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Velocity Measurements Using a Single Transmitted Linear Frequency-Modulated Chirp
Ultrasound in Med. & Biol., Vol. 33, No. 5, pp. 768 –773, 2007 Copyright © 2007 World Federation for Ultrasound in Medicine & Biology Printed in the USA. All rights reserved 0301-5629/07/$–see front matter
doi:10.1016/j.ultrasmedbio.2006.11.013
● Original Contribution VELOCITY MEASUREMENTS USING A SINGLE TRANSMITTED LINEAR FREQUENCY-MODULATED CHIRP YOAV LEVY and HAIM AZHARI Faculty of Biomedical Engineering, Technion, IIT, Haifa, Israel (Received 19 June 2006, revised 30 October 2006, in final form 7 November 2006)
Abstract—Velocity measurement is a challenge for a variety of remote sensing systems such as ultrasonic and radar scanners. However, current Doppler-based techniques require a comparatively long data acquisition time. It has been suggested to use coded signals, such as linear frequency-modulated signals (chirp), for ultrasonic velocity estimation by extracting the needed information from a set of several sequential coded pulses. In this study, a method for velocity estimation using a single linear frequency-modulated chirp transmission is presented and implemented for ultrasonic measurements. The complex cross-correlation function between the transmitted and reflected signals is initially calculated. The velocity is then calculated from the phase of the peak of the envelope of this cross-correlation function. The suggested method was verified using computer simulations and experimental measurements in an ultrasonic system. Applying linear regression to the data has yielded very good correlation (r ⴝ 0.989). With the suggested technique, higher frame rates of velocity mapping can be potentially achieved relative to current techniques. Also, the same data can be utilized for both velocity mapping and image reconstruction. (E-mail: [email protected]) © 2007 World Federation for Ultrasound in Medicine & Biology. Key Words: Velocity measurement, Coded excitation, Linear frequency modulated chirp.
resolution, by using coded excitation, the high spatial resolution can be recovered using an appropriate signal processing algorithm (e.g., matched filter (Misaridis and Jensen 2005)). It has been suggested to use coded signals, such as linear frequency-modulated signals (chirp), for ultrasonic velocity estimation by extracting the needed information from a set of several sequential coded pulses (Wilhjelm and Pedersen 1993). In this study, we present a method for velocity estimation using a single coded pulse transmission.
INTRODUCTION Measurement of velocity is a challenge for a variety of remote sensing systems such as ultrasonic and radar scanners. Commonly, the Doppler frequency shift caused by a moving reflector is measured and converted into velocity estimation. This method is well established and has been implemented using many techniques. However, current Doppler-based techniques require either the transmission of a long continuous wave, which sacrifices axial resolution, or the acquisition of echoes from several pulses to generate a velocity map of each region in the image. Therefore, both methods require a comparatively long data acquisition time, typically on the order of the period of the Doppler frequency shift. Coded excitation methodology (Misaridis and Jensen 2005) is used in ultrasonic imaging systems to improve signal-to-noise ratio (SNR). In this methodology, a long coded signal is used to transmit high energy while preserving low-intensity constraints. Although, typically, a long pulse duration leads to poor spatial
THEORY A chirp from f0 to f1 whose length is Tm can be represented by the following formula (Jensen 1996, eqn (9.20)) e(t) ⫽ sin(2 f 0t ⫹ S0t2); 0 ⱕ t ⱕ Tm
(1)
where f0 is the start frequency, f1 is the end frequency and S0 is the sweep rate of the signal
Address correspondence to: Haim Azhari D.Sc., Faculty of Biomedical Engineering, Technion IIT, Haifa 32000, Israel. E-mail: [email protected]
S0 ⫽ 768
f1 ⫺ f0 . Tm
Velocity measurements using frequency-modulated chirp ● Y. LEVY and H. AZHARI
The instantaneous frequency of the signal is (Wilhjelm and Pedersen 1993) f(t) ⫽ f 0 ⫹ S0t.
(2)
The time of appearance of each frequency t(f) is t(f) ⫽
f ⫺ f0 . S0
(3)
Other properties of the chirp signal are ⌬f ⫽ f 1 ⫺ f 0
(5)
c⫺v c⫹v
(6)
where a is a reflection coefficient (a frequency-independent reflection is assumed), c is the acoustic velocity in the medium and t⬘ is a time shift that is related to the path from the transducer to the moving target. The signal’s intensity is not important in the following discussion; therefore, we set a ⫽ 1. When c ⬎⬎ 2v c⫺v 2v ⬇1⫺ c⫹v c
(7)
2v 1 ␣⬇1⫹ . c
Ⲑ
The instantaneous frequency f ⬘(t) of the reflected signal in its corresponding coordinates (setting t ⫽ 0 for the signal front end) is given by
f ⬘ (t) ⫽ f 0 ⬘ ⫹ S0 ⬘ t where
冦
兰
2 f(t)dt.
(9)
t共 f Ⲑ ␣兲
For cases in which the parameter ␣ is approximately unity (␣ ⬃ 1), the frequency can be taken as constant over the integration range: f(t) ⬇ f.
(10)
Therefore, using the approximation in eqn (10) and using eqn (3), the integral in eqn (9) can be solved:
rs(t) ⫽ a · e(␣t ⫺ t ⬘ )
␣⫽
t(f)
⌬(f) ⫽
where ⌬f is the frequency bandwidth of the chirp and fm is the center instantaneous frequency. A received signal rs(t) from a moving reflector with a velocity v along the beam axis can be represented by (Jensen 1996, eqn (9.21))
␣⫽
whose instantaneous frequency is f/␣. The phase gap between the phase of the point whose instantaneous frequency is f in the transmitted signal and the point in the received signal that has the same instantaneous frequency (⌬(f)) is equal to the phase gap between the points for which the instantaneous frequencies are f and f/␣ in the transmitted signal (within the range where the transmitted and received bands overlap). The phase gap ⌬(f) can be calculated by the expression:
(4)
f1 ⫹ f0 fm ⫽ 2
769
f0 ⬘ ⫽ ␣f0 f1 ⬘ ⫽ ␣f1 S0 ⬘ ⫽ ␣2S0
(8)
Ⲑ
0 ⱕ t ⱕ Tm ␣
Consider an ultrasonic chirp signal that was reflected from a moving target. During the reflection, the instantaneous frequency of each point of the transmitted chirp is changed, according to the Doppler frequency shift, from f to ␣ f. On the other hand, the phase of each point is preserved. Therefore, the phase of the point in the received signal whose instantaneous frequency is f is equal to the phase of the point in the transmitted signal
⌬(f) ⬇ 2 f · (t(f) ⫺ t(f ⁄ ␣)) ⬇ 2 f ·
冉 冊 冉 冊
⬇ 2 f 2 ·
f ⫺ f ⁄␣ S0
1⫺1⁄␣ . S0
(11)
Substituting eqn (7) into eqn (11), the phase gap between corresponding points in the chirps for a specific instantaneous frequency f is given by the expression ⌬(f) ⬇ 2 f 2
⫺2v . S0c
(12)
Under typical physiologic blood flow conditions, the changes in f0 and S0 as a result of the reflection from the moving blood (calculated in eqn (8)) are too small for a reliable Doppler velocity estimation using a single pulse transmission (Wilhjelm and Pedersen 1993). However, as shown below, the cross-correlation function of the transmitted and received chirp signals in the time domain is sensitive to the resulting changes in the start frequency and the sweep rate and hence, can be utilized for velocity estimation using a single transmission. Consider the chirp e(t), which was defined in eqn (1). This chirp can be turned into a phase-encoded chirp e(t,) (Ha et al. 1991), where is the encoded phase e(t, ) ⫽ sin(2 f 0t ⫹ S0t2 ⫹ ).
(13)
The corresponding approximated cross-correlation between e(t,0) and e(t,) is given by Ha et al. (1991):
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Ultrasound in Medicine and Biology
R() ⬇ sinc共⌬f · 兲cos共2 f m ⫹ ).
(14)
The cross-correlation function R() is comprised of an envelope (the sinc function) and a carrier frequency equal to the center instantaneous frequency fm. The phase of the carrier frequency at the peak of the cross-correlation envelope ( ⫽ 0) is . Two additional approximations should be done to use Ha et al. approximation as a cross-correlation function between a transmitted chirp pulse and its corresponding reflected chirp signal: a. Use the original spectrum for determination of ⌬f and fm. b. Approximate the phase gap between the chirps (which is a function of the phase) by the phase gap between corresponding points in the transmitted and reflected chirps having the center instantaneous frequency fm. Those approximations are valid as long as ␣ ⬇ 1 (i.e. c Ⰷ 2v). Substituting eqn (12) into eqn (14), an approximation for the cross-correlation function between a transmitted chirp pulse and its corresponding reflected chirp signal is
冉
R() ⯝ sinc共⌬f · 兲cos 2 f m ⫹ 2 f m2
冊
2v . S0c
(15)
One can note that the cross-correlation function envelope peaks at ⫽ 0 and ⫽ 2 f m2 ⫺ 2v⁄S0c will be the peak’s phase. Let us define the phase at the peak of the crosscorrelation envelope as the “optimal-correlation phase” (OCP). This phase is measured by taking the phase of RH() at its maximal absolute value, where RH() is the analytical signal RH() ⫽ R() ⫹ iRˆ() where Rˆ() is the Hilbert transform of the cross-correlation function between the transmitted and reflected chirp waves. In case of a stationary reflector, the OCP is zero, because there is no frequency shift, i.e., v ⫽ 0. Using eqn (15), the velocity of the moving reflector along the beam can be determined from the OCP, OCP, by the expression v⫽
⫺OCPS0c . 4 f m2
(16)
Volume 33, Number 5, 2007
OCP by eqn (16) for each combination of chirp length and reflector velocity. The transmitted signals were simulated using eqn (1) and the echoes were calculated according to eqn (8). The synthetic signals were multiplied by a Hamming window to emulate a realistic situation in which the transducer’s impulse response modulates the signal. Parabolic interpolation was used to determine the accurate peak of the correlation function. Finally, a map depicting the relative error in velocity estimation, as a function of target velocity and chirp length, was generated from these simulation results. In addition, the effect of noise on velocity estimation was evaluated. SNR was varied from ⫺20 dB to ⫹20 dB by adding white Gaussian noise to the simulated data. At each SNR level, 100 simulations were conducted. The estimated velocity was normalized to the accurate value and its mean and standard deviation (SD) were calculated as a function of SNR. EXPERIMENTAL METHODS A transducer (Panametrics, GE Sensing, Billerica, MA, USA, 5 MHz, diameter of 6.3 mm) was placed in a water-bath in front of a computer-controlled moving target. The target was a stainless steel cube that could be moved at velocities of up to several cm/s, defined by the user (these values served as a “gold-standard”). Chirp signals were generated by a Tabor 8026 (Tabor Electronics, Tel Hanan, Israel) arbitrary wave-form, generator and a Panametrics 5800 pulser/receiver was used as a receiver. A Gage CompuScope 12100 (Gage Applied Technologies, Lockport, IL, USA), one-channel 100MHz mode, 12-bit A/D converter was used digitally to store the detected waves. A schematic depiction of the experimental system used here is shown in Fig. 1. The reflection from a static target was initially recorded and served as a reference signal, which represents the transmitted signal. The target was then moved at constant velocities ranging from ⫺50 mm/s to 50 mm/s and incremented by 10 mm/s. At each velocity, several reflections of chirp signals were recorded. Each recorded signal was correlated with the reference signal and the velocity (calculated from the OCP, eqn (16)) vs. the target velocity was plotted. RESULTS
SIMULATION METHODS A numeric computer simulation was written to verify that the approximations that were made during the theoretical derivation of eqn (16) are acceptable. During these simulations, transmitted chirps in varying lengths were correlated with a synthetic set of echoes that represent reflections from targets with varying velocities. The velocity of the moving reflector along the beam is determined from the
Simulations The relative error map that was generated by the numerical computer simulations is depicted in Fig. 2. The map presents the normalized error of the velocity estimation for a moving target reflector, whose velocity ranged from 0.1 m/s to 1 m/s. The chirp had a frequency sweep from 3 MHz to 5 MHz and its length ranged from 10 s to 40 s. The velocity estimation error obtained by
Velocity measurements using frequency-modulated chirp ● Y. LEVY and H. AZHARI
771
Receiver
Signal
Signal Generator
Velocity Control
Trigger A/D Data
Fig. 1. A schematic depiction of the ultrasonic experimental set-up used in this study.
these simulations was ⬍10%. As can be noted, the accuracy improves for longer chirp signal. The correlation function (eqn (15)) for the stationary target is depicted in Fig. 3a; for comparison, the correlation function obtained from a moving target (1 m/s) is depicted in Fig. 3b. As can be observed, the correlation function for the moving target is asymmetric and distorted relative to the stationary case. The simulated effect of noise on the velocity estimation (chirp frequency ranged from 3 MHz to 5 MHz, its length was 40 s and target velocity was 0.25 m/s) was also evaluated by varying the SNR from –20 dB to ⫹20 dB. The results are depicted in Fig. 4. The error bars depict the mean and SD of the estimated velocity normalized to the accurate value. As can be noted, the mean velocity estimates are fairly stable throughout the SNR range. Measurements Using an ultrasonic-transmitted chirp signal 40 s in length and a frequency sweep ranging from 3 MHz to 5 MHz, the velocity of the moving metal target was evaluated experimentally as explained previously. The results obtained from the set of measurements are depicted in Fig. 5. In this figure, the estimated velocity (calculated from the OCP, eqn (16)) is plotted vs. the target’s velocity set by the controller. The error bars represent the standard deviation for each measured value. Applying linear regression to the data has yielded the regression line of: VMeasured ⫽ Vtarget 0.8478 ⫺0.0005 [m/s], (r ⫽ 0.989), where VMeasured represents the values obtained by the suggested method and Vtarget is the velocity set be the motion control system.
DISCUSSION In this study, a method for velocity estimation is introduced. The main advantage of the technique is its ability to obtain the velocity estimation using a single pulse (chirp) transmission. With this technique, the same chirp signal can potentially be used for both imaging and velocity estimation. This would enable the velocity map to be generated at the same frame rate as the standard
Fig. 2. A map depicting the relative error in velocity estimation of a moving target obtained by the computer simulations. The velocities ranged from 0.1 m/s to 1 m/s. The chirp frequency ranged from 3 MHz to 5 MHz and its length ranged from 10 s to 40 s. As can be noted, the error is smaller for longer chirp signal lengths.
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Ultrasound in Medicine and Biology
Volume 33, Number 5, 2007
Fig. 3. (a) The correlation function derived for the stationary target (chirp frequency ranged from 3 MHz to 5 MHz and its length was 40 s). (b) The correlation function derived for a moving target (same chirp, velocity ⫽ 1 m/s). As can be noted, the correlation function for the moving target is asymmetric and distorted relative to the stationary case.
ultrasonic image, which can also be reconstructed from the same dataset. Another advantage offered by the suggested method is an adjustable dynamic range. The dynamic range of the estimation can be determined from eqn (16) by substituting ⫾ instead of . vmax ⫽
冏 冏
S0c . 4f m2
for which the acoustic impedance is lower than the medium’s acoustic impedance. In the later case, the signal is reflected with a phase shift of and, therefore, the corresponding average phase gap between the transmitted and reflected chirps becomes: ⫹ . For a system that contains targets with a variety of imped-
(17)
For comparison, in the common Doppler shift method, the maximal detectible velocity vmax is determined by (Jensen 1996, eqn (6.45)), vmax ⫽
c · f PRF 4f 0
(18)
where fPRF is the pulse repetition frequency and f0 is the central transmitted frequency. This imposes limitation on the common method from two aspects: First, fPRF is limited by the distance to the target. Second, decreasing f0 is commonly associated with a decrease in the axial resolution. With the suggested technique, on the other hand, the maximal detectable velocity can be adjusted by either changing S0 (the sweep rate) and/or fm (the center instantaneous frequency). This offers the operator more freedom in setting the measurement system. It should be noted, though, that the mathematical derivation outlined above was done for a case in which the phase of the reflected signal is not inverted. This is not the case for an ultrasonic reflection from a target
Fig. 4. Simulated effect of noise on the velocity estimation. SNR was varied from –20 dB to ⫹20 dB. The error bars depict the mean and SD of the estimated velocity relative to the accurate value. As can be noted, the mean velocity estimates are fairly stable throughout the SNR range.
Velocity measurements using frequency-modulated chirp ● Y. LEVY and H. AZHARI
773
Fig. 5. Measured velocity (using the OCP, eqn (16)) vs. the target velocity set by the motion control system. The error bars represent one SD. The solid line represents the regression line obtained for these data.
ances (higher and lower than the medium), the dynamic range is further limited to an absolute phase change of no more than /2. Hence, the chirp’s parameters should be set to vmax ⫽
冏 冏
S0c . 8f m2
(19)
Studying the regression line obtained in the experimental part, it can be noted that the slope differs from unity, i.e., VMeasured ⫽ Vtarget 0.8478. This may stem from the following reasons. (i) The central instantaneous frequency fm actually represents the combined effect of all the frequencies in the transmitted band. Hence, the changes in fm caused by the moving target are not identical to the changes that a single frequency would experience as a result of the Doppler effect. (ii) The actual transmitted signal is not an ideal linear frequency-modulated chirp, but is distorted by the impulse response of each element in the transmission system. (iii) The lower frequencies are dominant in the received signal (because of the frequency-dependency
of the attenuation) and, therefore, the OCP tends to be smaller than expected. Nevertheless, this problem can be simply overcome by using a calibration process. In conclusion, the suggested method can be used to estimate the velocity of a moving target using a single transmitted linear frequency-modulated chirp. This may potentially yield high frame rate of velocity estimations. The method was verified using computer simulations and experimental measurements with an ultrasonic system. REFERENCES Ha STT, Sheriff RE, Gardner GHF. Instantaneous frequency, spectral centroid, and even wavelets, Geophys Res Lett 1991;18:1389 – 1392. Jensen JA. Estimation of Blood Velocities Using Ultrasound. Cambridge: Cambridge University Press; 1996. Misaridis T, Jensen JA. Use of modulated excitation signals in medical ultrasound. Part I: Basic concepts and expected benefits. IEEE Trans Ultrason Ferroelectr Freq Control 2005;52:177–191. Wilhjelm JE, Pedersen PC. Target velocity estimation with FM and PW echo ranging Doppler systems—Part I: Signal analysis. IEEE Trans Ultrason Ferroelectr Freq Control 1993;40:366 –372.
doi:10.1016/j.ultrasmedbio.2006.11.013
● Original Contribution VELOCITY MEASUREMENTS USING A SINGLE TRANSMITTED LINEAR FREQUENCY-MODULATED CHIRP YOAV LEVY and HAIM AZHARI Faculty of Biomedical Engineering, Technion, IIT, Haifa, Israel (Received 19 June 2006, revised 30 October 2006, in final form 7 November 2006)
Abstract—Velocity measurement is a challenge for a variety of remote sensing systems such as ultrasonic and radar scanners. However, current Doppler-based techniques require a comparatively long data acquisition time. It has been suggested to use coded signals, such as linear frequency-modulated signals (chirp), for ultrasonic velocity estimation by extracting the needed information from a set of several sequential coded pulses. In this study, a method for velocity estimation using a single linear frequency-modulated chirp transmission is presented and implemented for ultrasonic measurements. The complex cross-correlation function between the transmitted and reflected signals is initially calculated. The velocity is then calculated from the phase of the peak of the envelope of this cross-correlation function. The suggested method was verified using computer simulations and experimental measurements in an ultrasonic system. Applying linear regression to the data has yielded very good correlation (r ⴝ 0.989). With the suggested technique, higher frame rates of velocity mapping can be potentially achieved relative to current techniques. Also, the same data can be utilized for both velocity mapping and image reconstruction. (E-mail: [email protected]) © 2007 World Federation for Ultrasound in Medicine & Biology. Key Words: Velocity measurement, Coded excitation, Linear frequency modulated chirp.
resolution, by using coded excitation, the high spatial resolution can be recovered using an appropriate signal processing algorithm (e.g., matched filter (Misaridis and Jensen 2005)). It has been suggested to use coded signals, such as linear frequency-modulated signals (chirp), for ultrasonic velocity estimation by extracting the needed information from a set of several sequential coded pulses (Wilhjelm and Pedersen 1993). In this study, we present a method for velocity estimation using a single coded pulse transmission.
INTRODUCTION Measurement of velocity is a challenge for a variety of remote sensing systems such as ultrasonic and radar scanners. Commonly, the Doppler frequency shift caused by a moving reflector is measured and converted into velocity estimation. This method is well established and has been implemented using many techniques. However, current Doppler-based techniques require either the transmission of a long continuous wave, which sacrifices axial resolution, or the acquisition of echoes from several pulses to generate a velocity map of each region in the image. Therefore, both methods require a comparatively long data acquisition time, typically on the order of the period of the Doppler frequency shift. Coded excitation methodology (Misaridis and Jensen 2005) is used in ultrasonic imaging systems to improve signal-to-noise ratio (SNR). In this methodology, a long coded signal is used to transmit high energy while preserving low-intensity constraints. Although, typically, a long pulse duration leads to poor spatial
THEORY A chirp from f0 to f1 whose length is Tm can be represented by the following formula (Jensen 1996, eqn (9.20)) e(t) ⫽ sin(2 f 0t ⫹ S0t2); 0 ⱕ t ⱕ Tm
(1)
where f0 is the start frequency, f1 is the end frequency and S0 is the sweep rate of the signal
Address correspondence to: Haim Azhari D.Sc., Faculty of Biomedical Engineering, Technion IIT, Haifa 32000, Israel. E-mail: [email protected]
S0 ⫽ 768
f1 ⫺ f0 . Tm
Velocity measurements using frequency-modulated chirp ● Y. LEVY and H. AZHARI
The instantaneous frequency of the signal is (Wilhjelm and Pedersen 1993) f(t) ⫽ f 0 ⫹ S0t.
(2)
The time of appearance of each frequency t(f) is t(f) ⫽
f ⫺ f0 . S0
(3)
Other properties of the chirp signal are ⌬f ⫽ f 1 ⫺ f 0
(5)
c⫺v c⫹v
(6)
where a is a reflection coefficient (a frequency-independent reflection is assumed), c is the acoustic velocity in the medium and t⬘ is a time shift that is related to the path from the transducer to the moving target. The signal’s intensity is not important in the following discussion; therefore, we set a ⫽ 1. When c ⬎⬎ 2v c⫺v 2v ⬇1⫺ c⫹v c
(7)
2v 1 ␣⬇1⫹ . c
Ⲑ
The instantaneous frequency f ⬘(t) of the reflected signal in its corresponding coordinates (setting t ⫽ 0 for the signal front end) is given by
f ⬘ (t) ⫽ f 0 ⬘ ⫹ S0 ⬘ t where
冦
兰
2 f(t)dt.
(9)
t共 f Ⲑ ␣兲
For cases in which the parameter ␣ is approximately unity (␣ ⬃ 1), the frequency can be taken as constant over the integration range: f(t) ⬇ f.
(10)
Therefore, using the approximation in eqn (10) and using eqn (3), the integral in eqn (9) can be solved:
rs(t) ⫽ a · e(␣t ⫺ t ⬘ )
␣⫽
t(f)
⌬(f) ⫽
where ⌬f is the frequency bandwidth of the chirp and fm is the center instantaneous frequency. A received signal rs(t) from a moving reflector with a velocity v along the beam axis can be represented by (Jensen 1996, eqn (9.21))
␣⫽
whose instantaneous frequency is f/␣. The phase gap between the phase of the point whose instantaneous frequency is f in the transmitted signal and the point in the received signal that has the same instantaneous frequency (⌬(f)) is equal to the phase gap between the points for which the instantaneous frequencies are f and f/␣ in the transmitted signal (within the range where the transmitted and received bands overlap). The phase gap ⌬(f) can be calculated by the expression:
(4)
f1 ⫹ f0 fm ⫽ 2
769
f0 ⬘ ⫽ ␣f0 f1 ⬘ ⫽ ␣f1 S0 ⬘ ⫽ ␣2S0
(8)
Ⲑ
0 ⱕ t ⱕ Tm ␣
Consider an ultrasonic chirp signal that was reflected from a moving target. During the reflection, the instantaneous frequency of each point of the transmitted chirp is changed, according to the Doppler frequency shift, from f to ␣ f. On the other hand, the phase of each point is preserved. Therefore, the phase of the point in the received signal whose instantaneous frequency is f is equal to the phase of the point in the transmitted signal
⌬(f) ⬇ 2 f · (t(f) ⫺ t(f ⁄ ␣)) ⬇ 2 f ·
冉 冊 冉 冊
⬇ 2 f 2 ·
f ⫺ f ⁄␣ S0
1⫺1⁄␣ . S0
(11)
Substituting eqn (7) into eqn (11), the phase gap between corresponding points in the chirps for a specific instantaneous frequency f is given by the expression ⌬(f) ⬇ 2 f 2
⫺2v . S0c
(12)
Under typical physiologic blood flow conditions, the changes in f0 and S0 as a result of the reflection from the moving blood (calculated in eqn (8)) are too small for a reliable Doppler velocity estimation using a single pulse transmission (Wilhjelm and Pedersen 1993). However, as shown below, the cross-correlation function of the transmitted and received chirp signals in the time domain is sensitive to the resulting changes in the start frequency and the sweep rate and hence, can be utilized for velocity estimation using a single transmission. Consider the chirp e(t), which was defined in eqn (1). This chirp can be turned into a phase-encoded chirp e(t,) (Ha et al. 1991), where is the encoded phase e(t, ) ⫽ sin(2 f 0t ⫹ S0t2 ⫹ ).
(13)
The corresponding approximated cross-correlation between e(t,0) and e(t,) is given by Ha et al. (1991):
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Ultrasound in Medicine and Biology
R() ⬇ sinc共⌬f · 兲cos共2 f m ⫹ ).
(14)
The cross-correlation function R() is comprised of an envelope (the sinc function) and a carrier frequency equal to the center instantaneous frequency fm. The phase of the carrier frequency at the peak of the cross-correlation envelope ( ⫽ 0) is . Two additional approximations should be done to use Ha et al. approximation as a cross-correlation function between a transmitted chirp pulse and its corresponding reflected chirp signal: a. Use the original spectrum for determination of ⌬f and fm. b. Approximate the phase gap between the chirps (which is a function of the phase) by the phase gap between corresponding points in the transmitted and reflected chirps having the center instantaneous frequency fm. Those approximations are valid as long as ␣ ⬇ 1 (i.e. c Ⰷ 2v). Substituting eqn (12) into eqn (14), an approximation for the cross-correlation function between a transmitted chirp pulse and its corresponding reflected chirp signal is
冉
R() ⯝ sinc共⌬f · 兲cos 2 f m ⫹ 2 f m2
冊
2v . S0c
(15)
One can note that the cross-correlation function envelope peaks at ⫽ 0 and ⫽ 2 f m2 ⫺ 2v⁄S0c will be the peak’s phase. Let us define the phase at the peak of the crosscorrelation envelope as the “optimal-correlation phase” (OCP). This phase is measured by taking the phase of RH() at its maximal absolute value, where RH() is the analytical signal RH() ⫽ R() ⫹ iRˆ() where Rˆ() is the Hilbert transform of the cross-correlation function between the transmitted and reflected chirp waves. In case of a stationary reflector, the OCP is zero, because there is no frequency shift, i.e., v ⫽ 0. Using eqn (15), the velocity of the moving reflector along the beam can be determined from the OCP, OCP, by the expression v⫽
⫺OCPS0c . 4 f m2
(16)
Volume 33, Number 5, 2007
OCP by eqn (16) for each combination of chirp length and reflector velocity. The transmitted signals were simulated using eqn (1) and the echoes were calculated according to eqn (8). The synthetic signals were multiplied by a Hamming window to emulate a realistic situation in which the transducer’s impulse response modulates the signal. Parabolic interpolation was used to determine the accurate peak of the correlation function. Finally, a map depicting the relative error in velocity estimation, as a function of target velocity and chirp length, was generated from these simulation results. In addition, the effect of noise on velocity estimation was evaluated. SNR was varied from ⫺20 dB to ⫹20 dB by adding white Gaussian noise to the simulated data. At each SNR level, 100 simulations were conducted. The estimated velocity was normalized to the accurate value and its mean and standard deviation (SD) were calculated as a function of SNR. EXPERIMENTAL METHODS A transducer (Panametrics, GE Sensing, Billerica, MA, USA, 5 MHz, diameter of 6.3 mm) was placed in a water-bath in front of a computer-controlled moving target. The target was a stainless steel cube that could be moved at velocities of up to several cm/s, defined by the user (these values served as a “gold-standard”). Chirp signals were generated by a Tabor 8026 (Tabor Electronics, Tel Hanan, Israel) arbitrary wave-form, generator and a Panametrics 5800 pulser/receiver was used as a receiver. A Gage CompuScope 12100 (Gage Applied Technologies, Lockport, IL, USA), one-channel 100MHz mode, 12-bit A/D converter was used digitally to store the detected waves. A schematic depiction of the experimental system used here is shown in Fig. 1. The reflection from a static target was initially recorded and served as a reference signal, which represents the transmitted signal. The target was then moved at constant velocities ranging from ⫺50 mm/s to 50 mm/s and incremented by 10 mm/s. At each velocity, several reflections of chirp signals were recorded. Each recorded signal was correlated with the reference signal and the velocity (calculated from the OCP, eqn (16)) vs. the target velocity was plotted. RESULTS
SIMULATION METHODS A numeric computer simulation was written to verify that the approximations that were made during the theoretical derivation of eqn (16) are acceptable. During these simulations, transmitted chirps in varying lengths were correlated with a synthetic set of echoes that represent reflections from targets with varying velocities. The velocity of the moving reflector along the beam is determined from the
Simulations The relative error map that was generated by the numerical computer simulations is depicted in Fig. 2. The map presents the normalized error of the velocity estimation for a moving target reflector, whose velocity ranged from 0.1 m/s to 1 m/s. The chirp had a frequency sweep from 3 MHz to 5 MHz and its length ranged from 10 s to 40 s. The velocity estimation error obtained by
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Receiver
Signal
Signal Generator
Velocity Control
Trigger A/D Data
Fig. 1. A schematic depiction of the ultrasonic experimental set-up used in this study.
these simulations was ⬍10%. As can be noted, the accuracy improves for longer chirp signal. The correlation function (eqn (15)) for the stationary target is depicted in Fig. 3a; for comparison, the correlation function obtained from a moving target (1 m/s) is depicted in Fig. 3b. As can be observed, the correlation function for the moving target is asymmetric and distorted relative to the stationary case. The simulated effect of noise on the velocity estimation (chirp frequency ranged from 3 MHz to 5 MHz, its length was 40 s and target velocity was 0.25 m/s) was also evaluated by varying the SNR from –20 dB to ⫹20 dB. The results are depicted in Fig. 4. The error bars depict the mean and SD of the estimated velocity normalized to the accurate value. As can be noted, the mean velocity estimates are fairly stable throughout the SNR range. Measurements Using an ultrasonic-transmitted chirp signal 40 s in length and a frequency sweep ranging from 3 MHz to 5 MHz, the velocity of the moving metal target was evaluated experimentally as explained previously. The results obtained from the set of measurements are depicted in Fig. 5. In this figure, the estimated velocity (calculated from the OCP, eqn (16)) is plotted vs. the target’s velocity set by the controller. The error bars represent the standard deviation for each measured value. Applying linear regression to the data has yielded the regression line of: VMeasured ⫽ Vtarget 0.8478 ⫺0.0005 [m/s], (r ⫽ 0.989), where VMeasured represents the values obtained by the suggested method and Vtarget is the velocity set be the motion control system.
DISCUSSION In this study, a method for velocity estimation is introduced. The main advantage of the technique is its ability to obtain the velocity estimation using a single pulse (chirp) transmission. With this technique, the same chirp signal can potentially be used for both imaging and velocity estimation. This would enable the velocity map to be generated at the same frame rate as the standard
Fig. 2. A map depicting the relative error in velocity estimation of a moving target obtained by the computer simulations. The velocities ranged from 0.1 m/s to 1 m/s. The chirp frequency ranged from 3 MHz to 5 MHz and its length ranged from 10 s to 40 s. As can be noted, the error is smaller for longer chirp signal lengths.
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Fig. 3. (a) The correlation function derived for the stationary target (chirp frequency ranged from 3 MHz to 5 MHz and its length was 40 s). (b) The correlation function derived for a moving target (same chirp, velocity ⫽ 1 m/s). As can be noted, the correlation function for the moving target is asymmetric and distorted relative to the stationary case.
ultrasonic image, which can also be reconstructed from the same dataset. Another advantage offered by the suggested method is an adjustable dynamic range. The dynamic range of the estimation can be determined from eqn (16) by substituting ⫾ instead of . vmax ⫽
冏 冏
S0c . 4f m2
for which the acoustic impedance is lower than the medium’s acoustic impedance. In the later case, the signal is reflected with a phase shift of and, therefore, the corresponding average phase gap between the transmitted and reflected chirps becomes: ⫹ . For a system that contains targets with a variety of imped-
(17)
For comparison, in the common Doppler shift method, the maximal detectible velocity vmax is determined by (Jensen 1996, eqn (6.45)), vmax ⫽
c · f PRF 4f 0
(18)
where fPRF is the pulse repetition frequency and f0 is the central transmitted frequency. This imposes limitation on the common method from two aspects: First, fPRF is limited by the distance to the target. Second, decreasing f0 is commonly associated with a decrease in the axial resolution. With the suggested technique, on the other hand, the maximal detectable velocity can be adjusted by either changing S0 (the sweep rate) and/or fm (the center instantaneous frequency). This offers the operator more freedom in setting the measurement system. It should be noted, though, that the mathematical derivation outlined above was done for a case in which the phase of the reflected signal is not inverted. This is not the case for an ultrasonic reflection from a target
Fig. 4. Simulated effect of noise on the velocity estimation. SNR was varied from –20 dB to ⫹20 dB. The error bars depict the mean and SD of the estimated velocity relative to the accurate value. As can be noted, the mean velocity estimates are fairly stable throughout the SNR range.
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Fig. 5. Measured velocity (using the OCP, eqn (16)) vs. the target velocity set by the motion control system. The error bars represent one SD. The solid line represents the regression line obtained for these data.
ances (higher and lower than the medium), the dynamic range is further limited to an absolute phase change of no more than /2. Hence, the chirp’s parameters should be set to vmax ⫽
冏 冏
S0c . 8f m2
(19)
Studying the regression line obtained in the experimental part, it can be noted that the slope differs from unity, i.e., VMeasured ⫽ Vtarget 0.8478. This may stem from the following reasons. (i) The central instantaneous frequency fm actually represents the combined effect of all the frequencies in the transmitted band. Hence, the changes in fm caused by the moving target are not identical to the changes that a single frequency would experience as a result of the Doppler effect. (ii) The actual transmitted signal is not an ideal linear frequency-modulated chirp, but is distorted by the impulse response of each element in the transmission system. (iii) The lower frequencies are dominant in the received signal (because of the frequency-dependency
of the attenuation) and, therefore, the OCP tends to be smaller than expected. Nevertheless, this problem can be simply overcome by using a calibration process. In conclusion, the suggested method can be used to estimate the velocity of a moving target using a single transmitted linear frequency-modulated chirp. This may potentially yield high frame rate of velocity estimations. The method was verified using computer simulations and experimental measurements with an ultrasonic system. REFERENCES Ha STT, Sheriff RE, Gardner GHF. Instantaneous frequency, spectral centroid, and even wavelets, Geophys Res Lett 1991;18:1389 – 1392. Jensen JA. Estimation of Blood Velocities Using Ultrasound. Cambridge: Cambridge University Press; 1996. Misaridis T, Jensen JA. Use of modulated excitation signals in medical ultrasound. Part I: Basic concepts and expected benefits. IEEE Trans Ultrason Ferroelectr Freq Control 2005;52:177–191. Wilhjelm JE, Pedersen PC. Target velocity estimation with FM and PW echo ranging Doppler systems—Part I: Signal analysis. IEEE Trans Ultrason Ferroelectr Freq Control 1993;40:366 –372.