CLASSICAL UNCERTAINTY PRINCIPLE

© Michael Bliznetsov

Contact to the author: bliznetsovmt@gmail.ru　

The modified Fourier analysis has constant resolution along the frequency axis and solves the problem of the standard Fourier spectra shift towards low frequencies and the problem of infinitely low frequencies in the Fourier spectra of physically observed wavelets. The product of duration and spectral width for the standard and modified Fourier spectra of elementary causal pulses is studied in the paper. The transients of an electrical oscillatory contour are elementary causal pulses. The rigorous proof of the classical uncertainty relationship is carried out on the basis of the modified Fourier spectra of oscillating functions. The time-bandwidth product is constant for oscillating elementary causal pulses. The extent-bandwidth product is the velocity of a wavelet. It’s proved on the basis of the modified Fourier spectra that the more relative bandwidth of the unidirectional (nonoscillatory) pulses is; the more is the time-bandwidth product and the extent-bandwidth product. Unidirectional causal pulses are shock waves.

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Key words: spectral analysis resolution, unidirectional pulse, infinitely low frequency, time-bandwidth product, extent-bandwidth product

The classical uncertainty principle is a direct consequence of the Fourier transform and relates to wave properties. The Heisenberg uncertainty principle in quantum physics results from the Plank and de Broglie hypotheses and relates to the properties of particles and waves.

There is a principle of reciprocal correspondence between ∆t time interval and ∆ω frequency interval. The classical uncertainty principle shows that the lower the pulse duration, the greater the spectrum width of the pulse, and vice versa. In the general case the relation between the pulse duration and the frequency interval is given by [4]

∆t ∆ ≥ 2, (1)

or passing from the radian measure to the frequency one

∆t ∆f ≥ 1 . (2)

The inequality sign refiects the fact that as a result of superposition of the harmonic oscillations ensemble occupying the ∆f frequency range, we get the pulse with the duration

∆t ≈ 1/∆f . (3)

only at a certain choice of the summed oscillations phases. In an extreme case when phases have random values ∆t duration may become arbitrarily large.

The basic characteristic of the energy wave motion is the wave packet propagation velocity. For a wave packet with extension ∆x the uncertainty relationship (2) is

∆õ ∆f ≥ , (4)

where is the wave packet propagation velocity.

The product of the wave packet extension in space and the spectrum bandwith is equal at least to the wavelet velocity. Note that the velocity of the wave motion of energy is a constant value in the nondispersive medium and can’t be arbitrary large depending on the phase spectrum. The sign of inequality in the formula (4) does not have any physical explanation.

In order to answer the question what a wave motion of energy is we need to answer the question what a non-wave motion of energy is. It is known that the shock waves propagate through media at a higher velocity than a normal wave, but the formula (4) doesn’t refer to shock waves. It’s known that a shock wave exited by an explosion in the atmosphere has a steep leading edge and a fiat trailing edge [6]. While moving away from the source the shock wave transforms into a normal wave or a wavelet, which propagation velocity is a constant value that is equal to the velocity of sound in the atmosphere. The wavelet pressure profiles have a simple shape in the atmosphere. But the effort to answer a simple question: if the pressure wavelet in the atmosphere can be a unidirectional (nonoscillatory) pulse shows the paradox of infinitely low frequencies. It is known that the absolute value of the Fourier spectrum of any unidirectional pulse contains low and infinitely low frequencies, which amplitude increases when decreasing the frequency and the maximum value of the spectrum amplitude is at zero hertz. A unidirectional pulse has finite time duration but the wavelength of the pulse dominating frequency is infinite in space, independent on the wave propagation velocity. The Fresnel zone, or the wave transmission volume, is also equal to infinity for unidirectional pulses.

The main characteristic of the spectral analysis is resolution, which can be low or high. But the resolution of the spectral analysis must be constant along the frequency axis [1, 2, 3]. The reason of the Fourier spectra shift towards low frequencies is variable resolution of the Fourier analysis along the frequency axis, and the lower frequency, the less resolution. The Fourier spectrum multiplied by frequency is a modified Fourier spectrum. The modified Fourier analysis is equivalent to the physical spectral analyzer using constant Q-factor resonators that have constant resolution along the frequency axis. The continuous analysis or the analysis of the current spectrum of nonstationary processes by the spectral analyzer with constant resolution along the frequency axis is a continuous wavelet analysis [3].

In the given paper the time-bandwidth product for the standard and modified Fourier spectrum of elementary causal pulses is studied. The transients of an electrical oscillatory contour are elementary causal pulses. The strict proof of the classical uncertainty relationship is carried out on the basis of the modified Fourier spectra of oscillating functions. The time-bandwidth product is equal to 1 for the oscillating elementary causal pulses. The extent-bandwidth product is the velocity of the wavelet. It is proved on the basis of the modified Fourier spectra that the more relative bandwidth of the unidirectional pulses the more the time-bandwidth product and the extent-bandwidth product. Unidirectional causal pulses are shock waves which propagation velocity is higher than the velocity of the energy wave motion in the given medium.

On the basis of the standard Fourier spectra the time-bandwidth product is a variable value and only when increasing the duration of the oscillating pulses this product tends to 1. The time-bandwith product for unidirectional pulses is lower than 1 on the basis of the standart Fourier spectra.

It is known that the resolution of the physical spectral analyzer consisting of a set of narrowband resonators with the passband ∆ω and the resonant frequency ω₀ will be constant if the resonators have equal Q-factor

Q= (5)

The resolution will be variable along the frequency axis if the analyzer consists of resonators with a constant passband

, (6)

and in this case the Q-factor of the resonators will be variable along the frequency axis and when _î → ∞, Q → ∞ and when _î → 0, Q → 0 i.e. the resolution decreases when decreasing the frequency.

Consider the reason of the Fourier spectra shift towards low frequencies and the reason of the infinitely low frequencies occurrence in the Fourier spectra. The inverse Fourier transform is the following

, (7)

where f(t) is time function, F(ω) is amplitude spectrum.

In the formula (7) the expression F(ω)dω is the amplitude which corresponds to the given frequency interval. The frequency interval dω is an infinitely small and constant value. Passing on to finite values this condition will be ∆ω=const, i.e. as a condition (6) in the spectral analyzer with variable resolution along the frequency axis.

For formula (7) we stipulate (5). It follows from (5) that the absolute value of the frequency interval is variable and proportional to ω₀. It is possible to write this in expression (7) as the following: F(ω)ω₀dω or ω₀F(ω)dω i.e. this is the Fourier spectrum multiplied by the frequency. There is a Fourier transform theorem that is known as the derivative theorem: differentiation of a time function corresponds to multiplication of its spectrum to frequency ω

f′(t)=ω|F(ω)| . (8)

Thus, the Fourier spectrum multiplied by the frequency or the Fourier spectrum of the signal derivative corresponds to the spectral analysis with identical resolution along the frequency axis from the condition (5). The suggested modified Fourier spectral analysis solves the reason of the Fourier spectra shift towards low frequencies and the problem of the infinitely low frequencies occurrence in the Fourier spectra. There are two basic conclusions from the offered Fourier spectral analysis.

1. The Heaviside unit step function has an infinitely wide modified spectrum equal to 1 along the whole range of frequencies. The Dirac delta pulse has no spectra in the area of physically observed frequencies because an infinitely short in time pulse is a physically unobservable value and its spectrum cannot contain physically observable harmonic components. Physical resonators do not respond to the Dirac delta pulse.

2. The modified Fourier spectra of the transition functions of an electrical oscillatory contour are equal to the amplitude-frequency characteristic of the oscillatory contour. The transition function of the oscillatory contour is a time function at the output of the contour after the unit step function was put into it. It is known that the amplitude spectrum of the output signal A(f) is determined by the formula

A(f)=H(f)K(f), (9)

where H(f) is the amplitude spectrum of the unit step function, and K(f ) is the amplitude-frequency characteristic of the contour. The modified Fourier spectrum of the unit step function is equal to 1(f). As a result we have: A(f)=1(f)K(f), i.e. the spectrum of the transition function is equal to the frequency characteristic of the electrical contour.

The modified Fourier spectrum of the step function will have zero frequency only in a limiting case when the time duration of such pulse will be infinite. Such presentation is physically contensive because for measurement of the harmonical component frequency with the period, for example, 100 years we need time not less than 100 years and the infinite time is necessary for measuring the oscilation amplitude of the infinite period.

The modified Fourier spectrum of the unit step function has infinitely high frequencies only in a limiting case when the leading edge has an instantaneous function jump that involves infinite propagation velocity of information and energy. Physically observable signals cannot contain both zero and infinitely high frequency in their spectrum.

In the class of all causal signals with zero value before zero time and with identical amplitude spectra, the minimum-phase signal has the least duration and is characterized by the fastest energy event. The limiting case of a causal and minimum-phase pulse with the modified Fourier spectrum equal to 1 along the whole frequency range is the Heaviside unit step function. Elementary causal pulses are transition functions of the electrical oscillatory LCR-circuit, where L is inductance, C is capacity, and R is resistance. The components L, C and R are connected in series and the transition function is measured at R resistance [7]. Values L, C and R define the amplitude-frequency characteristic K(ω), the resonant frequency ω₀ and the Q-factor of an oscillating contour. The inverse value of the Q- factor characterises the relative bandwith of the circuit 1/Q=∆ω/ω₀ calculated at the level =0.707... Note that the frequency characteristic of the circuit is the modified Fourier spectrum of the transition function.

There are three analytical expressions for defining the transition function h(t) depending on ∆ω/ω₀

1. ∆ω/ω₀ > 2

h₁(t)=, (10)

where α=R/2L (attenuation constant);

;

.

2. ∆ω/ω₀=2

h₂(t)= (11)

 

　

3. ∆ω/ω₀ < 2

h₃(t)=, (12)

where is the frequency of natural oscillations.

Note that the transition function is a causal pulse with minimum phase spectrum.

4 The measurement of the duration of the elementary causal pulses

For studies of ∆t∆f product we need the analytical definition of the duration ∆t of elementary causal pulses. It is well known from the theory of electrical oscillatory circuits that for the pulses with ∆f/f₀ < 2 representing damped sinusoid time functions (12) the process duration is estimated by the time constant τ that is an inverse value to the attenuation constant α [7]

. (13)

The time constant establishes that after time t=τ has elapsed the amplitude of the exponential component of a damped sinusoid is e times less than the maximum value, where e=2.718. ... It is known [7] that the Q-factor is also determined by the expression Q=ω₀L/R, so L/R=Q/ω₀ . After substituting this relation into expression (13) we have

. (14)

Now from (14) we can obtain an expression for the uncertainty relationship

. (15)

It is proved that the product τ∆f is a constant value for oscillating elementary pulses with ∆f/f₀<2. It is a strict proof of the classical uncertainty relationship performance on the basis of the modified Fourier spectrum. We equate the right part of the expression (15) to 1, i.e. multiply the left and the right parts of the formula (15) by π. Such an operation is equivalent to the statement that the total pulse duration ∆t is determined as

∆t=πτ, (16)

and the amplitude is or 23.14 ... times less than the maximum value of the exponential component in (12). That’s why we have to measure the total duration of elementary pulses from the time of the maximum amplitude of the exponentially damped component to the time of =0.043 ... amplitude level. Thus the uncertainty relationship is true for pulses with ∆f/f₀< 2.

∆t∆f=1, (17)

 

∆t=, (18)

It follows from the formulas (10,11), that the greater ∆ω/ω₀ of unidirectional pulses the more abrupt their forward front and the longer their duration. The total duration ∆t of unidirectional pulses (10,11) is also determined by the duration of the exponentially damped component [1,2]

∆t=. (19)

where f_l is the lower boundary frequency at the 0.707... level of the modified Fourier spectrum, ∆f is the bandwidth of the modified Fourier spectrum at the level 0.707...

The modified Fourier spectrum of the transitive function of an electrical contour LCR is equal to the spectral characteristic of the contour. On the basis of the Q-factor value of the contour Q=f₀ /∆f and the given resonant frequency f₀ , the bandwidth of the modified spectrum ∆f at the level 0.707... is known. The modified Fourier spectrum is a standard Fourier spectrum multiplied by the frequency (8) and on the contrary, the standard Fourier spectrum is the modified Fourier spectrum divided by the frequency. That’s why the bandwidth of the standard Fourier spectra at the level 0.707... we will determine by a numerical way.

We accept that the circuit resonant frequency is f₀=40Hz. In Fig. 1a the transition function of the electrical circiut with the relative bandwidth ∆f/f₀=0.2 (12) and its standard and modified Fourier spectra are shown (Fig. 1b).

The maximum of the modified spectrum is at 40 Hz frequency. The bandwidth of the modified Fourier spectrum at level 0.707 . .. is known and has the value of ∆f=8Hz. The value of the pulse duration from the formulas (16) or (18) is ∆t=0.125s. The uncertainty relationship for the modified Fourier spectrum is carried out precisely and is ∆t∆f = 0.125s·8Hz = 1. The standard spectrum has infinitely low frequencies and is shifted towards lower frequences by an insignificant value. The bandwidth of the standard spectrum insignificantly exceeds 8Hz and also we accept that ∆f ≈ 8Hz. Therefore it is possible to write down ∆t∆f=0.125s·8Hz≈ 1 for the standard Fourier spectrum. Note that the more the oscillating pulses duration the smaller the difference of the standard and modified Fourier spectra bandwidths.

In Fig. 2a the oscillatory pulse (12) with ∆f /f₀=1 and its standard and modified Fourier spectra are shown (Fig. 2b).

The bandwidth of the modified Fourier spectrum is known and has the value ∆f=40Hz. The duration of the pulse from the formulas (16) or (18) is ∆t=0.025s. The uncertainty relationship for the modified Fourier spectrum is carried out precisely and is ∆t∆f=0.025s·40Hz=1. The standard Fourier spectrum and its maximum are shifted towards low frequencies. The standard Fourier spectrum contains harmonical components in the field of superlow frequencies and has a significant amplitude at frequency of f=0Hz. The bandwidth of the standard Fourier spectrum at level 0.707... has the value of ∆f=46.75Hz. The time-bandwidth product for the standard Fourier spectrum is more than 1 and has the value of ∆t∆f=0.025s·46.75Hz=1.168.

In Fig. 3a the unidirectional pulse (11) with ∆f/f₀=2 and its standard and modified Fourier spectra are shown (Fig. 3b).

The bandwidth of the modified spectrum is ∆f=80Hz. The pulse duration from the formula (19) is ∆t=0.01767s. For unidirectional pulses and the modified Fourier spectra the time-bandwidth product is more than 1 and in this case has the value ∆t∆f=0.01767s·80Hz==1,4142... The standard spectrum is shifted towards low and superlow frequencies and has the maximum amplitude at frequency f=0Hz. The bandwidth of the standard Fourier spectrum at level 0.707... has the value ∆f=25.7Hz. The time-bandwidth product for the standard Fourier spectrum is less than 1 and is ∆t∆f=0.01767s·25.7Hz=0.4543.

In Fig. 4a the unidirectional pulse (10) with ∆f/f₀=10 and its standard and modified Fourier spectra are shown (Fig. 4b).

The bandwidth of the modified spectrum ∆f=400Hz. Such expansion of a spectrum has a physical explanation. The the forward front steepness of the pulse with ∆f/f₀=10 (Fig. 4a) is much more than that of the pulse with ∆f/f₀=2 (Fig. 3a) that is the reason of the spectrum expansion to high frequencies (Fig. 4b).

The duration of the unidirectional pulse with ∆f/f₀=10 is much more than the duration of the unidirectional pulse with ∆f/f₀=2 that is the reason of the spectrum expansion to low frequencies (Fig. 4b). The maximum of the modified spectrum f₀=40Hz. The duration of a pulse from the formula (19) has the value ∆t=0.1237...s. For the modified Fourier spectra the time-bandwidth product is ∆t∆f=0.1237...s·400Hz=49.49... . The greater ∆f/f₀ is_, the greater is ∆t∆f for the unidirectional pulses. The standard spectrum is shifted towards superlow frequencies and has the maximal amplitude at frequency f=0Hz. The bandwidth of the standard spectrum has the value of ∆f=4Hz, that is 100 times less than the bandwidth of the modified spectrum. For the standard spectra ∆t∆f=0.1237 .. .s· 4Hz=0.4949 ...s.

So for the standard Fourier spectra the time-bandwidth product is a variable value. For unidirectional pulses this value is less than 1. For oscillating pulses this value is more than 1 and ∆t∆f → 1 when increasing the duration of these pulses.

The time-bandwidth product is a constant value for the modified Fourier spectrum of the oscillating functions with ∆f/f₀ < 2. The time-frequency uncertainty relationship is carried out precisely on the basis of the modified Fourier spectra received with identical resolution along the frequency axis.

The classical uncertainty relationship relates to wave properties. The basic characteristic of the energy wave motion is the propagation velocity of the wave packet. The propagation velocity of a wave packet in a given non-dispersive medium is a constant value. For oscillating functions with ∆f/f₀ < 2 we write the following expression

, (20)

where T ₀ is a dominant period or an inverse value from f₀ ;

λ ₀ is a dominant wave length;

∆x is the extention of a wavelet in space.

It’s known that

The classical uncertainty relationship is a direct consequence of the modified Fourier expansions and relates to wave properties. The basic characteristic of the energy wave motion is the wavelet velocity. The wavelet velocity in a given medium is constant. The time-bandwidth and the extent-bandwidth products are constants for oscillating elementary causal pulses. The extent-bandwidth product is the velocity of a wavelet in a given medium.

On the basis of the standard Fourier spectra the time-bandwidth and extent-bandwidth products are variable values and only with increasing the duration of oscillating pulses the time-bandwidth and extent-bandwidth products aspire to the constant value.

On the basis of the modified Fourier spectra it is proved that the larger the relative spectrum width of unidirectional pulses the larger the time-bandwidth and the extent- bandwidth products. Unidirectional waves are shock waves. Shock waves are a nonwave form of the energy motion. The Heaviside unit step function is a limiting case of a shock wave with infinite propagation velocity.

The performance of the classical uncertainty relationship corresponds to the wave form of energy motion. The unperfomance of the classical uncertainty relationship corresponds to the shock-wave form of the energy motion.

[1] M. T. Bliznetsov, “Elementary wavelet” Geofizika, 1, 52-60 (2001) in Russian.

[2] M. T. Bliznetsov, “Spectral analysis resolution and study the uncertainty relationship” Physics Essays 18, 63 – 80 (2005)

[3] M. T. Bliznetsov, “Modified Fourier analysis and wavelet analysis” Geofizika 3, 3 – 8 (2006) in Russian

[4] F. S. Grawford, Waves, McGraw-Hill, New York (1967).

[5] E. R. Kanasevich, Time sequence analysis in Geophysics, The University of Alberta Press (1981).

[6] G. P. Kinney, Explosive shocks in air, Macmillan Co., New York (1962).

[7] E. I. Minaev, Bases of Radioelectronics, Radio and Communication, Moscow (1985) in Russian.

[8] J. A. Sharpe, “The production of elastic waves by explosion pressure” Geophysics 7, 144 - 154 (1942).