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On asymmetric analysis windows for detection of closely spaced signal components
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Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 20 (2006) 702–717 www.elsevier.com/locate/jnlabr/ymssp
On asymmetric analysis windows for detection of closely spaced signal components Miroslav Zivanovic, Alfonso Carlosena Universidad Pu´blica de Navarra, Department of Electrical and Electronic Engineering, Campus de Arrosadı´a, s/n, E-31006 Pamplona, Spain Received 24 May 2004; received in revised form 5 April 2005; accepted 12 April 2005 Available online 13 June 2005
Abstract Discrimination capacity, i.e. the capacity to resolve signal components, is probably one of the most important properties of the analysis windows. For components separated by more than a DFT bin the discrimination capacity is determined by the bandwidth and shape of the window’s spectral main lobe and the side lobe fall-off rate. The new classical paper of Harris is a milestone where a plethora of windows is discussed and compared through the aforementioned parameters. However, those parameters fail to describe situations given by components clustered within a DFT bin bandwidth. In those cases the discrimination capacity is almost completely determined by the spectral phase of the analysis window, while the shape of the window’s magnitude spectrum has little effect. This paper is devoted to a specific class of asymmetric analysis windows capable of resolving components separated by less than a DFT bin. Those windows have recently been proposed and analytically discussed. However, a number of important properties have not been tackled. Herein, the asymmetric analysis windows are thoroughly described and compared to the classical analysis windows in various aspects. The main benefits of the proposed windows are extended discrimination capacity, robustness against additive noise and simple generation. These properties make them very good candidates for narrow-band spectrum analysis of mechanical systems. Results for some real-world examples are presented, showing the capability of the new windows to resolve closely spaced vibration modes when there is a lack of spectral resolution due to the finite register length. r 2005 Elsevier Ltd. All rights reserved. Keywords: FFT; Spectrum resolution; Analysis windows; Tone separation
Corresponding author. Tel.: +34 948 169 024; fax: +34 948 169 720.
E-mail address: [email protected] (M. Zivanovic). 0888-3270/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2005.04.003
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1. Introduction There are a number of DFT-based signal processing algorithms devoted to harmonic detection and estimation of finite-register data. Consequently, the most important issue to be dealt with is interplay between the computational and physical resolution of the discrete spectrum. Computational resolution is defined [1] as a distance between the DFT bins at which the discrete time Fourier transform (DTFT) is computed. This parameter can be constant in the whole spectrum [2] or in a particular frequency range [3,4]. Other approaches are designed to yield nonuniform bin spacing on the linear frequency grid, but those methods seem to be less known [5–7]. A critical comparison for the above methods has been described by the authors, what sheds new light on their properties and applicability [8]. Physical resolution (also known as a minimum resolution bandwidth) is defined [9] as a minimum separation between two equal-strength signal components such that for arbitrary spectral locations their respective main lobes can be resolved. This parameter depends on the analysis window and is typically determined, according to several criteria, from the bandwidth and the shape of the main lobe in the window’s magnitude spectrum [10,11]. The most common criteria like 3 dB, 6 dB, equivalent noise bandwidth (ENBW), have been applied in Ref. [9] to establish a comparison among the classical analysis windows like the rectangular, triangular, families of raised cosine, etc. This study reveals that the rectangular window has the smallest resolution bandwidth, which corresponds to the narrowest main lobe. However, those criteria neglect the main lobe’s spectral phase, which is an important factor in addition of complex spectra. All classical analysis windows have the same spectral phase, due to the causality-imposed time shift; consequently, the determining factor in the physical resolution will still be the bandwidth and the shape of the main lobe. However, an arbitrary non-symmetric time weighing could exhibit quite different spectral phase; thus, the classical criteria would simply not be adequate for window comparison. We have demonstrated [12] that for estimating the physical resolution, spectral phase is a stronger condition and should be the principal criterion, complemented by the classical magnitude-based criteria. In a view of that, we have proposed a class of asymmetric windows [12] which can be generated from the classical analysis windows. A particular asymmetric analysis window is characterised by a broad main lobe which, according to the classical criteria, should provide for a quite large resolution bandwidth. On the other hand, the phase of the main lobe is such that the margins for component discrimination, imposed by the rectangular window, are extended. Hence, the proposed asymmetric windows outperform the classical analysis windows in terms of physical resolution. The goal of this paper is to give a more thorough insight into the properties of the proposed asymmetric windows and to show their effectiveness in dealing with real-world signals/systems. We shall start with a brief summary of the previous work, while for more detailed explanations and the analytical background the reader is invited to consult [12]. Next, we shall discuss the most prominent properties of the proposed windows in terms of spectral analysis and physical resolution. Also, a noise study will be presented in order to quantify the robustness of the new windows against additive noise. Possible applications are related to a number of practical problems, like [13], which arise frequently in modal analysis. An extensive experimental part will show the benefits of using the asymmetric analysis windows in real-world applications. Finally, we stress that throughout the paper a critical comparison with the classical analysis windows in several aspects was done.
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2. The new class of asymmetric windows 2.1. Summary of the previous work As a measure of physical resolution for an arbitrary analysis window, we defined a parameter denominated discrimination bandwidth. It was referred to as a minimum frequency distance between two signal components that produces amplitude zero crossing between the corresponding spectral peaks. We have shown that this parameter is inversely proportional to the slope of the main lobe’s spectral phase, whenever that phase can be conceived as a linear function of frequency. As mentioned in Introduction, all classical analysis windows, which we denominated S-class windows, share the same spectral phase slope; it is proportional to N/2, where N is the size of the window: Discrimination bandwidth ¼
p ðradÞ. N=2
(1)
Therefore, all S-class windows will have the same discrimination bandwidth and a comparison among them should always be accomplished by means of the classical magnitude-based criteria. Once the basis for comparison among arbitrary analysis windows has been established, we proposed and discussed a specific class of asymmetric windows, which we denominated $-class windows. A $-class window is generated by taking the first N samples of a 2N-sample S-class window. In order to best describe the properties of the $-class windows, we define also the unilateral window obtained by simply reversing the order of samples of a $-class window. The three above-mentioned window shapes are shown in Fig. 1. We are now interested in determining the slope of the spectral phase in the main lobe. However, the spectral phase of a $-class window is in general a non-linear function of frequency. Nevertheless, we have shown that within the main lobe bandwidth the spectral phase can be well approximated by a linear segment, the fact that facilitates comparison with the S-class windows in terms of physical resolution. In particular, we have found that the discrimination bandwidth for an arbitrary N-sample $-class window is given by Discrimination bandwidth ¼
p ðradÞ. N 1m
(2)
The parameter m is determined by the shape of the corresponding 2N-sample S-class window and its value is always between 0 and N/2. The former corresponds to a $-class window generated from the 2N-sample unite impulse shifted N samples to the right of the origin; the letter corresponds to a $-class window generated from the 2N-sample rectangular window. We note that the delayed unite impulse is treated here as a special case of S-class window. The rest of the $-class windows are comprehended between those two limiting cases and achieve discrimination bandwidth less than p=ðN=2Þ. Therefore, the $-class of windows reduces the discrimination bandwidth imposed by the S-class windows and thus improve the physical resolution. In the following section, we will discuss more thoroughly on the nature of the parameter m and how it is related to the properties of the $-class.
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2.2. Some properties of the $-class of windows Let us first examine, through the following example, how windows with the same main lobe width can produce very different spectral phase. The spectra for the 100-point Hanning window, 100-point unilateral Hanning window and 100-point asymmetric Hanning window are shown in Fig. 2. In terms of the ENBW, all three windows provide for the same physical resolution equal to 1.5 bins. However, their spectra are quite different as the real and imaginary parts are concerned. The unilateral Hanning and asymmetric Hanning windows have the same magnitude spectra, but very different real and imaginary components. As both windows are generated from the 200sample Hanning window (see Fig. 1) and recalling that the discrimination bandwidth is closely related to the time shift of the analysis window, we guess that the unilateral Hanning window should have the least pronounced spectral phase, while the asymmetric Hanning window should exhibit the spectral phase characteristic with the largest slope within the main lobe. Hence, from Eq. (2) we deduce that the asymmetric Hanning window will have the smallest m and thus the smallest discrimination bandwidth. The largest m will correspond to the unilateral Hanning window, while the S-class window (Hanning window) will be in the middle point with m ¼ N=2 1. The graphical comparison among the corresponding spectral phase curves is shown in Fig. 3. Recall that for both unilateral Hanning and asymmetric Hanning windows the phase is quasi-linear with frequency within the main lobe. We have already said that the parameter m for an N-sample $-class window depends on the shape of the corresponding 2N-sample S-class window. The stronger the tapering, the smaller m,
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being theoretically 0 for the unite impulse. This effect can also be explained in the frequency domain: the smallest m will correspond to the S-class window with the broadest main lobe and vice versa. In order to support this statement, we have compared a number of asymmetric windows generated from different S-class windows, namely the triangular, Hanning, Blackman and Nuttall, in terms of spectral phase. It can be seen in Fig. 4 that, as we expected, the spectral phase slope varies coherently according to the main lobe’s bandwidth (expressed here through the ENBW). The Nuttall window achieves the largest slope, having the widest main lobe, while the smallest slope is assigned to the triangular window. The same four windows will be compared in terms of discrimination bandwidth in Section 3. Another way to achieve small discrimination bandwidth will be introduced through the following argumentation. The slope of the spectral phase of an N-sample $-class window is a result of the combined action between the time shift and the tapering of the corresponding
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2N-sample S-class window. In the preceding paragraph we have shown the influence of the tapering, being constant the time shift equal to N. Now, we can fix the tapering and vary the time shift, which can be done in practice through larger S-class windows. By taking the first N samples from an LN-sample S-class window, where L42, we obtain a particular non-symmetric window
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which we denominated L$-class window. It is to be expected that the time shift of LN=2 produces spectral phase slope proportional to LN=2 1 m and thus very small discrimination bandwidth for an L$-class window. This, however, will not be completely satisfied, especially for large L; an explanation lies in the frequency representation of an arbitrary non-symmetric window. Recall that the most prominent S-class windows are raised-cosine windows. The DTFT of a raised-cosine window is an addition of the central Dirichlet kernel (spectrum of the rectangular window) and a number of pairs of translated kernels located at the zeros of the central kernel, in order to partially cancel the side lobe structure. However, in the DTFT of an N-sample $-class window, the translated kernels are shifted in frequency by p=N towards the central kernel. If the translated kernels were also frequency-compressed by a factor of 2, then the obtained DTFT would correspond exactly to a 2N-sample S-class window. Although this is obviously not the case, the slope of the spectral phase is still proportional to the time shift. For an L$-class window, the strong overlap among the central and the translated kernels would not permit the spectral phase slope increase in a linear fashion. Nevertheless, an improvement can indeed be achieved, as it is shown in Fig. 5, where we can see that the slope variation is still coherent with increasing time shift. This section can be complemented by examining the influence of the $-class windowing on harmonic amplitudes. For an N-point $-class window, the amplitude A of a single harmonic component is computed by taking into account the contributions of all the Dirichlet kernels at the harmonic frequency: " # K=2 n p o X 2 Dð0Þ 2 i , A¼ < ai D N N i¼1
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Fig. 5. Spectral phase characteristics for the $-class, 4$-class and 8$-class Hanning windows; the size of the windows is 100 samples.
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where D is the Dirichlet kernel, K is the number of kernels in the window’s spectrum, ai are the kernels’ weighs and N is the size of the window. However, the effect of the asymmetric windowing on harmonic amplitudes is of little interest. This is because the $-class windows are aimed at resolving components clustered within a DFT bin, where heavy interharmonic interference provokes gross errors in parameter estimation, even if all the components have been correctly detected. In such circumstances it is quite difficult to quantify the difference between the true harmonic parameters and the parameters estimated from the spectral peaks. Furthermore, and to the best of our knowledge, a number of papers that deal with harmonic estimation (e.g. [14–17]) assume the interharmonic distance (separation between the signal components) is large enough to disregard the interference between the corresponding spectral peaks. In spite of the above argumentation, we will present a study on harmonic parameter estimation errors for different analysis windows in Section 3. 2.3. Influence of noise A very important aspect of window’s performance is robustness against additive noise. More specifically, we are interested in how the detection of closely spaced signal components varies as the noise power changes. As a measure of detection we use the parameter we denominated discrimination depth. For two arbitrary-strength components it is defined as a magnitude difference between the smaller spectral peak and the crossover point. For equal-strength components the crossover point is typically half-way between the peaks. Hence, this parameter shows how close to the limit of resolvability (two components still represented as two peaks) the corresponding components are. When some quantity of noise is added to a deterministic signal, the discrimination depth becomes a random variable and thus can be characterised by its probabilistic properties: the first moment (mean and bias) and the second moment (variance and standard deviation). We have established a particular case scenario in order to compare different analysis windows. The test signal was composed of two tones and a zero mean uniformly distributed white noise r(n) xðnÞ ¼ 2 cos½ð2pf 1 n þ 0:7Þ þ cos½ð2pf 2 n þ 0:2Þ þ rðnÞ. The normalised frequencies f1 and f2 have been adjusted for each analysis window in order to achieve the maximum discrimination depth in absence of noise. Note the amplitude and phase differences, which have been set in order to obtain more realistic conditions for component discrimination. As a measure of noise contribution to the signal we have used the signal-to-noise ratio (SNR) in Decibels. Each time the SNR were modified, 100 experiments were realised and the bias and normalised standard deviation were calculated for the rectangular, Hanning and asymmetric Hanning windows, and the corresponding curves were plotted in Fig. 6. In general, all three windows exhibit quite similar behaviour in presence of noise. By taking a closer look, we can see that for SNR ¼ 5 dB the asymmetric Hanning window presents the largest relative bias, although it remains below 20% approximately. As SNR increases, all three curves tend to zero while the discrimination depth approaches to its nominal value. On the other hand, the asymmetric Hanning window is between the rectangular and Hanning window, as the standard deviation is concerned. Like the normalised bias, the differences among the curves are more pronounced for low SNR. Therefore, from the above analysis based on the particular case
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Fig. 6. Relative bias and normalised standard deviation of the discrimination depth as a function of SNR for the rectangular, Hanning and asymmetric Hanning windows. The curves are generated by calculating the statistics on a two-component signal whose parameters are: A1 ¼ 2, j1 ¼ 0:8 , A2 ¼ 1, j2 ¼ 0:1 ; the normalised frequencies f1 and f2 have been adjusted for each analysis window in order to achieve the maximum discrimination depth in absence of noise.
scenario, we can say that even for low SNR, the discrimination capacity of both $-class and Sclass windows is still preserved.
3. Experimental results 3.1. Comparative study In this subsection, we compare different analysis windows in terms of discrimination depth as a function of interharmonic distance for two equal-strength components. For a particular window, this analysis provides for the discrimination bandwidth and the resolution limit, corresponding to the maximum and minimum discrimination depth, respectively. The first comparative study, whose results are shown in Fig. 7, comprehends the four $-class windows already treated in Section 2: the triangular, Hanning, Blackman and Nuttall. As it was expected, according to the discussion in Section 2, the Nuttall and triangular window are on the margins of physical resolution, while the Hanning and Blackman window lie in between. The Nuttall window achieves the smallest discrimination bandwidth around 0.65 bins but it loses its discrimination capability for interharmonic distance greater than 0.95 bins. The triangular window has the largest discrimination bandwidth of 0.75 approximately; however, its discrimination depth is around 0.30 for interharmonic distance equal to 1 bin. As the resolution limit is about, there is only a 0.03 bins difference between the Nuttall and triangular window. Therefore, a choice of a $-class window is a trade-off between discrimination bandwidth and resolution limit. Similarly, we have compared the three L$-class Hanning windows from Section 2 and shown the results in Fig. 8. As L increases the discrimination bandwidth gets smaller, what is coherent with the qualitative explanation given in Section 2. However, this process is not linear in the sense that for large L the discrimination
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Fig. 8. Relative discrimination depth as a function of interharmonic distance for two equal-strength components through the following L$-class windows: $-class, 2$-class and 8$-class Hanning.
bandwidth keeps almost constant, due to the effect of the strong Dirichlet kernel overlap already discussed. This situation is well reflected in the curves for the 4$-class and 8$-class Hanning windows, respectively. Finally, we have compared windows from different classes: the rectangular, Hanning and asymmetric Hanning, with the SNR as a parameter. We have chosen the rectangle
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Table 1 Window
Discrimination BW (bins)
Frequency bias (bins)
Amplitude bias
Asymmetric Hanning Rectangular
0.71 1.00 1.00
0.49 0.46 0.20
0.40 0.35 0.10
window because it is a particular case of the S-class, the Hanning window being the simplest raised cosine S-class window and the asymmetric Hanning window, belonging to the $-class and being generated from the Hanning window. From Fig. 9 (SNR ¼ 30 dB) we observe that the asymmetric Hanning window has the discrimination bandwidth around 0.71 bins and the resolution limit below 0.60 bins. Both the rectangular and Hanning windows have the same discrimination bandwidth equal to 1 bin, which arise from the fact that they are S-class windows. The rectangular window performs better than the Hanning window, because the latter has broader main lobe. This fact is captured through the resolution limit, which is around 0.65 bins for the rectangular and approximately 0.75 bins for the Hanning window. However, up to 0.80 bins, the asymmetric Hanning window has the largest discrimination capacity. Similar conclusions can be inferred from Fig. 9b, where the comparison has been done for SNR ¼ 10 dB. The preceding paragraph is complemented by the following study on harmonic parameter estimation errors. In particular, we are interested in quantifying the bias in amplitude and frequency estimation for two equal-strength components separated by the discrimination bandwidth for a given analysis window. In absence of additive noise the bias is due only to the overlap between the main lobes. The amplitude and frequency bias are measured for the rectangular, Hanning and asymmetric Hanning windows and given in Table 1. The smallest bias corresponds to the rectangular window while the asymmetric Hanning window introduces the
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largest estimation errors. Note the similarity between the Hanning and asymmetric Hanning windows; the latter has only slightly larger bias but, in turn, smaller discrimination bandwidth. This effect is detected in all raised-cosine S-class windows and their $-class counterparts, where larger main lobe is traduced in larger amplitude and frequency bias. 3.2. Detection of strongly coupled vibration modes Very often the goal is to detect high-order vibration modes, masked by the spectral contribution of the principal mode [13], in order to identify as good as possible a model for the system under test. Therefore, it can be expected that the best-choice analysis would be a combined action of both S-class and $-class windows: the former to isolate well-separated modes and the letter to discriminate the closely spaced ones (relative to the long and near-term leakage, respectively). The performance of herein proposed asymmetric windows has been tested in a coin verification system fully described in Ref. [18]. A brief discussion of the system follows, supported by Fig. 10. The nucleus of the system is a sensor assembly comprising a miniature piezoelectric accelerometer mounted on an inertial mass, which has a form of a 25 mm-long and 7 mm-wide steel cylinder. The vibrations originated from the coin impacts against the inertial mass surface are transformed into equivalent electrical signal generated by the accelerometer. This signal is then transferred to the coin selector where it is processed by means of different coin verification algorithms.
Fig. 10. Miniature piezoelectric accelerometer mounted on a 25 mm-long and 7 mm-wide steel cylinder. This device is referred to as sensor assembly and is a heart of the coin verification system.
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Fig. 11. Influence of the analysis window on the response of the sensor assembly for the plastic selector: (a) FFT of the 270-sample response signal (a) with no windowing (represented up to 700 kHz) and (b) for different analysis windows and factor-32 zero padding (represented up to 200 kHz).
The device has been tested for two different types of selector, namely plastic and methacrylate. The goal was to check the presence of the vibration modes specific for each of the said materials, which could interfere with the coin impact signal. It has been done through low-pass filtering and windowing of the impulse response of each system. The final step was to calculate the FFT with some zero padding to smooth out the frequency response. The impulse response has been obtained by loosing a 1 mm-diameter steel ball from approximately 15 cm height with respect to the cylinder and acquiring the output signal of the sensor assembly by a digital oscilloscope at the rate of 4 MS/s. In the case of the plastic selector, we aimed to detect the 35 kHz secondary mode strongly masked by the 17 kHz principal mode. The corresponding interharmonic distance is equal to 1.14 DFT bins. Due to important damping of the secondary mode, both rectangular and Hanning window detect only the strongest mode. On the contrary, the asymmetric Hanning window exhibits a deep dip in the crossover point between the modes (Fig. 11). This observation is coherent with the expression (3) which states that the discrimination bandwidth does not depend on the magnitude but only on the phase. We observe that the local maxima of the spectrum do not coincide with the mode frequencies because of the strong interharmonic interference. For the methacrylate selector the goal was to discriminate two closely spaced high-order vibration modes at 34 and 35 kHz, respectively. This time the modes’ damping coefficients are similar and the corresponding interharmonic distance is equal to 1.10 DFT bins. In order to improve the discrimination capacity even more the asymmetric window applied has been constructed from an 8N-point S-class Hanning window. The classical windows exhibit one peak at approximately 35 kHz and consequently fail to detect the closely spaced modes. However, the asymmetric window achieves positive discrimination depth and manages to discern the modes (Fig. 12). Again, the harmonic parameters can only be roughly estimated from the spectrum, due to the strong overlap between the modes.
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M. Zivanovic, A. Carlosena / Mechanical Systems and Signal Processing 20 (2006) 702–717
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Fig. 12. Influence of the analysis window on the response of the sensor assembly for the methacrylate selector: (a) FFT of the 4400-sample response signal (a) with no windowing (represented up to 350 kHz) and (b) for different analysis windows and factor-32 zero padding (represented up to 50 kHz).
4. Conclusions This paper, together with the previous work [12], represents a general framework for characterising analysis windows in terms of spectrum physical resolution. In [12], we have already concluded that both spectral magnitude and phase of an analysis window are indispensable for describing its capacity for resolving closely spaced signal components. We have demonstrated that for interharmonic distances less than a DFT bin, the variation of the spectral phase in the main lobe is more critical parameter than the shape of the magnitude spectrum. Assuming linear the spectral phase in the main lobe, it was shown that larger the slope, better the discrimination capacity. In a view of that, we have proposed a particular class of asymmetric windows which can be easily generated from the classical windows. The proposed windows are characterised by a very steep spectral phase in the main lobe, which makes them extend the physical resolution limits imposed by the classical windows. In this paper we have given a more thorough insight into the properties of the proposed asymmetric windows through several comparative studies in terms of physical resolution. As a measure of resolution, we have defined the discrimination bandwidth as a two equal-strength line separation which produces zero-magnitude crossover point between the corresponding spectral peaks. We have shown that, while all the classical windows have the same discrimination bandwidth equal to 1 bin, the proposed asymmetric windows will always have smaller discrimination bandwidth, even in presence of additive noise. How small the discrimination bandwidth will be, depends on two parameters of the classical window used as a generating sequence: the tapering and the causality-imposed time shift. Strong tapering means wide main lobe but steep spectral phase; consequently, windows like e.g. the Nuttall with ENBW ¼ 2 can achieve the discrimination depth as low as 0.65 bins. Further improvement in physical resolution is achieved by making the spectral phase slope even more pronounced. This is done by simply
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using larger classical windows; the corresponding non-symmetric window will exhibit the discrimination bandwidth reduction of 0.05 bins approximately. We have also discussed the influence of additive noise through the parameter denominated discrimination depth, defined for two arbitrary-strength components as a magnitude difference between the smaller spectral peak and the crossover point. It has been shown that the relative bias and the normalised standard deviation of the discrimination depth vary with the SNR similarly for the classical and asymmetric windows. Both classes of windows are quite robust against additive noise, because even for low SNR they maintain their discrimination properties. For a particular case scenario, we show that for SNR ¼ 5 dB the discrimination depth for the asymmetric Hanning window is expected to vary as much as 0.36. The overall analyses have been supported by the coin verification mechanical system example, where the asymmetric windowing has proven to be the most adequate for resolving closely spaced vibration modes with either the same or very different damping coefficients. Finally, we note that the proposed windows can easily be implemented as a complementary tool for band-selectable analysis in many dynamic signal analysers.
Acknowledgements This work has been partially supported by the Gobierno de Navarra.
References [1] S.J. Orfanidis, Introduction to Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1996, p. 491. [2] S. Gade, H. Herlufsen, Frequency Analysis, Bruel and Kjaer, 1985. [3] L.R. Rabiner, R.W. Schafer, C.M. Rader, The chirp-z transform algorithm, IEEE Transactions on Audio Electroacoustics 17 (1969) 86–92. [4] B. Liu, F. Mintzer, Calculation of narrow-band spectra by direct decimation, IEEE Transactions on Acoustics, Speech and Signal Processing ASSP-26 (6) (1978) 529–534. [5] A.V. Oppenheim, D.H. Johnson, Discrete representation of signals, Proceedings of IEEE 60 (1972) 681–691. [6] C. Braccini, A.V. Oppenheim, Unequal bandwidth spectral analysis using digital frequency warping, IEEE Transactions on Acoustics, Speech and Signal Processing ASSP-22 (1974) 236–244. [7] A. Makur, S.K. Mitra, Warped discrete Fourier transform: theory and applications, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 48 (9) (2001). [8] M. Zivanovic, A. Carlosena, Non-parametric interpolation methods: a comparative study, IEEE Transactions on Instrumentation and Measurement 50 (5) (2001) 1127–1132. [9] F.J. Harris, On the use of windows for harmonic analysis with the discrete Fourier transform, IEEE Proceedings 66 (1) (1978). [10] S. Gade, H. Herlufsen, Windows to FFT Analysis, Part I and II, Bruel & Kjaer Technical Review, No. 4, 1987. [11] S.E. Vidovin, et al., The frequency resolution of digital spectrum analyzers, Telecommunication and Radio Engineering 45 (2) (1990) 69–72. [12] M. Zivanovic, A. Carlosena, Extending the limits of resolution for narrow-band harmonic and modal analysis: a non-parametric approach, IOP Measurement Science and Technology 13 (2002) 2082–2089. [13] O. Dossing, A Method of Determining the Modal Frequencies of Structures with Coupled Modes, Bruel & Kjaer Technical Review, No. 3, 1986. [14] V.H. Jain, W.L. Collins, D.C. Davis, High accuracy analog measurements via interpolated FFT, IEEE Transactions on Instrumentation and Measurement IM-28 (1979) 113–122.
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[15] T. Grandke, Interpolation algorithms for discrete Fourier transform of weighted signals, IEEE Transactions on Instrumentation and Measurement IM-32 (1983) 350–355. [16] G. Andria, M. Savino, A. Trotta, Windows and interpolation algorithms to improve electrical measurement accuracy, IEEE Transactions on Instrumentation and Measurement IM-38 (4) (1989) 856–863. [17] J. Schoukens, R. Pintelon, H. Van Hamme, The interpolated fast Fourier transform: a comparative study, IEEE Transactions on Instrumentation and Measurement IM-41 (2) (1992) 226–232. [18] J.L. Pina, L.S. Echavarri, A. Carlosena, A.J. Lo´pez-Martı´ n, Device and procedure for verification of coins, Patent Application EP1378867, July 2002.
Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 20 (2006) 702–717 www.elsevier.com/locate/jnlabr/ymssp
On asymmetric analysis windows for detection of closely spaced signal components Miroslav Zivanovic, Alfonso Carlosena Universidad Pu´blica de Navarra, Department of Electrical and Electronic Engineering, Campus de Arrosadı´a, s/n, E-31006 Pamplona, Spain Received 24 May 2004; received in revised form 5 April 2005; accepted 12 April 2005 Available online 13 June 2005
Abstract Discrimination capacity, i.e. the capacity to resolve signal components, is probably one of the most important properties of the analysis windows. For components separated by more than a DFT bin the discrimination capacity is determined by the bandwidth and shape of the window’s spectral main lobe and the side lobe fall-off rate. The new classical paper of Harris is a milestone where a plethora of windows is discussed and compared through the aforementioned parameters. However, those parameters fail to describe situations given by components clustered within a DFT bin bandwidth. In those cases the discrimination capacity is almost completely determined by the spectral phase of the analysis window, while the shape of the window’s magnitude spectrum has little effect. This paper is devoted to a specific class of asymmetric analysis windows capable of resolving components separated by less than a DFT bin. Those windows have recently been proposed and analytically discussed. However, a number of important properties have not been tackled. Herein, the asymmetric analysis windows are thoroughly described and compared to the classical analysis windows in various aspects. The main benefits of the proposed windows are extended discrimination capacity, robustness against additive noise and simple generation. These properties make them very good candidates for narrow-band spectrum analysis of mechanical systems. Results for some real-world examples are presented, showing the capability of the new windows to resolve closely spaced vibration modes when there is a lack of spectral resolution due to the finite register length. r 2005 Elsevier Ltd. All rights reserved. Keywords: FFT; Spectrum resolution; Analysis windows; Tone separation
Corresponding author. Tel.: +34 948 169 024; fax: +34 948 169 720.
E-mail address: [email protected] (M. Zivanovic). 0888-3270/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2005.04.003
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1. Introduction There are a number of DFT-based signal processing algorithms devoted to harmonic detection and estimation of finite-register data. Consequently, the most important issue to be dealt with is interplay between the computational and physical resolution of the discrete spectrum. Computational resolution is defined [1] as a distance between the DFT bins at which the discrete time Fourier transform (DTFT) is computed. This parameter can be constant in the whole spectrum [2] or in a particular frequency range [3,4]. Other approaches are designed to yield nonuniform bin spacing on the linear frequency grid, but those methods seem to be less known [5–7]. A critical comparison for the above methods has been described by the authors, what sheds new light on their properties and applicability [8]. Physical resolution (also known as a minimum resolution bandwidth) is defined [9] as a minimum separation between two equal-strength signal components such that for arbitrary spectral locations their respective main lobes can be resolved. This parameter depends on the analysis window and is typically determined, according to several criteria, from the bandwidth and the shape of the main lobe in the window’s magnitude spectrum [10,11]. The most common criteria like 3 dB, 6 dB, equivalent noise bandwidth (ENBW), have been applied in Ref. [9] to establish a comparison among the classical analysis windows like the rectangular, triangular, families of raised cosine, etc. This study reveals that the rectangular window has the smallest resolution bandwidth, which corresponds to the narrowest main lobe. However, those criteria neglect the main lobe’s spectral phase, which is an important factor in addition of complex spectra. All classical analysis windows have the same spectral phase, due to the causality-imposed time shift; consequently, the determining factor in the physical resolution will still be the bandwidth and the shape of the main lobe. However, an arbitrary non-symmetric time weighing could exhibit quite different spectral phase; thus, the classical criteria would simply not be adequate for window comparison. We have demonstrated [12] that for estimating the physical resolution, spectral phase is a stronger condition and should be the principal criterion, complemented by the classical magnitude-based criteria. In a view of that, we have proposed a class of asymmetric windows [12] which can be generated from the classical analysis windows. A particular asymmetric analysis window is characterised by a broad main lobe which, according to the classical criteria, should provide for a quite large resolution bandwidth. On the other hand, the phase of the main lobe is such that the margins for component discrimination, imposed by the rectangular window, are extended. Hence, the proposed asymmetric windows outperform the classical analysis windows in terms of physical resolution. The goal of this paper is to give a more thorough insight into the properties of the proposed asymmetric windows and to show their effectiveness in dealing with real-world signals/systems. We shall start with a brief summary of the previous work, while for more detailed explanations and the analytical background the reader is invited to consult [12]. Next, we shall discuss the most prominent properties of the proposed windows in terms of spectral analysis and physical resolution. Also, a noise study will be presented in order to quantify the robustness of the new windows against additive noise. Possible applications are related to a number of practical problems, like [13], which arise frequently in modal analysis. An extensive experimental part will show the benefits of using the asymmetric analysis windows in real-world applications. Finally, we stress that throughout the paper a critical comparison with the classical analysis windows in several aspects was done.
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2. The new class of asymmetric windows 2.1. Summary of the previous work As a measure of physical resolution for an arbitrary analysis window, we defined a parameter denominated discrimination bandwidth. It was referred to as a minimum frequency distance between two signal components that produces amplitude zero crossing between the corresponding spectral peaks. We have shown that this parameter is inversely proportional to the slope of the main lobe’s spectral phase, whenever that phase can be conceived as a linear function of frequency. As mentioned in Introduction, all classical analysis windows, which we denominated S-class windows, share the same spectral phase slope; it is proportional to N/2, where N is the size of the window: Discrimination bandwidth ¼
p ðradÞ. N=2
(1)
Therefore, all S-class windows will have the same discrimination bandwidth and a comparison among them should always be accomplished by means of the classical magnitude-based criteria. Once the basis for comparison among arbitrary analysis windows has been established, we proposed and discussed a specific class of asymmetric windows, which we denominated $-class windows. A $-class window is generated by taking the first N samples of a 2N-sample S-class window. In order to best describe the properties of the $-class windows, we define also the unilateral window obtained by simply reversing the order of samples of a $-class window. The three above-mentioned window shapes are shown in Fig. 1. We are now interested in determining the slope of the spectral phase in the main lobe. However, the spectral phase of a $-class window is in general a non-linear function of frequency. Nevertheless, we have shown that within the main lobe bandwidth the spectral phase can be well approximated by a linear segment, the fact that facilitates comparison with the S-class windows in terms of physical resolution. In particular, we have found that the discrimination bandwidth for an arbitrary N-sample $-class window is given by Discrimination bandwidth ¼
p ðradÞ. N 1m
(2)
The parameter m is determined by the shape of the corresponding 2N-sample S-class window and its value is always between 0 and N/2. The former corresponds to a $-class window generated from the 2N-sample unite impulse shifted N samples to the right of the origin; the letter corresponds to a $-class window generated from the 2N-sample rectangular window. We note that the delayed unite impulse is treated here as a special case of S-class window. The rest of the $-class windows are comprehended between those two limiting cases and achieve discrimination bandwidth less than p=ðN=2Þ. Therefore, the $-class of windows reduces the discrimination bandwidth imposed by the S-class windows and thus improve the physical resolution. In the following section, we will discuss more thoroughly on the nature of the parameter m and how it is related to the properties of the $-class.
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200−sample S−class window 1 0.8 0.6 0.4 0.2 0 0
50
100
100−sample unilateral window 1
150
100−sample asymmetric window 1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0 0
50 Sample
100
200
0 0
50 Sample
100
Fig. 1. Basic window shapes treated throughout this paper.
2.2. Some properties of the $-class of windows Let us first examine, through the following example, how windows with the same main lobe width can produce very different spectral phase. The spectra for the 100-point Hanning window, 100-point unilateral Hanning window and 100-point asymmetric Hanning window are shown in Fig. 2. In terms of the ENBW, all three windows provide for the same physical resolution equal to 1.5 bins. However, their spectra are quite different as the real and imaginary parts are concerned. The unilateral Hanning and asymmetric Hanning windows have the same magnitude spectra, but very different real and imaginary components. As both windows are generated from the 200sample Hanning window (see Fig. 1) and recalling that the discrimination bandwidth is closely related to the time shift of the analysis window, we guess that the unilateral Hanning window should have the least pronounced spectral phase, while the asymmetric Hanning window should exhibit the spectral phase characteristic with the largest slope within the main lobe. Hence, from Eq. (2) we deduce that the asymmetric Hanning window will have the smallest m and thus the smallest discrimination bandwidth. The largest m will correspond to the unilateral Hanning window, while the S-class window (Hanning window) will be in the middle point with m ¼ N=2 1. The graphical comparison among the corresponding spectral phase curves is shown in Fig. 3. Recall that for both unilateral Hanning and asymmetric Hanning windows the phase is quasi-linear with frequency within the main lobe. We have already said that the parameter m for an N-sample $-class window depends on the shape of the corresponding 2N-sample S-class window. The stronger the tapering, the smaller m,
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1 Magnitude Real Imaginary
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(c)
0
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1
1.5
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Frequency [bin]
Fig. 2. Magnitude, real and imaginary spectrum for the following 100-point analysis windows: (a) Hanning, (b) unilateral Hanning and (c) asymmetric Hanning.
being theoretically 0 for the unite impulse. This effect can also be explained in the frequency domain: the smallest m will correspond to the S-class window with the broadest main lobe and vice versa. In order to support this statement, we have compared a number of asymmetric windows generated from different S-class windows, namely the triangular, Hanning, Blackman and Nuttall, in terms of spectral phase. It can be seen in Fig. 4 that, as we expected, the spectral phase slope varies coherently according to the main lobe’s bandwidth (expressed here through the ENBW). The Nuttall window achieves the largest slope, having the widest main lobe, while the smallest slope is assigned to the triangular window. The same four windows will be compared in terms of discrimination bandwidth in Section 3. Another way to achieve small discrimination bandwidth will be introduced through the following argumentation. The slope of the spectral phase of an N-sample $-class window is a result of the combined action between the time shift and the tapering of the corresponding
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4.5 4
Phase [rad]
3.5 3 2.5 2 1.5 1 0.5 0 0
0.2
0.4 0.6 Frequency [bin]
0.8
1
Fig. 3. Spectral phase characteristics for the Hanning, unilateral Hanning and asymmetric Hanning windows within a DFT bin; the size of the windows is 100 samples.
4 Triang (ENBW = 1.33) Hanning (ENBW = 1.50) Blackman (ENBW = 1.73) Nuttall (ENBW = 2.00)
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Phase [rad]
2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3
0.4
0.5
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Frequency [bin]
Fig. 4. Spectral phase characteristics for the following $-class windows: triangular, Hanning, Blackman and Nuttall within a DFT bin; the size of the windows is 100 samples.
2N-sample S-class window. In the preceding paragraph we have shown the influence of the tapering, being constant the time shift equal to N. Now, we can fix the tapering and vary the time shift, which can be done in practice through larger S-class windows. By taking the first N samples from an LN-sample S-class window, where L42, we obtain a particular non-symmetric window
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which we denominated L$-class window. It is to be expected that the time shift of LN=2 produces spectral phase slope proportional to LN=2 1 m and thus very small discrimination bandwidth for an L$-class window. This, however, will not be completely satisfied, especially for large L; an explanation lies in the frequency representation of an arbitrary non-symmetric window. Recall that the most prominent S-class windows are raised-cosine windows. The DTFT of a raised-cosine window is an addition of the central Dirichlet kernel (spectrum of the rectangular window) and a number of pairs of translated kernels located at the zeros of the central kernel, in order to partially cancel the side lobe structure. However, in the DTFT of an N-sample $-class window, the translated kernels are shifted in frequency by p=N towards the central kernel. If the translated kernels were also frequency-compressed by a factor of 2, then the obtained DTFT would correspond exactly to a 2N-sample S-class window. Although this is obviously not the case, the slope of the spectral phase is still proportional to the time shift. For an L$-class window, the strong overlap among the central and the translated kernels would not permit the spectral phase slope increase in a linear fashion. Nevertheless, an improvement can indeed be achieved, as it is shown in Fig. 5, where we can see that the slope variation is still coherent with increasing time shift. This section can be complemented by examining the influence of the $-class windowing on harmonic amplitudes. For an N-point $-class window, the amplitude A of a single harmonic component is computed by taking into account the contributions of all the Dirichlet kernels at the harmonic frequency: " # K=2 n p o X 2 Dð0Þ 2 i , A¼ < ai D N N i¼1
(3)
4 $−class 4$−class 8$−class
3.5 3
Phase [rad]
2.5 2 1.5 1 0.5 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Frequency [bin]
Fig. 5. Spectral phase characteristics for the $-class, 4$-class and 8$-class Hanning windows; the size of the windows is 100 samples.
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where D is the Dirichlet kernel, K is the number of kernels in the window’s spectrum, ai are the kernels’ weighs and N is the size of the window. However, the effect of the asymmetric windowing on harmonic amplitudes is of little interest. This is because the $-class windows are aimed at resolving components clustered within a DFT bin, where heavy interharmonic interference provokes gross errors in parameter estimation, even if all the components have been correctly detected. In such circumstances it is quite difficult to quantify the difference between the true harmonic parameters and the parameters estimated from the spectral peaks. Furthermore, and to the best of our knowledge, a number of papers that deal with harmonic estimation (e.g. [14–17]) assume the interharmonic distance (separation between the signal components) is large enough to disregard the interference between the corresponding spectral peaks. In spite of the above argumentation, we will present a study on harmonic parameter estimation errors for different analysis windows in Section 3. 2.3. Influence of noise A very important aspect of window’s performance is robustness against additive noise. More specifically, we are interested in how the detection of closely spaced signal components varies as the noise power changes. As a measure of detection we use the parameter we denominated discrimination depth. For two arbitrary-strength components it is defined as a magnitude difference between the smaller spectral peak and the crossover point. For equal-strength components the crossover point is typically half-way between the peaks. Hence, this parameter shows how close to the limit of resolvability (two components still represented as two peaks) the corresponding components are. When some quantity of noise is added to a deterministic signal, the discrimination depth becomes a random variable and thus can be characterised by its probabilistic properties: the first moment (mean and bias) and the second moment (variance and standard deviation). We have established a particular case scenario in order to compare different analysis windows. The test signal was composed of two tones and a zero mean uniformly distributed white noise r(n) xðnÞ ¼ 2 cos½ð2pf 1 n þ 0:7Þ þ cos½ð2pf 2 n þ 0:2Þ þ rðnÞ. The normalised frequencies f1 and f2 have been adjusted for each analysis window in order to achieve the maximum discrimination depth in absence of noise. Note the amplitude and phase differences, which have been set in order to obtain more realistic conditions for component discrimination. As a measure of noise contribution to the signal we have used the signal-to-noise ratio (SNR) in Decibels. Each time the SNR were modified, 100 experiments were realised and the bias and normalised standard deviation were calculated for the rectangular, Hanning and asymmetric Hanning windows, and the corresponding curves were plotted in Fig. 6. In general, all three windows exhibit quite similar behaviour in presence of noise. By taking a closer look, we can see that for SNR ¼ 5 dB the asymmetric Hanning window presents the largest relative bias, although it remains below 20% approximately. As SNR increases, all three curves tend to zero while the discrimination depth approaches to its nominal value. On the other hand, the asymmetric Hanning window is between the rectangular and Hanning window, as the standard deviation is concerned. Like the normalised bias, the differences among the curves are more pronounced for low SNR. Therefore, from the above analysis based on the particular case
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0.2
0.1 asymmetric hannning rectangular
0.08
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0.06
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0.14 0.12 0.1 0.08 0.06
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0.18
5
10
15
(a)
SNR [dB]
20
25
0
30
(b)
5
10
15
20
25
30
SNR [dB]
Fig. 6. Relative bias and normalised standard deviation of the discrimination depth as a function of SNR for the rectangular, Hanning and asymmetric Hanning windows. The curves are generated by calculating the statistics on a two-component signal whose parameters are: A1 ¼ 2, j1 ¼ 0:8 , A2 ¼ 1, j2 ¼ 0:1 ; the normalised frequencies f1 and f2 have been adjusted for each analysis window in order to achieve the maximum discrimination depth in absence of noise.
scenario, we can say that even for low SNR, the discrimination capacity of both $-class and Sclass windows is still preserved.
3. Experimental results 3.1. Comparative study In this subsection, we compare different analysis windows in terms of discrimination depth as a function of interharmonic distance for two equal-strength components. For a particular window, this analysis provides for the discrimination bandwidth and the resolution limit, corresponding to the maximum and minimum discrimination depth, respectively. The first comparative study, whose results are shown in Fig. 7, comprehends the four $-class windows already treated in Section 2: the triangular, Hanning, Blackman and Nuttall. As it was expected, according to the discussion in Section 2, the Nuttall and triangular window are on the margins of physical resolution, while the Hanning and Blackman window lie in between. The Nuttall window achieves the smallest discrimination bandwidth around 0.65 bins but it loses its discrimination capability for interharmonic distance greater than 0.95 bins. The triangular window has the largest discrimination bandwidth of 0.75 approximately; however, its discrimination depth is around 0.30 for interharmonic distance equal to 1 bin. As the resolution limit is about, there is only a 0.03 bins difference between the Nuttall and triangular window. Therefore, a choice of a $-class window is a trade-off between discrimination bandwidth and resolution limit. Similarly, we have compared the three L$-class Hanning windows from Section 2 and shown the results in Fig. 8. As L increases the discrimination bandwidth gets smaller, what is coherent with the qualitative explanation given in Section 2. However, this process is not linear in the sense that for large L the discrimination
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Interharmonic distance [bin]
Fig. 7. Relative discrimination depth as a function of interharmonic distance for two equal-strength components through the following $-class windows: triangular, Hanning, Blackman and Nuttall.
1 $−class 4$−class 8$−class
0.9
Relative discrimination depth
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.6
0.65
0.7 0.75 0.8 0.85 Interharmonic distance [bin]
0.9
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1
Fig. 8. Relative discrimination depth as a function of interharmonic distance for two equal-strength components through the following L$-class windows: $-class, 2$-class and 8$-class Hanning.
bandwidth keeps almost constant, due to the effect of the strong Dirichlet kernel overlap already discussed. This situation is well reflected in the curves for the 4$-class and 8$-class Hanning windows, respectively. Finally, we have compared windows from different classes: the rectangular, Hanning and asymmetric Hanning, with the SNR as a parameter. We have chosen the rectangle
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Relative discrimination depth
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0.7 0.6 0.5 0.4 0.3
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(a)
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0 0.6
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(b)
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Interharmonic distance [bin]
Fig. 9. Relative discrimination depth as a function of interharmonic distance for two equal-strength components through the rectangular, Hanning and asymmetric Hanning windows: (a) SNR ¼ 30 dB and (b) SNR ¼ 10 dB.
Table 1 Window
Discrimination BW (bins)
Frequency bias (bins)
Amplitude bias
Asymmetric Hanning Rectangular
0.71 1.00 1.00
0.49 0.46 0.20
0.40 0.35 0.10
window because it is a particular case of the S-class, the Hanning window being the simplest raised cosine S-class window and the asymmetric Hanning window, belonging to the $-class and being generated from the Hanning window. From Fig. 9 (SNR ¼ 30 dB) we observe that the asymmetric Hanning window has the discrimination bandwidth around 0.71 bins and the resolution limit below 0.60 bins. Both the rectangular and Hanning windows have the same discrimination bandwidth equal to 1 bin, which arise from the fact that they are S-class windows. The rectangular window performs better than the Hanning window, because the latter has broader main lobe. This fact is captured through the resolution limit, which is around 0.65 bins for the rectangular and approximately 0.75 bins for the Hanning window. However, up to 0.80 bins, the asymmetric Hanning window has the largest discrimination capacity. Similar conclusions can be inferred from Fig. 9b, where the comparison has been done for SNR ¼ 10 dB. The preceding paragraph is complemented by the following study on harmonic parameter estimation errors. In particular, we are interested in quantifying the bias in amplitude and frequency estimation for two equal-strength components separated by the discrimination bandwidth for a given analysis window. In absence of additive noise the bias is due only to the overlap between the main lobes. The amplitude and frequency bias are measured for the rectangular, Hanning and asymmetric Hanning windows and given in Table 1. The smallest bias corresponds to the rectangular window while the asymmetric Hanning window introduces the
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largest estimation errors. Note the similarity between the Hanning and asymmetric Hanning windows; the latter has only slightly larger bias but, in turn, smaller discrimination bandwidth. This effect is detected in all raised-cosine S-class windows and their $-class counterparts, where larger main lobe is traduced in larger amplitude and frequency bias. 3.2. Detection of strongly coupled vibration modes Very often the goal is to detect high-order vibration modes, masked by the spectral contribution of the principal mode [13], in order to identify as good as possible a model for the system under test. Therefore, it can be expected that the best-choice analysis would be a combined action of both S-class and $-class windows: the former to isolate well-separated modes and the letter to discriminate the closely spaced ones (relative to the long and near-term leakage, respectively). The performance of herein proposed asymmetric windows has been tested in a coin verification system fully described in Ref. [18]. A brief discussion of the system follows, supported by Fig. 10. The nucleus of the system is a sensor assembly comprising a miniature piezoelectric accelerometer mounted on an inertial mass, which has a form of a 25 mm-long and 7 mm-wide steel cylinder. The vibrations originated from the coin impacts against the inertial mass surface are transformed into equivalent electrical signal generated by the accelerometer. This signal is then transferred to the coin selector where it is processed by means of different coin verification algorithms.
Fig. 10. Miniature piezoelectric accelerometer mounted on a 25 mm-long and 7 mm-wide steel cylinder. This device is referred to as sensor assembly and is a heart of the coin verification system.
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The device has been tested for two different types of selector, namely plastic and methacrylate. The goal was to check the presence of the vibration modes specific for each of the said materials, which could interfere with the coin impact signal. It has been done through low-pass filtering and windowing of the impulse response of each system. The final step was to calculate the FFT with some zero padding to smooth out the frequency response. The impulse response has been obtained by loosing a 1 mm-diameter steel ball from approximately 15 cm height with respect to the cylinder and acquiring the output signal of the sensor assembly by a digital oscilloscope at the rate of 4 MS/s. In the case of the plastic selector, we aimed to detect the 35 kHz secondary mode strongly masked by the 17 kHz principal mode. The corresponding interharmonic distance is equal to 1.14 DFT bins. Due to important damping of the secondary mode, both rectangular and Hanning window detect only the strongest mode. On the contrary, the asymmetric Hanning window exhibits a deep dip in the crossover point between the modes (Fig. 11). This observation is coherent with the expression (3) which states that the discrimination bandwidth does not depend on the magnitude but only on the phase. We observe that the local maxima of the spectrum do not coincide with the mode frequencies because of the strong interharmonic interference. For the methacrylate selector the goal was to discriminate two closely spaced high-order vibration modes at 34 and 35 kHz, respectively. This time the modes’ damping coefficients are similar and the corresponding interharmonic distance is equal to 1.10 DFT bins. In order to improve the discrimination capacity even more the asymmetric window applied has been constructed from an 8N-point S-class Hanning window. The classical windows exhibit one peak at approximately 35 kHz and consequently fail to detect the closely spaced modes. However, the asymmetric window achieves positive discrimination depth and manages to discern the modes (Fig. 12). Again, the harmonic parameters can only be roughly estimated from the spectrum, due to the strong overlap between the modes.
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Fig. 12. Influence of the analysis window on the response of the sensor assembly for the methacrylate selector: (a) FFT of the 4400-sample response signal (a) with no windowing (represented up to 350 kHz) and (b) for different analysis windows and factor-32 zero padding (represented up to 50 kHz).
4. Conclusions This paper, together with the previous work [12], represents a general framework for characterising analysis windows in terms of spectrum physical resolution. In [12], we have already concluded that both spectral magnitude and phase of an analysis window are indispensable for describing its capacity for resolving closely spaced signal components. We have demonstrated that for interharmonic distances less than a DFT bin, the variation of the spectral phase in the main lobe is more critical parameter than the shape of the magnitude spectrum. Assuming linear the spectral phase in the main lobe, it was shown that larger the slope, better the discrimination capacity. In a view of that, we have proposed a particular class of asymmetric windows which can be easily generated from the classical windows. The proposed windows are characterised by a very steep spectral phase in the main lobe, which makes them extend the physical resolution limits imposed by the classical windows. In this paper we have given a more thorough insight into the properties of the proposed asymmetric windows through several comparative studies in terms of physical resolution. As a measure of resolution, we have defined the discrimination bandwidth as a two equal-strength line separation which produces zero-magnitude crossover point between the corresponding spectral peaks. We have shown that, while all the classical windows have the same discrimination bandwidth equal to 1 bin, the proposed asymmetric windows will always have smaller discrimination bandwidth, even in presence of additive noise. How small the discrimination bandwidth will be, depends on two parameters of the classical window used as a generating sequence: the tapering and the causality-imposed time shift. Strong tapering means wide main lobe but steep spectral phase; consequently, windows like e.g. the Nuttall with ENBW ¼ 2 can achieve the discrimination depth as low as 0.65 bins. Further improvement in physical resolution is achieved by making the spectral phase slope even more pronounced. This is done by simply
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using larger classical windows; the corresponding non-symmetric window will exhibit the discrimination bandwidth reduction of 0.05 bins approximately. We have also discussed the influence of additive noise through the parameter denominated discrimination depth, defined for two arbitrary-strength components as a magnitude difference between the smaller spectral peak and the crossover point. It has been shown that the relative bias and the normalised standard deviation of the discrimination depth vary with the SNR similarly for the classical and asymmetric windows. Both classes of windows are quite robust against additive noise, because even for low SNR they maintain their discrimination properties. For a particular case scenario, we show that for SNR ¼ 5 dB the discrimination depth for the asymmetric Hanning window is expected to vary as much as 0.36. The overall analyses have been supported by the coin verification mechanical system example, where the asymmetric windowing has proven to be the most adequate for resolving closely spaced vibration modes with either the same or very different damping coefficients. Finally, we note that the proposed windows can easily be implemented as a complementary tool for band-selectable analysis in many dynamic signal analysers.
Acknowledgements This work has been partially supported by the Gobierno de Navarra.
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