Two-dimensional (2-D) maximally flat finite impulse response (FIR) digital filters have flat characteristics in both passband and stopband. 2-D maximally flat diamond-shaped half-band FIR digital filter can be designed very efficiently as a special case of 2-D half-band FIR filters. In some cases, this filter would require the reduction of the filter lengths for one of the axes while keeping the other axis unchanged. However, the conventional methods can realize such filters only if difference between each order is 2, 4 and 6. In this paper, we propose a closed-form frequency response of 2-D low-pass maximally flat diamond-shaped half-band FIR digital filters with arbitrary filter orders. The constraints to treat arbitrary filter orders are firstly proposed. Then, a closed-form transfer function is achieved by using Bernstein polynomial.

Linear phase maximally flat digital differentiators (DDs) with stopbands obtained by minimizing the Lp norm are filters with important practical applications, as they can differentiate input signals without distortion. Stopbands designed by minimizing the Lp norm can be used to control the relationship between the steepness in the transition band and the ripple scale. However, linear phase DDs are unsuitable for real-time processing because each group delay is half of the filter order. In this paper, we proposed a design method for a low-delay maximally flat low-pass/band-pass FIR DDs with stopbands obtained by minimizing the Lp norm. The proposed DDs have low-delay characteristics that approximate the linear phase characteristics only in the passband. The proposed transfer function is composed of two functions, one with flat characteristics in the passband and one that ensures the transfer function has Lp approximated characteristics in the stopband. In the optimization of the latter function, Newton's method is employed.

Maximally flat digital differentiators (MFDDs) are widely used in many applications. By using MFDDs, we obtain the derivative of an input signal with high accuracy around their center frequency of flat property. Moreover, to avoid the influence of noise, it is desirable to attenuate the magnitude property of MFDDs expect for the vicinity of the center frequency. In this paper, we introduce a design method of linear phase FIR band-pass MFDDs with an arbitrary center frequency. The proposed transfer function for both of TYPE III and TYPE IV can be achieved as a closed form function using Jacobi polynomial. Furthermore, we can easily derive the weighting coefficients of the proposed MFDDs using recursive formula. Through some design examples, we confirm that the proposed method can adjust the center frequency arbitrarily and the band width having flat property.

Two discrete-time Wirtinger-type inequalities relating the power of a finite-length signal to that of its circularly-convolved version are developed. The usual boundary conditions that accompany the existing Wirtinger-type inequalities are relaxed in the proposed inequalities and the equalizing sinusoidal signal is free to have an arbitrary phase angle. A measure of this sinusoidal signal's power, when corrupted with additive noise, is proposed. The application of the proposed measure, calculated as a ratio, in the evaluation of the power of a sinusoid of arbitrary phase with the angular frequency π/N, where N is the signal length, is thoroughly studied and analyzed under additive noise of arbitrary statistical characteristic. The ratio can be used to gauge the power of sinusoids of frequency π/N with a small amount of computation by referring to a ratio-versus-SNR curve and using it to make an estimation of the noise-corrupted sinusoid's SNR. The case of additive white noise is also analyzed. A sample permutation scheme followed by sign modulation is proposed for enlarging the class of target sinusoids to those with frequencies M π/N, where M and N are mutually prime positive integers. Tandem application of the proposed scheme and ratio offers a simple method to gauge the power of sinusoids buried in noise. The generalization of the inequalities to convolution kernels of higher orders as well as the simplification of the proposed inequalities have also been studied.

New explicit formulas for tap-coefficients of halfband low/high pass MAXFLAT non-recursive filters are presented by using their relationship with already presented maximally linear type IV differentiators. These formulas are modified to give a new class of narrow transition band filters, with a performance comparable to that of optimal filters.