An engineering approach to time-frequency uncertainty criteria/ Laiko ir daznio neapibreztumo kriterijaus inzinerinis nustatymas.
The uncertainty principle or duration-bandwidth theorem is a well known fundamental result in signal analysis and in quantum mechanics, see, e.g. reviews [1, 2] and references therein. In the information and communication technology a large number of theoretical and practical problems exists where estimation for timefrequency spreading product are needed to assess, for example, the width of spectral peaks by the duration of the measurement window [3, 4]. Often the principal limitations are not set separately to the time or frequency domain but for the time-frequency spreading product AtAm (below "uncertainty product"). In quantum mechanics, this kind of uncertainty criteria are of fundamental importance allowing to relate, e.g. uncertainty of particle momentum to the spatial spreading of wavefunction [1, 2].
In present study we will rely on the traditional complex Fourier transform s(t) [left and right arrow] s([omega]) [1-6]:
S([omega]) = [[integral].sup.+[infinity].sub.-[infinity]] s(t)exp(- j[omega]t)dt, (1)
s(t) = -1/2[pi] [[integral].sup.+[infinity].sub.-[infinity]] s([omega])exp(+ j[omega]t) d[omega], (2)
but at that we will compare two different methodologies for evaluation of time-limited signals:
1) The conventional theoretical approach based on standard deviations [DELTA]t, [DELTA][omega] of time and frequency [1, 2, 5-8] (see Appendix A) that yield the traditional inequality of the Fourier Uncertainty Principle (FUP)
[DELTA]t[DELTA][omega] [greater than or equal to] 1/2. (3)
2) An engineering approach that detects signal widths on certain relative levels and yields the uncertainty relations in the limit value form
[DELTA]'t[DELTA]'[omega] [right arrow] const, (4)
separately for any harmonic of signal if the number of signal oscillation periods within the time window increases. In particular, our pre-work  yielded an approximate result const [approximately equal to] 7.582 for the special case of rectangular time window and 50% detection height.
The problematic tasks for the conventional approach
The lower limit of inequality (3) characterizes exactly the single Gaussian pulses [1-6] and yields also rather acceptable uncertainty product estimation for other single pulses . However, already in the case of the most common tasks of the pulse train type, the conventional standard deviations yield an unreasonably growing uncertainty product [DELTA]t[DELTA][omega] with the increasing number of pulses N . Fig. 1 summarizes the typical problematic tasks for what the conventional standard deviations of frequency are not related to the widths of spectral peaks.
[FIGURE 1 OMITTED]
The first case in Fig. 1 considers the time-limited train of sine-pulses. The Fourier transform (1) yields at that two main spectral peaks at [+ or -][[omega].sub.1] = [+ or -]2[pi]/[T.sub.1] where [T.sub.1] is the single pulse repetition period. The standard deviation methodology (see Appendix A) that integrates the square of frequency deviation from the average value yields here a large frequency uncertainty [DELTA][omega] [approximately equal to] [[omega].sub.1] that is impractical for most of applications. In contrast, a relative definition [DELTA]'[omega], e.g., full width at half maximum (FWHM) of any of spectral main peaks, may offer a much more reasonable measure.
The second case in Fig.1 presents the polyharmonic signal that may also considered as an approximation for the arbitrary repeating pulse. As one can see, the conventional uncertainty of frequency [DELTA][omega] tends to include the full band of peaks corresponding to different harmonics and not the width of individual main peaks that remain nearly constant if the relative definition at 50% of height would be used.
The third case in Fig.1 shows the carrier wave with a modulating signal. At that the conventional [DELTA][omega] will be predominantly defined by the frequency of carrier wave in contrast to the modified definition [DELTA]'[omega] that is not influenced by the carrier wave.
In conclusion, due to artificially increased frequency spreading that follows the distances between spectral peaks and not the widths of individual peaks, the conventional methodology yields also the increasing uncertainty product [DELTA]t[DELTA][omega] if the number of pulses increases . An additional problem is that the signals with discontinuities on the time window border yield not enough rapidly vanishing spectra of the 1/[omega] type. At that the standard deviation calculation methodology (Appendix A) fails to give a finite result.
Due to the abovementioned problems, the straightforward use of FUP (3) has been rather limited for signal processing tasks. Instead of that in engineering the rough time-frequency uncertainty estimations of the form
[DELTA]t[DELTA]f [approximately equal to] 1, (5)
where f = [omega]/2[pi] is the ordinary frequency, have been
often employed, e.g. [10-12]. The background of uncertainty relation (5) relies on Fourier transform of the rectangular pulse signal [3,4] (see below Figs. 2 and 3).
In terms of angular frequency [omega] = f/2[pi], the uncertainty relation (5) must be written as
[DELTA]t[DELTA][omega] [approximately equal to] 2[pi] [approximately equal to] 6.28, (6)
that offers over 12 times larger uncertainty product than the lower limit of conventional FUP (3). The results for separate spectral peaks in Fig. 1 support clearly relation (6) rather than lower limit of (3). Below we will focus on the criteria in the form of (5) and (6).
It should be mentioned that to overcome the problem of double peaks (the first case in Fig. 1), earlier Kharkhevich has proposed a modified conventional methodology  that uses only positive frequencies for calculation of [DELTA][omega]. However, as the numerical calculations show , this truncated approach also does not yield the expected saturation of the uncertainty product at a certain level if the number of pulses increases. Additionally, the approach of Kharkhevich is not solving the problem of over increased [DELTA][omega] for the cases of polyharmonic and the carrier wave shifted signals (the second and third example in Fig. 1).
The engineering approach for time-frequency uncertainty
To overcome the abovementioned problems, associated with the conventional methodology of standard deviations (Appendix A), the more practical approaches may be developed on the basis of the next principles:
1) Considering separately the different harmonics of signal (assumption that frequencies of the possible coexisting harmonic signals are separated enough compared to characteristic frequency 1/T of the time window);
2) Discussing only one half of spectrum (assumption that harmonic frequency is enough high f >> 1/T that the influence of other spectrum half is negligible). This simplification relies on the fact the Fourier transform of time-windowed harmonic signal with frequency [[omega].sub.1] creates two halves of spectra that are shifted by [+ or -][[omega].sub.1] (see Appendix B and Fig. 2).
Next, in contrast to the conventional theoretical approach that uses the single Gaussian pulse as a primary model task, the practical engineering approach is reasonable to develop on the basis of the model task of rectangular pulse with a harmonic carrier wave. This task is explained below in Fig. 2 and Fig. 3 and by the Appendix B.
[FIGURE 2 OMITTED]
Fig. 2 explains one general property of Fourier transform: the harmonic signal within a time-window splits the window spectrum into two halves that are shifted by the signal frequency (formulae in Appendix B). If this frequency f is remarkably higher than the characteristic window frequency 1/T then the two halves of spectrum do not interfere and the analysis may focus only on the spectrum of the window function.
In the case of rectangular window model task
s(t) = A, -T/2 [less than or equal to] t [less than or equal to] T/2, (7)
the spectrum is the well-known sinc-function [3-6]
S(f) = AT sin([pi]fT)/[pi]fT. (8)
Properties of function (8) are explained by the Fig.3 below.
[FIGURE 3 OMITTED]
Fig. 3 explains the important features of rectangular pulse that cannot be analyzed by the conventional approach as the slow 1/f decreasing rate of spectral sidebands does not allow to obtain a finite standard deviation for frequency. Important feature of this particular spectrum is that the zero points are separated by the characteristic window frequency f1 = 1/T. This property is widely used in measurement applications [3, 4] and may be interpreted as a duration-resolution product relation [DELTA]t[DELTA]f = 1. From engineering viewpoint, the time duration (uncertainty) of this signal is well defined and equals to [DELTA]'t = 1. As one can see, different widths [DELTA]'f of the main spectral peak between 0 and 2/T may be obtained by varying the relative detection height parameter h. This corresponds to the range of uncertainty product [DELTA]'t[DELTA]'f between 0 and 2. However, the detection heights below the value 0.2172 that is defined by the maxima of side lobes may be considered impractical in the presence of noise.
It is not difficult to conclude from the analysis of the spectrum (8) that for the rectangular pulse (window) the equation for definition of uncertainty products reads
sin([pi][c.sub.u]/2) = h[pi][c.sub.u]/4, (9)
where the uncertainty product constant
[c.sub.u] = [DELTA]'t[DELTA]'f. (10)
In particular, the detection height h = 0.5 yields [c.sub.u] [approximately equal to] 1.206709. Multiplication of this number by 2[pi] gives the constant 7.581977 that agrees with the approximate results in .
Fig.4 shows the dependence of the uncertainty product [DELTA]'t[DELTA]'f on the selection of the detection height h. Additionally, the relative signal energy content [integral][absolute value of S].sup.2] df within frequency interval [DELTA]' f in the vicinity of main spectral peak is evaluated. As one can see, h = 50% corresponds to the energy content 84.1 %.
[FIGURE 4 OMITTED]
Thus, the full width at half maximum (FWHM) criterion yields here the uncertainty product [DELTA]'t[DELTA]'f = 1.206709. This may be a rather practical estimation for asynchronous (analog) spectral measurements where unknown but periodic sum of different harmonic signals is analyzed within a time window with abrupt borders. Use of the exact number [DELTA]'t[DELTA]'f = 2 that corresponds to the full width of spectral main peak at zero level may be questionable in the presence of noise.
Different time windows for the theoretical and engineering approaches
We proposed above that the engineering approach to uncertainty product criteria should rely on the model task of rectangular pulse that cannot be analyzed by the conventional approach. Now the questions arise what happens if the rectangular pulse is not ideal and, additionally, what results gives the FWHM criterion for the Gaussian pulse that is the primary model task of conventional theoretical approach.
To find answers for those questions we performed Fourier transform and uncertainty calculations moving from rectangular pulse through trapezoidal and triangle pulses towards more smooth cosine-signals and the
Gaussian pulse. The results are presented below in Fig. 5 and in Table 1.
[FIGURE 5 OMITTED]
The results in Table 1 support the usefulness of the offered here engineering approach. As already emphasized, the standard deviations based conventional approach cannot evaluate the rectangular pulse at all (Table 1, last row). Already the results for trapezoidal pulses start to increase rapidly if the side slopes of those pulses become more abrupt. Moreover, the disturbing one order of magnitude difference exists typically between conventional theoretical and practical uncertainty product values (difference between [DELTA]t[DELTA]f and [DELTA]'t[DELTA]'f columns). In contrast, the practical engineering approach yields very similar uncertainty product values around [DELTA]'t[DELTA]'f [approximately equal to] 1 if for both--time and frequency domain the detection criterion at 50% level is used (column [DELTA]'t[DELTA]'f ). Very interesting conclusion may be drawn by analyzing the last column in Table 1 where the time uncertainty is made equal to window width T (except for Gaussian pulse that has no a certain duration). In this comparison the rectangular pulse becomes the narrowest in terms of uncertainty and the obtained above numerical value 1.206709 marks the minimal limit. This result agrees with general concepts of spectral measurements that the rectangular pulse has the best resolution [3, 4, 14].
An application example--spectral resolution versus time-frequency uncertainty product
In the previous section we demonstrated that, for example, in the case of rectangular window of duration T the width of spectral peak as defined at 50% level should be [DELTA]'f [approximately equal to] 1.2067/T. The question arises, can this spectral spreading result for one signal employed also for resolution problem of two signals of close frequency within the same time window. This problem is analyzed below for two cos-signals of unit amplitude and different frequency [f.sub.1], [f.sub.2] in a unit window T = 1 [3, 4] (the frequencies must be high enough [f.sub.1], [f.sub.2] >> 1/T, for example, of 10 kHz range for millisecond window). Fig.6 shows the summary spectra for two opposite cases: the in-phase and reverse phase adding (phase difference in the centre of the time window).
[FIGURE 6 OMITTED]
This considered task is of the field of time-limited asynchronous spectral measurements. In digital (synchronized) measurements, if is possible to assure the integer number of oscillation periods within the time window, the resolution limit is defined as [DELTA]f = 1/T [3, 4, 14]. The calculation results in Fig.6 show that the possibility of detection of two frequencies by two separate peaks arises approximately starting from [f.sub.2] - [f.sub.1] > 1.5/ t . Quite reasonable detection with accuracy +13% (in-phase adding) and -16% (reverse phase adding) is obtained only starting from [f.sub.2] - [f.sub.1] > 2/T. Thus for the reliable resolution of two signals the greater values of numerical constants should be considered in comparison of 1.2067 that characterizes the spectral spreading of a single signal.
Applications in quantum mechanics
The present engineering approach may be extended also to quantum mechanics as the spatial frequency of oscillating wavefunctions represents the momentum of a particle [1, 2, 9, 15]. The standard coordinate-momentum Heisenberg's uncertainty relation reads
[DELTA]x[DELTA]p [greater than or equal to] 0.5h, (11)
that is the direct counterpart of FUP inequality (3) [2, 9, 15].
The given lower limit of uncertainty product characterizes fairly the lower states in quantum wells. For example, the ground state in deep square quantum wells with the coordinate-dependent wavefunction of the form of the half period of cos-function [9, 15], the uncertainty product equals [DELTA]x[DELTA]p [approximately equal to] 0.567h that may be also obtained from the column 2 of Table 1. However, the higher states in quantum wells that resemble the first case in Fig. 1, yield much higher uncertainty products if we follow the conventional methology. The latter accounts for distance between spectral peaks and not the width of individual main peaks (the first case in Fig. 1). To obtain more reasonable estimations for individual peaks in momentum domain (i.e. of the spatial spectrum), the developed here engineering approach with numerical constants from Table 1 may be applied. In particular, in the idealized model case of infinitely deep square quantum well where the coordinate-dependent wavefunctions of higher states consist of integer number of sine/cosine halfwaves [9, 15], the respective coordinate-momentum uncertainty relation, corresponding to the momentum peak width detection at 50% level, reads
[DELTA]'x[DELTA]'p [right arrow] 1.206709 * 2[pi]*h [approximately equal to] 7.581977 h. (12)
In the present study we discussed the Fourier transform and Fourier Uncertainty Principle and pointed out several drawback of the conventional methodology based on standard deviations:
a) The artificial increasing of spectrum spreading due to account for positive and negative halves of Fourier spectra;
b) Accounting of distances between of all spectral peaks in the case of multi-harmonic signal.
c) Infinite standard deviations of spectra in the case of rectangular pulses.
In order to overcome the mentioned problems and to present the time-frequency uncertainty criteria in a more suitable form for practical signal processing tasks, we offered "an engineering approach" based on the following principles:
a) Separated analysis of different harmonic signals;
b) Consideration of only main spectral peaks of different harmonic signals;
c) Consideration of only positive frequencies half of Fourier spectra;
c) Width detection at certain level (predominantly FWHM) for the width of signal in time and in frequency domains;
d) Use of rectangular time window as the basic model task (in contrast to the Gaussian signal of the conventional approach).
Appendix A: calculation of standard deviations
Standard deviations of signal spreading in time domain and in frequency domain, associated with the conventional formulation (3) of Fourier Uncertainty Principle, should be calculated as the square roots of the respective mathematical variances [1, 2, 5-8]:
[DELTA]t = [square rooot of (Var(t))], (13)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)
[DELTA][omega] = [square root of (Var([omega])], (16)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)
Note that if the decreasing rate of the spectrum is 1/[absolute value of [omega]] or slower then the final [DELTA][omega] cannot be obtained.
Appendix B: harmonic signal within a time window
Let us consider a time-limited harmonic signal cos([[omega].sub.0](t - [t.sub.0])) within a time window p(t)
s(t) = p(t)cos([[omega].sub.0]>(t - [t.sub.0] (19)
that is also
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)
If we denote the spectrum of the window function as P([omega]) = [integral] p(t)exp(-j[omega]t)dt then the full spectrum of the time-limited harmonic signal within that window becomes
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)
This is the halved spectrum of the window function: one half shifted to [omega] = [[omega].sub.0] with phase multiplier exp(-j[[omega].sub.0][t.sub.0]) and second half shifted to [omega] = -[[omega].sub.0] with phase multiplier exp(+j[[omega].sub.0][t.sub.0]) (Fig. 2).
This work has been partly supported by the Estonian Science Foundation Project 6914. Drs. Raul Land and Toomas Parve are greatly acknowledged for their valuable remarks.
Received 2011 09 19 Accepted after revision 2011 11 17
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A. Udal, V. Kukk
Department of Computer Control, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia, phone: +3726202110, e-mails: firstname.lastname@example.org; email@example.com
Table 1. Uncertainty products for different time-window signals, calculated by standard deviation methodology and by the 50% height methodology (T - width of time window) Theoretical approach Engineering approach (standard deviations) (FWHM detection) [DELTA]t [DELTA] [DELTA]t [DELTA]'t Task [omega] [DELTA]f [DELTA]'f T[DELTA]'f Gaussian 0.5 0.0796 0.8825 >2 ? pulse raised cos, 0.513 0.0816 1 2 one period cos- 0.567 0.0902 1.093 1.639 halfwave Triangle 0.545 0.0867 0.8859 1.7718 pulse Trapezium 0.694 0.1105 1.1592 1.5456 50% plateau Trapezium 1.191 0.1896 1.2013 1.3348 80% plateau Trapezium 1.719 0.2736 1.2055 1.2689 90% plateau Rectangular [infinity] [infinity] 1.2067 1.2067 pulse
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|Title Annotation:||SIGNAL TECHNOLOGY/SIGNALU TECHNOLOGIJA|
|Author:||Udal, A.; Kukk, V.|
|Publication:||Elektronika ir Elektrotechnika|
|Date:||Jan 1, 2012|
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