METHOD AND SYSTEM FOR CONVERTING PHYSIOLOGICAL SIGNALS

Abstract

A method for converting physiological signals includes: obtaining a first signal as a function of a time parameter, wherein the first signal represents electrocardiogram data; obtaining a second signal as a function of the time parameter, wherein the second signal represents physiological data different from the electrocardiogram data; mixing the first signal and the second signal to obtain a mixed signal; and generating a frequency spectrum pertaining to the mixed signal.

Claims

1. A method for converting physiological signals, comprising: obtaining a first signal (S.sub.1) as a function of a time parameter, wherein the first signal (S.sub.1) represents electrocardiogram data; obtaining a second signal (S.sub.2) as a function of the time parameter, wherein the second signal (S.sub.2) represents physiological data different from the electrocardiogram data; mixing the first signal (S.sub.1) and the second signal (S.sub.2) to obtain a mixed signal (M); and generating a frequency spectrum pertaining to the mixed signal (M).

2. The method according to claim 1, wherein the physiological data comprises rheogram data, in particular rheogram data of a blood vessel.

3. The method according to claim 1, further comprising: normalizing the first signal (S.sub.1) and/or normalizing the second signal (S.sub.2), prior to mixing the first signal (S.sub.1) and the second signal (S.sub.2).

4. The method according to claim 3, wherein: normalizing the first signal (S.sub.1) comprises dividing the first signal (S.sub.1) by a first maximum value attained by the first signal (S.sub.1) over a predetermined first time interval; and/or normalizing the second signal (S.sub.2) comprises dividing the second signal (S.sub.2) by a second maximum value attained by the second signal (S.sub.2) over a predetermined second time interval.

5. The method according to claim 1, wherein mixing the first signal (S.sub.1) and the second signal (S.sub.2) comprises linearly combining the first signal (S.sub.1) and the second signal (S.sub.2), in particular with equal weights of the first signal (S.sub.1) and the second signal (S.sub.2).

6. The method according to claim 1, wherein the mixed signal (M) is given in terms of a sum of the first signal (S.sub.1) and the second signal (S.sub.2).

7. The method according to claim 1, wherein generating the frequency spectrum pertaining to the mixed signal (M) comprises subjecting the mixed signal (M) to an integral transform.

8. The method according to claim 7, wherein the integral transform comprises a Fourier transform and/or a fractional Fourier transform and/or a Radon transform.

9. The method according to claim 1, further comprising: comparing the frequency spectrum pertaining to the mixed signal (M) with a reference frequency spectrum, and/or with a second frequency spectrum obtained according to the method according to any of the preceding claims.

10. A system for converting physiological signals, comprising: an acquisition unit adapted to obtain a first signal (S.sub.1) as a function of a time parameter, wherein the first signal (S.sub.1) represents electrocardiogram data; and adapted to obtain a second signal (S.sub.2) as a function of the time parameter, wherein the second signal (S.sub.2) represents physiological data different from the electrocardiogram data; a mixing unit adapted to mix the first signal (S.sub.1) and the second signal (S.sub.2) to obtain a mixed signal (M); and a transform unit adapted to generate a frequency spectrum pertaining to the mixed signal (M).

11. The system according to claim 10, further comprising: a normalization unit adapted to normalize the first signal (S.sub.1) and/or adapted to normalize the second signal (S.sub.2).

12. The system according to claim 10, wherein the mixing unit is adapted to linearly combine the first signal (S.sub.1) and the second signal (S.sub.2), in particular with equal weights of the first signal (S.sub.1) and the second signal (S.sub.2).

13. The system according to claim 10, wherein the transform unit is adapted to subject the mixed signal (M) to an integral transform to generate the frequency spectrum pertaining to the mixed signal (M).

14. The system according to claim 10, further comprising: an analysis unit adapted to compare the frequency spectrum pertaining to the mixed signal (M) with a reference frequency spectrum, and/or with a second frequency spectrum.

Description

BRIEF DESCRIPTION OF THE FIGURES

[0053] The features and numerous advantages of the techniques of the present disclosure will be best apparent from a detailed description of exemplary embodiments with reference to the accompanying drawings, in which:

[0054] FIG. 1 is a flow diagram illustrating a method for converting physiological signals according to an embodiment;

[0055] FIG. 2 is a schematic illustration of a system for converting physiological signals according to an embodiment;

[0056] FIG. 3 is a schematic illustration of a system for converting physiological signals according to another embodiment;

[0057] FIGS. 4a-4h illustrate the effects of the present disclosure based on experimental data of passive states and active states obtained from a mixture of hydraulic impedance data of the femoral artery and electrocardiogram data in rats; and

[0058] FIGS. 5a-5f illustrate corresponding experimental data processed by more conventional conversion techniques employing hydraulic impedance data of the femoral artery in the same rats.

DETAILED DESCRIPTION

[0059] The techniques of the present disclosure will now be described with respect to specific examples of analyzing the hydraulic impedance of blood vessels, such as arteries. However, the same techniques may be employed in the analysis of other physiological data.

[0060] FIG. 1 is a flow diagram illustrating a method for converting physiological signals according to an embodiment.

[0061] In a first step S10, the method involves obtaining a first signal as a function of a time parameter, when the first signal represents electrocardiogram data.

[0062] For instance, the first signal S.sub.1 may correspond to a time series of signal values S.sub.1(t.sub.1), S.sub.1(t.sub.2), S.sub.1(t.sub.3), . . . , S.sub.1(t.sub.n) for a positive integer number n, wherein each signal value S.sub.1(t), i=1, . . . , n represents an electrocardiogram value pertaining to the respective time such as an electrocardiogram value obtained from a pre-acquired or life electrocardiogram of a human or animal body at time t.sub.1.

[0063] In some embodiments, the first signal may also be represented in terms of a continuous function S.sub.1(t), for some real-valued time parameter t, t.sub.1≤t≤T.sub.1 within predetermined interval boundaries t.sub.1, T.sub.1.

[0064] Similarly, in a second step S12, a second signal is obtained as a function of the time parameter, wherein the second signal represents physiological data different from the electrocardiogram data.

[0065] For instance, the second signal S.sub.2 may correspond to a time series of signal values S.sub.2(t.sub.1), S.sub.2(t.sub.2), S.sub.2(t.sub.3), . . . , S.sub.2(t.sub.m) for a positive integer number m, which may or may not be equal to n, wherein each signal value S.sub.2(t.sub.j), j=1, . . . , m represents physiological data pertaining to the respective time t.sub.1, wherein the physiological data is different from the electrocardiogram data.

[0066] The physiological data may be any physiological data pertaining the human body or the animal body, such as an electrical impedance and/or a hydraulic impedance pertaining to a blood vessel of the human body or the animal body. For instance, the electrical impedance of the blood vessel may be obtained from electric resistance measurements along the length of the blood vessel. The hydraulic impedance is known to be functionally related to the electrical impedance. These techniques are generally known in the art, and do not form part of the present disclosure. Similar to the electrocardiogram data of the first signal, each signal value S.sub.2(t.sub.1), j=1, . . . , m may represent a pre-acquired or life measurement value of the electrical impedance or hydraulic impedance at time t.sub.j.

[0067] In some embodiments, the second signal may also be represented in terms of a continuous function S.sub.2(t), for some real-valued time parameter t, t.sub.2≤t≤T.sub.2 within predetermined interval boundaries t.sub.2, T.sub.2, which may or may not be equal to the pair of interval boundaries t.sub.1, T.sub.1.

[0068] In a third step S14, the first signal and the second signal are mixed to obtain a mixed signal.

[0069] In an example, the mixed signal M(t) may represent a linear combination of the first signal S.sub.1(t) and the second signal S.sub.2(t), in the form

M(t)=a S.sub.1(t)+b S.sub.2+C  (1)

with real-valued parameters a, b, c, wherein a and b are non-zero.

[0070] For instance, the mixture may correspond to a weighted average

M(t)=p S.sub.1(t)+(1−p)S.sub.2(t)  (2)

of the signals S.sub.1 and S.sub.2, with a positive weight coefficient 0<p<1.

[0071] The mixed signal M(t) may also correspond to sum of the first signal S.sub.1(t) and the second signal S.sub.2(t), in the form

M(t)=S.sub.1(t)+S.sub.2(t).  (3)

[0072] In some practically relevant scenarios, the first signal and the second signal may be signals of different dimensions, or may have amplitudes that may vary considerably in strength. It may thus be advantageous to normalize the first signal and/or the second signal prior to the mixing.

[0073] For instance, normalizing the first signal S.sub.1(t) may comprise dividing the first signal S.sub.1(t) by a first maximum value |S.sub.1| attained by the first signal over the first time interval t.sub.1≤t≤T.sub.1 with predetermined interval boundaries t.sub.1, T.sub.1. Similarly, normalizing the second signal S.sub.2(t) may comprise dividing the second signal S.sub.2(t) by a second maximum value |S.sub.2| attained by the second signal over the second time interval t.sub.2≤t≤T.sub.2 with predetermined interval boundaries t.sub.2, T.sub.2.

[0074] The mixing may then comprise mixing the normalized first signal and the normalized second signal to obtain the mixed signal.

[0075] For instance, when taking the normalization into account, the linear mixing according to Eq. (1) reads

[00001]$\begin{array}{cc}M\left(t\right)=a\frac{S_1\left(t\right)}{.Math.S_1.Math.}+b\frac{S_2\left(t\right)}{.Math.S_2.Math.}+c& \left(4\right)\end{array}$

with the real-valued parameters a, b, c, wherein a and b are non-zero. Similarly, the sum of Eq. (3) reads

[00002]$\begin{array}{cc}M\left(t\right)=\frac{S_1\left(t\right)}{.Math.S_1.Math.}+\frac{S_2\left(t\right)}{.Math.S_2.Math.}.& \left(5\right)\end{array}$

[0076] In a fourth step S16, the method involves generating a frequency spectrum pertaining to the mixed signal.

[0077] For instance, generating the frequency spectrum may involve applying an integral transform T to the mixed signal M(t), in the form

M.fwdarw.T[M]  (6)

to convert the time-dependent mixed signal M(t) into a frequency-dependent signal T[M](ω). For instance, the integral transform T may involve a Fourier transform or fractional Fourier transform, or a function of the Fourier transform or fractional Fourier transform, such as a square of the Fourier transform or fractional Fourier transform.

[0078] The frequency-dependent signal T[M](ω) may then be decomposed into individual frequency contributions to analyze the frequency spectrum.

[0079] The inventors found that analyzing the spectrum of the mixed signal may provide helpful additional insights into the physiological state of the human or animal body, as compared with an analysis of the spectrum of only the second signal S.sub.2 alone, or of only the first signal S.sub.1 alone.

[0080] Examples of different integral transforms that can be employed in the context of the present disclosure, as well as experimental data that illustrates the effect of analyzing the mixture of the first signal and the second signal as compared to only analyzing the signals individually will be described in further detail below with reference to FIGS. 4 and 5.

[0081] The flow diagram of FIG. 1 shows the method steps S10 to S16 in a certain order. However, this order is for illustration only, and one skilled in the art will appreciate that the order of some of the method steps may be reversed.

[0082] For instance, FIG. 1 shows the step S12 of obtaining the second signal after the step S10 of obtaining the first signal. However, in other examples the step of obtaining the second signal may precede the step of obtaining the first signal, or the first signal and the second signal may be obtained (approximately) simultaneously.

[0083] FIG. 2 is a schematic illustration of a corresponding system 10 for converting physiological signals according to an embodiment.

[0084] The system 10 comprises an acquisition unit 12, a mixing unit 14 communicatively coupled to the acquisition unit 12, and a transform unit 16 communicatively coupled to the mixing unit 14.

[0085] The acquisition unit 12 is adapted to obtain a first signal S.sub.1(t) as a function of a time parameter t, wherein the first signal S.sub.1(t) represents electrocardiogram data. The acquisition unit 12 is further adapted to obtain a second signal S.sub.2(t) as a function of the time parameter t, wherein the second signal S.sub.2(t) represents physiological data different from the electrocardiogram data.

[0086] As illustrated in FIG. 2, the acquisition unit 12 provides the first signal S.sub.1(t) and the second signal S.sub.2(t) to the mixing unit 14, which is adapted to mix the first signal S.sub.1(t) and the second signal S.sub.2(t) to obtain a time-dependent mixed signal M(t). For instance, the mixed signal M(t) may represent a sum of the first signal S.sub.1(t) and the second signal S.sub.2(t), in accordance with Eq. (3).

[0087] In the embodiment illustrated in FIG. 2, the mixing unit 14 provides the mixed signal M(t) to the transform unit 16, and the transform unit 16 may be adapted to generate a frequency spectrum pertaining to the mixed signal M(t). For instance, the transform unit 16 may be adapted to integral-transform the mixed signal in accordance with Eq. (6), and may decompose the resulting signal T[M](ω) into individual frequency contributions that permit a user or computer system to analyze the frequency spectrum.

[0088] FIG. 2 shows the acquisition unit 12, the mixing unit 14 and the transform unit 16 as separate and individual units that are communicatively coupled. However, one skilled in the art will understand that this represents just one exemplary configuration, and in other configurations all or some of the acquisition unit 12, the mixing unit 14 and the transform unit 16 may be combined into a single unit.

[0089] FIG. 3 is a schematic illustration of another system 10′ for converting physiological signals according to an embodiment.

[0090] The system 10′ shown in FIG. 3 generally corresponds to the system 10 shown in FIG. 2, and corresponding components are denoted with the same reference signs. However, the system 10′ additionally comprises a normalization unit 18 in between the acquisition unit 12 and the mixing unit 14, and an analysis unit 20 downstream of the transform unit 16.

[0091] The normalization unit 18 is adapted to normalize the first signal S.sub.1(t) and/or the second signal S.sub.2(t). For instance, the normalization unit 18 may be adapted to normalize the first signal S.sub.1(t) by dividing the first signal S.sub.1(t) by a first maximum value |S.sub.1| attained by the first signal S.sub.1(t) over the predetermined first time interval t.sub.1≤t≤T.sub.1. The normalization unit 18 may be further adapted to normalize the second signal S.sub.2(t) by dividing the second signal S.sub.2(t) by a second maximum |S.sub.2| value attained by the second signal S.sub.2(t) over the predetermined second time interval t.sub.2≤t≤T.sub.2.

[0092] The normalization unit 18 may then provide both the normalized first signal S.sub.1(t)/|S.sub.1| and the normalized second signal S.sub.2(t)/|S.sub.2| to the mixing unit 14, which may subsequently combine the normalized first signal S.sub.1(t)/|S.sub.1| and the normalized second signal S.sub.2(t)/|S.sub.2| into a time-dependent mixed signal M(t), such as in accordance with Eq. (4) or Eq. (5).

[0093] The mixing unit 14 may subsequently provide the time-dependent mixed signal M(t) to the transform unit 16, which may be adapted to generate a frequency spectrum pertaining to the mixed signal M(t), as described above with reference to FIG. 2.

[0094] The analysis unit 20 may be adapted to receive the frequency spectrum from the transform unit 16, and may be further adapted to compare the frequency spectrum pertaining to the mixed signal M(t) with a reference frequency spectrum, and/or with a second frequency spectrum.

[0095] For instance, the reference frequency spectrum may represent a previously acquired spectrum representing reference date of the human or animal body. By comparing the frequency spectrum pertaining to the mixed signal M(t) with the reference spectrum, such as by means of automated data analysis, distinctions may be identified that could be used to later classify the acquired frequency spectrum pertaining to the mixed signal M(t). These techniques may provide helpful insights into the current physiological state of the human or animal body.

[0096] Alternatively or additionally, the analysis unit 20 may be adapted to compare different frequency spectra generated according to the techniques described above with reference to FIGS. 1 to 3, but generated by means of a plurality of different integral transforms T, T′. A comparison of the corresponding frequency spectra of the frequency-dependent signals T[M](ω), T′[M](ω) may allow to determine the integral transform that is most useful for distinguishing between different physiological states of the human or animal body, in a given scenario.

[0097] Some exemplary integral transforms T[M] that may be employed in the context of the present disclosure will now be described in additional detail.

[0098] The Wigner function and the tomographic representation

[0099] One can assume that the totality of all signal values is represented by a set of vectors |f in the space of all possible values. Symbol f designates a signal amplitude in this case. This value can be a complex one in general. The space of states should be complete, i.e.,

f′|f=δ(f−f′) and f df|f(f|={circumflex over (1)}  (7)

[0100] This assumes that a signal amplitude can take continuous values in some interval [f.sub.min′ f.sub.max]. In other words, it has a continuous spectrum. A signal takes discrete values if it is represented in a digital form. In this case properties (7) can be rewritten as follows

f′|f=δ.sub.f,f′ and Σ.sub.f=fmin.sup.fmax|ff|={circumflex over (1)}.  (8)

[0101] Let us introduce a density matrix ρ.sup.s of a signal using quantum mechanics methods. It can be represented as a density matrix of a pure state in the form:

ρ.sup.s=|ff|.  (9)

[0102] This relation in a “time-representation” has the following expression:

ρ.sup.s(t,t′)=t|ff|t′≡f(t)f*(t′).  (10)

[0103] Let us define the Wigner-Ville function of a time-dependent signal ρ.sup.s(t) as follows:

[00003]$\begin{array}{cc}{W}^s\left(\omega ,t\right)={\int }_{-\infty }^{+\infty }dx{\rho }^s\left(t+\frac{x}{2},t-\frac{x}{2}\right)e^{-i\omega x}.& \left(11\right)\end{array}$

[0104] The reverse transform is determined in complete correspondence with the Wigner function of a quantum system:

[00004]$\begin{array}{cc}{\rho }^s\left(t,t^{\prime }\right)={\int }_{-\infty }^{+\infty }\frac{d\omega }{2\pi }{W}^s\left(\omega ,\frac{t+t^{\prime }}{2}\right)e^{i\omega \left(t-t^{\prime }\right)}.& \left(12\right)\end{array}$

[0105] A signal can be expressed through the Wigner function using Eq. (10) for t′=0:

[00005]$\begin{array}{cc}f\left(t\right)f^{*}\left(0\right)={\int }_{-\infty }^{+\infty }\frac{d\omega }{2\pi }{W}^s\left(\omega ,\frac{t}{2}\right)e^{i\omega t}.& \left(13\right)\end{array}$

[0106] The value of a signal at the initial time in any case can be represented as follows:

f*(0)=|f(0)|e.sup.−φ(0),  (14)

where a phase φ(0) at the initial time remains undetermined. Accordingly, we have

[00006]$\begin{array}{cc}f\left(t\right)=e^{i\phi \left(0\right)}\frac{\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }d\omega W^s\left(\omega ,\frac{t}{2}\right)e^{i\omega t}}{\sqrt{\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }d\omega W^s\left(\omega ,0\right)}}.& \left(15\right)\end{array}$

[0107] Now we can define a signal tomogram

[00007]$\begin{array}{cc}{w}^s\left(X,\mu ,v\right)=\int \int {\int }_{-\infty }^{\infty }\frac{dtd\omega du}{{\left(2\pi \right)}^2}{W}^s\left(\omega ,t\right)e^{-iu\left(X-\mu t-v\omega \right)}& \left(16\right)\end{array}$

Here X is an eigen value of a Hermitian operator

{circumflex over (X)}=μ{circumflex over (t)}+v{circumflex over (ω)},μ.sup.2+v.sup.2=1.  (17)

[0108] One takes as usual

μ=cos θ,v=sin θ,0≤0≤π/2.  (18)

[0109] The eigen value of g is located on the straight line of the first quadrant in the plane of variables (t, ω). The Wigner function can be written down explicitly through a signal function after substituting Eq. (16) into Eq. (12),

[00008]$\begin{array}{cc}{w}^s\left(X,\mu ,v\right)=\int \int \int \int \frac{dtd\omega dudy}{{\left(2\pi \right)}^2}f\left(t+\frac{y}{2}\right)f^{*}\left(t-\frac{y}{2}\right)e^{-i\omega y}{e}^{-iu\left(X-\mu t-v\omega \right)}.& \left(19\right)\end{array}$

[0110] One can integrate Eq. (19) over the ω variable

f.sub.−∞.sup.∞dωe.sup.−ψ(y−yv)=2πδ(y−vu).  (20)

[0111] Thus we have

[00009]$\begin{array}{cc}{w}^S\left(X,\mu ,v\right)=\frac{1}{2\pi }\int \int \int \mathrm{dtdudy}f\left(t+y/2\right)f^{*}\left(t-y/2\right)e^{-iu\left(X-\mu t\right)}\delta \left(y-\mathrm{vu}\right).& \left(21\right)\end{array}$

[0112] Performing the integration over u we obtain

[00010]$\begin{array}{cc}{w}^S\left(X,\mu ,v\right)=\frac{1}{2\pi .Math.V.Math.}\int \int \mathrm{dtdy}f\left(t+\frac{y}{2}\right)f^{*}\left(t-\frac{y}{2}\right)e^{-i\frac{y}{v}\left(X-\mu t\right)}.& \left(22\right)\end{array}$

[0113] Let us change variables in the integral of Eq. (22) as follows:

t=(z+q)/2,y=z−q.  (23)

[0114] The Jacobian of the transform of Eq. (23) is equal to −1, but the integration limits extend from −∞ to +∞ and we obtain

[00011]$\begin{array}{cc}\begin{array}{c}{W}^S\left(X,\mu ,v\right)=\frac{1}{2\pi .Math.v.Math.}\int \int dzdqf\left(z\right)f^{*}\left(q\right)e^{-i\frac{1}{v}\left(\left(z-q\right)X-\mu \left(z^2-q^2\right)/2\right)}\\ =\frac{1}{2\pi .Math.v.Math.}{.Math.{\int }_{-\infty }^{\infty }dzf\left(z\right)e^{\frac{i\mu }{2v}{Z}^2-\frac{iXz}{v}}.Math.}^2\end{array}.& \left(24\right)\end{array}$

[0115] Finally, let us rewrite a tomogram definition introducing a more conventional designation of the integration variable by letter t as follows:

[00012]$\begin{array}{cc}{w}^s\left(X,\mu ,v\right)=\frac{1}{2\pi .Math.v.Math.}{.Math.{\int }_{-\infty }^{\infty }f\left(t\right)\exp \left[\frac{i\mu }{2v}{t}^2-\frac{iXt}{v}\right]\mathrm{dt}.Math.}^2.& \left(25\right)\end{array}$

[0116] The obtained tomographic representation is connected with the so-called fractional Fourier transform (FRFT), which is widely used for example in quantum optics. On the other hand, it corresponds to the so-called Radon transform. As usual, the FRFT can be written in the form

[00013]$\begin{array}{cc}F\left(X,\alpha \right)=\sqrt{\frac{v-i\mu }{2\pi v}}\exp \left(\frac{i\mu }{2v}{X}^2\right){\int }_{-\infty }^{\infty }\exp \left[\frac{i}{2v}\left(-2Xt+\mu t^2\right)\right]f\left(t\right)\mathrm{dt}.& \left(26\right)\end{array}$

[0117] The dimensionless variable a may vary in the interval 0≤α≤1 and may be called the transform order. The transform order is functionally connected with the previously introduced parameters θ, μ and v by the relation α=2θ/π. In this case relation (18) can be rewritten a follows:

[00014]$\begin{array}{cc}\begin{array}{cc}\mu =\cos \frac{\pi }{2}\alpha ,& v=\sin \frac{\pi }{2}\alpha \end{array}& \left(27\right)\end{array}$

[0118] It is then easy to see that the tomographic representation is equal to the square of the FRFT modulus,

w.sup.s(X,μ,v)=|F(X,α)|.sup.2  (28)

[0119] Tomogram Calculation

[0120] The identical transform is realized at α=0 and at α=1 we obtain the Fourier transform. Numerical calculations of this kind of transform on a computer may be complicated in practice by two reasons. Firstly, the real signal (cardiogram) has a complex fine-scale structure and demands large numbers of grid nodes in a numerical representation. Secondly, the integral transform core in Eq. (26) is an imaginary exponent of a quadratic function. So, it is a frequently oscillating function and gives additional problems in approximation of the integral by quadrature formulae. One way to overcome these difficulties is a reduction of the FRFT to an integral of a convolution type. The following application of the fast Fourier transform algorithm makes the procedure more successful. Using the identity

μX.sup.2−2Xt+μt.sup.2=(μ−1)X.sup.2+(X−t).sup.2+(μ−1)t.sup.2  (29)

[0121] as presented by H. M. Ozaktas et al., IEEE Transactions on Signal Processing 44 (1996) 2141, one can transform Eq. (26) into the following form:

[00015]$\begin{array}{cc}F\left(X,\alpha \right)=\sqrt{\frac{v-i\mu }{2\pi v}}\exp \left(\frac{i\left(\mu -1\right)}{2v}{X}^2\right){\int }_{-\infty }^{\infty }\exp \left[\frac{i}{2v}{\left(X-t\right)}^2\right]\tilde{f}\left(t\right)\mathrm{dt}& \left(30\right)\end{array}$$\mathrm{with}$$\begin{array}{cc}\tilde{f}\left(t\right)=\left(\frac{i\left(\mu -1\right)}{2v}{t}^2\right)f\left(t\right).& \left(31\right)\end{array}$

[0122] The integral in Eq. (30) has a type of a convolution.

[0123] Generally, the transform order α may be chosen in the range 0≤α≤1, in particular α<1. In some embodiments, the transform order α may be chosen no smaller than 0.90, in particular no smaller than 0.95.

[0124] In exemplary numerical calculations, the interval 0≤α≤1 can be broken into 50 equal distances, and the transform order α can thus take 51 values including the boundaries of the interval. The quadrature formula for integral (30) may be constructed on a uniform grid. The grid size may be determined automatically for the given precision of the calculations. A permissible relative error of calculations may be taken to be of the order of 10.sup.−5. This precision may be achieved with a grid size up to 2.sup.24 knots. One tomogram calculation on a PC may be executed from few to tens of minutes of computer time, due to the Fast Fourier Transform (FFT) algorithm application.

[0125] It should be noted that a tomogram form essentially depends on a choice of a unit of time representation. In primary data received from a device, the time T may be given in seconds as usual. When a tomogram is calculated, a dimensionless time t=T/T.sub.0 may be used, where T.sub.0 may be a unit of time in seconds. A frequency scale in the tomogram cross-section at α=1 may be determined unambiguously by a choice of T.sub.0. Namely, a value ω=T.sub.0.sup.−1X may be a circular frequency, and it may have a dimension of rad/sec.

[0126] To illustrate the techniques of the present disclosure, FIGS. 4a-4h show an example of the original time series of the sum of normalized electrocardiogram data and the normalized hydraulic impedance of the femoral artery in rats determined according to Eq. (5) above (FIG. 4a), the corresponding squared Fourier spectrum (FIG. 4b) and two tomograms for different transform orders α=0.99 (FIG. 4c) and α=0.98 (FIG. 4d), for two different states of the artery (passive state and active state). The hydraulic impedance has been determined by means of electrical resistance measurements along the length of the femoral artery.

[0127] In the example of the frequency spectra shown in FIGS. 4a-h, in the passive state only two harmonics have a large amplitude, whereas the third harmonic is minimum or is equal to 0. This effect is revealed already in the squared fractional Fourier transform according to FIG. 4b. The distinctions are even more pronounced in the tomograms according to FIGS. 4c and 4d. The amplitudes of the small frequencies in the active state exceed the corresponding amplitudes of the small frequencies in the passive state by a factor of 5 to 7.

[0128] For comparison, FIGS. 5a-f show a corresponding example of the original time series of the hydraulic impedance only (FIG. 5a), without the mixture, the corresponding squared Fourier spectrum (FIG. 5b) and a tomogram (FIG. 5c) with transform order α=0.96 of the femoral artery in rats, again for two different states of the artery (passive state and active state).

[0129] A comparison reveals that the squared fractional Fourier transforms according to FIG. 5b are largely identical for the passive state and the active state, and the tomograms according to FIG. 5c likewise fail to provide a clear trend that permits to uniquely distinguish between the passive state and the active state.

[0130] The description of the embodiments and the figures merely serve to illustrate the techniques of the present disclosure and the beneficial effects associated therewith, but should not be understood to imply any limitation. The scope of the disclosure is to be determined from the appended claims.

REFERENCE SIGNS

[0131] 10, 10′ system for converting physiological signals [0132] 12 acquisition unit [0133] 14 mixing unit [0134] 16 transform unit [0135] 18 normalization unit [0136] 20 analysis unit