Elsevier

Signal Processing

Volume 87, Issue 5, May 2007, Pages 1003-1013
Signal Processing

Recursive Fourier transform algorithms with integrated windowing

https://doi.org/10.1016/j.sigpro.2006.09.005Get rights and content

Abstract

Recursive signal spectrum estimation algorithms based on the discrete Fourier transform (DFT) are presented in this paper. Unlike other methods of DFT evaluation, the recursive method updates the spectrum coefficients instead of calculating them from the beginning each time. Some difficulties with windowing are associated with this solution. A few developments of the classic recursive algorithm towards integration of the windowing process are proposed in this paper. They are based on a family of windows obtained by a multiple convolution of the rectangular window. Due to recursive realization the stability issues are taken in account. A solution which avoids this problem is included.

Introduction

The discrete Fourier transform (DFT) is one of the basic tools of digital signal processing. Nowadays there are many algorithms which enable it to be calculated in a more effective way than through direct use of the definition equation. The most popular are the block processing oriented fast Fourier transform (FFT) algorithms. An alternative to this group is the recursive algorithms. The first structure that calculated the DFT in the recursive manner was the Goertzel algorithm [1]. Its limitation was the processing of finite series of samples, whose length had to be equal to the transformation length. In 1966 the recursive Fourier transform (RFT) [2], without this limitation, was proposed. The RFT algorithm updates the spectrum coefficients in the recursive manner according to the new signal samples received. The train of samples being processed does not have to be finite. Formally it realizes the short time Fourier transform, thus some authors call it the moving window DFT algorithm [3] or sliding DFT. Due to the narrowband nature of the estimated spectrum coefficients, their updating in every sampling period is not required. A modification of the RFT that makes it possible to move the analysis window in a step larger than a sampling period, has been proposed [4]. But oversampling is not the only problem with the RFT. Problems with the effective correction of the spectrum leakage effect, windowing, are of no less importance. It is connected with the idea of the RFT, that assumes the updating of the spectrum coefficients instead of calculating them from the beginning each time. Implementation of the classic solution, such as multiplying a block of signal samples by a window in the time domain, is not possible in this case. The signal samples introduced to the RFT structure are not available for modification. Hence windowing in the frequency domain remains, but even when using the trigonometric windows [5] the number of arithmetic operations required to do this task is comparable to or greater than the number required by the RFT. The high computational efficiency of the algorithm is lost at the windowing stage. One of the opportunities to solve this problem is an integration of the RFT algorithm with windowing. A recursive structure with an integrated triangle window has been presented [6]. The frequency properties of the triangle window are better than those of the rectangular one, but are still poor. The goal of this paper is to check the possibilities of extending this idea to other windows created on the base of the rectangular window.

Section snippets

The RFT algorithm

The estimation of the k-signal spectrum coefficient in an m-period of time by means of the DFT is defined by the equation:Y(m,k)=1Nn=0N-1x(m-n)e-j2kπn/N.

The limitation of the number of signal samples to a finite number N requires applying a time window. In the simplest case (Eq. (1)) it is the rectangular window. Assuming its time shift in every sampling period, a Z-transform of Eq. (1) can be evaluated:Y(z)=Z{1Nn=0N-1x(m-n)e-j2kπn/N}=X(z)H(z),whereX(z)=Z{x(m)},H(z)=Z{1N(1,e-j2kπ/N,e-j4kπ/N,e-

The change of time window shape

The rectangular window which appears in the classic RFT has disadvantageous spectrum properties. It has a small side-lobes attenuation (−13 dB) and additionally they fall down slowly (6 dB/oct.). The change of the rectangular shape into another, more optimal shape, is easy for direct calculation of the DFT by definition or by using the FFT family algorithms. It is enough to multiply the signal samples x(m) by a new window function w(n). Eq. (1) is then modified as follows:Y(m,k)=n=0N-1x(m-n)w(n)e

A windows family for the RFT algorithm

Time window functions describe the impulse responses of narrowband low-pass filters. In the case of the recursive realization, their representations in the Z-domain are more suitable, because they enable us to obtain the realization structures directly. The equation for the rectangular window may be obtained from Eq. (4) after assuming the estimation of the spectrum coefficient with the index value k=0 (the spectrum coefficient for the lowest frequency):W(z)=1N1-z-N1-z-1.

By treating the

The modulation of the window samples

The common window Eq. (10) can be directly used for the estimation of a spectrum sample with the index k equal to zero only. The estimation of the remaining coefficients requires us to modulate the window samples by the values of the complex exponential function with arguments dependent on the k-index. To do this, window samples have to be fitted to the complex exponential function samples. A raising to a power in the Z-domain is equivalent to multiple (dependent on the index of the power J)

Obtaining the RFT equations with recursive windowing

Treating the window samples modulated by the samples of the complex exponent as FIR filter coefficients and then evaluating their Z-transform:HJ(z)=n=0hJ(n)z-n=e-j2kπ/N(JN+J-2)J(1+e-j2kπ/Nz-1+e-j4kπ/Nz-2++e-j2kπ(N-2)/NJz-(N-2)/J)J,enables us after recounting the sum of the series, to find the transfer function of the filter which realizes the DFT together with windowing in a recursive manner:HJ(z)=e-j2kπ/N(JN+J-2)J[1-(e-j2kπ/Nz-1)((N-2)/J)+11-e-j2kπ/Nz-1]J.

Unlike the classic RFT algorithm

The optimization of the recursive part

The single recursive loop with the multiplication inside the loop can be changed into a structure with two multiplications outside, as shown in Fig. 5. It can be proved that both structures have the same transfer function. The modulated by the complex exponential function signal samples at the input of the recursive part:Y0M(z)=Y0(e-j2kπ/Nz)are given in the adder of the recursive loop without the multiplication. The Y1M(z) signal is obtained at its output, hence:Y1(z)=Y1M(ej2kπ/Nz)=Y0(z)1-e-j2kπ

The optimization of the non-recursive part

When estimating single spectrum coefficients, optimizing the non-recursive part of the structure can further reduce the number of arithmetic operations. It is enough to move the modulation operation of the signal that is entered into the recursive block to the entry of the non-recursive part. Thus the analyzed signal samples x(m) will be directly modulated and then introduced to the structure. The general transfer function (Eq. (21)) has to be rewritten asHJ(ej2kπ/Nz)=e-j2kπ/N(JN+J-2)J(1-z-((N-2

Summary

The windows family achieved from the rectangular window by multiple convolution in the time domain allows us to create the recursive algorithm which connects the DFT with windowing. The choice of the window frequency parameters is translated into the complexity of the algorithm: the number of taps in the input delay line and the number of recursive loops. The achievement of higher attenuation values of side-lobes and their fall-off speed leads to the structures that use the higher-order

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There are more references available in the full text version of this article.

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