Dejan G. Ćirić

1D and 2D economical FIR filters generated by Chebyshev polynomials of the first kind

Christoffel–Darboux formula for Chebyshev continual orthogonal polynomials of the first kind is proposed to find a mathematical solution of approximation problem of a one-dimensional (1D) filter function in the z domain. Such an approach allows for the generation of a linear phase selective 1D low-pass digital finite impulse response (FIR) filter function in compact explicit form by using an analytical method. A new difference equation and structure of corresponding linear phase 1D low-pass digital FIR filter are given here. As an example, one extremely economic 1D FIR filter (with four adders and without multipliers) is designed by the proposed technique and its characteristics are presented. Global Christoffel–Darboux formula for orthonormal Chebyshev polynomials of the first kind and for two independent variables for generating linear phase symmetric two-dimensional (2D) FIR digital filter functions in a compact explicit representative form, by using an analytical method, is proposed in this paper. The formula can be most directly applied for mathematically solving the approximation problem of a filter function of even and odd order. Examples of a new class of extremely economic linear phase symmetric selective 2D FIR digital filters obtained by the proposed approximation technique are presented.

Generalised Christoffel–Darboux formula most directly applied in generating fully symmetric doubly resistively terminated LC lossless ladder filters

In this article, the approximation problem of a continuous analogue filter function of even and odd order is solved mathematically most directly applying the proposed generalised Christoffel–Darboux formula for two continuous orthogonal polynomials (Chebyshev first and second kind) on the equal finite segment of orthogonality in a compact explicit representative form. A set of analytic expressions of the proposed formula for the representative examples of odd-orders is given. Additionally, these expressions are applied in generating new excellent class all pole low-pass prototype analogue filter functions of odd orders (7, 9, 11, and 13th order). Based on the generated functions, the fully symmetric doubly resistively terminated LC ladder filter networks are realised by Darlington realisation. The generated filters are analysed, and their characteristics are presented. The effects of finite tolerance of a critical reactance of these ladder filters on the filter characteristics are considered. The filters generated by the proposed formula show excellent properties and possess important advantages in comparison to the corresponding filters generated by other methods.

Optimal Determination of the Truncation Point of Room Impulse Responses

A theoretically obtained optimal Truncation Point (TP) in the Room Impulse Response (RIR) is located at the intersection of the main decay slope and the noise floor. Precise determination of this point is difficult in practice. Namely, the deviation of the RIR decay from the exponential decay can lead to discrepancies in the determined TP from the optimal value. In order to increase the accuracy of TP determination, additional terms are incorporated in a simple algorithm. Besides, iterative algorithm for RIR truncation is proposed. These are compared with the existing one, using simulated and measured RIRs. The results confirm the validity of the proposed algorithms, indicating their advantages. In addition, the influences of the response parameters on RIR truncation are investigated.

Inaccuracies in sound pressure level determination from room impulse response

The integration over discrete squared room impulse response (RIR) leads to the sound pressure level (SPL). However, certain inaccuracies in SPL determination appear as a consequence of influences of finite integration upper limit (UL) and noise. Due to that, these influences are investigated using two types of RIRs: those generated by simulation and those measured in the room. The investigations show that for RIRs with sufficiently small noise floors (below -20 or -15 dB) acceptably small inaccuracies are obtained if the integration UL is placed at T/3 (alternatively at T/4). A more general solution is to set UL at the knee representing the point where main decay intersects noise floor. For even smaller inaccuracies, it would be better to set UL at the point somewhat before the knee called optimal UL. The influences of finite integration UL and noise can also be reduced by implementation of finite UL compensation and noise subtraction.

Coupling of earphones to human ear and to coupler

The use of a standardized acoustical coupler should enable a calibration of audiometric earphones which ensures that the thresholds determined in the audiometry will be independent of the earphone type. This requires that the coupler approximates the average human ear closely. Nevertheless, the differences among earphones as well as between human ears and the coupler affect the results of audiometric measurements inducing uncertainty. The influence of these differences is examined by investigating the coupling of different earphones to human ears and to the standardized coupler. This is done by measurement of the transfer functions from input voltage of the earphone terminals to the entrance of the ear canal in two situations: (1) open, and (2) blocked. Similar measurements were carried out with the coupler, but since the ‘‘ear‐canal entrance’’ is not well‐defined for the coupler, the mentioned measurements were done at different depths in the coupler. The earphone’s coupling to (i) human ears and to (ii)...

Automated estimation of the truncation of room impulse response by applying a nonlinear decay model

Noise represents one of the most significant disturbances in measured room impulse responses (RIRs), and it has a potentially large impact on evaluation of the decay parameters. In order to reduce noise effects, various methods have been applied, including truncation of an RIR. In this paper, a procedure for the response truncation based on a model of RIR (nonlinear decay model) is presented. The model is represented by an exponential decay plus stationary noise. Unknown parameters of the model are calculated by an optimization that minimizes the difference between the curve generated by the model and the target one of the response to be truncated. Different curves can be applied in the optimization-absolute value of the RIR, logarithmic decay curve, and Schroeder curve obtained by the backward integration of the RIR. The proposed procedure is tested on various synthesized and measured impulse responses. It is compared with the procedure taken from the literature, often applied in practice.

1D linear-phase band-pass multiplierless FIR Hilbert transformers and filters

ABSTRACT An original analytical method, based on modified Christoffel–Darboux formula, is used in the paper in order to synthesise a linear-phase band-pass finite impulse response (FIR) filter function that can have an effect of Hilbert transformer. New structure of the band-pass FIR filter in recursive realisation, together with the corresponding difference equation, is presented providing the efficient filter solution without multipliers. Several examples of filter types for different parity of two real free integer parameters, including a particular solution of Hilbert transformer, are considered in terms of required number of adders and values of cut-off frequencies of the pass and stop bands. A comparison of the proposed band-pass filter characteristics with those of a classical filter solution is provided in the paper.

Generating 2-D FIR filter functions by Christoffel–Darboux formula for Chebyshev polynomials of the second kind

ABSTRACT Global Christoffel–Darboux formula for different polynomials has already been used for the filter design. Here, this formula for orthonormal Chebyshev polynomials of the second kind and for two independent variables is applied in generating novel class linear-phase two-dimensional (2-D) finite impulse response (FIR) digital filter functions. In this way, 2-D filters with some specific features including economy, phase linearity, symmetry and selectivity are designed. Representative examples of the 2-D FIR digital filters of a new class obtained by the proposed approximation technique are given. A filter generated by the proposed approach is compared with the corresponding one generated by the procedure from literature.

Subband coding of audio signal with logarithmic compandors

This paper presents an audio compression system that applies the sub-band coding (SBC) technique with logarithmic compandors. SBC decomposes signal into the chosen number of frequency bands, which are coded separately. The goal is to design a SBC system that uses quasi-logarithmic quantizers and to find the optimal bit allocation. This would result in decrease of the average number of bits required and provide the gain in signal to quantization noise ratio (SQNR), compared to the pulse-code modulation (PCM) system.

Application of wavelet transform in reduction of noise in room impulse response

Wavelet analysis is one of the methods for solving difficult engineering problems that enables overcoming some deficiencies of classical Fourier analysis. This paper investigates possibilities for reducing noise in the measured room impulse responses using wavelet transform. In that, an approach based on discrete wavelet transform is applied. The effects of some wavelet parameters on noise reduction and impulse response distortion are analyzed.