Complementary Autocorrelation Research Articles

Two Indicators of Cross-Correlation for the Functions from Zqn to Z2q

Boolean functions play an important role in the design of secure cryptosystems and code division multiple access (CDMA) communication. Several possible generalizations of Boolean functions have been obtained in recent years. In this paper, we analyze the properties of functions from in terms of their Walsh-Hadamard transform (WHT). We provide a relationship between cross-correlation and the WHT of these functions. Also, we present a necessary and sufficient condition for the functions to have complementary autocorrelation. The Parseval’s identity for the current setup of these functions is obtained. Further, we obtained the modulus indicator (MI) and the sum-of-squares-modulus indicator (SSMI) of cross-correlation among two functions for the current setup.

Simultaneous Characterization of Two Ultrashort Optical Pulses at Different Frequencies Using a WS2 Monolayer.

The precise characterization of ultrashort laser pulses has been of interest to the scientific community for many years. Frequency-resolved optical gating (FROG) has been extensively used to retrieve the temporal and spectral field distributions of ultrashort laser pulses. In this work, we exploit the high, broad-band nonlinear optical response of a WS2 monolayer to simultaneously characterize two ultrashort laser pulses with different frequencies. The relaxed phase-matching conditions in a WS2 monolayer enable the simultaneous acquisition of the spectra resulting from both four-wave mixing (FWM) and sum-frequency generation (SFG) nonlinear processes while varying the time delay between the two ultrashort pulses. Next, we introduce an adjusted double-blind FROG algorithm, based on iterative fast Fourier transforms between two FROG traces, to extract the intensity distribution and phase of two ultrashort pulses from the combination of their FWM and SFG FROG traces. Using this algorithm, we find an agreement between the computed and observed FROG traces for both the FWM and SFG processes. Exploiting the broad-band nonlinear response of a WS2 monolayer, we additionally characterize one of the pulses using a second-harmonic generation (SHG) FROG trace to validate the pulse shapes extracted from the combination of the FWM and SFG FROG traces. The retrieved pulse shape from the SHG FROG agrees well with the pulse shape retrieved from our nondegenerate cross-correlation FROG measurement. In addition to the nonlinear parametric processes, we also observe a nonlinearly generated photoluminescence (PL) signal emitted from the WS2 monolayer. Because of its nonlinear origin, the PL signal can also be used to obtain complementary autocorrelation and cross-correlation traces.

Construction of Golay Complementary Matrices and Its Applications to MIMO Omnidirectional Transmission

Golay complementary sequences (GCS) have found wide applications in radar and communications thanks to their prominent property of complementary autocorrelation. This paper studies the extension of Golay complementary sequences to Golay complementary matrices (GCM), i.e, a pair of matrices whose two-dimensional autocorrelations add to be a two-dimensional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula> -function. While the construction of (real) binary GCM are available in the literature, how to construct the quaternary GCM (and the more general unimodular GCM) remains unknown. We put together this missing piece, and show the feasible size that allows for binary and quaternary GCM. The feasible sizes can be further expanded if the phases of the matrices’ entries can be arbitrary. In addition, we consider the extension of the GCM to the set of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N\geq 2$</tex-math></inline-formula> ) matrices, whose two-dimensional autocorrelations add to be a two-dimensional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula> -function, which we refer to as autocorrelation complementary matrices (ACM). For <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N=4$</tex-math></inline-formula> , the ACM allows for matrices of arbitrary size. The GCM/ACM can be applied to omnidirectional transmission for a multi-input multi-output (MIMO) communication system with a uniform rectangular array (URA), which is useful for broadcasting cell-wide common messages. We propose two schemes for omnidirectional transmission, one is based on space-time block code (STBC) and the other is based on the dual cross-polarization of the antenna array. Both schemes can be implemented in the radio frequency (RF) analog domain, generating (theoretically) perfect omnidirectional beampattern with low hardware cost.

Designing Complementary Sequence Sets With Impulse Like Complementary Auto-Correlation Function

In this letter, we address the problem of designing unit modulus complementary sequence set with impulse-like complementary auto-correlation function which has immense potential with applications in ultra wide-band (UWB) communication system, code-division multiple-access (CDMA) system and active sensing systems like MIMO RADAR and SONAR. We have formulated the design problem as the minimization of complementary peak side-lobe level and proposed a monotonic algorithm based on the majorization-minimization technique. Through numerical simulations, we show the construction of sequence sets (of different lengths) with impulse like complementary auto-correlation - which is first of its kind in the sequence design literature. We also discuss the computationally efficient implementation of the proposed algorithm via FFT, IFFT operations.

Design of Unimodular Sequences With Good Autocorrelation and Good Complementary Autocorrelation Properties

Complex sequences with constant magnitude in the time domain and good aperiodic autocorrelation properties are of fundamental interest due to its applications. Their design typically involves the minimization of a nonlinear function that strives to make equal to zero the correlation coefficients in a region of interest. In this letter, the design of unimodular sequences whose aperiodic autocorrelation and aperiodic complementary autocorrelation vanish for a given set of lags is proposed. The sequences are obtained by minimizing a generalized weighted integrated sidelobe level that depends on their complete second-order characterization. Numerical results show that the proposed sequences are ideally suited to identify strictly linear as well as widely linear systems.

A Novel Method for Phase Noise Measurement Based on Cyclic Complementary Autocorrelation

A novel digital signal processing-based method for phase noise measurement on sinusoidal signals is proposed. The method obtains the power spectral density of the phase noise by Fourier transforming its autocorrelation function. The autocorrelation function of the phase noise is in turn derived from the cyclic complementary autocorrelation function of the analytical version of the sinusoidal signal under test. The method proves that, taking advantage of the cyclostationarity properties of the signal under test, it is possible to improve the sensitivity in phase noise measurements by diminishing the effects of the instrumentation noise, which mainly limits the performance of the majority of the digital processing-based approaches.

Nonparametric Spectral Estimation of Phase Noise in Modulated Signals Based on Complementary Autocorrelation

A novel nonparametric spectral estimation technique for phase noise based on the complementary autocorrelation (CAC) is presented. Utilizing the fact that the CAC of phase noise is not zero, this technique offers a distinct advantage that the power spectrum of phase noise is estimated whereas that of other kinds of noise are suppressed as long as they are proper (second-order circularity). This property is most useful when estimating the phase noise spectrum directly from digitally-modulated signals because those signals are typically accompanied by various kinds of undesired noise that easily mask the phase noise of interest. A frequency-domain estimation method based on complementary spectrum is presented for practical computation of phase noise spectrum. For the method, a formula for the noise suppression factor is derived assuming the properness in noise. Furthermore, suppressibility analyses are conducted, which reveal that many of the noises associated with modulated signals are indeed proper, and can thus be suppressed. Numerical simulations demonstrate the method, and verify the noise suppression factor. It is believed that this technique is the first of its kind being able to extract phase noise spectrum while suppressing many other kinds of noise.

A complete description of Golay pairs for lengths up to 100

In his 1961 paper, Marcel Golay showed how the search for pairs of binary sequences of length $n$ with complementary autocorrelation is at worst a $2^{\frac {3n}{2}-6}$ problem. Andres, in his 1977 master’s thesis, developed an algorithm which reduced this to a $2^{\frac {n}{2}-1}$ search and investigated lengths up to 58 for existence of pairs. In this paper, we describe refinements to this algorithm, enabling a $2^{\frac {n}{2}-5}$ search at length 82. We find no new pairs at the outstanding lengths 74 and 82. In extending the theory of composition, we are able to obtain a closed formula for the number of pairs of length $2^kn$ generated by a primitive pair of length $n$. Combining this with the results of searches at all allowable lengths up to 100, we identify five primitive pairs. All others pairs of lengths less than 100 may be derived using the methods outlined.

Monitoring conformational dynamics of a single molecule by selective fluorescence spectroscopy.

A recently developed, real-time spectroscopic technique, burst-integrated fluorescence lifetime (BIFL), is shown to be well suited for monitoring the individual molecular conformational dynamics of a single molecule diffusing through the microscopic, open measurement volume (approximately 10 fl) of a confocal epi-illuminated set-up. In a highly diluted aqueous solution of 20-mer oligonucleotide strand of DNA duplex labeled with the environment-sensitive fluorescent dye tetramethylrhodamine (TMR), fluorescence bursts indicating traces of individual molecules are registered and further subjected to selective burst analysis. The two-dimensional BIFL data allow the identification and detection of different temporally resolved conformational states. A complementary autocorrelation analysis was performed on the time-dependent fluctuations in fluorescence lifetime and intensity. The consistent results strongly support the hypothesized three-state model of the conformational dynamics of the TMR-DNA duplex with a polar, a nonpolar, and a quenching environment of TMR.

Real-time long range complementary correlation optical time domain reflectometer

The improved performance of an optical time-domain reflectometer (OTDR) through the application of a correlation technique using codes with complementary autocorrelation properties is discussed. The theoretical foundations of the method are presented along with experimental results exhibiting the best one-way range reported to date for a practical long-haul, long-wavelength OTDR system.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>