Robust Symmetrical Number System Research Articles

Wideband direction finding using a photonic robust symmetrical number system technique

Dual electrode Mach–Zehnder modulators (DE-MZMs) are used to conduct phase detection for direct wideband direction finding (DF) of microwave signals. It is demonstrated theoretically and through simulation and experimentation that the normalized magnitude of the output signal phase detector circuit is equal to |sin(ψ/2)|, where ψ is the phase difference between the plane waves arriving at the reference and measurement antennas of a linear DF array. A four-element wideband photonic DF system with robust symmetrical number system preprocessing is presented. Simulation and experimental testing results are provided to demonstrate the theoretical concept. The results demonstrate a direct DF receiver using DE-MZMs that achieves fine angular resolution using a much smaller array size than is typically required for linear arrays employing super-resolution signal processing techniques.

Extended Closed-Form Expressions for the Robust Symmetrical Number System Dynamic Range and an Efficient Algorithm for Its Computation

The robust symmetrical number system (RSNS) is a number theoretic transform based on N ≥ 2 sequences that can extract the maximum amount of information from symmetrical folding waveforms. The sequences, based on coprime moduli, exhibit an integer gray code property making the RSNS well suited for many applications that benefit from an inherent error detection and correction capability, such as analog-to-digital converters, direction finding arrays, and radar waveform design. To use the RSNS, it is necessary to know the greatest length of combined sequences without ambiguities, called the dynamic range M, for which only a few closed-form expressions currently exist. In this paper, an efficient algorithm for computing M and its position within the combined set of sequences is presented and shown to be independent of the size of the moduli. The algorithm is used to generate the equations for several groups of additional moduli arrangements. Closed-form expressions for M are conjectured and proved using the obtained congruence equations that define the ambiguity locations.

Increasing the Flux Measurement Range of an RF-SQUID Resonant Detection Circuit Using the Robust Symmetrical Number System

The design and simulation of a new low temperature Niobium radio frequency superconducting quantum interference devices (RF-SQUID) flux measurement architecture concept that uses N = 3 RF-SQUID rings (or channels) to significantly extend the range of the flux measurement capability beyond ±Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> /4 is presented. A resonant detection method based on a robust symmetrical number system (RSNS) preprocessing technique is shown to provide a large increase in the flux measurement range while producing a high-resolution representation of the input magnetic field. The RSNS preprocessing is a modular scheme in which a modulus number of comparators is used at the output of each RF SQUID. The number of comparators with logic 1 in each channel represents the integer values within each RSNS modulus sequence. When considered together, the integers within each sequence change one at a time at the next code position, resulting in an integer Gray code property. We show that the RSNS preprocessing has the feature that the maximum nonlinearity is less than a least significant bit.

Photonic analog-to-digital converter based on the robust symmetrical number system

A novel approach to realizing photonic analog-to-digital conversion with Gray-code-like property is proposed and demonstrated. Instead of using Mach–Zehnder modulators (MZMs) with different half-wave voltages, an array of MZMs with identical half-wave voltages are applied to realize quantization and encoding, which greatly simplifies the implementation. Multiple comparators with preset thresholds are applied at the output of each MZM to improve the number of bits. Through properly setting the bias voltages of the MZMs, a photonic analog to digital converter (ADC) based on the robust symmetrical number system (RSNS) coding method is realized. As an example, a 3-channel structure with maximum quantization level of 17 (corresponding to 4.09 bits) is investigated in detail. We show that the differential encoding technique can be applied in the proposed structure, which increases the equivalent number of bits of the ADC system.

Extending the unambiguous range of polyphase P4 CW radar using the robust symmetrical number system

Polyphase continuous waveform radar systems often use polyphase P4 modulation because of the periodic autocorrelation zero sidelobes and good Doppler tolerance. The P4&apos;s unambiguous detection range is equal to the number of subcodes within a code period. To extend the unambiguous detection range beyond a single-code period, this study develops a novel relationship between the polyphase P4 code and the robust symmetrical number system (RSNS) where the P4 phase values within the code period match the symmetrical residues within an RSNS sequence. By transmitting N≥2 coprime P4 code periods, the unambiguous range is extended by considering the paired values from each sequence. With this new approach, the unambiguous range is extended to the greatest length of combined coprime phase sequences that contain no repeated paired terms. The flexibility of the RSNS allows the unambiguous range to be extended using a fewer number of subcodes within each code period. The combined phase sequences also have an inherent integer Gray code property to control range detection errors. Examples are shown to demonstrate the feasibility of the approach and to compare the unambiguous detection range to that of a single P4 code period.

Robust symmetrical number system preprocessing for minimizing encoding errors in photonic analog-to-digital converters

A photonic analog-to-digital converter (ADC) preprocessing architecture based on the robust symmetrical number system (RSNS) is presented. The RSNS preprocessing architecture is a modular scheme in which a modulus number of comparators are used at the output of each Mach-Zehnder modulator channel. The number of comparators with a logic 1 in each channel represents the integer values within each RSNS modulus sequence. When considered together, the integers within each sequence change one at a time at the next code position, resulting in an integer Gray code property. The RSNS ADC has the feature that the maximum nonlinearity is less than a least significant bit (LSB). Although the observed dynamic range (greatest length of combined sequences that contain no ambiguities) of the RSNS ADC is less than the optimum symmetrical number system ADC, the integer Gray code properties make it attractive for error control. A prototype is presented to demonstrate the feasibility of the concept and to show the important RSNS property that the largest nonlinearity is always less than a LSB. Also discussed are practical considerations related to multi-gigahertz implementations.

Error Detection and Correction in Communication Channels Using Inverse Gray RSNS Codes

A novel number theoretic transform called Inverse Gray Robust Symmetrical Number System (IGRSNS) is proposed for error control coding in this paper. IGRSNS is obtained by modifying Robust Symmetrical Number System (RSNS) that was proposed earlier, using Inverse Gray code property. Due to ambiguities present in each residue, RSNS has a short dynamic range (DR) compared to that in other number systems. The short DR of RSNS enables it to be effectively used for error detection without the addition of any redundant modulus as in Redundant Residue Number System. Although RSNS has a large redundant range, its detection ability is not optimal due to the Gray code property associated with it. In an attempt to overcome this limitation, we have proposed Inverse Gray coding to be combined with RSNS in increasing its effectiveness in error detection. The resulting IGRSNS thus inherits properties of both RSNS and Gray code. Analysis and simulations show that IGRSNS has a near-optimal error detection ability. The potential of IGRSNS for error correction is also investigated. Further, a novel error correction algorithm using one redundant modulus is proposed. Simulations show that the proposed algorithm performs well under all cases of single bit errors.

A robust symmetrical number system based parallel communication system with inherent error detection and correction

This paper presents a novel robust number theoretic transform called inverse gray robust symmetrical number system (IGRSNS) and proposes its application for CDMA systems. The transceiver structure for three moduli IGRSNS-CDMA with one redundant modulus is proposed. The proposed system is simulated for different values of moduli sets under various channel conditions. Simulation results show that the BER performance of IGRSNS-CDMA outperforms that of RNS-CDMA under all circumstances.

RSNS-to-Binary Conversion

The robust symmetrical number system (RSNS) is a symmetrical number system formed such that only one element in each RSNS vector changes between code transitions. This integer Gray-code property makes the RSNS useful in analog-to-digital converters (ADC), direction finding antenna architectures and electro-optical ADCs since it eliminates any encoding errors that can occur when the input signal lies about any code transition point. One of the fundamental difficulties in a hardware application of the RSNS, such as an ADC, is the conversion of the RSNS residue vectors into a more convenient representation such as a binary number. By exploiting the one-to-one correspondence between the RSNS and the residue number system, an efficient process for conversion of the symmetrical residues is formed. The conversion process presented in this paper produces a hardware implementation that is at least an order of magnitude smaller in terms of transistor count than read-only-memory conversion methods, and is easily pipelined to achieve fast conversion speeds.

$N$-Sequence RSNS Ambiguity Analysis

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The robust symmetrical number system (RSNS) is a modular system formed using <emphasis><formula formulatype="inline"><tex>$N \geq 2$</tex></formula></emphasis> integer sequences and ensures that two successive RSNS vectors (paired terms from all <emphasis><formula formulatype="inline"><tex>$N$</tex></formula></emphasis> sequences) differ by only one integer. This integer Gray-code property reduces the possibility of encoding errors and makes the RSNS useful in applications such as folding analog-to-digital converters (ADCs), direction finding antenna architectures, and photonic processors. This paper determines the length of combined sequences that contain no vector ambiguities. This length or longest run of distinct vectors we call the RSNS dynamic range <emphasis><formula formulatype="inline"> <tex>$(\mathhat{M})$</tex></formula></emphasis>. The position of <emphasis><formula formulatype="inline"><tex>$\mathhat{M}$</tex></formula></emphasis> which is the starting point in the sequence is also derived. Computing <emphasis><formula formulatype="inline"><tex>$\mathhat{M}$</tex></formula></emphasis> and the position of <emphasis><formula formulatype="inline"><tex>$\mathhat{M}$</tex> </formula></emphasis> allows the integer Gray-code properties of the RSNS to be used in practical applications. We first extend our two-sequence results to develop a closed-form expression for <emphasis><formula formulatype="inline"> <tex>$\mathhat{M}$</tex></formula></emphasis> for a three-sequence RSNS with moduli of the form <emphasis><formula formulatype="inline"><tex>$2^r -1, 2^r, 2^r+1$</tex></formula></emphasis>. We then extend the results to solving the <emphasis><formula formulatype="inline"><tex>$N$</tex></formula></emphasis> -sequence RSNS ambiguity locations in general. </para>

Two-channel RSNS dynamic range

Symmetrical number systems have been explored for many applications in both analog and digital signal processing due to the common availability of symmetrical folding waveforms (e.g., cos/sup 2/). The robust symmetrical number system (RSNS) is a modular scheme in which the integer values within each modulus, when considered together, change one at a time at the next code position (Gray code properties). Although the RSNS has a smaller dynamic range than the optimum symmetrical number system, the RSNS Gray code properties make it particularly attractive for error control. In the past, computer search algorithms have been used to determine the RSNS dynamic range (length of unambiguous vectors). In this brief, we define the two-channel RSKS and present a theorem that gives its dynamic range for relatively prime moduli m/sub 1/, m/sub 2/, 5 /spl les/ m/sub 1/ /spl les/ m/sub 2/.

High-resolution phase sampled interferometry using symmetrical number systems

This paper identifies a new phase sampling interferometer approach that can he easily incorporated into the established techniques to provide a high resolution, small-baseline array with a fewer number of phase sampling comparators. The approach is based on preprocessing the received signal using symmetrical number systems (SNS). Antennas based on both an optimum symmetrical number system (OSNS) and a robust symmetrical number system (RSNS) are investigated. The SNS preprocessing is used to decompose the spatial filtering operation into a number of parallel suboperations (moduli) that are of smaller computational complexity. A much higher direction finding (DF) spatial resolution is achieved after the N different moduli are used and the results of these low precision suboperations are recombined. By incorporating the OSNS or RSNS preprocessing concept, the field of view of a specific configuration of interferometers and phase sampling comparator arrangements can be analyzed exactly. The OSNS gives the maximum dynamic range or number of spatial resolution bins while the RSNS reduces considerably the number of possible encoding errors. Experimental results for both a 5-bit OSNS and a 6-bit RSNS array are compared. The errors in the encoding of the direction of arrival are quantified for both architectures.

A folding ADC preprocessing architecture employing a robust symmetrical number system with gray-code properties

A folding analog-to-digital converter (ADC) preprocessing architecture based on a new robust symmetrical number system (RSNS) is presented. The RSNS preprocessing architecture is a modular scheme in which the integer values within each modulus (comparator states), when considered together, change one at a time at the next position (gray-code properties). Although the observed dynamic range of the RSNS ADC is less than the optimum symmetrical number system ADC, the RSNS gray-code properties make it particularly attractive for error control. With the RSNS preprocessing, the encoding errors due to comparator thresholds not being crossed simultaneously are eliminated. As a result, the interpolation circuits can be removed and only a small number of comparators are required. Computer generated data is used to help determine the properties of the RSNS. These properties include the dynamic range (largest number of distinct consecutive vectors) and the location of the dynamic range within the number system. Closed-form expressions for the dynamic range are also presented for channel moduli of the form m/sub 1/=2/sup k/-1, m/sub 2/=2/sup k/, m/sub 3/=2/sup k/+1. RSNS ADC circuit design principles are presented. To compare the advantages of the RSNS ADC with previously published results, the transfer function of a 3-channel architecture (k=2) is evaluated numerically using SPICE.