, we investigate the relation between chaotic kneading sequences and linear maximum-length shift-register sequences. Theoretical and numerical evidence that the set of the maximum-length shift-register sequences is a subset of the set of the universal sequence of one-dimensional chaotic unimodal maps is given. By stabilizing unstable periodic orbits on superstable periodic orbits, we also develop techniques to control the generation of long binary sequences.

The peak sidelobe level of families of binary sequences

A numerical investigation is presented for the peak sidelobe level (PSL) of Legendre sequences, maximal length shift register sequences (m-sequences), and Rudin-Shapiro sequences. The PSL gives an alternative to the merit factor for measuring the collective smallness of the aperiodic autocorrelations of a binary sequence. The growth of the PSL of these infinite families of binary sequences is tested against the desired growth rate o(/spl radic/nlnn) for sequence length n. The claim that the PSL of m-sequences grows like O(/spl radic/n), which appears frequently in the radar literature, is concluded to be unproven and not currently supported by data. Notable similarities are uncovered between the PSL and merit factor behavior under cyclic rotations of the sequences.

Stabilizing long-period orbits via symbolic dynamics in simple limiter controllers

We present an efficient approach to determine the control parameter of simple limiter controllers by using symbolic dynamics of one-dimensional unimodal maps. By applying addition- and subtraction-symbol rules for generating an admissible periodic sequence, we deal with the smallest base problem of the digital tent map. The proposed solution is useful for minimizing the configuration of digital circuit designs for a given target sequence. With the use of the limiter controller, we show that one-dimensional unimodal maps may be robustly employed to generate the maximum-length shift-register sequences. For an arbitrary long Sarkovskii sequence, the control parameters are analytically given.

A New ICI Matrices Estimation Scheme Using Hadamard Sequence for OFDM Systems

The intercarrier interference (ICI) matrix for the orthogonal frequency division multiplexing (OFDM) systems usually has a fairly large dimension. The traditional least-square solution based on the pseudo-inverse operation, therefore, has its limitation. In addition, the provision of a sufficiently long training sequence to estimate the complete ICI matrix is not feasible, since it will result in severe throughput reduction. In this paper, we derive a lower bound for the mean-square estimation error among the least-square ICI matrix estimators using different training sequences and prove that the minimum mean-square error (MMSE) optimality is attained when the training sequences in different OFDM blocks are orthogonal to each other, regardless of the sequence length. We also prove that the asymptotical mean-square estimation error using the maximal-length shift-register sequences (m-sequences) as in the existing communication standards is 3 dB larger than that using the perfectly orthogonal sequences for ICI matrix estimation. Thus, we propose to employ the training sequences based on the Hadamard matrix to achieve a highly efficient and optimal ICI matrix estimator with minimum mean-square estimation error among all least-square ICI matrix estimators. Meanwhile, our new scheme involves only square computational complexity, while other existing least-square methods require the complexity proportional to the cube of the ICI matrix size. Analytical and experimental comparisons between our new scheme using Hadamard sequences and the existing method using m-sequences (pseudo-random sequences) show the significant advantages of our new ICI matrix estimator. The proposed method is most suitable for OFDM systems with large amount of subcarriers, using high order of subcarrier modulation, and designed for high-end of RF frequency band, where accurate ICI estimation is crucial.

On the peak-to-average power ratio of M-sequences

We prove that in the finite field F = F q , q = 2 m , with a primitive element α , there exists a nonzero element β such that max t ∈ [ 0 , 1 ) ∑ k = 0 q - 2 ( - 1 ) Tr ( β α k ) e 2 π ikt ⩾ 1 π 2 q ln ln q . As an application of this result we show that the peak-to-average power ratio of the maximal-length shift-register sequences ( M-sequences) tends to infinity when their length grows.

Comparison between constant feedback and limiter controllers

Using symbolic dynamics of the one-dimensional unimodal map, the chaos stabilization mechanics of the feedback and limiter control schemes are considered. For feedback control, it is found that the control strength can be efficiently obtained from the superstable parameter of the embedded periodic orbits, and the scaling of the control-space period-doubling bifurcation cascade still obeys the Feigenbaum law. For Sarkovskii orbits, the scaling is also consistent with that of the original chaotic system. For limiter control, a single critical point in the unimodal map is extended to a superstable periodic window and a simple approach for determining the value of the control plateau is found. The scaling in the control space of the period-doubling bifurcation cascade is indeed superexponential. A different scaling for the fine structure of the Sarkovskii sequence is also found. Simple one-dimensional unimodal maps can also be used to generate maximum-length shift-register sequences.

Efficient Design of Event-Related fMRI Experiments Using M-Sequences

Rapid event-related fMRI (erfMRI) allows estimation of the shape of hemodynamic responses (HDR) associated with transient brain activation evoked by various sensory, motor, and cognitive events. Choosing a sequence of events that maximizes efficiency of estimating the HDR is essential for conducting event-related brain imaging experiments, since increasing efficiency is essentially equivalent to reducing scanning time or increasing the strength of the principal magnetic field. The efficiency of an erfMRI design depends critically on the temporal arrangement of the sequence of events and the noise in the fMRI signal. We introduce to erfMRI a simple method for generating efficient event sequences based on maximum-length shift register sequences, or m-sequences. We show that under the assumption of white uncorrelated MRI noise, efficiency of erfMRI experimental designs that employ m-sequences exceeds efficiency of the best randomly generated sequences. This is true for single and multiple event type experiments, which allow either parallel events (overlapping events design) or designs in which only one event occurs at a time (nonoverlapping events design). HDR estimation efficiency afforded by m-sequences grows with the number of event types, and is greatest when event sequences are relatively short, albeit within commonly used scan times (i.e., 63–255 total events per scan). The improvement in efficiency, however, comes at a cost of constraints imposed by m-sequence generation rules, such as predetermined sequence lengths; for nonoverlapping events design m-sequence-based designs are not available for all possible numbers of event types. Nevertheless, designs that are available with m-sequences cover a large subset of commonly used erfMRI experimental designs. Under conditions of characteristic time-correlated fMRI noise, randomly generated sequences may yield efficiencies that exceed those afforded by m-sequences for single event-type designs, since in this case one can generate random sequences that partially decorrelate MRI noise by chance. Our simulations suggest that for designs of realistic sequence lengths that use more than one event type, m-sequence based designs tend to outperform random designs, thus making the knowledge of noise inessential. Finally, within an r-th order m-sequence (generated by a shift register of length r) all possible combinations of subsequences of length r occur, and thus these subsequences are exactly counterbalanced. This property is essential for minimizing effects of psychological and neuronal adaptation and expectation.

On differentially encoded M-sequences

We show that a differentially encoded maximal-length shift register sequence (or M-sequence) is also an M-sequence or the logical complement of an M-sequence. The differentially encoded sequence is a cyclically shifted version of the original M-sequence or its complement.

A pseudorandom pink noise for the computer-based measurements of linear responses

We propose a nonbinary pseudorandom sequence for the measurement of linear responses. Compared with a maximal length shift register sequence (m sequence), this sequence is more suitable for digital processing based on computers; with the use of discrete Fourier transforms, the response functions are reproduced without approximations from digitally sampled data because the input power spectrum is pink, i.e., completely flat in a given frequency range and 0 otherwise. In practice, the new sequence is simply the sum of harmonics with arbitrary phases and is produced readily with digital wave form generators. For reducing the peak power of this sequence, the amplitude distribution is better to be bimodal rather than Gaussian. For demonstrating its feasibility with common digital hardware, the magnetic resonance of Rb atoms in a sub-MHz region was measured successfully. With the use of the fast Fourier transform algorithm, our software task was only 0.4% of that for the cross-correlational calculation based on an m sequence.

M-sequences for OFDM peak-to-average power ratio reduction and error correction

A block coding scheme using the maximum-length shift-register sequences (m-sequences) is suggested for orthogonal frequency division multiplexing (OFDM). The scheme, with a very simple implementation, can significantly reduce the peak-to-average power ratio of OFDM signals and provide error-correcting capability at the same time.

Novel techniques in high speed modem implementation using programmable DSPs

We present the design and performance of a high speed MSK modem that incorporates digital heterodyne processing. A novel table look-up procedure for direct IF/RF modulation, based on a suitable maximal-length shift register sequence, is described. High performance, linear phase, low-complexity mixing and filtering in the receiver is achieved using multirate IFIR filters, where the shaping filter may possibly beM-th-band. The configuration of the IFIR filters is chosen based on the carrier frequency and the filtering requirements.

GENERATION OF PSEUDO-CHAOTIC SEQUENCES

This paper deals with the spectral properties of pseudo-random signals generated in maximum-length shift registers. It is shown that infinitely long registers are comparable in behavior with one-dimensional discrete-time chaotic maps. Therefore the theory, results and tools developed for chaotic systems can be applied to the design of the spectral properties of maximum-length shift register sequences.