Dolph-Chebyshev Windows Research Articles

New methods for calculation of Dolph-Chebyshev and Barsilon-Temes window functions and their modifications

Results of detailed theoretical studies of widely used Dolph-Chebyshev and Barsilon-Temes windows are presented. It is proved that the normalized spectrums of these windows are identically determined by a finite number of cosine functions. In the case of application of Chebyshev functions of the first and second kinds of even order, cosine functions of even arguments are used, and cosine function of odd arguments are used for Chebyshev functions of an odd order. Using the sum of Dolph-Chebyshev and BarsilonTemes window functions defined by a sequence of Chebyshev functions of adjacent orders, new window functions with a standard main lobe and essentially suppressed sidelobes are developed.

Efficacy of the frequency and damping estimation of a real-value sinusoid Part 44 in a series of tutorials on instrumentation and measurement

Sinusoidal and damped sinusoidal signals are common in different branches of engineering. Estimation of frequency and damping is very important in many fields [1]–[15], e.g., linear system identification, speech processing, image processing, transient analysis, electric power system analysis, radar and sonar systems, communication, nuclear magnetic resonance spectroscopy (NMRS), mechanical spectroscopy, economics, seismology, and others. The literature on the subject is very rich. Recently an interesting introduction to frequency and damping estimation methods was published in the IEEE Instrumentation & Measurement Magazine [15]. In this paper, we focus on the methods with the best performance for a single component, damped, real-valued sinusoidal signal disturbed by additive white Gaussian noise. In our previous works [14] and [16]–[18], we investigated these estimation methods, two-point and three-point interpolated Discrete Fourier Transform (IpDFT) for an undamped sinusoidal signal analyzed with Rife-Vincent class I, Kaiser-Bessel and Dolph-Chebyshev windows, methods of Bertocco, Yoshida and Bertocco-Yoshida family for orders 0-3, the Agrez algorithm, IpDFT for damped sinusoidal signal analyzed with Rife-Vincent class I windows, time domain and frequency domain Least Squares (LS) optimization, Hilbert transform, covariance, Prony, iterative Steiglitz-McBride (STMB), Kumaresan-Tufts, total least squares, Matrix Pencil (MatPen), and Pisarenko. For this presentation, we choose only the best methods from the above set, i.e. the Bertocco-Yoshida order 1 (BY1) STMB and MatPen. We also describe the iterative leakage correction algorithm [19] with application to BY1, and compare all results with the well-known Aboutanios and Mulgrew algorithm (AbM) [13]. Our intention is to describe and compare the best existing algorithms. We consider a real-valued signal because it is a typical measurement case. The DFT spectrum of this signal is disturbed by a spectral leakage [5].

New Window Functions for the Design of Narrowband Lowpass Differentiators

New window functions are studied for the design of FIR narrowband lowpass differentiators (FNLD). The ideal lowpass differentiator is truncated by window functions. Using the sinc sum function, the frequency responses of the truncated lowpass differentiators are changed to a new form, which contains two parts. The first part is the integral of the sum of weighted sinc sum functions and the second one is the direct sum of other weighted sinc sum functions. According to the properties of the sinc sum function, the properties of the two parts are analyzed and differentiator errors are mainly dictated by the second part. As a result, the stopband error of FIR lowpass differentiators is approximately proportional to the cutoff frequency of their corresponding ideal differentiators. It is deduced that window constants can be derived according to the second part. Examples are given on how to get optimized windows for the design of FNLD. Three new windows as a family are obtained. The performances of FNLD obtained by using these new windows are concluded. Consequently, FNLD can be designed conveniently. Comparisons show that theses new windows are much better than the Hanning, Hamming, Blackman, and Dolph–Chebyshev windows and more optimized than the Kaiser window in terms of the performances of FNLD.

Signal to Noise Ratio(SNR) improvement of atmospheric signals using Variable windows

Windows reduce the sidelobe leakages from the spectrum when applied to finite length time domain sequence. Adjustable windows can be used to control the amplitude of sidelobes with respect to main lobe. This paper presents the effect of window parameter in Dolph-chebyshev, Kaiser, Gaussian and Tukey windows on the Signal to Noise Ratio(SNR) of radar returns and proposes an optimum value of window parameters with which data may be weighed. A comparative study is made on the improvement of SNR using these variable windows.

DFT Interpolation Algorithm for Kaiser–Bessel and Dolph–Chebyshev Windows

This paper describes the discrete Fourier transform (DFT) interpolation algorithm for arbitrary windows and its application and performance for optimal noncosine Kaiser-Bessel and Dolph-Chebyshev windows. The interpolation algorithm is based on the polynomial approximation of the window's spectrum that is computed numerically. Two- and three-point (2p and 3p) interpolations are considered. Systematic errors and noise sensitivity are analyzed for the chosen Kaiser-Bessel and Dolph-Chebyshev windows and compared with Rife-Vincent class I windows.

New efficient window function, replacement for the Hamming window

A new simple window function is presented, which for the same window order (M), has a main-lobe width less than or equal to that of the Hamming window, while offering about 2–4.5 dB smaller peak side-lobe amplitude. Furthermore, just like the Hamming window, it is computationally efficient for signal spectrum analysis; this is because of the fact that, the sum of window coefficients with its shifted version by M/2 samples (i.e. 50% overlap) is constant for the overlapped region. The new window is obtained by adding the third harmonic of the cosine function to the Hamming window, and finding the appropriate amplitudes of DC term, cosine function, and its third harmonic to minimise the peak side-lobe amplitude. A comparison with the Kaiser and Dolph–Chebyshev windows is also performed. Finite impulse response filters designed by windowing method show the efficiency of the new window.

A new efficient approach to the design of parametric windows with arbitrary sidelobe profiles

Based on a simple variable transformation in the spectral domain, a new method to parametrize any window in the same way as Kaiser-Bessel or Dolph-Chebyshev windows is presented, allowing a very flexible trade-off between the width of the main lobe and the height of the sidelobes. The derivation of the method is a generalization of the procedure used for the design of Kaiser-Bessel window and van der Maas function, which is related to the Dolph-Chebyshev window. The properties of this spectral transformation are studied. New bases of functions are presented, which can be used to easily design families of windows having a specific asymptotic decrease rate of the sidelobes. Links are made between this work and windows found in the literature.

A new efficient approach to the design of parametric windows with arbitrary sidelobe profiles

Based on a simple variable transformation in the spectral domain, a new method to parametrize any window in the same way as Kaiser–Bessel or Dolph–Chebyshev windows is presented, allowing a very flexible trade-off between the width of the main lobe and the height of the sidelobes. The derivation of the method is a generalization of the procedure used for the design of Kaiser–Bessel window and van der Maas function, which is related to the Dolph–Chebyshev window. The properties of this spectral transformation are studied. New bases of functions are presented, which can be used to easily design families of windows having a specific asymptotic decrease rate of the sidelobes. Links are made between this work and windows found in the literature.

Time domain window

A new window formula in the time domain is presented. It has mediate properties (figures of merit) into the Dolph-Chebyshev and Kaiser-Bessel windows, and its form is simple. The proposed window has four parameters, which allow its properties to be shaped for more applications.

Optimum approximations to Dolph-Chebyshev data windows

The narrow main lobe widths of Dolph-Chebyshev (1946) spectral windows can be approximated to within about 4%, with sidelobe depression to at least -98 dB, by windows defined with four Fourier series coefficients. Two windows may be particularly attractive, the opt59 and opt71 windows, since the true value of the fourth coefficient is zero.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Approximations to Dolph-Cbebyshev data windows

A family of optimum data windows was presented recently [J. C. Burgess, J. Acoust. Soc. Am. 62, S51(A) (1977)]. These windows are optimum in the sense that their spectral representations have the narrowest main lobes possible for a specified ratio of maximum side lobe amplitude to main lobe amplitude subject to the condition that the windows are described by Fourier series having very few nonzero coefficients. The process by which the Fourier coefficients are determined is equivalent to fixing the amplitude of one lobe in the spectral representation for each nonzero coefficient. A Fourier series having M + 1 coefficients thus controls the amplitudes of M side lobes. The Dolph-Chebyshev windows have the narrowest main lobes possible, and all side lobes for each window have the same level. The windows described by Burgess can thus be thought of as approximations to Dolph-Chebyshev windows. For M + 1 = 4 (i.e., four nonzero Fourier coefficients), the approximations have main lobe widths less than 5% wider than the corresponding Dolph-Chebyshev windows.