Perfect Periodic Sequences Research Articles

HRTFs Measurement Based on Periodic Sequences Robust towards Nonlinearities in Automotive Audio.

The head related transfer functions (HRTFs) represent the acoustic path transfer functions between sound sources in 3D space and the listener's ear. They are used to create immersive audio scenarios or to subjectively evaluate sound systems according to a human-centric point of view. Cars are nowadays the most popular audio listening environment and the use of HRTFs in automotive audio has recently attracted the attention of researchers. In this context, the paper proposes a measurement method for HRTFs based on perfect or orthogonal periodic sequences. The proposed measurement method ensures robustness towards the nonlinearities that may affect the measurement system. The experimental results considering both an emulated scenario and real measurements in a controlled environment illustrate the effectiveness of the approach and compare the proposed method with other popular approaches.

Robust Room Impulse Response Measurement Using Perfect Periodic Sequences for Wiener Nonlinear Filters

The paper discusses a measurement approach for the room impulse response (RIR), which is insensitive to the nonlinearities that affect the measurement instruments. The approach employs as measurement signals the perfect periodic sequences for Wiener nonlinear (WN) filters. Perfect periodic sequences (PPSs) are periodic sequences that guarantee the perfect orthogonality of a filter basis functions over a period. The PPSs for WN filters are appealing for RIR measurement, since their sample distribution is almost Gaussian and provides a low excitation to the highest amplitudes. RIR measurement using PPSs for WN filters is studied and its advantages and limitations are discussed. The derivation of PPSs for WN filters suitable for RIR measurement is detailed. Limitations in the identification given by the underestimation of RIR memory, order of nonlinearity, and effect of measurement noise are analysed and estimated. Finally, experimental results, which involve both simulations using signals affected by real nonlinear devices and real RIR measurements in the presence of nonlinearities, compare the proposed approach with the ones that are based on PPSs for Legendre nonlinear filter, maximal length sequences, and exponential sweeps.

Orthogonal Periodic Sequences for the Identification of Functional Link Polynomial Filters

The paper introduces a novel family of deterministic periodic signals, the orthogonal periodic sequences (OPSs), that allow the perfect identification on a finite period of any functional link polynomials (FLiP) filter with the cross-correlation method. The class of FLiP filters is very broad and includes many popular nonlinear filters, as the well-known Volterra and the Wiener nonlinear filters. The novel sequences share many properties of the perfect periodic sequences (PPSs). As the PPSs, they allow the perfect identification of FLiP filters with the cross-correlation method. But, while PPSs exist only for orthogonal FLiP filters, the OPSs allow also the identification of non-orthogonal FLiP filters, as the Volterra filters. In OPSs, the modeled system input can be any persistently exciting sequence and can also be a quantized sequence. Moreover, OPSs can often identify FLiP filters with a sequence period and a computational complexity much smaller that PPSs. The provided experimental results, involving the identification of real devices and of a benchmark model, highlight the potentialities of the proposed OPSs in modeling unknown nonlinear systems.

Nonlinear system identification using Wiener basis functions and multiple-variance perfect sequences

Abstract The paper addresses nonlinear identification using the Wiener series. Differently from the traditional approach, the truncated Wiener series is expressed as a linear combination of basis functions, which are orthogonal for white Gaussian inputs. The coefficients of the basis functions are efficiently estimated with the cross-correlation method, computing the cross-correlation between the basis functions and the system output. Perfect periodic sequences (PPSs), which are periodic sequences guaranteeing the perfect orthogonality of the basis functions over a period, are also developed. The PPSs allow to avoid the estimation problems experienced with the cross-correlation method using stochastic inputs. The Wiener series formulation in terms of basis functions allows also to develop a novel, more efficient, multiple-variance identification method. Multiple-variance methods exploit input signals with multiple variances for estimating the Volterra kernels. They overcome the problem of locality of the solution, i.e., the fact that the identified model well approximates the nonlinear system only for input signal variances close to that used for the identification. Optimal values of the multiple variances are also studied in the paper. Experimental results, involving the identifications of real devices, show that the proposed approach can accurately model the identified system on a wide range of input variances.

Periodic Sequences Modulated Filter Banks

This letter studies nearly perfect reconstruction periodic sequences modulated filter banks, which includes many useful examples, e.g., the discrete Fourier transform (DFT), generalized DFT, and cosine-modulated ones. A general framework is developed to meet arbitrary, but feasible design constraints, e.g., length of filters, system delay, decimation ratio, phase linearity, etc., and it also reveals new design freedoms, i.e., phase shifts in the modulation sequences. An efficient Newton algorithm is proposed, and a MATLAB/Octave package with diverse design examples is provided. Numerical results show that better designs can be obtained by searching for the solutions in a larger space than the one from traditional frameworks.

Robust Room Impulse Response Measurement Using Perfect Sequences for Legendre Nonlinear Filters

The paper proposes a novel approach for measuring the room impulse response that is robust toward the nonlinearities affecting the power amplifier or the loudspeaker. The approach is implemented by modeling the acoustic path as a Legendre nonlinear filter and by measuring the first-order kernel using perfect periodic sequences and the cross-correlation method. Perfect sequences for Legendre filters are periodic sequences that guarantee the orthogonality of the Legendre basis functions over a period. They ensure the robustness of the first kernel measurement toward nonlinear distortions. The paper also explains how perfect periodic sequences for Legendre filters that are suitable for room impulse response identification can be developed. Experimental results involving both simulated and real environments illustrate the effectiveness and the robustness of the proposed approach.

A study about Chebyshev nonlinear filters

The paper studies a novel family of nonlinear filters based on Chebyshev polynomials of the first kind, the Chebyshev nonlinear filters. This family shares many of the characteristics of the recently introduced Legendre and even mirror Fourier nonlinear filters, but has also peculiar properties. Chebyshev nonlinear filters belong to the class of linear-in-the-parameters nonlinear filters. Their basis functions are polynomials, specifically, products of Chebyshev polynomial expansions of the input signal samples. According to the Stone–Weierstrass theorem, they are universal approximators for causal, time-invariant, finite-memory, continuous, nonlinear systems. Their basis functions are mutually orthogonal for white input signals with a particular nonuniform distribution. They admit perfect periodic sequences, i.e., periodic input sequences that guarantee the mutual orthogonality of the basis functions on a finite period. Using perfect periodic input signals, an unknown nonlinear system and its most relevant basis functions can be identified with the cross-correlation method. It is shown in the paper that the perfect periodic sequences of Chebyshev nonlinear filters are simply related to those of even mirror Fourier nonlinear systems. Experimental results involving a real nonlinear system illustrate the potentialities of these filters.

Nonlinear system identification using quasi-perfect periodic sequences

Perfect periodic sequences are currently used for modeling linear and nonlinear systems. A periodic sequence, applied as input to a linear or nonlinear system, is called perfect if the basis functions of the modeling filter are orthogonal to each other, and thus the auto-correlation matrix is diagonal. In this paper, we introduce quasi-perfect periodic sequences for a sub-class of linear-in-the-parameters nonlinear filters, called functional link polynomial filters, which is derived by using the constructive rule of Volterra filters. A periodic sequence is defined as quasi-perfect for a nonlinear filter if the resulting auto-correlation matrix is block-diagonal and highly sparse. Moreover, the samples of the sequence are represented by only a few discrete levels. It is shown in the paper that quasi-perfect periodic sequences for third-order systems can be obtained by means of a simple combinatorial rule. The derived sequences, which are the same for all functional link polynomial filters, allow an efficient implementation of the least-squares approximation method. Simulation results and a real-world experiment show good performance of the proposed identification method.

Legendre nonlinear filters

The paper discusses a novel sub-class of linear-in-the-parameters nonlinear filters, the Legendre nonlinear filters. The novel sub-class combines the best characteristics of truncated Volterra filters and of the recently introduced even mirror Fourier nonlinear filters, in particular: (i) Legendre nonlinear filters can arbitrarily well approximate any causal, time-invariant, finite-memory, continuous, nonlinear system; (ii) their basis functions are polynomials, specifically, products of Legendre polynomial expansions of the input signal samples; (iii) the basis functions are also mutually orthogonal for white uniform input signals and thus, in adaptive applications, gradient descent algorithms with fast convergence speed can be devised; (iv) perfect periodic sequences can be developed for the identification of Legendre nonlinear filters. A periodic sequence is perfect for a certain nonlinear filter if all cross-correlations between two different basis functions, estimated over a period, are zero. Using perfect periodic sequences as input signals permits the identification of the most relevant basis functions of an unknown nonlinear system by means of the cross-correlation method. Experimental results involving identification of real nonlinear systems illustrate the effectiveness and efficiency of this approach and the potentialities of Legendre nonlinear filters.

Perfect periodic sequences for even mirror Fourier nonlinear filters

The paper deals with the identification of a class of nonlinear filters recently introduced in the literature, the even mirror Fourier nonlinear filters, and shows that perfect periodic sequences can be developed for these filters. A periodic sequence is perfect for a nonlinear filter if all cross-correlations between two different basis functions, estimated over a period, are zero. By applying perfect periodic sequences as input signals to even mirror Fourier nonlinear filters, it is possible to model unknown nonlinear systems using the cross-correlation method. Moreover, the most relevant basis functions, i.e., those that guarantee the most compact representation of the nonlinear system according to some information criterion, can be easily estimated. Experimental results on the identification of real nonlinear systems illustrate the main advantage of the proposed method which is the remarkable reduction in the computational complexity.

Optimal design of perfect DFT sequences

In this paper, we propose a novel scheme to construct a large set with N(N+1) perfect sequences of length N, derived from an Inverse Discrete Fourier Transform (IDFT) of Chu and maximum-length sequences. This optimum set of perfect periodic autocorrelation sequences has a maximum absolute value of periodic cross-correlation strictly lower than N and close to the well known lower bound SQRT(N). Moreover, we present a method to transform these perfect sequences into orthogonal sequences. A similar method is also proposed to obtain optimum bipolar codes derived from an alternative set of N2 perfect sequences.In the design of perfect sequences, the difficulty is to achieve both low cross-correlation and low peak-to-average power ratio (PAPR). Many of the proposed perfect DFT sequences should have low PAPR and thus can be applied in an OFDM–CDMA (Orthogonal Frequency-Division Multiplexing–Code Division Multiple Access) communication system or in a simple CDMA communication system. Alternatively, the proposed perfect DFT sequences may be useful for pre-coded mapping in OFDM communication systems or for the design of radar waveform diversity sets.

Orthogonal perfect discrete Fourier transform sequences

In this study, the authors propose a method to obtain many different sets of N orthogonal perfect periodic autocorrelation sequences, with length N, derived from Walsh-Hadamard codes and orthogonal Gold codes. Moreover, these orthogonal perfect discrete Fourier transform (DFT) sequences can be transformed into real orthogonal perfect sequences or into bipolar codes with better correlation properties than other well known bipolar codes used in code division multiple access (CDMA) systems.

Efficient NLMS and RLS Algorithms for Perfect and Imperfect Periodic Sequences

The paper discusses computationally efficient NLMS and RLS algorithms for perfect and imperfect periodic excitation sequences. The most interesting aspect of these algorithms is that they are exact LMS and RLS algorithms suitable for identification and tracking of every linear system and they require a real-time computational effort of just a multiplication, an addition and a subtraction per sample time. Moreover, the algorithms have convergence and tracking properties that can be better than or comparable with the NLMS algorithm for white noise input. The transient and steady state behavior of the algorithms and their tracking properties are also studied in the paper.

Perfect and almost perfect sequences

The autocorrelation function of a sequence is a measure for how much the given sequence differs from its translates. Periodic binary sequences with good correlation properties have important applications in various areas of engineering. In particular, one needs sequences with a two-level autocorrelation function, that is, all nontrivial autocorrelation coefficients equal some constant γ. The case where γ is as small as theoretically possible in absolute value has turned out to be especially useful; such sequences are called perfect. Unfortunately, in many cases perfect sequences cannot exist, and so one has also considered “almost perfect” sequences, where one allows one nontrivial autocorrelation coefficient to be different from γ. In this paper, we concentrate on the existence problem for perfect and almost perfect binary periodic sequences; such sequences are actually equivalent to certain cyclic difference sets and cyclic divisible difference sets, respectively, structures which have been studied in Design Theory for a long time. This connection allows one to obtain strong results on the existence of (almost) perfect sequences. We also discuss some related questions, namely the aperiodic case, ternary perfect sequences, binary perfect arrays, and the merit factor of a sequence. Finally, we conclude the paper with a simplified description of an interesting real-world application, namely “coded aperture imaging”. Our paper serves mainly as a survey of the area just outlined, but it also contains some new results.

Perfect periodic correlation sequences

Sequences with the property of perfect periodic autocorrelation are used in various fields such as communication and radar systems. Various methods were introduced to construct such sequences. Most of these methods concentrated on binary or N-phase sequences. In this paper two methods to construct general phase sequences are introduced. The methods are based upon the properties of the Fourier transform of perfect periodic autocorrelation sequences. The sensitivity of these codes to errors is also elaborated.