Second-order Resonators Research Articles

Finite-precision Goertzel filters used for signal tone detection

The effect of coefficient quantization on the output of a second-order Goertzel filter tone-detector is analyzed in order that the Goertzel filter may be systematically designed for minimum complexity. Three alternative configurations are identified for the Goertzel filter's component second-order digital resonator, which are especially suited to VLSI implementation, and which complement each other on the zero-to-Nyquist frequency interval. The Goertzel filter's tone response is formulated using the Fourier summation transform (and shown to depend on both the frequency and phase of the incident tone). Hence analytic expressions are derived for the tone response of a fully finite-precision Goertzel filter when used with either a rectangular or half-end-point, input data window. These formulas are amenable to optimization, and are applied to improve a DTMF receiver design which is subject to the use of low-complexity second-order resonators.

A sliding Goertzel algorithm

This paper presents a sliding Goertzel algorithm to accurately estimate the Fourier coefficients of multifrequency (MF) sinusoidal signals buried in noise. The algorithm is based on second-order digital resonators that are tuned at the desired frequencies. The proposed method provides the following advantages when compared with the conventional Goertzel algorithm. Firstly, it computes Fourier coefficients in less than one signal period. Therefore, faster detection time is achieved, particularly when the greatest common divisor (GCD) of the input frequencies is small. Secondly, it is less prone to numerical overflow problems in fixed-point arithmetic implementation. Thirdly, the algorithm is quite suitable for time varying sinusoidal signal estimation. An analysis is undertaken to provide additional insight into the issue of required acquisition time versus the desired accuracy for the proposed algorithm. Extensive simulation tests are also included to demonstrate its performance.

Notch Fourier transform

This paper presents a new computational algorithm for the Fourier transform at arbitrary frequencies. This algorithm, if an input signal has r frequencies, can be implemented by one FIR filter composed of r second-order digital notch filters (2-DNF's; <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_{p}(z) = 1 - 2 \cos (2\pif_{p}/f_{s}) z^{-1} + z^{-2}</tex> ), and r second-order digital resonators (2- DR's; H' <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> (z)= 1/H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> (z)) where frequencies f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> (p = 1, ..., r) are arbitrary and f <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</inf> is a sampling frequency. This algorithm is referred to as the notch Fourier transform (NFT). This NFT algorithm has the following main advantages, 1) It can be applied to the input signal composed of arbitrary frequencies. 2) In advance of the end of one period of the signal, Fourier coefficients can be computed. 3) Although we can analyze the input signal composed of r arbitrary frequencies by solving simultaneous equations, the amount of processing in the NFT is smaller by the order of magnitude N where N = 2r is the number of sample data. At every sampling time after one block of data, the NFT can compute all Fourier coefficients with the number of multiplications proportional to N. But for the exact computation of Fourier coefficients, we must know the signal frequencies in advance.