A Reformulative Retouch on the Fourier Transform – “Unprincipled” Uncertainty Principle

According to a widespread notion, a monochromatic harmonic temporal wave’s frequency ω becomes inherently “uncertain” to a degree ω Δ if the wave lasts for only a limited time t Δ owing to the “Uncertainty Principle” which states that constant ≥ ω Δ ⋅ Δt (i.e. ω Δ cannot be zero if t Δ is finite). This frequency uncertainty is commonly thought of as also being reflected in the fact that the (“sinc”-shaped) frequency spectrum obtained by Fouriertransforming a finite-duration monochromatic wave spreads over a nonzero frequency range. Since, as often argued, the Fourier spectrum represents the ”frequency components” of a temporal function (or signal), the nonzero width of the spectrum tells us that the ”nominally” monochromatic ω frequency of a time-limited sinusoid is “effectively” polychromatic, with frequencies continuously distributed over a ω Δ range in accord with the Uncertainty Principle.


Introduction
According to a widespread notion, a monochromatic harmonic temporal wave's frequency ω becomes inherently "uncertain" to a degree ω Δ if the wave lasts for only a limited time t Δ owing to the "Uncertainty Principle" which states that constant ≥ ω Δ ⋅ Δt (i.e.ω Δ cannot be zero if t Δ is finite).This frequency uncertainty is commonly thought of as also being reflected in the fact that the ("sinc"-shaped) frequency spectrum obtained by Fouriertransforming a finite-duration monochromatic wave spreads over a nonzero frequency range.Since, as often argued, the Fourier spectrum represents the "frequency components" of a temporal function (or signal), the nonzero width of the spectrum tells us that the "nominally" monochromatic ω frequency of a time-limited sinusoid is "effectively" polychromatic, with frequencies continuously distributed over a ω Δ range in accord with the Uncertainty Principle.
A typical area where this argument shows up is Pulsed Fourier-Transform (PFT) nuclear magnetic resonance (NMR) spectroscopy.In NMR a molecular ensemble is placed in a strong homogeneous static magnetic field, whereupon the ensemble's NMR-active nuclei (those that act like spinning atomic magnets, say protons) will precess about the direction of the magnetic field.Those "spins" in the ensemble which precess (Larmor precession) with identical L ω frequencies form a net macroscopic magnetization which is normally aligned along the external field, but which will also (Larmor-)precess with a frequency L ω following a transient perturbation of the magnetization from its equilibrium state.Typically there are several such bundles of homogeneous spins in the analyzed sample, covering a range of Larmor frequencies L ω Δ .PFT-NMR works on the principle that the sample's bulk magnetizations of interest can be excited into precession-mode (made to "resonate") simultaneously in a wide L ω Δ ("off-resonance") frequency range when perturbed by a "driving" electromagnetic wave of a single well-defined monochromatic radio-frequency (RF) D ω if the wave is applied as a "pulse" in a sufficiently short time t Δ .Because this phenomenon may appear counter-intuitive, the fundamental NMR literature is abundant in attempts to offer a preliminary explanation for why pulsed excitation works, typically invoking the "Uncertainty Principle" and the "sinc"-shaped FT-spectrum of the pulse according to the notions outlined above.Some representative examples from authoritative www.intechopen.comFourier Transform -Signal Processing 212 sources are as follows: "…although the applied excitation may be precisely centred at a frequency D ω , […], our act of turning the excitation power on at time zero and and off at time Δt effectively broadens the spectral range of the excitation (to a bandwidth of ~1/Δt)."(Marshall and Verdun, 1990); "A pulse of monochromatic RF with a rectangular envelope can be described in the frequency domain as a band of frequencies centred at the RF frequency.The Heisenberg principle states that there is a minimum uncertainty in the simultaneous specification of […] the frequency of a system and the duration of the measurement.[…] this means that the […] irradiation is spread over a wide frequency band.
[…] the 'sinc' Fourier spectrum of a rectangular RF pulse shows that a shorter pulse gives a wider 'sinc' band and a longer pulse gives a narrower 'sinc' band."(King and Williams, 1989); "…if the pulse is made shorter, we will no longer have a truly monochromatic Fourier spectrum even though the source is still monochromatic.This is because many different frequencies have to be combined in order to form the rising and falling edges of the rectangular pulse."(Fukushima and Roeder, 1981); "… as the Uncertainty Principle indicates, a pulse [of carrier frequency D ω ] will contain, in effect, a range of frequencies centred on D ω .
[…] the distribution of RF magnetic field amplitudes takes the 'sinc' form […] which is the frequency-domain equivalent of a short pulse in the time domain.The two domains are connected by the Fourier transform."(Harris, 1983); "… the RF source […] is monochromatic, so we have to work out a way of using a single frequency to excite multiple frequencies.To see how this can be done we take our cue from the Uncertainty Principle.If the irradiation is applied for a time Δt, then […] the nominally monochromatic irradiation is uncertain in frequency by about 1/Δt."(Derome, 1987) One should quickly note that at a macroscopic level the basic NMR phenomenon is a purely classical effect (Hoult, 1989 andHanson, 2008) involving interactions between the oscillating magnetic component of the RF pulse and the sample's bulk magnetizations.Indeed, the fact that a monochromatic pulse can excite spins in a wide frequency range can be well explained classically using the laws of driven mechanical harmonic oscillators and magnetism (Bloch, 1946) without resorting to the vague idea that the D ω RF pulse frequency should be "uncertain" or "effectively" polychromatic according to its Fourier spectrum.Furthermore, except for one case the above arguments are unclear on whether by "Uncertainty Principle" they mean the Heisenberg Uncertainty Principle (HUP) which is a probabilistic, quantum-mechanical statement, or the Fourier Uncertainty Principle (FUP) which is a deterministic theorem in the discipline of time-frequency analysis.
In a series of recent papers (Szántay, 2007(Szántay, , 2008a(Szántay, , 2008b(Szántay, , 2008c) ) the author of the present work explored the above problem in detail, showing that a) the idea that the frequency of a time-limited sinusoid would be uncertain in the Heisenberg sense is flawed; b) it is the FUP (and not the HUP) that is of relevance to the problem; c) the notion that a monochromatic pulse "effectively" comprises of a range of physically existing frequency components is erroneous.During that discourse the need had come up to contemplate the essence of the Fourier Transform (FT) itself, which may appear somewhat odd considering the fact that the FT is ubiquitously applied in a broad range of sciences and technology, and works explaining its mathematics and its manifold physical applications are available in almost countless abundance.However, it turns out that the FT is so commonplace, it has become so much consolidated into many fields of science and engineering, its basic principles and formulas are so commonly taken for granted, that, paradoxically, this can easily stop people from thinking more deeply about how and why the FT works as a mathematical tool and what exactly it means when evoked to interpret physical phenomena.One of the central features of the present author's approach in discussing the FT was the use of a "touched up" mathematical formalism to provoke a new way of looking at some key ideas of the FT and to avoid some hidden ambiguities in meaning that are otherwise inherent to the conventional formalism and can give rise to the misconceptions mentioned above.Besides coming out with a disambiguative but symbolically elaborate formalism for the FT involving the depiction of the domain intervals on which functions "exist" and introducing the concept of thinking about the FT in either "trigonometric-frequency-space" or "phasor-frequencyspace", those articles also expounded, from a new perspective, on several apparently simple but in practice elusive related concepts such as "periodicity", "harmonic oscillation", "phasor", "negative frequency", "uncertainty", etc.; much emphasis was placed also on a comparison of Fourier analysis and the Fourier transform, on the pros and cons of metaphoric thinking and the dangers of unrecognized linguistic ambiguities (e.g., words and phrases like "uncertainty", "nominally monochromatic" and "effectively polychromatic" deceptively appear to be meaningful, but on deeper examination their meaning is ambiguous and notoriously misunderstood).
Owing to the volume and thematic complexity of those papers the need has repeatedly arisen to present its main thrusts in a more compact and symbolically simplified form.The present work therefore attempts to condense the original discussion into a shorter communication by focusing only on two themes, namely the reformulation of the primary FT equations (other closely related topics such as Fourier analysis will be left out) and the main conclusions on the problem of the "Uncertainty Principle".These thoughts are hoped to provide some fresh insights and a deeper understanding of the essence of the FT and the Uncertainty Principles.In that regard it should be emphasized that the applied mathematical formalism is not a l'art pour l'art design: it serves to facilitate a more conscientious way of thinking about the pertinent topics.For the full technical and philosophical details behind the ideas presented here the reader is encouraged to consult the original articles.

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The term Fourier analysis will be used to refer to the Fourier series expansion of periodic functions and the term Fourier transform (FT) will refer to the generalization of Fourier analysis including nonperiodic functions.The FT will be discussed only in the context of the Fourier-pair variables time (t) and frequency (ω).The word "spectrum" will in genereal mean the ω-dimension (ω-D) representation of a time-dimension (t-D) function, more specifically as it has been obtained by the FT.

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Sets will be represented by bold, italicized letters while vectors will be denoted by bold, non-italicized letters.If in a set x its members x can assume values on a continuous scale, I will use the symbol "@" as a primary generic index to tag an arbitrary value as " @ x " so as to emphasize that we are focusing on that particular value in the set ("@" can be interpreted as: "at a given value"., we will consider five main interval types labeled as follows: is open and unbounded on both sides; x is closed and bounded (compact) at arbitrary @ x and @' x values on both sides; will be associated with a function which has a finite function value at will be used in connection with a Dirac delta whose function value is infinite at Underlined symbols are used to denote complex quantities having a real (ℜ) and an imaginary (ℑ) component, so a complex constant ) , ( is defined and symbolized as will be referred to as a phasor where the phase ϕ is measured from the positive real axis (+ℜ) with a counterclockwise angle being positive and a clockwise angle being negative.A non-underlined character, if lacking the tags ℜ or ℑ, represents a real-valued quantity by default.Likewise, a function in general is represented as if it is real-valued and as ) (x f if it is complexvalued.

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The action of a mathematical formula (rule, operator, etc…) upon a set will be denoted as set Formula .Thus, a rule acting on a variable x to give a function ) (x f is written as , and an operator that converts a function is the set of all allowable inputs that the function's argument x may assume.However, ) (x f can of course be restricted to or can simply be of interest to us over a subset x of the natural domain, and herein x will be called the operative interval of ) (x f .A function's operative interval will be denoted as and should be interpreted either as having no values or zero function values for any x ∉ x . For example, for a single rectangular "bump" along x we have corresponding to the rectangle's width, while for a function that starts with a finite value at x = 0 but decays exponentially to zero at x = ∞ we have ranging from zero to positive infinity.Care must be taken not to confuse the operative interval (which is always indicated as a subscript following a bar www.intechopen.complaced after the function's name or rule of association) with the function's natural domain.For example, for a function the natural domain may be the operative interval is ⇔ x , implying that the natural domain must also be ⇔ x .A proper understanding of the FT requires that one should always be conscious of what domains and intervals are relevant to the time scales and frequency ranges in the pertinent mathematical expressions.These aspects are often far from being self-evident from the formulas themselves in their conventional notation, and are usually present as hidden assumptions or unstated premises.
To overcome this problem herein the pertinent interval types will be represented as , otherwise zero, and , otherwise zero.If the properties of ) (x f are considered on different intervals simultaneously, the pertinent interval symbols will be separated by semicolons, such as x .

Trigonometric and phasor frequencies
The symbol ω will be used as a general representation of (angular) frequency such that the oscillation frequencies associated with a temporal function have constant values ( = ω ) (t constant) over the operative interval t of the function.In the customary notation a sinusoid (trigonometric) harmonic wave with a complex amplitude (i.e., a 1D oscillation in the complex plane that gives a 2D wave as a function of time) has the general form and a harmonic phasor (i.e., a vector A rotating uniformly either clockwise or anti-clockwise in the complex plane, and describing a 3D helix as a function of time) has the general form , where ϕ is the wave's initial phase at t = 0.For the harmonic phasor the phase angle is measured from the positive real axis (+ℜ) with a counterclockwise angle being positive and a clockwise angle being negative.For simplicity both a sinusoid wave and a harmonic phasor will be referred to as a "harmonic wave".
The sinusoid and phasor forms of a harmonic wave are connected by Euler's famous identity, written in its conventional form as , from which the trigonometric terms can be expressed by rearrangement as . Using this conventional notation the concept of the sign of ω, in particular the idea of a negative frequency, can be a confusing issue.Thus, based on making a distinction between "trigonometric frequency" and "phasor frequency" the following notation is introduced: a trigonometric frequency (the frequency of a sinusoid oscillation) will be labeled as ϖ ∈ ϖ ; a phasor frequency (the frequency of harmonic phasor) will be labeled as . Accordingly, a sinusoid harmonic wave ) (t h ϖ and a harmonic phasor ) ( ω t h / take the following general forms: Note from (2) that the phasor's complex amplitude A always corresponds to t = 0 since A is defined by its magnitude (absolute value) and the phasor's initial phase, which are both time-independent quantities.From (2) we see that the phasor can be equivalently viewed also as a sum of two phasors of the same frequency, one with a real amplitude A ℜ , and the other with a pure imaginary amplitude A i ℑ ⋅ .
For the trigonometric wave ) t ( h ϖ the meaning of ϖ seems straightforward: it expresses the number of full periods (multiplied by 2π) completed by the wave in unit time, whereby ϖ is defined as having positive numerical values only since it would not make sense to speak of a negative number of oscillations per unit time.The algebraically positive nature of ϖ merits emphasis because in the context of the FT tigonometric frequencies are sometimes (mistakenly) given negative values in the literature [e.g. the fact that neither the cosine transform, nor the sine transform alone can distinguish between the two possible rotational directions of a harmonic phasor is often explained by the formal truths that ].However, one should appreciate that the ) does not make any physical sense.This issue is partly rooted in the fact that the literature makes no symbolic distinction between " ϖ " and " ω / ", hence the intended meaning (phasor vs. trigonometric) of "frequency", labeled as " ω ", is typically an implicit consequence of the context in which it appears but is not implied or conveyed by the symbol " ω " itself.may easily escape mental objection when speaking about positive and negative phasor frequencies when the latter are also labeled as ω + and ω − .There is in fact no need to allow sinusoid frequencies to be negative (see below) and we can adhere to the physically appropriate definition that trigonometric frequencies are always positive.
A harmonic phasor ) ( ω t h / has either a positive or a negative frequency + ω / and − ω / depending on the direction of "skew" (or "helicity") of the 3D helix described by the phasor as a function of time.I employ the convention according to which for positive ω / the phasor ) ω exp( ) ( direction as time progresses, while for negative ω / the phasor ) represents rotation in the opposite sense.
Note that according to the above considerations there is an intrinsic difference between the concepts of the "sign" of the frequency for a harmonic sinusoid wave and a harmonic phasor.If, in a ) , , ( t ℑ ℜ space, we associate a positive skew with + ω / and a negative skew with − ω / (and this is what we mean by the positive and negative sign of ω / ), then a trigonometric frequency ϖ , which is "skew-less", should also be "sign-less".Thus our definition of ϖ being always positive carries the implicit understanding that a numerically positive frequency value requires a different physical interpretation for ϖ and All subsequent equations will be formulated in line with the above definitions on the concept of frequency.Forcing oneself to accordingly rethink the common text-book equations related to the FT is an instructive endeavour yielding added levels of insight.
Furthermore, the concept of a "frequency sweep" is introduced as follows: the symbols ω / , ω / and ω / emphasize the fact that we take into account all values for the variable ω / in the continuous sets ω , respectively.Thus we may think of ω / , ω / and ω / as representing a "sweep" of the frequency value from −∞ to +∞ , from 0 to +∞ , and from 0 to −∞ , respectively, in phasor-frequency space.Likewise, the symbol ϖ designates a trigonometric-frequency "sweep" of ϖ from 0 to +∞ in the set ⇒ ϖ .Frequency-sweep notation will be used, when needed for emphasis, in expressions describing temporal harmonic waves as well as in the function names of FT-spectra.For example, when stressing that we are considering the set of all sinusoid harmonic waves ) (t h ϖ covering the entire trigonometric frequency range ⇒ ϖ where each wave lasts from −∞ to +∞ , we write . Similarly, the set of all perpetual harmonic phasors in the infinite phasor-frequency range ⇔ / ω can be written as . If however we want to focus on a harmonic wave at a given frequency value we write

FT-contextualized function notation in the time and frequency dimensions
The ways in which the forward and inverse FT operators , respectively.The so-called Fourier Inversion Theorem (FIT) (see below) states that the FT is an invertible transform: there is a one-to-one correspondence between all temporal and spectral function pairs that we consider here; to reflect this concept the forward and backward FT operator symbols are combined into the symbol ↔ FT , so the link between the two dimensions can be depicted as: In the time dimension we generally consider complex-valued temporal functions to hold, the forward FT must be performed by accounting for all possible frequencies in the infinite natural domains ⇔ / ω or ⇒ ϖ , depending on whether we want to express the spectrum in ω / -space or ϖ -space.To reflect this concept the notation for the pertinent frequency sweep (as described in the previous section) will be indicated in the spectrum's name.Thus, spectra with an infinite support are written as An ω / -space spectrum can always be equated with a suitable, complex combination of two ϖ -space spectra, one obtained through the "cosine transform" and one through the "sine transform" of . Phasor representation usually being more convenient to work with, herein preference will generally be given to ω / -space formalism (however, the trigonometric form will also be discussed to some extent since this gives some interesting added insight into the nature of the FT).In all, the FT relationship will most generally be thought of as: Finally, we will find that some spectra of the type are centred about a specific frequency @ ω / .Although in such cases the value of @ ω / is contained in the relevant rule, , it will often prove informative to indicate @ ω / in the function's name as well, which will be done by writing

Euler's identities
With the above considerations in mind we can reformulate Euler's identities as follows.For a given frequency we have On the other hand the entire positive, entire negative, or complete positive and negative phasor frequency range can be embraced via positive trigonometric frequency notation which may be represented by writing Equations ( 4) -(6) as These equations reflect the concept that we can describe a harmonic phasor's frequency and sign of rotation in both phasor notation and complex trigonometric form, so long as in the latter case we use both the (real) cosine and (imaginary) sine waves -both of which have algebraically positive ϖ frequencies.There is no contradiction in the fact that we can obtain a negative phasor frequency − ω / by combining a real and an imaginary sinusoid, both of which have positive ϖ trigonometric frequencies.Equations (5,6) and (8,9) show that while a given sinusoid has no frequency sign ambiguity (it is always positive) in ϖ -space, it exhibits a sign duality when expressed in ω / -space, i.e., a sinusoid is always equated with the sum of two phasors, one with a positive and one with a negative frequency.

Interrelationships in temporal and spectral representations
In our formalism a key stepping stone to the FT involves the consideration of Euler's identities from both a t-D and ω-D perspective.Let us take a harmonic phasor , called herein the principle phasor, which can be viewed as a vector rotating in time as shown in temporal representation for a positive phasor Figure 1(a).We can also find it informative to represent our phasor as an ω-D entity in Figure 2(A).The temporal and spectral representations emphasize different aspects of the phasor: the temporal form shows its progression in time, from which the numerical value of ± ω / @ is not directly accessible, while the spectral representation displays the ± ω / @ value in ω / -space, but offers no direct sense of the phasor's temporal behavior or of its operative interval t .Considering the principle phasor in the light of ( 4)-( 6), we can formulate the following scheme: Equation ( 10t) tells us that we can decompose the principle phasor into a sum of two component phasors, , having at 0 = t a real and a pure imaginary amplitude, respectively, as shown for a positive phasor in Figure 1(b,c).In spectral form Eq. ( 10t) becomes (10s), and . This tells us that if we take the real and imaginary parts of the principle phasor's complex spectrum and the imaginary part Neither the real, nor the imaginary spectrum are normally represented in a 3D complex amplitude vs. frequency graph as in Figure 2(B,C), but as 2D real-amplitude vs. frequency, or imaginary-amplitude vs. frequency graphs [the analogous graphical schemes for a negative principle phasor can be seen in the original work (Szantay, 2007)].Although in this representation the concepts of "real" and "imaginary" spectrum may seem almost selfevident, one should note that the ordinate value at a given frequency of the real and imaginary spectrum is often erroneously understood as meaning the amplitude of a cosine and sine trigonometric harmonic wave, respectively.

Consider the decomposition of
into a cosine and a sine wave according to Eq. (11t).Because their amplitudes are related as @ A and @ A i ⋅ , the planes of oscillation of the cosine and sine wave are perpendicular to each other in the complex plane as shown in Figure 1(d,e).In spectral form [Expression (11s)] the respective complex amplitudes are depicted in ϖ -space according to Figure 2(D,E).Note that when we translate (11t) into spectral form, (11s) ceases to be an equality.This is because: a) the spectral amplitudes @ @ • ϖ A for the cosine and sine components [Figure 2(D,E)] belong to two different mathematical "species", therefore they cannot simply be added; b) the spectrum of the principle phasor [Figure 2(A)] is a function in ω / -space, while the spectra of its component sinusoids [Figures 2(D,E)] are functions in ϖ -space.(Again, this is a fundamental difference which can easily escape attention in the absence of a symbolism that differentiates between phasor and trigonometric frequencies.)Note also that we have no analogous problem with the relevant temporal forms of Figures 1(a,d,e), since they have a common argument: time.Thus the amplitudes depicted in Figure 2(D,E) do not add up to those shown in Figure 2(A).In order to bring

Temporal representation
onto a "common ground" we need to translate the ϖ -space spectra into ω / -space.In the temporal domain this means that the principle phasor's cosine component [Figure 1(d)] and sine component [Figure 1(e)] are decomposed into a sum of oppositely rotating phasors [Figure 1(f,h)] according to (12t).The decomposition of the cosine term reflects Equations ( 5) while that of the sine term corresponds to (6) multiplied by "i".If we now add up the four phasors in (12t) we re-gain our principle phasor [Figures 1(a)]: the two negative phasor-frequency components have identical amplitudes but are in opposite phase [Figures 1(h,i)] and hence cancel each other at every point in time, while the two positive phasor-frequency components, both having an amplitude @ 5 .0 A , are in-phase [Figure 1(f,g)] and thus reinforce each other to give the original principle phasor.
In spectral notation the relationships of (12t) are expressed according to (12s) and the pertinent ω / -space spectra are shown in Figure 2(F,G,H,I).In analogy to the correlation between the left and right sides of (11s), there is a correlation, but not an equation, between the pertinent terms on the right side of ( 11s) and (12s).
By adding up the four amplitudes in [12s] we obtain the principle phasor's spectrum From a broader viewpoint, the whole basis of the FT rests on the idea that a temporal function can be constructed from suitable harmonic "principle phasors".Any given point on the complex FT-spectrum will represent a constituent harmonic principle phasor.It is according to the above scheme that we can conceptually envisage the way the spectrum of a temporal function (or signal) is obtained.

The Fourier Transform
The FT is based on the theorem that t ) (t f can be uniquely expressed in terms of its projections onto an infinite frequency set of basis phasors  7)-( 9)], the essence of the FT can conceptually be written as: Employing the nomenclature and concepts outlined above, the Fourier integrals can be expressed as follows: Equations ( 15) and ( 14) express the forward and inverse FT, respectively.In the spectra , and this expression offers little in the way of showing us that even if ) t ( f is finite in time, the basis phasors Further insight may be gained into the FT by noting that the right-hand side of Equation (14b) represents an infinite collection of eternal "principle phasors" on the phasor frequency scale.Thus, in analogy to Equations ( 10)-( 12) we can formulate the following scheme: From Equations ( 4) and ( 5) we can express the forward FT for the phasor-space spectral form also as: In Equations ( 16) -( 21) the relationships between the various ϖ -space and ω / -space spectral forms are as follows: From this formalism one can draw a number of subtle insights into the FT as was detailed in the original papers which also included a discussion on a geometrical interpretation of the FT and on its capability to unravel the oscillating frequency components that are otherwise obscured in a complicated time signal (Szántay, 2007(Szántay, , 2008a(Szántay, , 2008b(Szántay, , 2008c)).These aspects will not be elaborated here.Suffice to note only that if a signal, represented by the function , comprises of harmonic phasor components (such is the case with the NMR "free induction decay" except that it contains exponentially damped harmonic phasors), then the signal must be detected from two orthogonal directions simultaneously to be able to tell the sense of rotation of the pertinent phasors.Letting these directions correspond to the +ℜ and +ℑ axes we obtain the functions A few basic FT relationships, represented graphically as well as in terms of function rules written in the style discussed above are shown in Figure 3 (all of these formulas are of course available in conventional notation from standard textbooks).

The Heisenberg and Fourier Uncertainty Principles
With reference to the issue of whether or not the frequency of a time-limited sinusoid wave becomes inherently "fuzzy" due to the Uncertainty Principle (as was discussed in the Introduction), let us now take a closer look at the meaning of the term "Uncertainty Principle".One mental trap here is that for most people the phrase "Uncertainty Principle" invokes a single famous theorem, while there are actually several uncertainty principles that have come to constitute their independent branches of mathematics, physics and signal analysis theories.In the context of our present discussion we can speak of two main types of uncertainty principle: Heisenberg's Uncertainty Principle (HUP) which is a quantummechanical concept grounded in the physical world, and the Fourier Uncertainty Principle (FUP) which is a classical concept and stems from a mathematical necessity inherent to the way the FT works.
Within our framework of discussion the HUP can be written for the noncommuting variables time and frequency through the expression: where the symbol sd Δ denotes the standard deviation of frequency.
In the classical realm of time-frequency signal analysis there exists another fundamental inequality commonly referred to as the Fourier Uncertainty Principle (FUP).In qualitative terms the FUP states that the "broader"  definition that would adequately quantify or delineate the concept of a function's "breadth" (Cohen, 1995 andClaerbout, 2004), we adopt from Claerbout the generic symbol "∧", herein called "characteristic span", connoting the "compressedness" or "elongatedness" of a function along its domain, the idea being that "∧" symbolizes any mathematical definition, including sd Δ , that may be suitably applied to the specific function at hand.Using the symbol "¬" to denote "of", so " x , the FUP can thus be expressed as where TBP is the so-called "time-bandwidth product", Ξ is the minimum value of the TBP, and ( 24b) is just a short-hand for (24a The FUP and HUP share a common mathematical form and terminology which is why, as it was pointed out by Cohen (Cohen, 1995), they are easily mixed up, especially since both are often written simply in the form " constant ω ≥ Δ ⋅ Δt ".However, the physical interpretations of ( 23) and ( 24) are different: in the HUP the word "uncertainty" reflects probabilistic uncertainties in the measurement of quantum-mechanical physical observables, while the FUP is a deterministic theorem resulting from the innate nature of the FT, as was illustrated geometrically earlier (Szántay, 2008c).
The notion that by time-limiting a monochromatic sine wave we impose some "uncertainty" on its frequency stems either from confusing the HUP and the FUP, or from misunderstanding the FUP itself.With regard to the deterministic sine wave (such as the NMR pulse), it is obviously the FUP that is of relevance and not the HUP.As for the FUP, one reason it is often misinterpreted is because "uncertainty" is a dubious word with several possible interpretations in various situations, as was discussed in detail before (Szántay, 2008c).The concept of the characteristic span simply reflects "broadness" which is not a probabilistic quantity per se; just because a function has a large characteristic span, it is not "uncertain".In that sense, referring to (24) as an "Uncertainty Principle" is a misnomer, as was also emphasized by Cohen (Cohen, 1995).
Again, in the frequency @ ϖ has an exactly known value.Since now

H
must also broaden to some finite value.
However, just because we have time-windowed

H
> 0 is not a reflection of any "uncertainty" in @ ϖ but a mathematical necessity ensuing from the fact that all information that specifies the temporal function or signal must be uniquely coded into the spectrum.One way of thinking about this is that while for ⇔ ϖ t ) ( @ t h there was no need for the spectrum to code for the Fig. 4. Simple time-windowed monochromatic temporal waves and their ω / -space spectra (for simplicity, only the positive phasor-frequency side of the spectra is shown).Each wave's frequency is the same value, @ ϖ , but their duration, time location, and initial phase are different.The value of @ ϖ "codes into" the frequency location of the center of the sincshaped spectra, therefore this location is the same in each spectrum.Differences in the spectral profiles reflect differences in the waves' other features, because those are coded exactly by the FT into the (complex) FT-spectrum.Spectral broadening is due to that fact that the FT-spectrum carries all that coding information, and not to any "uncertainty" in @ ϖ duration and time location of the wave (which is why the spectrum is a Dirac-delta a Dirac-delta would not provide enough degrees of freedom in its spectral features to be able to code for the wave's duration and time location.It is, then, this encoding of the "window" into the FT-spectrum that manifests itself in a spectral profile that covers the infinite frequency set ⇔ / ω , and not some kind of "uncertainty" in @ ϖ .
Thinking in terms of "curves", we can render to the curve of [Figure 4(a)] some distinct physical "aspects", such as a "frequency aspect" defined through the term ) ( cos @ t ϖ , and a "window aspect" defined through the location and duration of the operative interval , while the "window aspect" translates into the broadened complex profile of the spectrum.One must therefore understand that the scaled basis elements do not necessarily represent physical vibrations that are actually or "effectively" present in . Rather, each eternal basis harmonic wave's (complex) amplitude carries a single piece of the entire set of codes that is needed to reconstruct the original temporal function . The physical interpretation that we should, then, correctly assign to is simply that the frequency of the central basis element of the spectrum corresponds to the monochromatic frequency of , and the rest of the basis elements code for all of its other features such as its "window aspect".
The popular notion that the forward FT disseminates a temporal function (signal) into its "frequency components" is incorrect if by "frequency component" one understands a physical vibration contained in, and lasting for the duration of, the signal.Rather, the spectrum represents a transposition of all properties of the temporal signal into frequency space, and some of those properties may happen to be periodic events whose frequency will show up as peaks in the spectrum, but other features of the signal also become scrambled into the spectrum.Claiming therefore that any given point of the FT-spectrum represents a "frequency component" of the temporal signal is misleading and violates the LCPE.Indeed, Figures 4(b-e) illustrate this point by showing that if we keep the monochromatic "frequency aspect" of constant but vary its other "aspects", the spectral profiles will change so as to code for every alteration in the temporal curve, but the (unchanged) value of @ ϖ remains "coded into" the same frequency position for each spectrum.

Back to NMR
If one reconsiders the statements on PFT-NMR quoted in the Introduction, the flaws in those arguments should now be evident.An RF pulse can be treated as a deterministic, timewindowed harmonic phasor in that it acts as a classical magnetic field vector rotating with a specific monochromatic frequency.The assertion that "a nominally monochromatic short pulse effectively has a (Heisenberg) uncertainty-broadened frequency band" simply reflects a misuse of the HUP and/or a physical misinterpretation of the FUP.The FUP does not warrant the causal inference that "wide frequency band" should mean a wide range of physically existing driving RF oscillations, as is often deemed necessary for explaining the phenomenon of off-resonance excitation.Actually, according to the LCPE the sinc-shaped spectral profile of the pulse is synonymous in physical meaning with its temporal form, therefore the statement that a "short pulse has a wide frequency band" does not tell us any more "physics" about the pulse than saying that a "short pulse is a short pulse".In fact there is nothing paradoxical about the fact that a monochromatic pulse can excite off-resonant spins: the phenomenon is solely due to the pulse's large amplitude, akin to the way a classical mechanical resonator also gives nonzero response to a sufficiently large off-resonance driving force.Nota bene: a) the common notion that "a shorter pulse excites a wider frequency band" is misleading in that it is valid if by 'excites' one specifically means 'gives absorption-shaped measured signals', but is not valid, and thus does not conflict with the above statements, if by 'excites' one just means 'perturbs' in general (Szántay, 2008c); b) the response of a linear system to an input signal can in fact be calculated from the spectrum of the input signal by way of the so-called Superposition Principle (the NMR system is nonlinear in general, and the linearity approximation is valid only for special limiting conditions), but that concept should not be confused with the ideas discussed above.

Conclusion
The Fourier Transform (FT) is often thought of as a mathematical device capable of "revealing" the "frequency components" of a temporal signal.Although this notion is correct with regard to any sinusoid oscillatory feature that may be present in the signal, it is deceptive in general, easily leading to the belief that the overall spectral profile represents all the physical "frequency components" that are present in the signal.This idea can give rise to false interpretations such as the concept that the frequency of a time-limited monochromatic sinusoid is "uncertain" or "effectively polychromatic", in line with the "Uncertainty Principle".In fact the Heisenberg Uncertainty Principle (HUP) has nothing to do with this problem; it is the Fourier Uncertainty Principle (FUP) that is of relevance, but the word "uncertainty" is also misleading in that regard: the FUP does not imply that timewindowing a deterministic wave with a monochromatic frequency makes its frequency "uncertain" or that the forward FT "reveals" such uncertainty just because the spectrum has a profile that covers a broad range of frequencies.Whatever physical attributes we associate with a temporal signal (such as that it has an exact monochromatic frequency) must be carried over exactly into the frequency dimension according to the "Law of Conjugate Physical Equivalence" (LCPE).

Acknowledgement
The author is indebted to Dr. Zsuzsanna Sánta for her critical and constructive mathematical insights.
zero on the right and open and unbounded on the left;

Fig. 1 .
Fig. 1.Graphical representation of Equations (10t)-(12t) for a positive principle phasor.For added clarity the side-view projections of the principle phasor onto the real and imaginary axis are shown in (a') )].We may think of these "secondary-phasor" spectra [Figures2(F,G,H,I)] as "interfaces" that allow us to connect the principle phasor's ω / -space spectrum [Figures2(A)] with its ϖ -space spectra[Figures 2(D,E)].

[
of these features are coded exactly into the shape of the spectrum Figure4(a)].The "frequency aspect" of ).The actual values of TBP and Ξ are specific to the