JP6897336B2 - Signal processing device - Google Patents

Description

The present invention relates to a signal processing device that corrects a power spectrum of a pulse train obtained from an input signal such as music, and the corrected pulse signal can be exerted on a living body as an electrical stimulus by outputting the corrected pulse signal to, for example, a pair of electrodes. it can.

Conventionally, as a method of relieving pain by electrical stimulation, a pulsed current is passed from a peripheral nerve or an electrode placed near the posterior cord of the spinal cord, and the nerve group of the inhibitory system existing in the spinal cord gland and the upper central nervous system is used. A method of activation has been used.

Further, for the purpose of sustaining the effect of stimulation for a long time, a method of irregularly changing the pulse frequency with time has also been proposed (see Patent Document 1 below). In addition, the change in pulse frequency can be felt as a moderately fresh and pleasant stimulus without giving an excessive abrupt sensation to the living body when the fluctuation power spectrum follows the so-called "1 / f fluctuation rule". Has been known.

By the way, as one method of generating the change of the pulse train according to this 1 / f fluctuation rule, the temporal change of the number of zero crosses at a fixed time interval representing the frequency fluctuation of the classic music known as a masterpiece is divided by a frequency divider circuit or the like. Methods such as converting to a frequency suitable for percutaneous electrical stimulation are used, and in particular, by listening to the music that is the source of the pulse train extraction through headphones or speakers, the relaxing effect received from hearing can be expected. , A technique for adjusting the temporal relationship between the audio signal and the pulse train extracted from the audio signal has also been proposed (see Patent Document 2 below).

However, since each person has different tastes for music, in order to maximize the relaxing effect of the listener, even if the frequency fluctuation of the music does not follow the so-called "1 / f fluctuation rule", it is listened to. There was a dilemma in music selection that it was preferable to prioritize the preference of the person. In fact, in some music that intentionally aimed at a random stimulus effect, the frequency fluctuation is close to white noise, and when the pulse train generated from this is given to the living body as an electrical stimulus, it is excessively abrupt. It gives a feeling and may induce a feeling of fear in some cases, so improvement is desired. However, even the lowest-order low-pass filter normally used for the purpose of removing fluctuations in the high frequency range has 1 / f ² characteristics, and it is not always easy to output 1 / f fluctuation characteristics. Absent.

Japanese Unexamined Patent Publication No. 54-1792 Jikkai Sho 62-74442

T. Musha, H. Takeuchi and T. Inoue, 1 / f Fluctuations in the Spontaneous Spike Discharge Intervals of a Giant Snail Neuron, IEEE Trans. Biomed.Eng., Vol.30, no.3, pp.194-197 (1983) ) T. Musha, Y. Kosugi, G. Matumoto and M. Suzuki: Modulation of thetime Relation of Action Potential Impulses Propagating Along an Axon, IEEETrans. Biomed. Eng., Vol.28, no.9, pp.616-623 ( 1981)

An object of the present invention is to provide a means for correcting the power spectrum characteristics so as to be preferable when the power spectrum of the frequency fluctuation of the pulse train obtained from the input signal deviates significantly from the desired characteristics.

For example, it has been pointed out that stimuli with a frequency fluctuation power spectrum of 1 / f-like characteristics have a high affinity with living organisms, but such a tendency tends to occur in human heart rate variability and brain waves in a resting state. It has been pointed out that not only the α wave component is observed, but also the activity of lower-order nervous systems such as the nervous system of Giant African Snail is observed (see Non-Patent Document 1 above). Furthermore, the peripheral nerves themselves may also have a mechanism that favors the propagation of stimuli with such 1 / f-like properties, and that of Yariika, one of the most primitive nerves. It has been found that a pulse train propagating a giant axon also exhibits such a 1 / f-like characteristic after propagating a certain length (see Non-Patent Document 2 above). The cause of this is that the fatigue of the preceding propagating pulse on the axon activity does not recover immediately, but affects the propagation speed of the subsequent pulse, and the two pulses travel at extremely short intervals at the input section. The propagation velocity of the pulse on the rear side of the pulse is reduced, and the interval between the two pulses is widened. On the contrary, the rear pulse of the two pulses applied at a wide interval at the time of input advances while maintaining the distance from the preceding pulse, and the front pulse of the two pulses is larger than that. If the speed is slowed down by the influence of the previous pulse, it acts to close the interval between the two pulses.

Conventionally, in this type of analysis, an attempt has been made to explain the velocity reduction depending only on the interval from one preceding pulse, but the phenomenon observed in the living body has not been explained. That is, in the actual squid giant nerve axon, white noise is applied and propagated to observe 1 / f-like characteristics, whereas in computer simulation, extreme density modulation occurs and the spectrum. The slope of 1 / f became extremely steep and deviated significantly from the 1 / f-like characteristics (see FIG. 5 (a)). The inventors are based on the recognition that axon fatigue in the living body is not actually determined only by the interval with the preceding pulse, but is influenced by the accumulation of multiple pulses further preceding it. By expressing this effect as a parameter in the form of "average frequency", the behavior of the pulse train propagating in the nerve axon is modeled more accurately, and as a result, the power spectrum characteristic of the frequency fluctuation obtained from the density modulation of the pulse train is obtained. We have found that the desired characteristics, for example, 1 / f-like characteristics, can be brought closer to each other, and have completed the present invention.

That is, in the present invention, the pulse interval of the pulse train obtained from the music input signal is sequentially updated in chronological order, the power spectrum of the pulse train is corrected as a pulse train having characteristics close to 1 / f-like characteristics, and the corrected pulse signal is used as a living body. A signal processing device that outputs for a stimulator
In at least three pulses that are adjacent in time series in the pulse train, the weighted average frequency is calculated by reducing the weight of the leading pulse from the last row pulse and at least two leading pulses preceding the last row pulse. The average frequency is calculated, and the calculated average frequency is converted into the propagation speed by using the relationship between the average frequency and the propagation speed in a predetermined pulse propagation model that simulates a pulse propagating on the nerve axon. performs processing for obtaining the time required for the pulse propagation model when a pulse is propagated by a predetermined distance in the propagation velocity as the delay time, the last by the Turkey be added to the delay time at the time of the last row pulse It is characterized in that it is configured to update the time of a row pulse.

In this way, in at least three pulses that are adjacent in time series in the pulse train, the average frequency is calculated from the last row pulse and at least two preceding pulses preceding the last row pulse, and the calculated average frequency is obtained. By performing the process of obtaining the corresponding delay time and adding the delay time to the last row pulse, the nerve fatigue effect associated with the propagation of multiple preceding pulses can be expressed as the delay time according to the average frequency. By appropriately adjusting the delay time according to this average frequency as one of the parameters, it becomes possible to design the power spectrum of the output pulse train in a shape that suits the purpose, and the pulse train extracted from music as an input signal. For example, even if the deviation deviates from the 1 / f-like characteristic, the deviation can be corrected and used for biological stimulation as a pulse train close to the 1 / f-like characteristic.

The signal processing apparatus of the present invention is preferably configured to update the time of the last line pulse by performing the process recursively a plurality of times. In updating the pulse time, it is possible to reflect the speed that can change from moment to moment during the propagation of the last row pulse, and the power spectrum of the output pulse train can be made closer to the desired shape.

Further, in the signal processing apparatus of the present invention, the average frequency is calculated from the last-row pulse and at least two leading pulses preceding the last-row pulse by reducing the weighting of the leading pulse. The weighted average frequency . In this way, it is possible to reflect the phenomenon that the effect of the delay on the last pulse due to nerve fatigue becomes smaller as the pulse precedes, and the power spectrum of the output pulse train can be made closer to the desired shape. ..

Further, in the signal processing apparatus of the present invention, it is preferable that the last row pulse is erased from the pulse train when the average frequency exceeds a predetermined value. It is possible to reflect the phenomenon that the active potential is not excited when two current pulses are applied to the nerve axon (when the frequency is high) at short time intervals, and the purpose is to power spectrum of the output pulse train. It can be closer to the shape that fits.

Further, in the signal processing apparatus of the present invention, the delay time according to the average frequency is calculated by using a propagation model of a pulse propagating on a certain nerve axon. As the pulse propagation model, for example, a pulse frequency propagation velocity characteristic diagram modeled from a value measured from an existing squid can be used. More specifically, the pulse propagation model preferably includes a characteristic that the delay time increases as the frequency increases. For example, in the frequency-propagation velocity characteristic diagram (see FIG. 4) of a pulse modeled on a giant nerve axon of a real spear squid, a sparse pulse having a frequency less than a certain value c is propagated at a maximum velocity v (c). After that, the velocity decreases linearly with the increase of the frequency up to a certain constant value b, and when it exceeds the certain constant value b, it is not propagated at all (the pulse disappears).

According to the present invention, when the power spectrum of the frequency fluctuation of the pulse train obtained from the input signal deviates significantly from the desired characteristic, it is possible to provide a means for correcting the power spectrum characteristic so as to have a preferable shape.

It is a block block diagram of the signal processing apparatus of one Embodiment of this invention. It is a calculation flowchart which is executed sequentially for each pulse of a pulse train in a pulse propagation simulation part. It is a schematic diagram about each time interval of the last line pulse and three preceding pulses read into a pulse propagation simulation part. It is a frequency-propagation velocity characteristic diagram of a pulse that models a giant squid nerve axon. (A) is an output when white noise having a Gaussian distribution is applied to a pulse propagation simulation unit of a comparative example having the same configuration as that of the embodiment except that the frequency is obtained only from the interval with one preceding pulse. (B) is the power spectrum in the unit, and (b) is the power in the output unit when white noise having a Gaussian distribution is applied to the pulse propagation simulation unit of the embodiment using the average frequency obtained from the interval between the two preceding pulses. It is a spectrum. In each figure, the horizontal axis represents frequency and the vertical axis represents power. It is a figure which showed the power power spectrum before and after processing the pulse train generated by Gaussian distribution and the pulse train generated from the actual music data (NAT project PR video prelude) by the signal processing apparatus of embodiment, (a) is a figure which showed. , The power spectrum of the initial pulse generated by the average pulse interval of 8 ms and the standard deviation of the pulse interval of 2 ms. In (b), the pulse train of (a) was passed through the pulse propagation simulation unit of the present invention by 0.5 m. The latter power spectrum, (c) is the power spectrum of the initial pulse train extracted by thinning out the zero cross pulse at the beginning of the above-mentioned music data into pulses every 10 times, and (d) is (c). ) Is the power spectrum after passing the pulse train of 0.5 m through the pulse propagation simulation unit of the present invention. In each figure, the horizontal axis represents frequency and the vertical axis represents power. (A) to (d) are time-series data of pulse intervals corresponding to FIGS. 6 (a) to 6 (d), the horizontal axis represents time (seconds), and the vertical axis represents pulse intervals (m). Gaussian pulse sequence generated by the distribution Upon pulse propagation simulation (average pulse interval 8 ms, the standard deviation 2ms pulse interval), changing the number of terms of the weighted mean frequency f _i to determine when the average frequency of the (inverse of the pulse interval) a view showing the average of 1000 times of the processing result when, (a) represents, calculated a weighted mean frequency f _i by using a pulse of up to two before the last line pulse to be time update, the pulse time update process the represents the average of the power spectrum in the case of performing, (b) obtains a weighted mean frequency f _i by using a pulse of up to three before the last line pulse to be time update, performing a pulse time update process The average of the power spectra in the case of In each figure, the horizontal axis represents frequency and the vertical axis represents power.

Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings. As shown in FIG. 1, the signal processing device 10 of the embodiment includes a signal acquisition unit 12, a monaural conversion unit 14, a silent section removal unit 16, a pulse generation unit 18, a pulse section reduction unit 20, and pulse thinning. The adjustment unit 22 and the pulse propagation simulation unit 24 are provided, and each of these units 12, 14, 16, 18, 20, 22, and 24 is executed by the CPU (Central Processing Unit) reading a program from the ROM. It may be executed by an integrated circuit such as FPGA (Field Programmable Gate Array) or ASIC (Application Specific Integrated Circuit).

The signal acquisition unit 12 acquires voice data such as music as a signal to be processed.

The monaural conversion unit 14 converts the audio data acquired by the signal acquisition unit 12 into a monaural waveform signal.

The silent section removing unit 16 detects and removes a silent section from the waveform signal converted to monaural. The silent section is a section in which the acquired voice data remains silent for a predetermined time. The silence section removing unit 16 determines as a silence section and removes the section when the magnitude of the waveform signal is equal to or less than the silence determination threshold value (for example, −60 dB) for, for example, 3 seconds.

The pulse generation unit 18 generates a pulse train synchronized with the time of zero crossing of the waveform signal.

The pulse section reduction unit 20 detects a section having a large pulse interval from the generated pulse train and reduces the pulse interval to a predetermined interval. The pulse section reduction unit 20 reduces the pulse interval of, for example, 20 ms or more to 20 ms.

Since there are many dense sections in which the pulse interval is 1 ms or less in the pulse train generated from general music data, the pulse thinning adjustment unit 22 thins out the pulses so that, for example, 10 pulses become 1 pulse in the dense section. Do.

The pulse trains adjusted at appropriate intervals in this way are input to the pulse propagation simulation unit 24. The pulse propagation simulation unit 24 simulates and outputs a change in each pulse interval when each pulse of the pulse train propagates on, for example, a giant nerve axon of a spear squid. Specifically, first, the spear squid A pulse propagation model is created from the giant nerve axon, etc. of the above, and the time required for the pulse to propagate by a predetermined total distance X using this pulse propagation model is added to the time of the original pulse as the delay time and updated. As a result, each pulse interval of the input pulse train is sequentially adjusted.

FIG. 2 is a calculation flowchart in which the pulse propagation simulation unit 24 sequentially executes each pulse in the pulse train. Pulse Propagation simulation unit 24 in step S1, time update to be pulses (hereinafter also referred to as the last row pulse.) And the time t _i of pulses of at least two preceding outermost trailing pulse (in Figure 2 three) (Hereinafter, also referred to as a preceding pulse.) Time ... Reads ti _-3 , ti _-2 , and ti _-1 . Figure 2 shows an example of reading the last time the row pulse _{t i} and three leading pulse time _{t i-3 of, t i-2, t i-1,} and the time _{t i} of the last row pulse time _t i-2 of the two preceding _{pulses, t i-1} and may be those read, of the trailing pulse time _{t i} and four more leading pulses the time _{· · · t i-4, t i-3} , T _i-2 , and t _i-1 may be read.

In step S2, the pulse propagation simulation unit 24, the read time _{··· t i-3, t i} -2, t i-1, each from the _{t i} pulse interval _{··· τ i-2, τ i} -1 , Τ _i is calculated. FIG. 3 is a schematic diagram showing a case where three pulse intervals are obtained from the last row pulse and the three preceding pulses. Pulse interval _{··· τ i-2, τ i} -1, τ i is obtained by subtracting the pulse time of the preceding pulse time of the trailing _{(e.g., τ i = t i -t i} -1).

In step S3, the pulse propagation simulation unit 24 calculates an average frequency f _i from the read pulse in step S1. In the embodiment, as shown in the following equation, _{the inverse values of the pulse intervals τ i-2} , τ _i-1 , and τ _i are multiplied by a coefficient that decreases by a constant rate a according to the depth of the time series history of nerve fatigue. linear combination Te, and calculating a weighted mean frequency f _i.
f _i = {1 / τ _i + a (1 / τ _i-1 ) + a ² (1 / τ _i-2 ) + ・ ・ ・} / (1 + a + a ² + ・ ・ ・)
Note that a is an arbitrary number greater than 0 and less than 1, and can be, for example, a = 1 / e = about 0.367. e is the base of the natural logarithm. As a, a numerical value larger than 1 / e (for example, 0.4, 0.5, 0.6, etc.) or a numerical value smaller than 1 / e (0.3, 0.2, 0.1, etc.) may be adopted.

In step S4, it converts the propagation velocity v _i when the pulse of the weighted mean frequency f _i obtained in step S2 is propagating on nerve axons. FIG. 4 is a conversion model between the pulse frequency and the propagation velocity, which is modeled from the values measured from the giant squid giant nerve axons. As shown in this graph, in this model, dense pulses whose frequency exceeds a certain value b are not propagated on the nerve axon and disappear. Further, a sparse pulse whose frequency is below a certain value c is propagated at the maximum velocity v (c). Then, a pulse having an intermediate frequency of a constant value c or more and a constant value b or less has a relationship in which the propagation velocity linearly decreases as the frequency increases. Therefore, in step S4, when the weighted mean frequency f _i exceeds a predetermined value b, after removal of the pulse at time t _i in step S5, the process returns to step S1, a time t _{i + 1} of the next pulse, to time of the preceding pulse and t _i-3, reads the _{t i-2, t i-} 1, calculates the weighted mean frequency f _i and a pulse interval and the preceding pulse interval according to the following pulse, of Fig. 4 converting the weighted mean frequency _{f i + 1} to the propagation velocity _{v i + 1} from the model.

In step S6, the propagation velocity v _i obtained in step S4, the last row pulse to calculate the time Delta] t _i required on nerve axons to being small distance Δx propagation. The minute distance Δx can be a value obtained by dividing the total distance X by an arbitrary natural number. As an example, when the total distance X is 0.5 m and the number of repetitions of steps S2 to S8 is 10 times, the minute distance Δx becomes 0.05 m.

In step S7, determine a new time t _i of the last row pulse by adding the time Delta] t _i obtained in step S6 at time t _i of the last row pulse.

At the next step S8, it is determined whether the integrated value of a minute distance Δx reaches total distance X, the process returns to step S2, in the case of "NO", the same calculation using the new time t _i Repeat. On the other hand, if "YES" in step S9, and updates the initial time t _i of the last row pulse in the new time t _i obtained in step S7. The pulse train whose time is sequentially updated in this way and whose pulse interval is adjusted is output to a biostimulator such as a percutaneous mounting electrode or a magnetic generation coil via a pulse shaping circuit or a power amplification circuit, or to a storage device (not shown). Can be memorized.

According to the above processing system, the equivalent propagation velocity of a pulse traveling the pulse propagation simulation unit 24 is a progressive filter (pulse time t _i) is immediately before the applied pulse (time t _i-1 of pulse) not only can be expressed by a parameter called plurality preceding pulse (time _{t i-3, t i-} 2, t i-1 pulse) weighted mean frequency f _i fatigue effects associated with the progress of the By adjusting this parameter appropriately, it becomes possible to design the power spectrum of the output pulse train in a shape that suits the purpose. For example, when the pulse train extracted from music deviates from the 1 / f-like characteristics, for example. It can also be used for biological stimulation as a series in which the deviation is corrected. FIG. 5B shows the power spectrum of the pulse train in the output unit when white noise having a Gaussian distribution is applied to the pulse propagation simulation unit 24, and shows 1 / f and characteristics (1 / f) by the weighted average frequency method. You can see that it is approaching (dotted line).

Further, in the signal processing device 10 of the present embodiment, the process of obtaining the time required for the pulse to be propagated by the total distance X is performed recursively a plurality of times by dividing the pulse into a minute distance Δx. Upon updating the time t _i of the last row pulse, it is possible to reflect the rate that can constantly change during propagation of the last row pulse, the power spectrum of the output pulse train can be made closer to the shape suitable for the purpose.

Further, in the signal processing apparatus 10 of the present embodiment, as the average frequency f _i, and at least two leading pulse preceding the last row pulse and outermost trailing pulse, to reduce the weight as the preceding pulse calculated by using the weighted mean frequency f _i that is, as the preceding pulse, it is possible to reflect the phenomenon that the influence of propagation delay to the last row pulse caused by nerve fatigue is reduced, matching the purpose of the power spectrum of the output pulse train It can be closer to the shape.

Further, in the signal processing apparatus 10 of the present embodiment, when the average frequency f _i of the last row pulse exceeds a predetermined value b, since outermost trailing pulse is configured to be erased from the pulse train , When two current pulses are applied to the nerve axon at short time intervals (when the frequency is high), the phenomenon that the active potential is not excited can be reflected, and the purpose is the power spectrum of the output pulse train. You can get closer to the shape that fits.

FIG. 6 is a diagram showing frequency characteristics before and after processing the pulse train generated by the Gaussian distribution and the pulse train generated from the actual music data (NAT project PR moving image prelude) by the signal processing device 10 of the embodiment. FIG. 6A shows the frequency characteristics of the initial pulse generated by the average pulse interval of 8 ms and the standard deviation of the pulse interval of 2 ms, and shows characteristics close to white noise. On the other hand, FIG. 6C shows the frequency characteristics of the initial pulse train extracted by thinning out the zero cross pulse at the beginning of the above-mentioned music data into pulses every 10 times, and is a pulse train having a slight inclination than white noise. is there.

The pulse trains of FIGS. 6 (a) and 6 (c) having differences in these initial characteristics are passed through the pulse propagation simulation unit 24 as a propagation filter, which is modeled on the pulse propagation characteristics of the giant squid nerve axon. The characteristics change as shown in 6 (b) and (d). In the pulse propagation simulation unit 24, the weighted mean frequency f _i is used and the pulse interval tau _i of the last row pulse, and a pulse interval tau _i-1 of the _{previous, f i = {1 / τ} i + a It was obtained from (1 / τ _i-1 )} / (1 + a). Here, a = 1 / e = 0.367, and the sample was removed as a sample exceeding the maximum frequency b (about 260 Hz) shown in FIG. Further, the total distance X through which the pulse propagates was set to 0.5 m, the minute distance Δx was set to 0.05 m, and steps S2 to S8 in the calculation flowchart of FIG. 2 were repeated 10 times to update the pulse time. As a result of passing through such a pulse propagation simulation unit 24, as shown in FIGS. 6 (b) and 6 (d), the ratio of the reciprocal of the frequency and the power becomes almost 1, and the original pulse deviates from the 1 / f characteristic. Even if it is, it is possible to produce 1 / f characteristics. For reference, FIG. 7 shows a part of the time-series data of the pulse interval corresponding to FIG.

Figure 8 is a section of the pulse train generated by Gaussian Upon pulse time update process (average pulse interval 8 ms, the standard deviation 2ms pulse interval), (reciprocal of the pulse spacing) average frequency for obtaining the weighted mean frequency f _i a view showing the average of 1000 times the processing result obtained by changing the number, FIG. 8 (a), the weighted mean frequency f _i by using a pulse of up to two before the last line pulse to be updated calculated shows the frequency characteristic in the case of performing pulse time update process, FIG. 8 (b), obtains a weighted mean frequency f _i by using a pulse of up to three before the last line pulse to be updated, the pulse The frequency characteristics when the time update processing is performed are shown. In the case of using the weighted mean frequency f _i obtained from the pulse up to two before the last line pulse to be updated, a characteristic close to the power spectrum 1 / f of the pulse train as seen from FIG. 8 (a) It is in a high frequency region of what is while its characteristics are lost, in case of using the weighted mean frequency f _i obtained from the pulse up to three before the last line pulse to be updated, 8 As can be seen from (b), a power spectrum closer to the 1 / f characteristic is obtained including the high frequency region.

Although the present invention has been described above based on the illustrated examples, the present invention is not limited to the embodiments, and changes, modifications, and additions can be made within the scope of the claims .

Thus, according to the signal processing apparatus of the present invention, when the power spectrum of the frequency fluctuation of the pulse train obtained from the input signal deviates significantly from the desired characteristic, the power spectrum characteristic can be corrected to a preferable shape. Is.

10 Signal processing device 12 Signal acquisition unit 14 Monaural conversion unit 16 Silent section removal unit 18 Pulse generation unit 20 Pulse section reduction unit 22 Pulse thinning adjustment unit 24 Pulse propagation simulation unit

Claims (4)

The pulse interval of the pulse train obtained from the music input signal is sequentially updated in chronological order, the power spectrum of the pulse train is corrected as a pulse train having characteristics close to 1 / f, and the corrected pulse signal is output for the biostimulator. It is a signal processing device that
In at least three pulses that are adjacent in time series in the pulse train, the weighted average frequency is calculated by reducing the weight of the leading pulse from the last row pulse and at least two leading pulses preceding the last row pulse. The average frequency is calculated, and the calculated average frequency is converted into the propagation speed by using the relationship between the average frequency and the propagation speed in a predetermined pulse propagation model that simulates a pulse propagating on the nerve axon. performs processing for obtaining the time required for the pulse propagation model when a pulse is propagated by a predetermined distance in the propagation velocity as the delay time, the last by the Turkey be added to the delay time at the time of the last row pulse A signal processing device characterized in that it is configured to update the time of a row pulse. The signal processing apparatus according to claim 1, wherein the time of the last line passle is updated by performing the process recursively a plurality of times. The signal processing apparatus according to claim 1 or 2 , wherein when the average frequency exceeds a predetermined value, the last row pulse is eliminated from the pulse train. The signal processing apparatus according to any one of claims 1 to 3, wherein the pulse propagation model includes a characteristic that a delay time increases as a frequency increases.

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