Quadratic Residue Sequence Research Articles

The Distribution Properties of Consecutive Quadratic Residue Sequences

We consider any prime number p . Let k , s be two positive integers. We are interested in the arithmetic progressions (sequences) with the common difference s and length k , where the sequence entries are from the set of quadratic residue modulo p or the set of quadratic nonresidue modulo p . The numbers of such sequences are denoted as N p k , s and N p ′ k , s , respectively. In this paper, we apply analytic number theory methods, in particular, properties of Legendre’s symbol modulo p and character sums, to study the numbers N p k , s and N p ′ k , s . Exact formulas are given for certain values of k and s under some restrictions. In addition, estimation formulas in other cases are given.

Design and verification of ultrathin Schroeder diffuser based on membrane-type resonators

We present the design and experimental verification of a membrane-type resonator-based Schroeder diffuser (MRSD). The membrane-type resonator consists of a decorated membrane and a rigid back cavity. By combining multiple resonators with the phase shift profile based on the quadratic residue sequence, the MRSD can produce a high efficiency diffuse sound field. Furthermore, the designed MRSD has a thickness with deep-subwavelength scale, which is 1 order of magnitude thinner than the conventional diffuser constructed with periodical wells. The proposed MRSD has advantages in low frequency applications and offers versatility for novel manipulations of sound.

An ultrathin planar acoustic metasurface diffuser with narrowband uniform reflection

Diffuse reflection of sound is desirable in many practical scenarios, such as architectural acoustics, but most existing designs of acoustic diffusers have bulky size, corrugated profile, or limited spatial resolution. We design an ultrathin planar acoustic diffuser for producing narrowband diffuse reflection with high uniformity via precise modulation of reflected wavefront and propose a metasurface-based implementation comprising a monolayer of Helmholtz-like resonators much smaller than the working wavelength in all three dimensions. Our design is benchmarked with a commercialized Schroeder diffuser and is numerically proven to be capable of scattering the illuminating wave more uniformly than the conventional mechanisms based on the quadratic residue sequence over wide incident angles. We anticipate our design with simplicity and capability to find important applications in diverse scenarios.

Study of LDPC decoders with quadratic residue sequence for communication system

This paper shows an effective combination of LDPC codes along with DSSS technique to achieve the most reliable and an efficient transmission of information signal. Here the LogSPA and SSD decoders are selected for the decoding task and spreading of signal in DSSS is achieved by using quadratic residue sequences that are based on prime numbers as pseudo-noise sequences (PN sequence). In particular the PN sequences being used here are Legendre and Weil sequences. Due to the significant increase in the trend of PN sequence's application in global positioning system (GPS) and satellite communication systems, in the present work an attempt is done to show suitability of PN sequences based on quadratic residues with LDPC codes which have gained substantial importance in recent advancements due to their excellent error-correcting capabilities.

Study of LDPC decoders with quadratic residue sequence for communication system

This paper shows an effective combination of LDPC codes along with DSSS technique to achieve the most reliable and an efficient transmission of information signal. Here the LogSPA and SSD decoders are selected for the decoding task and spreading of signal in DSSS is achieved by using quadratic residue sequences that are based on prime numbers as pseudo-noise sequences (PN sequence). In particular the PN sequences being used here are Legendre and Weil sequences. Due to the significant increase in the trend of PN sequence's application in global positioning system (GPS) and satellite communication systems, in the present work an attempt is done to show suitability of PN sequences based on quadratic residues with LDPC codes which have gained substantial importance in recent advancements due to their excellent error-correcting capabilities.

Sound absorbers with a micro-perforated panel backed by an array of parallel-arranged sub-cavities at different depths

This paper presents a type of wide band micro-perforated panel (MPP) absorber with an array of parallel arranged sub-cavities at different depths based on the quadratic residue sequence (QRS). An analytical prediction model is proposed for evaluating the normal incidence sound absorption coefficients of this type of MPP absorber in its design stage. Next, simulations with a finite element procedure and experimental measurements were conducted to provide cross-validations of the proposed analytical model and to facilitate the discussions on the sound absorption performance of this type of MPP absorber. The results show that the analytical model provides simple yet accurate prediction on the sound absorption performance of this MPP absorber; the absorption bandwidth of this type of MPP absorber can be significantly enhanced through the fine design on the sub-cavity depth sequence. It is also shown that the normal incidence absorption coefficient of this type of MPP absorber can remain beyond 0.5 in the frequency range of 450–3500 Hz, with the maximum value of 0.98.

Designing of variable Schroeder diffusers using digital architecture techniques

Schroeder diffusers consist of a series of wells of the same width and different depths. The wells are separated by thin fins. The depths of the wells are determined by a mathematical number sequence, such as the quadratic residue sequence. The sequence number for the nth well, Sn, is given by1: Sn = n2modulo N where modulo indicates the least non-negative remainder and is often written as mod for short. N is the number generator which in this case is also a prime and the number of wells per period. The well width follow the simple equation as follow in which λmin is the minimum wavelength before cross-modes in the wells appear1: W = λmin / 2 In this paper it is intended to build a physical model of a N = 7 diffuser and make the depth of each well separately variable by means of digital architecture tools. The tools which is referred to would be Arduino kit micro controller and 7 stepper motors and 7 drivers. The idea behind this practice is twofold. First is to come up with a variable diffuser which works in several (preferably 5) design frequencies and hence their integer multiples. The user can make this adjustment via a simple computer program. Second, to try other mathematical number sequence and observe the diffusion characteristics of new products using the analyses of acoustic measurement results. 1. Acoustic Absorbers and Diffusers Theory, Design and Application, T. J. Cox and P. D’Antonio, 2nd ed.

Improving the indoor sound quality by using cymatic shapes

Acoustic diffusers are important components in enhancing the quality of room acoustics. This paper investigates a new type of 2D diffusers obtained by the Cymatics phenomena. Cymatics is the study of sound and vibration made visible, typically on the surface of a plate, diaphragm, or membrane. Four shapes of the diffusers were designed by the Cymatic shapes and modeled by using a quadratic residue sequence. The polar response of the diffusers was measured using DIRAC software. Polar response results were generally consistent with expectations. This type of diffusers can generate a uniform polar response over the frequency range we are interested in (400–4000 Hz). It is found that this type of acoustic diffusers can be used to maintain the acoustic energy in a room and at the same time can treat unwanted echoes and reflections by scattering sound waves in many directions.

Improvement of Mode Distribution in a Triangular Prism Reverberation Chamber by QRS Diffuser

This paper presents the field uniformity characteristics in a triangular prism reverberation chamber that can be substituted for an open area test site or an anechoic chamber to measure electromagnetic interference. To improve size problems of a stirrer that is an official unit to generate a uniform field in the reverberation chamber, we suggest a diffuser of Quadratic Residue Sequence method. To validate the substitution of a diffuser for a stirrer, a diffuser is designed for 1-3GHz, and three types of equilateral triangular prism reverberation chambers are modeled. Afterwards, the field distributions in these three reverberation chambers are both simulated and tested. Using XFDTD 6. 2 of finite difference time domain method, field deviations of each structure are simulated and compared to each other. An evaluation of field uniformity is done by cumulative probability distribution which is specified in the IEC 61000-4-21. The result shows that the field uniformity in the chamber is within ±6dB tolerance and also within ±3dB standard deviation, which means a diffuser can satisfy the requirement of international standards.

Optimization of Directivity of Point Source Array by Using Pseudostochastic Sequences

We investigate the methods to optimize the directivity of point source array by using pseudostochastic sequences. Maximum-length sequence (MLS) array and Quadratic-residue sequence (QRS) array are theoretically analysed, and their simulated responses are shown. The results indicate that pseudostochastic sequences can be used to optimize the directivity of point source array.

Complex-valued coded excitations for acoustical channel measurements

Coded signals have been increasingly applied in acoustical/bio-ultrasonic measurements in order to increase the measurement quality or to cope with demanding requirements [Xiang and Schroeder (2003)]. Increasing interest in coded signals as acoustical excitations has prompted development of efficient correlation algorithms [Daigle and Xiang, J. Acoust. Soc. Am. (2006)]. These algorithms allow efficient processing of complex-valued sequences as needed in critical acoustical measurements, such as using simultaneous sources. Exploiting excellent correlation properties of some coded signals, this work applies complex-valued quadratic residue (QR) sequences and Zech-logarithm sequences in both base-band acoustical channel measurements using sequential excitations and band-limited acoustical channel measurements using amplitude-modulation schemes. Numerical simulations and laboratory experiments carried out in this work demonstrate feasibility of the techniques. This paper discusses advantages and challenges when using coded excitations in potential acoustical applications.

Polyphase power–residue sequences

Polyphase versions of the well–known binary Legendre or quadratic–residue sequences were first described and analysed by Sidelnikov over thirty years ago, but have since received very little attention. Here, it is shown that these q –phase sequences of prime length L can also be constructed from the index sequence of length L or, equivalently, from the cosets of q th power residues and non–residues. These sequences are also shown to fall into two classes, each of which has well–defined periodic autocorrelation functions and merit factors. Class–I sequences exist for prime L ≡ q + 1 mod 2 q and q even, whereas class–II sequences correspond to primes of the form L ≡1 mod 2 q and are available for all values of q . Class–I sequences have complex out–of–phase correlation values with magnitudes less than or equal to 5 , whereas for class–II sequences these are purely real with magnitude less than or equal to 3. Some other properties are also investigated.

New designs for signal sets with low cross correlation, balance property, and large linear span: GF(p) case

New designs for families of sequences over GF(p) with low cross correlation, balance property, and large linear span are presented. The key idea of the new designs is to use short p-ary sequences of period /spl upsi/ with the two-level autocorrelation function together with the interleaved structure to construct a set of long sequences with the desired properties. The resulting sequences are interleaved sequences of period /spl upsi//sup 2/. There are /spl upsi/ cyclically shift distinct sequences in each family. The maximal correlation value is 2/spl upsi/ + 3 which is optimal with respect to the Welch bound. Each sequence in the family is balanced and has large linear span. In particular, for binary case, cross/out-of-phase autocorrelation values belong to the set {1, -/spl upsi/, /spl upsi/ + 2, 2/spl upsi/ + 3, -2/spl upsi/ - 1}, any sequence where the short sequences are quadratic residue sequences achieves the maximal linear span. It is shown that some families of these sequences can be implemented efficiently in both hardware and software.

Linear complexity of polyphase power residue sequences

The well known family of binary Legendre or quadratic residue sequences can be generalised to the multiple-valued case by employing a polyphase representation. These p-phase sequences, with p prime, also have prime length L, and can be constructed from the index sequence of length L or, equivalently, from the cosets of pth power residues and non-residues modulo-L. The linear complexity of these polyphase sequences is derived and shown to fall into four classes depending on the value assigned to b0, the initial digit of the sequence, and on whether p belongs to the set of pth power residues or not. The characteristic polynomials of the linear feedback shift registers that generate these sequences are also derived.

Modified Jacobi sequences

Quadratic residue and twin prime sequences are well known types of binary sequence with ideal periodic autocorrelation functions. These are just special cases of much larger families of sequences referred to here as Legendre sequences and modified Jacobi sequences. Although these more general forms are suboptimal and do not have ideal autocorrelation functions, nevertheless they do exhibit out-of-phase autocorrelation values which are independent of their lengths, and so the longer sequences may be useful. The structure and properties of these sequences are investigated, and a two-dimensional array representation that enables the sequences and their autocorrelation functions to be derived in a simple and compact manner are employed.

On the bandwidth of diffusors based upon the quadratic residue sequence

The lower and upper frequency boundary are discussed within which a diffusor based upon the quadratic residue sequence shows its typical operation. Using a qualitative model of the diffusor’s way of operation these limits are found to be related with the diffusor’s width and well width, respectively. Therefore, the prime number N on which the underlying quadratic residue sequence is constructed is an indicator for the bandwidth. The choice of the design frequency is discussed.

A reflection grating that nullifies the specular reflection: A cone of silence

This paper describes the geometry of an acoustical diffusor that nullifies the specular reflection of plane sound waves whose frequencies include a design frequency and all its harmonics up to a certain limit. The diffusor is a specially designed acoustical phase grating whose phase cancellation properties are governed by concepts from pure number theory going back to the great mathematician Karl Friedrich Gauss in the early 1800s and in the late 1920s to applications of the theory by Manfred R. Schroeder of Götingen and Bell Laboratories. Presented here is an exposition of Schroeder’s ground‐breaking quadratic residue diffusor, using sufficient mathematical rigor to prove its astounding uniform reflection property, originally conceived for improved concert hall acoustics. It then becomes clear how to modify the quadratic residue sequences to what is called modified primitive root sequences to achieve the nullification of the specular reflection. Since the method, for any angle of incidence, results in cancellation in the specular direction and throughout an angular region on each side of the specular direction, the cancellations occur in a cone‐shaped region. In particular, for normal incidence the sender of a signal sits in the cone—a cone of silence.

A new number-theoretic acoustical concrete masonry unit

A new number-theoretic structural acoustic concrete masonry unit will be described. These new acoustical building blocks uniformly disperse reflections created by the flat surfaces of conventional absorbing block over a broad range of frequencies, while simultaneously providing extended broadband low-frequency absorption, load-bearing reinforced construction, fire-safety, high transmission loss, economy, and an aesthetic appearance. Random-incidence acoustical testing yielded an NRC of 0.85 for the unpainted block and 0.40 for the painted block, with 100% efficiency at 100 Hz. The STC for painted block was 55. Structural testing for both compressive strength and flexural mid-span deflection were in compliance with ASTM C90-90. A novel cavity design, which provided extended low-frequency absorption will be described. The system consists of three blocks. The A and B blocks facing the noise source each contains a partial quadratic residue sequence and the rear mounting C block forms a structural cavity. The acoustical properties of the system address the needs of all musical facilities, movie theaters, auditoriums, school rehearsal rooms, and noise control applications including highway barriers, airports, power plant transformer substations, etc.

Cinder block modular diffusor

The present invention relates to an acoustical diffusor device which is made up of a plurality of specially designed and shaped cinder or concrete blocks which may be assembled together through the use of mortar to provide a diffusor of desired shape and configuration. Each diffusor includes a plurality of wells, the depths of which are determined through the use of number theory sequences, such as the quadratic-residue sequence developed by Karl Frederick Gauss. These surface irregularities are unique in that they provide a flat power spectrum and constant scattered energy in the diffraction directions.

Cyclically orthogonal sequences with low multiplicity of values derived from quadratic residues

AbstractOrthogonal cyclic sequences with low multiplicity of values are given for application to spread spectrum communications systems. Two‐, three‐, and five‐valued orthogonal cyclic sequences are obtained by representing real orthogonal cyclic sequences of length N by cosine series possessing phase constant of odd function, or even function with values {0, }, and by relating the phase constant to quadratic residues of N. We use the fact that an extended quadratic residue sequence taking values {0, +1, ‐1} becomes a sequence of odd function satisfying homogeneous equation of discrete sine transform when N is a prime number of the form N =4° ‐ 1 ( is a natural number). When N is a prime number of the form N =4° + 1, the sequence becomes that of even function satisfying a homogeneous equation of discrete cosine transform. Three‐valued orthogonal cyclic sequences are obtained by adding a dc component, to quadratic residue sequences of these odd and even functions. A two‐valued orthogonal cyclic sequence is obtained when a phase constant is given by a quadratic residue sequence of odd function multiplied by a constant. Moreover, when the phase constant is a monomial of odd power and is related to the quadratic residue sequence, a three‐valued orthogonal cyclic sequence is obtained for the prime N =4° ‐ 1, and a five‐valued orthogonal cyclic sequence is obtained for the prime N =4° + 1. Since N sequences obtained by shifting one of those sequences form an orthogonal system, cross‐talk does not occur when these sequences are applied to synchronous spread spectrum multiple access.