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Windowing applied to a given signal is a technique commonly used in signal processing in order to reduce spectral leakage in a signal with many data. Several windows are well known: hamming, hanning, beartlett, etc. The selection of a window is based on its spectral characteristics. Several papers that analyze the amplitude and width of the lobes that appear in the spectrum of various types of window have been published. This is very important because the lobes can hide information on the frequency components of the original signal, in particular when frequency components are very close to each other. In this paper it is shown that the size of the window can also have an impact in the spectral information. Until today, the size of a window has been chosen in a subjective way. As far as we know, there are no publications that show how to determine the minimum size of a window. In this work the frequency interval between two consecutive values of a Fourier Transform is considered. This interval determines if the sampling frequency and the number of samples are adequate to differentiate between two frequency components that are very close. From the analysis of this interval, a mathematical inequality is obtained, that determines in an objective way, the minimum size of a window. Two examples of the use of this criterion are presented. The results show that the hiding of information of a signal is due mainly to the wrong choice of the size of the window, but also to the relative amplitude of the frequency components and the type of window. Windowing is the main tool used in spectral analysis with nonparametric periodograms. Until now, optimization was based on the type of window. In this paper we show that the right choice of the size of a window assures on one hand that the number of data is enough to resolve the frequencies involved in the signal, and on the other, reduces the number of required data, and thus the processing time, when very long files are being analyzed.

One of the most important tools in signal processing is the Nyquist theorem. Many of the processing tools are meaningless if the theorem is not satisfied. To date, the Nyquist theorem is often used in such a way that the acquisition of a signal is made with an excessive sampling frequency.

Sometimes, an overly large amount of samples is chosen. One of the most used tools to remedy the effect of oversampling is the use of windows that reduce noise and spectral leakage. Windows are used in non-parametric estimators and even in spectrograms. In 1978 Fredric J. Harris published his article “On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform” [

For over 30 years, research on the characteristics of the windows that appear in the article by Harris has not changed significantly. Many authors present new algorithms that allow for improvements in the lobes, both lateral and central, in the same direction as Harris [

As a complement to all previous work, the authors of this paper use the frequency resolution

to determine the minimum number of samples required in a window.

Due importance has not been given to (1) even though it is fundamental in the analysis as well as in the acquisition of a signal. Without the adequate resolution, the frequency information, important to a particular phenomenon, might be hidden. The evaluation of the frequency resolution, before acquiring a signal or in the process of analyzing it, allows the making of decisions about the use of certain tools, such as the minimum size of a window.

The main contribution of this paper is the possibility of making a precise choice on the number of data that ensures the resolution between two very close frequencies, and diminishes the processing time by reducing the number of data required if the analysis is made before acquisition.

To date, little is known about what the minimum size of a window should be. Usually, the ad-hoc choice depends on the flair and experience of the user.

Harris mentions in his article: “The two operations to which we subject the data are sampling and windowing. These operations can be performed in either order. Sampling is well understood, windowing is less so, and sampled Windows for DFT’s significantly less so!” [

Several processing tools like periodograms, spectrograms [

Examples with experimental and simulated signals, that show the importance of considering

A monochromatic signal with a frequency

In both

To emphasize the importance of the width of the peaks, a signal was acquired with four frequency components: 1 MHz, 1.01 MHz, 1.05 MHz and 1.1 MHz. The following parameters were used: sampling frequency

The resolution in the frequency domain is given by

and

Even though

Based on

There is a great variety of factors due to which the information in a given signal cannot be clearly observed, such as the noise of the devices used in an experiment, the experiment itself and even the software used to analyze the acquired signal. The instruments with which signals are acquired usually do so at high sampling rates with a small number of samples, regardless of the type of signal. In general, instruments only allow the manipulation of the sampling frequency in within a set of choices provided by the manufacturer. As a result, once the signal is acquired, nothing can be done about the resolution attained. Sometimes, processing techniques are used as remedial tools, but they cannot extract information that does not exist in the acquired signal.

This work focuses on clarifying that the hiding of information in a signal depends, not only on the lobes of the spectra produced by the windows, but also on the fact that the frequency resolution

To understand the importance of the frequency resolution

An important feature to be noted in

In the analysis of different graphs, it was observed that the minimum size of a window was controlled by the size of

where

where

In the example we have

By applying this result the graphs shown in

Inequality (3) allows the objective evaluation of the minimum size of a window.

Equation (2) and Equation (3) provide the minimum size of a window very accurately when components we want to differentiate have very similar amplitudes.

In this section the effect of the size of a window versus the use of the type of window is analyzed.

Different windows are used on a signal with two frequency components

With the above parameters, the minimum size of a window is calculated using Equation (2) and Equation (3),

Hence

As can be seen in

In the following example a signal with two components, but with a difference in amplitude of 40 dBs is considered. Three types of windows are used in particular because they tend to hide information [

with amplitudes of 1 and 0.01 volts respectively,

With the same sampling parameters, but slightly changing one of the frequencies (as

in [

However, an increase in the size of the window allows us to see the component with the smaller amplitude;

So far it has been observed that with a frequency resolution of

The importance of relative amplitudes of the components can be further analyzed. Analogous to Harris, three windows, rectangular, Poisson and Hanning-Poisson will be

applied to a signal with two frequency components,

amplitudes of 0 dBs, 2) with a difference in amplitudes of 20 dBs, and 3) with a difference in amplitudes of 40 dBs. The results are shown in Figures 15-17.

Figures 15-17 show that the spectral behavior of the windows has little influence on the observation of the frequency components, whether closely spaced components or with a large difference of amplitude.

The results shown so far allow for a more objective use of nonparametric periodograms. These are processing tools used to reduce significantly the signal leakage by applying spectral windowing, [

The Welch parametric periodogram was applied to the same signal considered in

with rectangular windows of 1024 samples.

The decrease in spectral leakage is remarkable in the previous figures, just as the theory predicts. It is clear that, the lower the number of samples in the spectral window used, the more the leakage decreases. However, by choosing a window with few samples, wrong results could be obtained, as can be seen in

It is clear that increasing the number of samples in the window will bring us closer to the original signal, but since one of the objectives is to decrease nonparametric periodogram spectral leakage of a signal, it is desirable to have a window with the fewest possible samples but that provides relevant information about the original signal.

Prabhu [

Even though there are no clear methods to determine the minimum size of a window, the 2013 version of Matlab in the path Signal Processing Toolbox/User Guide/ Statistical Signal Processing/Spectral Analysis/Nonparametric Methods states that

Resolution refers to the ability to discriminate spectral features, and is a key concept on the analysis of spectral estimator performance.

In order to resolve two sinusoids that are relatively close together in frequency, it is necessary for the difference between the two frequencies to be greater than the width of the mainlobe of the leaked spectra for either one of these sinusoids. The mainlobe width is defined to be the width of the mainlobe at the point where the power is half the peak mainlobe power (i.e., the 3 dB width). This width is approximately equal to

In other words, for two sinusoids of frequencies

If the Matlab suggestion is applied to the example of two sinusoids separated by 10 KHz, the value obtained for L is

However, as

In this paper, an inequality is proposed to determine objectively the minimum size of a window, instead of the trial and error technique commonly used. The results can be applied in particular to certain spectral estimators, better known as nonparametric periodograms.

It is also shown that the minimum size of a window is required to observe all the frequency components of a given signal; it is necessary that the frequency resolution should be considered when a signal is acquired and not only the Nyquist theorem.

Once the minimum size of a window has been evaluated, the relative amplitude of the frequency components and window type would be factors to be considered depending on the leakage they produce.

This work leaves behind the subjectivity to determine the minimum size of a window, merely by considering the desired resolution, which is now possible to assess objectively by controlling the number of samples and the sampling frequency.

The resolution

The consideration and evaluation of

Harris [

This work was supported by DGAPA; PAPIME PE110216 Project “Propagación de ondas en medios sólidos, fluidos y gases”.

Alvarado R., J.M and Stern F., C.E. (2016) Evaluation of the Minimum Size of a Window for Harmonics Signals. Journal of Signal and Information Processing, 7, 175-191. http://dx.doi.org/10.4236/jsip.2016.74017