Ultrasound in Med. & Biol., Vol. 33, No. 5, pp. 768–773, 2007Copyright © 2007 World Federation for Ultrasound in Medicine & Biology

Printed in the USA. All rights reserved0301-5629/07/$–see front matter

doi:10.1016/j.ultrasmedbio.2006.11.013

● Original Contribution

VELOCITY MEASUREMENTS USING A SINGLE TRANSMITTED LINEARFREQUENCY-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 andradar 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 ultrasonicvelocity estimation by extracting the needed information from a set of several sequential coded pulses. In thisstudy, a method for velocity estimation using a single linear frequency-modulated chirp transmission is presentedand implemented for ultrasonic measurements. The complex cross-correlation function between the transmittedand reflected signals is initially calculated. The velocity is then calculated from the phase of the peak of theenvelope of this cross-correlation function. The suggested method was verified using computer simulations andexperimental measurements in an ultrasonic system. Applying linear regression to the data has yielded very goodcorrelation (r � 0.989). With the suggested technique, higher frame rates of velocity mapping can be potentiallyachieved relative to current techniques. Also, the same data can be utilized for both velocity mapping and imagereconstruction. (E-mail: haim@bm.technion.ac.il) © 2007 World Federation for Ultrasound in Medicine &Biology.

Key Words: Velocity measurement, Coded excitation, Linear frequency modulated chirp.

INTRODUCTION

Measurement of velocity is a challenge for a variety ofremote sensing systems such as ultrasonic and radarscanners. Commonly, the Doppler frequency shift causedby a moving reflector is measured and converted intovelocity estimation. This method is well established andhas been implemented using many techniques. However,current Doppler-based techniques require either thetransmission of a long continuous wave, which sacrificesaxial resolution, or the acquisition of echoes from severalpulses to generate a velocity map of each region in theimage. Therefore, both methods require a comparativelylong data acquisition time, typically on the order of theperiod of the Doppler frequency shift.

Coded excitation methodology (Misaridis andJensen 2005) is used in ultrasonic imaging systems toimprove signal-to-noise ratio (SNR). In this methodol-ogy, a long coded signal is used to transmit high energywhile preserving low-intensity constraints. Although,typically, a long pulse duration leads to poor spatial

Address correspondence to: Haim Azhari D.Sc., Faculty of Bio-

medical Engineering, Technion IIT, Haifa 32000, Israel. E-mail:haim@bm.technion.ac.il

768

resolution, by using coded excitation, the high spatialresolution can be recovered using an appropriate signalprocessing algorithm (e.g., matched filter (Misaridis andJensen 2005)). It has been suggested to use coded sig-nals, such as linear frequency-modulated signals (chirp),for ultrasonic velocity estimation by extracting theneeded information from a set of several sequentialcoded pulses (Wilhjelm and Pedersen 1993).

In this study, we present a method for velocityestimation using a single coded pulse transmission.

THEORY

A chirp from f0 to f1 whose length is Tm can berepresented by the following formula (Jensen 1996, eqn(9.20))

e(t) � sin(2�f0t � �S0t2); 0 � t � Tm (1)

where f0 is the start frequency, f1 is the end frequency andS0 is the sweep rate of the signal

f1 � f0

S0 �

Tm.

Velocity measurements using frequency-modulated chirp ● Y. LEVY and H. AZHARI 769

The instantaneous frequency of the signal is(Wilhjelm and Pedersen 1993)

f(t) � f0 � 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 � f1 � f0 (4)

fm �f1 � f0

2(5)

where �f is the frequency bandwidth of the chirp and fmis the center instantaneous frequency.

A received signal rs(t) from a moving reflector witha velocity v along the beam axis can be represented by(Jensen 1996, eqn (9.21))

rs(t) � a · e(�t � t � )

� �c � v

c � v(6)

where a is a reflection coefficient (a frequency-indepen-dent reflection is assumed), c is the acoustic velocity inthe medium and t� is a time shift that is related to the pathfrom the transducer to the moving target. The signal’sintensity is not important in the following discussion;therefore, we set a � 1.

When c �� 2v

� �c � v

c � v� 1 �

2v

c

1�� � 1 �2v

c.

(7)

The instantaneous frequency f �(t) of the reflected signalin its corresponding coordinates (setting t � 0 for thesignal front end) is given by

f � (t) � f0 � � S0 � t where �f0 � � �f0

f1 � � �f1

S0 � � �2S0

0 � t � Tm��

(8)

Consider an ultrasonic chirp signal that was reflectedfrom a moving target. During the reflection, the instan-taneous frequency of each point of the transmitted chirpis changed, according to the Doppler frequency shift,from f to � f. On the other hand, the phase of each pointis preserved. Therefore, the phase of the point in thereceived signal whose instantaneous frequency is f is

equal to the phase of the point in the transmitted signal

whose instantaneous frequency is f/�. The phase gapbetween the phase of the point whose instantaneousfrequency is f in the transmitted signal and the point inthe received signal that has the same instantaneous fre-quency (�(f)) is equal to the phase gap between thepoints for which the instantaneous frequencies are f andf/� in the transmitted signal (within the range where thetransmitted and received bands overlap). The phase gap�(f) can be calculated by the expression:

�(f) � �t�f � ��

t(f)

2�f(t)dt. (9)

For cases in which the parameter � is approximatelyunity (� � 1), the frequency can be taken as constantover the integration range:

f(t) � f. (10)

Therefore, using the approximation in eqn (10) and usingeqn (3), the integral in eqn (9) can be solved:

�(f) � 2�f · (t(f) � t(f ⁄ �))

� 2�f · � f � f ⁄ �

S0�

� 2�f 2 · �1 � 1 ⁄ �

S0�. (11)

Substituting eqn (7) into eqn (11), the phase gap betweencorresponding points in the chirps for a specific instan-taneous frequency f is given by the expression

�(f) � 2�f 2�2v

S0c. (12)

Under typical physiologic blood flow conditions, thechanges in f0 and S0 as a result of the reflection from themoving blood (calculated in eqn (8)) are too small for areliable Doppler velocity estimation using a single pulsetransmission (Wilhjelm and Pedersen 1993). However,as shown below, the cross-correlation function of thetransmitted and received chirp signals in the time domainis sensitive to the resulting changes in the start frequencyand the sweep rate and hence, can be utilized for velocityestimation using a single transmission.

Consider the chirp e(t), which was defined in eqn(1). This chirp can be turned into a phase-encoded chirpe(t,) (Ha et al. 1991), where is the encoded phase

e(t, ) � sin(2�f0t � �S0t2 � ). (13)

The corresponding approximated cross-correlation be-

tween e(t,0) and e(t,) is given by Ha et al. (1991):

770 Ultrasound in Medicine and Biology Volume 33, Number 5, 2007

R() � sinc��f · �cos�2�fm � ). (14)

The cross-correlation function R() is comprised of anenvelope (the sinc function) and a carrier frequencyequal to the center instantaneous frequency fm. The phaseof the carrier frequency at the peak of the cross-correla-tion envelope ( � 0) is .

Two additional approximations should be done touse Ha et al. approximation as a cross-correlation func-tion between a transmitted chirp pulse and its corre-sponding reflected chirp signal:

a. Use the original spectrum for determination of �f andfm.

b. Approximate the phase gap between the chirps (whichis a function of the phase) by the phase gap betweencorresponding points in the transmitted and reflectedchirps having the center instantaneous frequency fm.

Those approximations are valid as long as� � 1 (i.e. c � 2v).

Substituting eqn (12) into eqn (14), an approxima-tion for the cross-correlation function between a trans-mitted chirp pulse and its corresponding reflected chirpsignal is

R() � sinc��f · �cos�2�fm � 2�fm2

2v

S0c�. (15)

One can note that the cross-correlation function envelopepeaks at � 0 and � 2�fm

2 � 2v⁄S0c will be the peak’sphase. Let us define the phase at the peak of the cross-correlation envelope as the “optimal-correlation phase”(OCP). This phase is measured by taking the phase ofRH() at its maximal absolute value, where RH() is theanalytical signal RH() � R() � iR̂() where R̂() is theHilbert transform of the cross-correlation function bet-ween the transmitted and reflected chirp waves. In caseof a stationary reflector, the OCP is zero, because there isno frequency shift, i.e., v � 0.

Using eqn (15), the velocity of the moving reflectoralong the beam can be determined from the OCP, OCP,by the expression

v ��OCPS0c

4�fm2 . (16)

SIMULATION METHODS

A numeric computer simulation was written to verifythat the approximations that were made during the theoret-ical derivation of eqn (16) are acceptable. During thesesimulations, transmitted chirps in varying lengths were cor-related with a synthetic set of echoes that represent reflec-tions from targets with varying velocities. The velocity of

the moving reflector along the beam is determined from the

OCP by eqn (16) for each combination of chirp length andreflector velocity. The transmitted signals were simulatedusing eqn (1) and the echoes were calculated according toeqn (8). The synthetic signals were multiplied by a Ham-ming window to emulate a realistic situation in which thetransducer’s impulse response modulates the signal. Para-bolic interpolation was used to determine the accurate peakof the correlation function. Finally, a map depicting therelative error in velocity estimation, as a function of targetvelocity and chirp length, was generated from these simu-lation results.

In addition, the effect of noise on velocity estima-tion was evaluated. SNR was varied from �20 dB to�20 dB by adding white Gaussian noise to the simulateddata. At each SNR level, 100 simulations were con-ducted. The estimated velocity was normalized to theaccurate 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 awater-bath in front of a computer-controlled movingtarget. The target was a stainless steel cube that could bemoved at velocities of up to several cm/s, defined by theuser (these values served as a “gold-standard”). Chirpsignals were generated by a Tabor 8026 (Tabor Electron-ics, Tel Hanan, Israel) arbitrary wave-form, generatorand a Panametrics 5800 pulser/receiver was used as areceiver. A Gage CompuScope 12100 (Gage AppliedTechnologies, Lockport, IL, USA), one-channel 100-MHz mode, 12-bit A/D converter was used digitally tostore the detected waves. A schematic depiction of theexperimental system used here is shown in Fig. 1.

The reflection from a static target was initially re-corded and served as a reference signal, which representsthe transmitted signal. The target was then moved atconstant velocities ranging from �50 mm/s to 50 mm/sand incremented by 10 mm/s. At each velocity, severalreflections of chirp signals were recorded. Each recordedsignal was correlated with the reference signal and thevelocity (calculated from the OCP, eqn (16)) vs. thetarget velocity was plotted.

RESULTS

SimulationsThe relative error map that was generated by the

numerical computer simulations is depicted in Fig. 2.The map presents the normalized error of the velocityestimation for a moving target reflector, whose velocityranged from 0.1 m/s to 1 m/s. The chirp had a frequencysweep from 3 MHz to 5 MHz and its length ranged from

10 �s to 40 �s. The velocity estimation error obtained by

asonic

Velocity measurements using frequency-modulated chirp ● Y. LEVY and H. AZHARI 771

these simulations was 10%. As can be noted, the ac-curacy improves for longer chirp signal. The correlationfunction (eqn (15)) for the stationary target is depicted inFig. 3a; for comparison, the correlation function obtainedfrom a moving target (1 m/s) is depicted in Fig. 3b. As canbe observed, the correlation function for the moving targetis asymmetric and distorted relative to the stationary case.

The simulated effect of noise on the velocity estima-tion (chirp frequency ranged from 3 MHz to 5 MHz, itslength was 40 �s and target velocity was 0.25 m/s) was alsoevaluated by varying the SNR from –20 dB to �20 dB. Theresults are depicted in Fig. 4. The error bars depict the meanand SD of the estimated velocity normalized to the accuratevalue. As can be noted, the mean velocity estimates arefairly stable throughout the SNR range.

MeasurementsUsing an ultrasonic-transmitted chirp signal 40 �s

in length and a frequency sweep ranging from 3 MHz to5 MHz, the velocity of the moving metal target wasevaluated experimentally as explained previously.

The results obtained from the set of measurementsare depicted in Fig. 5. In this figure, the estimated ve-locity (calculated from the OCP, eqn (16)) is plotted vs.the target’s velocity set by the controller. The error barsrepresent the standard deviation for each measured value.Applying linear regression to the data has yielded theregression line of: VMeasured � Vtarget 0.8478 �0.0005[m/s], (r � 0.989), where VMeasured represents the valuesobtained by the suggested method and Vtarget is the

Signal Ge

Receiver

A/D

Trigger

Signal

Data

Signal Ge

Receiver

A/D

Trigger

Signal

Data

Fig. 1. A schematic depiction of the ultr

velocity set be the motion control system.

DISCUSSION

In this study, a method for velocity estimation isintroduced. The main advantage of the technique is itsability to obtain the velocity estimation using a singlepulse (chirp) transmission. With this technique, the samechirp signal can potentially be used for both imaging andvelocity estimation. This would enable the velocity mapto be generated at the same frame rate as the standard

Fig. 2. A map depicting the relative error in velocity estimationof a moving target obtained by the computer simulations. Thevelocities ranged from 0.1 m/s to 1 m/s. The chirp frequencyranged from 3 MHz to 5 MHz and its length ranged from 10 �sto 40 �s. As can be noted, the error is smaller for longer chirp

Velocity ControlVelocity Control

experimental set-up used in this study.

neratornerator

signal lengths.

t is as

772 Ultrasound in Medicine and Biology Volume 33, Number 5, 2007

ultrasonic image, which can also be reconstructed fromthe same dataset.

Another advantage offered by the suggested methodis an adjustable dynamic range. The dynamic range ofthe estimation can be determined from eqn (16) by sub-stituting � instead of .

vmax � S0c

4fm2. (17)

For comparison, in the common Doppler shift method,the maximal detectible velocity vmax is determined by(Jensen 1996, eqn (6.45)),

vmax �c · fPRF

4f0(18)

where fPRF is the pulse repetition frequency and f0 is thecentral transmitted frequency. This imposes limitation onthe common method from two aspects: First, fPRF islimited by the distance to the target. Second, decreasingf0 is commonly associated with a decrease in the axialresolution. With the suggested technique, on the otherhand, the maximal detectable velocity can be adjusted byeither changing S0 (the sweep rate) and/or fm (the centerinstantaneous frequency). This offers the operator morefreedom in setting the measurement system.

It should be noted, though, that the mathematicalderivation outlined above was done for a case in whichthe phase of the reflected signal is not inverted. This is

Fig. 3. (a) The correlation function derived for the stationits length was 40 �s). (b) The correlation function deriv

be noted, the correlation function for the moving targe

not the case for an ultrasonic reflection from a target

for which the acoustic impedance is lower than themedium’s acoustic impedance. In the later case, thesignal is reflected with a phase shift of � and, there-fore, the corresponding average phase gap between thetransmitted and reflected chirps becomes: � �. Fora system that contains targets with a variety of imped-

Fig. 4. Simulated effect of noise on the velocity estimation.SNR was varied from –20 dB to �20 dB. The error bars depictthe mean and SD of the estimated velocity relative to theaccurate value. As can be noted, the mean velocity estimates

rget (chirp frequency ranged from 3 MHz to 5 MHz anda moving target (same chirp, velocity � 1 m/s). As canymmetric and distorted relative to the stationary case.

ary taed for

are fairly stable throughout the SNR range.

sents t

Velocity measurements using frequency-modulated chirp ● Y. LEVY and H. AZHARI 773

ances (higher and lower than the medium), the dy-namic range is further limited to an absolute phasechange of no more than �/2. Hence, the chirp’s pa-rameters should be set to

vmax � S0c

8fm2. (19)

Studying the regression line obtained in the experi-mental part, it can be noted that the slope differs fromunity, i.e., VMeasured � Vtarget 0.8478. This may stemfrom the following reasons. (i) The central instanta-neous frequency fm actually represents the combinedeffect of all the frequencies in the transmitted band.Hence, the changes in fm caused by the moving targetare not identical to the changes that a single frequencywould experience as a result of the Doppler effect. (ii)The actual transmitted signal is not an ideal linearfrequency-modulated chirp, but is distorted by theimpulse response of each element in the transmissionsystem. (iii) The lower frequencies are dominant in the

Fig. 5. Measured velocity (using the OCP, eqn (16)) vs.bars represent one SD. The solid line repre

received signal (because of the frequency-dependency

of the attenuation) and, therefore, the OCP tends to besmaller than expected. Nevertheless, this problem canbe simply overcome by using a calibration process.

In conclusion, the suggested method can be used toestimate the velocity of a moving target using a singletransmitted linear frequency-modulated chirp. This maypotentially yield high frame rate of velocity estimations.The method was verified using computer simulations andexperimental measurements with an ultrasonic system.

REFERENCES

Ha STT, Sheriff RE, Gardner GHF. Instantaneous frequency, spectralcentroid, and even wavelets, Geophys Res Lett 1991;18:1389–1392.

Jensen JA. Estimation of Blood Velocities Using Ultrasound. Cam-bridge: Cambridge University Press; 1996.

Misaridis T, Jensen JA. Use of modulated excitation signals in medicalultrasound. Part I: Basic concepts and expected benefits. IEEETrans Ultrason Ferroelectr Freq Control 2005;52:177–191.

Wilhjelm JE, Pedersen PC. Target velocity estimation with FM and PWecho ranging Doppler systems—Part I: Signal analysis. IEEE Trans

get velocity set by the motion control system. The errorhe regression line obtained for these data.

the tar

Ultrason Ferroelectr Freq Control 1993;40:366–372.