Patent ID: 12241957

DETAILED DESCRIPTION

The disclosed method may, using the multicarrier phase difference (MCPD) ranging principle, for a range determination, i.e., a distance determination, between a first radio signal transceiver, device A, and a second radio signal transceiver, device B, use as input a first set of measurement results and a second set of measurement results, wherein the first set of measurement results is acquired by the first radio signal transceiver, i.e., device A, based on signals transmitted from the second radio signal transceiver, i.e., device B, and the second set of measurement results is acquired by the second radio signal transceiver, i.e., device B, based on signals transmitted from the first radio signal transceiver. Each set of measurement results comprises, for each of a plurality of frequencies, a measurement pair of a phase measurement and a signal strength measurement.

Acquiring of the measurement results may start with the two devices A and B agreeing on the ranging parameters, align their frequencies (e.g. using carrier frequency offset (CFO) estimation and calibration) and realize coarse time synchronization, i.e. both A and B start a (digital) counter, i.e, clock at, e.g., the transmission/reception of a start frame delimiter (SFD) which both devices A and B use to control a local state machine. The state machine controls when which transceiver is doing what.

As illustrated inFIG.1, the measurements may be performed in the following steps:0. Device A and Device B setting their respective local oscillators (LOs) to a predetermined frequency and setting the loop counter k=0.1. Device A transmitting its LO signal and Device B performing a phase measurement ([k]) relative to its own LO. Further, device B may perform a received signal strength indication measurement RSSIB[k] (not shown). Alternatively, device B may perform a measurement of the Cartesian IB[k] (in-phase) and QB[k] (quadrature) components of the signal received at B, relative to its own LO.2. Device A and B changing transmit direction, allowing a guard time for stabilizing the LO.3. Device B transmitting its LO signal and Device A performing a phase measurement (ϕA[k]) relative to its own LO. Further, device A may perform a received signal strength indication measurement RSSLA[k] (not shown). Alternatively, device A may perform a measurement of the Cartesian IA[k] (in-phase) and QA[k](quadrature) components of the signal received at A, relative to its own LO.4. Device A and Device B increasing the frequency of their respective LOs by a predetermined frequency spacing Δƒand go back to step 1. This loop is repeated a predetermined number of times (Kƒ), resulting in measurements at Kƒdifferent frequencies with a spacing Δƒand ordered in frequency according to their respective frequency index k. For example, measurements may be performed with a 1 MHz frequency spacing of an 80 MHz band at 2.4 GHz.

Device A and device B have respective phase-locked loops (PLLs) to generate their respective local oscillator (LO) signals. When switching from transmit to receive or vice-versa, for each single frequency k, the PLLs remain on to allow for continuous phase signals.

FIG.2summarizes steps of the method. In block2ofFIG.2, once the measurements have been carried out, the method for determining the distance is not very time-critical. Therefore, it may be computed on a third device/entity with more processing power, which is, e.g., in the cloud, assuming the entity has access to the measurement data from both transceivers. Thus, the method may either be performed on device A and/or B, but may also be collected on a third Device C, which can then calculate the distance between A and B, where device C may be in the cloud. If a device is not to carry out the method, it may transmit, or cause to be transmitted its measurement results to the device that is to carry out the method. Thus, for example, if the method is to be carried out on device C, device B may transmit a frame with all its phase measurements to Device C (ϕB[0:Kƒ−1]) and device A may transmit a frame with all its phase measurements to Device C (ϕA[0:Kƒ−1]). Similarly, the RSSI measurements RSSLA[0:Kƒ−1] and, optionally, RSSIB[0:Kƒ−1] will be transmitted to the device carrying out the method, for example device C. Thus, the device carrying out the method receives the first set of measurement results, comprising ϕA[0:Kƒ−1] and RSSIA[0:Kƒ−1] (or IA[0:Kƒ−1] and QA[0:Kƒ−1]) and the second set of measurement results, comprising ϕB[0:Kƒ−1] and, optionally, RSSIB[0:Kƒ−1] (or IB[0:Kƒ−1] and QB[0:Kƒ−1].

Alternatively, device A will carry out the method and may then comprise a measurement unit configured to acquire the first set of measurement results based on signals transmitted from the second radio signal transceiver, i.e., device B, as per the above. It may further comprise a receiver configured to receive the second set of measurement results acquired by the second radio signal transceiver, i.e., device B, based on signals transmitted from the first radio signal transceiver, i.e., device A. Further, device A may comprise a processing unit for carrying out the steps of the method, as will be described below.

For each frequency and set of measurements, a complex number may be formed, proportional to the one-way frequency domain response, where the modulus represents an amplitude corresponding to the signal strength measurement and the argument of the complex number represents the phase measurement:
HA[k]=AA[k]exp(jϕA[k])
HB[k]=AB[k]exp(jϕB[k])where AA[k] and AB[k] are values proportional to signal amplitude, obtainable, for example, by taking the square root of the corresponding RSSI values.

Alternatively, in the case of measurement of the I and Q components of the signal, HA[k] and HB[k] may be formed thus:
HA[k]=IA[k]+jQA[k]
HB[k]=IB[k]+jQB[k]

In the absence of thermal or phase-noise, these measured magnitudes and phases at the kth frequency are related to the actual channel responses H[k] as follows
HA[k]∝H[k]exp(j2πθ[k])where θ[k] denotes a phase offset between A and B during the measurement of the kth frequency and the symbol cc denotes proportionality, i.e., a[k]∝b[k] means that a[k]=c b[k] for all values of k, where c is an unknown complex-value, but the same for all k.

Contrary to the method disclosed in EP3502736A1, there is no restriction on θ[k], which can be allowed to vary arbitrarily from frequency to frequency k.

Similarly, at B, we will measure
HB[k]∝H[k]exp(−j2πθ[k])

In block4ofFIG.2, an estimate X[k] of a value proportional to a two-way frequency domain channel response can be formed by multiplying the two values together, thereby canceling out the factors related to the offsets θ[k], since, as the measurements at A and B are taken shortly after each other, we may assume that θ[k] did not change:
X[k]=HA[k]HB[k]∝(H[k])2.

Thus, the calculation of the estimate of a value proportional to the two-way frequency domain channel response is based on the measurement pair from the first set of measurement results and the measurement pair from the second set of measurement results. Moreover, it comprises, or may be represented as comprising, multiplying the complex number representing the measurement pair from the first set of measurement results with the complex number representing the measurement pair from the second set of measurement results.

Alternatively, X[k] may, regarding amplitude, be calculated based on the measurement at A only:
X[k]=∥HA[k]∥2exp(ϕA[k]+ϕB[k])∝(H[k])2where ∥ ∥2denotes the absolute squared-operator. Note that ∥HA[k]∥2is equal to the RSSLA[k].

Thus, here, calculating the estimate of a value proportional to the two-way frequency domain channel response is based on the measurement pair from the first set of measurement results and the phase measurement from the second set of measurement results.

In the following, the one-way frequency-domain channel response H[k] will be reconstructed using a) X[k] and b) correlation properties for H[k] for adjacent frequencies.

A preliminary estimate Hsqrt[k] of the one-way frequency domain channel response H[k] is calculated by taking the square root of the estimate proportional to the two-way frequency domain channel response X[k]:
Hprelim[k]=Hsqrt[k]=√{square root over (X[k])}∝c[k]H[k]which is related to the true one-way frequency-domain channel response according to the proportionality above, where c[k] is either +1 or −1, caused by the inherent phase ambiguity of taking a complex square root. To estimate the values of c[k], we use the correlation properties for H[k].

Thus, starting from the estimated frequency-domain channel response Hprelim[k] that contains random phase reversals, i.e., sign flips at various frequency indices k, we want to detect those sign-flips (or, equivalently, the signs in c) and corrects them to restore the phase structure along the frequency dimension.

For the preliminary estimate, for example, solutions with the phase between −π/2 and π/2 may be selected, i.e., with a positive real part.

Thus, for each frequency, the preliminary estimate of the value proportional to the one-way frequency domain channel response is calculated based on the measurement pair from the first set of measurement results and the phase measurement, or optionally the measurement pair, from the second set of measurement results.

At block5a, optionally, each preliminary estimate Hprelim[k] may undergo a transformation, which may be frequency dependent, resulting in a representation Hprelim[k] of each preliminary estimate, as will be described further below.

Alternatively, as described below, calculations may be performed directly on values of Hprelim[k]. In that case, each representation Hprelim′[k] of a quantity is the quantity itself, i.e., Hprelim′[k]=Hprelim[k].

At block5b, for a frequency index k in the plurality of frequencies, a predicted estimate Hpred′[k] of H′[k] is calculated based values of Hprelim′[k] for frequency indices adjacent to k, i.e., Hprelim′[k−1], Hprelim′[k−2] . . . and/or Hprelim′[k+1], Hprelim[k+2] . . . .

In some cases, the predicted estimate Hpred′[k] of H′[k] may be calculated as the value of the estimate for the preceding frequency index k−1
Hpred′[k]=Hprelim′[k−1].

Alternatively, the predicted estimate may be based on extrapolation from two or more adjacent frequency indices, as will be described below.

At block6, a first metric distance dLbetween the predicted estimate Hpred′[k] and the preliminary estimate Hprelim′[k] is calculated:
dL=∥Hprelim′[k]−Hpred′[k]∥.

Still at block6, a second metric distance dL′ between the predicted estimate Hpred′[k] and a phase reversal −Hprelim′[k] of the preliminary estimate Hprelim′[k] may be calculated as:
dL=∥−Hprelim′[k]−Hpred′[k]∥=∥Hprelim′[k]+Hpred′[k]∥.

The metric according to which the first metric distance dLand the second metric distance dL′ is calculated may be the complex number norm, or some other geometric metric, i.e., distance measure.

At block8, a final estimate Hest[k] of the one-way frequency domain channel response is determined. The final estimate Hest[k] is based on a comparison of the predicted estimate Hpred′[k] with the preliminary estimate Hpred′[k] and its phase reversal −Hpred′[k], more specifically based on a comparison of the first metric distance dLand the second metric distance dL′, as calculated at block6.

Under an assumption of Gaussian-distributed random errors, dLis a measure for the likelihood given no sign-flip, i.e., no phase reversal is required in the final estimation relative to the preliminary estimation and dL′ is a measure for the likelihood that a phase reversal is required.

Thus, if dL′<dL, this indicates an incorrect sign in the preliminary estimate Hprelim[k] and the final estimate is determined to be
Hest[k]=−Hprelim[k].

Otherwise, the final estimate is determined to be
Hest[k]=Hprelim[k].

Thus, the final estimate Hest[k] is either a phase reversal −Hprelim[k] of the preliminary estimate, or the preliminary estimate Hprelim[k].

Further, the representation of the preliminary estimate may be updated
Hprelim′[k]=Hest′[k],and blocks5b,6, and8be repeated for a different value of k, using the updated preliminary estimate.

In particular, the estimation may be blocks5b,6, and8may be repeated in sequence for successive values of frequency indices k=1, 2, . . . Kƒ−1. In that case, in the case of a determination of Hest[k]=−Hprelim[k], i.e., a phase reversal required, for a frequency index k=m, all subsequent preliminary estimates may be phase reversed as well, i.e.,
Hprelim′[m]=−Hprelim′[m] form≥k,i.e., a sign-flip of all measurements stating at frequency index m. If two such sign changes are observed at say m and n with m<n, the values between m≤k<n are multiplied by −1 and the remainder by (−1)2=1. For three or more jumps, the procedure is simply extended. To limit the number of multiplications, first a mask can be created to keep track of which values should be sign-flipped and which not.

FIG.3Ashows a simulated example of Hsqrt[k] from one antenna at 80 different frequency channels that are spaced 1 MHz apart. The number markers are the frequency indices k.

It can be observed that between frequency25and26a jump is present. In fact, the frequency26(and all subsequent) is phase-inversed. In this case if frequency26were to be flipped back with respect to the origin, its distance to frequency25would be much smaller and thus more reasonable compared to the distance of other successive frequency pairs. Once the phase reversal is detected, the samples at frequency26and all subsequent frequencies may have their phase reversed.

FIG.3Bshows the final estimation Hest[k] determined according to the above. Compared to Hsqrt[k] ofFIG.3A, the jump between frequency indices25and26has been removed, indicating an improved reconstruction of the one-way frequency domain channel response.

For the optional transformation of block5a, an average phase advance between consecutive frequency indices k may be calculated as:

Δ⁢ϕave[k]=∑k=1Kf-1∠⁡(X[k+1]⁢X*[k])2⁢(Kf-1)where an asterisk denotes the complex conjugate and L denotes taking the argument of the complex number, i.e., the angle function.

Thus, the average phase advance is calculated by taking the argument of the product of the complex number representing the value proportional to the two-way frequency domain channel response for a frequency and the conjugate of the complex number representing the value proportional to the two-way frequency domain channel response for an adjacent frequency, summing over frequencies, and dividing by two times the number of frequency steps. In the case of non-uniform frequency spacing between successive frequency indices k, the formula may be modified accordingly.

Alternatively, as only the argument/phase of X[k] is used, the magnitude of this value may be omitted from the calculation, only using the argument of X[k].

Then, the transformation of block5amay be defined as
Hprelim′[k]=Hprelim[k]e−(k−1)Δϕave[k]
Hest′[k]=Hest[k]e−j(k−1)Δϕave[k]

This results in a down-mixing, removing an overlaid inherent phase advance between adjacent frequency indices. Thus, each such representation Hsqrt′[k] and Hest′[k] of a respective quantity Hsqrt[k] and Hest[k] is a frequency-dependent transformation compensating for an inherent phase advance between adjacent frequencies.

FIG.3Cshows the transformed Hprelim′[k]=Hsqrt′[k] based on the Hprelim[k]=Hsqrt[k] ofFIG.3A. As can be seen, the inherent overlaid phase rotation between successive frequency indices is removed. This increases the likelihood of an accurate predicted estimate at block5b. The predicted estimate at block5bmay either be an adjacent estimate according to the above, or an extrapolation as will be described below.

As mentioned above, the predicted estimate of block5b, for a frequency index k, may be an extrapolation from two or more estimates for two or more respective adjacent frequencies to the frequency. Extrapolation may be performed using estimates Hprelim′[k] as transformed by the transformation described above, or directly on the preliminary estimates Hprelim[k], i.e., with Hprelim′[k]=Hprelim[k].

To simplify notation, define a vector
h′=[Hprelim′[0],Hprelim′[1], . . . ,Hprelim′[Kƒ]],and let hk′ denote the kth element of h′, and ha:b′ a sub-vector of h′ starting from the ath element and ending, and including the bth element, in that specific order.

Extrapolation may be performed at an order M, where M signifies the number of adjacent points used for the extrapolation, where M=2, 3, 4 . . . .

An extrapolation function ƒ(x) may be defined, where x signifies an M-dimensional vector of complex values from which the extrapolations should be performed.

For example, ƒ(x) may be a linear extrapolation function, corresponding to order M=2. Such a function may be written
ƒ([ab])=b+(b−a)=2b−aand will be used below.

As another example, ƒ(x) may be a cubic extrapolation function, corresponding to M=3.

The predicted estimate of block5b(see above) may then be calculated as
Hpred′[k]=Hpred,L′[k]=ƒ(hk−M:k−1′)and the first metric distance d1and the second metric distance d2calculated at block6as described above.

Here, the two or more respective adjacent frequencies to the frequency are lower than the frequency corresponding to frequency index k, as signified by the sub-script L.

Additionally, extrapolation may be double-sided, both from below and from above in frequency. Then, a second predicted estimate of the representation of the value proportional to the one-way frequency domain channel response at frequency index k−1 may be calculated as
Hpred,H′[k−1]=ƒ(hk+M−1:−1:k′)where “:−1:” indicates a reversal of the elements of the sub-vector. Thus, Hpred,H′[k−1] is based on an extrapolation from one or more estimates for frequency index k and one or more respective frequency indices k+1, k+2, . . . , corresponding to frequencies adjacent to and higher than the frequency corresponding to frequency index k, as signified by the sub-script H.

Similar to the first metric distance dLand the second metric distance dL′, a third metric distance dHmay be calculated, still at block6, between the second predicted estimate Hpred,H′[k−1] and the representation Hprelim[′k−1] of the preliminary estimate as
dH=∥Hprelim′[k−1]−Hpred,H′[k−1]∥and a fourth metric distance dH′ may be calculated between the second predicted estimate Hpred,H′[k−1] and a phase reversal −Hprelim′[k−1] of the representation of the preliminary estimate as
dH=∥−Hprelim′[k−1]−Hpred,H′[k−1]∥=∥Hprelim′[k−1]+Hpred,H′[k−1]∥,where the condition dH′<dHindicates an incorrect sign in the preliminary estimate Hprelim[k], as it indicates a sign flip between Hprelim[k−1] and Hprelim[k].

The information from comparing, respectively, dLand dL′, and dHand dH′ may be combined. Thus, for example, at block8, the final estimation may be
Hest[k]=−Hprelim[k]if and only if dL′<dLand dH′<dH, and
Hest[k]=Hprelim[k]otherwise.

Thus, in this case of double-sided extrapolation, the final estimate Hest[k] of the value proportional to the one-way frequency domain channel response is based on comparison of the first metric distance dL, the second metric distance dL′, the third metric distance dH, and the fourth metric distance dH′.

The functioning of the extrapolation from both below and above in frequency may be better understood with reference toFIGS.3B,3C, and4, illustrating a limitation of reconstruction without extrapolation.

With reference toFIG.3B, even though the jump at frequency index26is corrected successfully, frequency index7was mistakenly phase reversed, as is evident from the apparent discontinuity of the otherwise smooth progression of the direction of the tangent of the resulting curve, due to that points small magnitude compared to its distance to the point corresponding to the previous frequency index6. Such a phenomenon may for example occur when experiencing deep fading at certain frequencies, due to multipath propagation.FIG.3Cshows that this discontinuity is present as well, in this example, after the transformation to compensate for the inherent phase advance.

FIG.4is a close-up look of the area around the point corresponding to frequency index7inFIG.3Cand illustrates linear extrapolation according to the above for frequency index k=7.

Shown with plus symbols and frequency indices are the transformed representations Hprelim′[k] of the preliminary estimates Hprelim[k].

A phase reversal −Hprelim′[7] (corresponding to reflection in the origin) of the representation of the preliminary estimate Hprelim′[7] for frequency index7is marked7A. In the same way, a phase reversal −Hprelim′[6] of the representation of the representation of the preliminary estimate Hprelim′[6] for frequency index6is marked6A.

Further, shown with an unfilled (white) circle is a prediction Hpred,L′[7] based on a linear extrapolation from lower frequencies, viz., from preliminary estimates Hprelim′[5] and Hprelim′[6].

Metric distances in the complex plane dL—between Hprelim[7] (“7”) and Hpred,L′[7] (unfilled circle)—and dL′ between −Hprelim′[7] (“7A”) and Hpred,L′[7] (unfilled circle) are further shown.

Since, in this example, dL′<dL, the phase-inversed preliminary estimate −Hprelim[7] becomes the final estimate, i.e., Hest[7]=−Hprelim[7].

Further, extrapolation from higher frequency is shown. The prediction Hpred,H′[6] from a linear extrapolation from preliminary estimates Hprelim′[8] and Hprelim[7] is shown with a filled (black circle).

Metric distances in the complex plane dH—between Hprelim′[6] (“6”) and Hpred,H′[6] (filled circle)—and dH′ between −Hprelim[6] (“6B”) and Hpred,H′[6] (filled circle) are further shown.

Thus, in this example, dH′<dH. This further indicated that a phase reversal has occurred for the preliminary estimate between frequency indices6and7, indicating the need to for a phase reversal in the final estimate for frequency index7in compensation.

Thus, with both dL′<dLand dH′<dH, also when performing double-sided extrapolation, Hest[7]=−Hprelim[7].

Finally, in block10ofFIG.2, the distance between the first and the second radio signal transceiver may be determined based on the final estimates. For example, an inverse fast Fourier transform (IFFT) can be used but also more advanced signal processing techniques typically referred to as super-resolution algorithms, as described in, e.g.,Schmidt: IEEE Transactions on Antennas and Propagation, Vol AP-34, No. 3, pp. 276-280, March 1986;Sakar: IEEE Antennas and Propagation Magazine, Vol. 37, No. 1, pp. 48-55, February 1995; andRoy: IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 37, No. 7, July 1989.

The reconstructed one-way frequency-domain channel response H[k] allows most ranging algorithms to mitigate more interference from multipath, as the order of the problem/number of components is reduced. In the presence of multipath, the number of components interfering with the estimation of the delay of the line-of-sight (LOS) component will be reduced and ranging and localization will be more accurate.

A computer program product comprising a computer-readable medium may store computer-readable instructions such that when executed on a processing unit the computer program product will cause the processing unit to perform the method according to the above.

The method may be performed in a processing unit, which may be arranged in a device A, B or C as discussed above.

The processing unit may be implemented in hardware, or as any combination of software and hardware. At least part of the functionality of the processing unit may, for instance, be implemented as software being executed on a general-purpose computer. The system may thus comprise one or more processing units, such as a central processing unit (CPU), which may execute the instructions of one or more computer programs in order to implement various desired functionality.

The processing unit may alternatively be implemented as firmware arranged e.g. in an embedded system, or as a specifically designed processing unit, such as an Application-Specific Integrated Circuit (ASIC) or a Field-Programmable Gate Array (FPGA).

The correlation properties for the one-way frequency domain channel response for adjacent frequencies will naturally vary depending on the exact environment. There may be a range of frequency step sizes where the method disclosed herein works increasingly well as the frequency stepping is reduced, but where no hard upper limit of applicability can be defined.

The concept of coherence bandwidth is a statistical measurement that is approximately the maximum frequency interval over which two signals at two frequencies experience correlated amplitude fading. An approximation of the coherence bandwidth over which the amplitude correlation is lower than 0.5 is

Bc=15⁢σTmwhere σTmis the root mean square (RMS) delay spread (Goldsmith, Andrea. Wireless communications. Cambridge university press, 2005.)

According to a measurement campaign at 2.4 GHz in a 7.8 m-by-10 m room where benches and laboratory equipment scatter around, the RMS delay spread ranges from 20 ns to 30 ns depending on the relative distance between Tx and Rx (Zepernick, H. J., & Wysocki, T. A. Multipath channel parameters for the indoor radio at 2.4 GHz ISM band. In 1999 IEEE 49th Vehicular Technology Conference, May 1999, Vol. 1, pp. 190-193). This indicates a coherence bandwidth in such an environment be around 8 MHz (with σTm=25 ns).

A typical ranging and direction-finding system may operate in the ISM band with a frequency step in the order of 1 MHz. According to the above, this this frequency step is small enough that the one-way frequency domain channel responses are correlated.

Additionally, the phase of frequency responses that are at different frequencies are correlated as well, because the slope of the phase along frequency is proportional to the distance between the initiator and reflector in a line-of-sight (LOS) channel. The amplitude coherency and phase coherency across frequency samples of the wireless channel when observed at every 1 MHz ensures that the complex frequency response rotates progressively rather than arbitrarily.

In the following, results validating the channel reconstruction of the one-way frequency domain channel response according to the present disclosure, including the transformation and double-sided linear extrapolation as detailed above.

Multi-path channels were generated using ray tracing. In the ray tracer, a 11 m-by-7 m meeting room without furniture was defined, as illustrated inFIG.5. An initiator was put in one of the corners while the reflector was moved around the room at 112 distinct locations forming the shape of an Arabic numeral 8. The scene is shown inFIG.5. The simulation contained up to 3rd order reflections. Random phase offsets were selected uniformly randomly from 0 to 2π radians and added to the channel response each time the carrier frequency was switched. The phase offset was kept constant when switching the direction of transmission. White Gaussian noise was added such that the SNR per antenna was 20 dB.

An indicator of reconstruction reliability was defined in the form of a reconstructed-channel quality indicator Q, describing the similarity between the reconstructed channel and the actual channel. It is defined by:

Q=❘"\[LeftBracketingBar]"he⁢s⁢tH⁢hhH⁢h⁢he⁢s⁢tH⁢he⁢s⁢t❘"\[RightBracketingBar]"where hestand h are column vectors, which are the estimated channel and the actual channel vectors, respectively, and H denotes the operation of conjugate transpose. The closer Q is to 1, the better the reconstruction matches the actual channel.

FIGS.6A-6Drespectively show the quality indicator Q for Rician-K values 10, 2, 5, and 1. The higher the Rician-K value, the stronger the line-of-sight component is with respect to multipath propagation. Each of the figures shows the Q value for a number of simulated radio channels as described above.

One can see that for high K values, the channel reconstruction is flawless. If the Rician K value reduces, performance of the algorithm somewhat reduces, while still a good reconstruction is realized for a significant part of the channels.

In the above, embodiments have mainly been described with reference to a limited number of examples. However, as is readily appreciated, other examples than the ones disclosed above are equally possible within the scope of the disclosure.

For example, the extrapolation can be done linearly, as described throughout, or in more sophisticated manners, e.g. using higher-order, non-linear extrapolation.

Instead of calculating an average phase advance transformation compensating for the inherent phase advance between adjacent frequencies may be provided through a tracking algorithm.