Balbir Kumar

Design of efficient second and higher order FIR digital differentiators for low frequencies

Digital differentiators of orders greater than unity are often required in processing various types of data in a number of practical applications. In many of these, the vital information in the signal is contained in the low frequency range. We propose, in this paper, a new and efficient design of FIR digital differentiators of second and higher orders for the low frequency range. Mathematical relations have been established between the weighting coefficients of the maximally linear (first order) FIR digital differentiators and those of the proposed (second and higher order) differentiators. Recursive as well as explicit formulas for the weighting coefficients of the second order differentiators have also been derived. Extremely high accuracies are obtainable from the proposed differentiators, for attractively low orders of the filters.

On the design of multi notch filters

A methodology for designing FIR multi notch filters (NFs) derived from second-order prototype IIR NFs is suggested. Rejection bandwidth for the designed filter can be controlled by suitable choice of ‘r’, the pole radius of the IIR prototype NFs. The suggested multi NF can also be adapted to eliminate second-, third- and fourth-order harmonics of periodic noise besides the fundamental noise frequency component. A special case when two notch frequencies ω1 and ω2 are such that [(cosω1)(cosω2) = − 1/2] has also been discussed. The IIR multi NF design for this special case results in reduction of the number of multipliers without affecting the response of the desired NF. For the aforereferred condition, the required coefficients of impulse response of FIR multi notch filter get reduced to almost half in number resulting in reduced computations. The number of zero coefficients further reduces with increase in ‘r’ value. In addition, the frequency response becomes better, with reduced ripples in the pass bands, when ‘r’ is increased and length ‘L’ of the FIR NF is chosen appropriately. Copyright

Fast communication: Design of maximally flat linear phase FIR notch filter with controlled null width

A technique for designing a maximally flat, linear phase FIR notch filter with controlled null width is presented in this paper. Design analysis is carried out with first-, third- and fifth-order zero derivative constraints of the amplitude response of the FIR filter at notch frequency. An algorithm is presented for designing maximally flat, linear phase FIR notch filter with specified null width around specified notch frequency with third-order zero derivative constraint at the notch frequency. Detailed analysis in this paper shows that the null width of a maximally flat, linear phase FIR notch filter can be controlled by suitable selection of individual zero-odd-order derivatives and also by the successive addition of zero-odd-order derivatives at the notch frequency.

On the design of efficient second and higher degree FIR digital differentiators at the frequency p/(any integer)

In a number of signal processing applications, digital differentiators (DD) of degree greater than unity performing over a narrow band of frequencies are required. The minimax relative error DDs are especially suitable for broad band applications, but they become inefficient when adopted for narrow band situations. This paper proposes second and higher degree DDs which are maximally accurate at the spot frequency: π/(any integer). Mathematical relations have been established between the weighting coefficients of the first degree FIR digital differentiators which are maximally linear at the frequency π/(any integer) and those of the proposed (second and higher degree) differentiators. It has been shown that very high accuracies in the frequency response of the approximation are achievable with attractively low order of the structure for the suggested differentiators. As an example, with just 16 multiplications per input sample of the signal, it is possible to obtain a third degree differentiator over a frequency bandwidth of 0.20π centred around ω = π/3, with an accuracy no worse than 99.999%. The phase error is zero over the entire frequency band 0⩽ω⩽π of operation.

Letter to the editor: Design of FIR notch filters by using Bernstein polynomials

In this paper, Bernstein polynomials have been used to derive an explicit formula for the coefficients of linear phase FIR notch filters which are maximally flat at ω=0 and π. The approach is relatively simple and enables us to design the filter for a specific notch frequency and bandwidth.

Interrelations between the coefficients of FIR digital differentiators and other FIR filters and a versatile multifunction configuration

Abstract Interrelationships between seven types of digital FIR filters, viz. the differentiators (DD), Hilbert transformers, half-band lowpass/highpass filters, bandpass/bandstop filters and frequency discriminators, have been established. It has been shown that all the aforementioned filters can be derived from the design of a DD. If we use the optimality criteria of maximal linearity of the frequency response of a differentiator, the proposed relations yield maximally flat/linear frequency responses for the remaining six filters. Similarly, an equiripple relative error design of a DD gives equiripple filters through the use of the suggested interrelations. A versatile, multipurpose, FIR configuration has also been proposed which yeilds optimal frequency response for all the seven types of digital filters. The suggested design is shown to be particularly useful for operation over the frequency ranges 0 ⪕ ω ⪕ 0.50π (0.5π ⪕ ω ⪕ π) for lowpass (highpass) filters and 0.25π ⪕ ω ⪕ 0.75π for the remaining ones. Mathematical formulas for computing the weights of aforementioned filters have also been derived. Possible uses of the suggested design have been given.

Unified Recursive Structure for Forward and Inverse Modified DCT/DST/DHT

Abstract This paper proposes implementation of the Modified Discrete Sine Transform (MDST) and Inverse MDST (IMDST) using recursive structures. The formulae required for recursive structures have been derived. Discrete Hartley Transform (DHT), a real-valued transform, is closely related to Discrete Fourier Transform (DFT) of a real-valued sequence and hence its use as an alternative to the Fourier Transform avoids complex arithmetic. This paper presents Modified Discrete Hartley Transform (MDHT)/Inverse MDHT (IMDHT) using Modified Discrete Cosine Transform (MDCT)/ Inverse MDCT (IMDCT) and MDST/Inverse MDST (IMDST) recursive structures. The proposed structures are used for simultaneous computation of MDCT/MDST/MDHT of length N (divisible by four) and their Inverse (IMDST/IMDHT). The proposed structures are parallel, simple, regular and therefore highly suitable for VLSI implementation.

Design of universal, variable frequency range FIR digital differentiators

An efficient design of digital differentiators (DD), with a variable frequency range of operation at low and midband frequencies, has been proposed. Implemented as a Taylor structure, a DD designed for orderN can be made to function as a universal configuration giving optimal transfer functions for all possible ordersN, without changing the coefficients.

Semi-analytic method for the design of digital FIR filters with specified notch frequency

Abstract A new semi-analytic method for designing digital FIR notch filters for a specified notch frequency (ω d ) has been proposed. A computer program based on the proposed algorithm has been developed and tested for filter lengths up to 65. Illustrative examples, confirming the new approach, have also been given.

Design of linear-phase, FIR integrators of degreer, r=1, 2, 3,..., for midband frequency range

In this paper a novel analytical technique for designing linear-phase, FIR integrators of first degree has been proposed by taking the optimality criteria of maximal flatness (in the Butterworth sense) of the amplitude response, at midband frequency:ω=π/2. Mathematical formulas to compute exact weights for the design of first-degree integrators (i.e., frequency response 1/(jω)) have been derived. Thereafter, the design has been extended to approximate linear-phase integrators for any generic degreer, r>1. The performance characteristics of the first-, second- and third-degree integrators have been given to highlight the efficacy of the designs. The suggested designs are particularly suitable for the midband frequency range of operations.