use of modulated excitation signals in ultrasound....

192 ieee transactions on ultrasonics ferroelectrics and frequency control vol 52 no 2 february 2005

Use of Modulated Excitation Signals in Medical Ultrasound

Part II Design and Performance for Medical Imaging Applications

Thanassis Misaridis and Joslashrgen Arendt Jensen Senior Member IEEE

AbstractmdashIn the first paper the superiority of linear FM signals was shown in terms of signal-to-noise ratio and roshybustness to tissue attenuation This second paper in the series of three papers on the application of coded excitashytion signals in medical ultrasound presents design methods of linear FM signals and mismatched filters in order to meet the higher demands on resolution in ultrasound imagshying It is shown that for the small time-bandwidth (TB) products available in ultrasound the rectangular spectrum approximation is not valid which reduces the effectiveness of weighting Additionally the distant range sidelobes are associated with the ripples of the spectrum amplitude and thus cannot be removed by weighting Ripple reduction is achieved through amplitude or phase predistortion of the transmitted signals Mismatched filters are designed to effishyciently use the available bandwidth and at the same time to be insensitive to the transducerrsquos impulse response With these techniques temporal sidelobes are kept below 60 to 100 dB image contrast is improved by reducing the energy within the sidelobe region and axial resolution is preserved

The method is evaluated first for resolution performance and axial sidelobes through simulations with the program Field II A coded excitation ultrasound imaging system based on a commercial scanner and a 4 MHz probe driven by coded sequences is presented and used for the clinishycal evaluation of the coded excitationcompression scheme The clinical images show a significant improvement in penshyetration depth and contrast while they preserve both axial and lateral resolution At the maximum acquisition depth of 15 cm there is an improvement of more than 10 dB in the signal-to-noise ratio of the images

The paper also presents acquired images using compleshymentary Golay codes that show the deleterious effects of attenuation on binary codes when processed with a matched filter also confirmed by presented simulated images

I Introduction

It was established in the first paper of this series [1] that the linear frequency modulated (FM) signal has the

best and most robust performance for SNR improvement

Manuscript received July 31 2002 accepted August 10 2004 This work was supported by grant 9700883 and 9700563 from the Danish Science Foundation and by B-K Medical AS

T Misaridis is currently with the National Technical University of Athens IASA PO Box 17214 10024 Athens Greece (email thmiiasagr)

J A Jensen is with the Center for Fast Ultrasound Imaging OslashrstedbullDTU Bldg 348 Technical University of Denmark DK-2800 Lyngby Denmark (email jajoersteddtudk)

and attenuation effects This is due to its unique symmeshytry properties and their implications on pulse compression which will be discussed in details in Section IV Nonlinear chirps and most binary codes do not possess this linearshyity and therefore tend to be more sensitive to frequency shifts from the medium to be imaged a property which is of great importance in ultrasound imaging Coded excitashytion in medical ultrasound can be used for improving the signal-to-noise ratio (SNR) andor the penetration depth as long as both sidelobe level and energy are kept below the limits of the typical dynamic range of an ultrasound image This paper primarily focuses on optimized design guidelines of pulse compression schemes that can meet the demands on resolution of high-performance coded ultrashysound systems

This paper is organized as follows In Section II the compression and resolution properties of the linear FM signal are given Section III deals with mismatched filtershying techniques as well as the effect of the transducer on pulse compression In Section IV the spectrum of the linshyear FM signal is derived analytically revealing the symmeshytry properties and the Fresnel distortions Based on this analysis FM signals and mismatched filters are designed (Section IV) yielding compression outputs with good axshyial resolution and very low sidelobes The proposed excitashytion scheme is evaluated through simulations and clinical images in Section VI and Section VII respectively Comshyplementary Golay codes also are considered and discussed in Section VIII Section IX gives a short discussion on the achievable resolution on the application of codes in array imaging and on flow coded imaging Section X summashyrizes the main findings of this work

II Compression Properties of the Linear FM Signal

The general linear FM (or chirp) signal can be expressed in complex notation as

B T T ψ(t) = a(t) middot exp j2π f0t + t2 minus le t le

2

2T 2 (1)

where f0 is the center frequency T is the signal durashytion and B is the total bandwidth that is swept The time

c0885ndash3010$2000 copy 2005 IEEE

193 misaridis and jensen use of modulated excitation signals in medical ultrasound part ii

derivative of the phase is defined as the instantaneous freshyquency fi and is given by

1 dΦ(t) d f0t + B t2

B fi = = 2T = f0 + t (2)

2π dt dt T

The linear FM signal has a quadratic phase modulation function and therefore a linear instantaneous frequency function of time Although the term instantaneous freshyquency contradicts with Fourier theorymdashin which a signal has to be infinite in time in order to be band-limitedmdashin practical terms fi indicates the spectral band in which the signal energy is concentrated at the time instant t The pashyrameter k = BT is referred to as the FM slope or the rate of FM sweep The signal sweeps linearly the frequencies in the interval [f0 minus B

2 f0 + B ]2 For digital generation of the FM signal an alternative

expression that has no negative times has the form 2ψ(t) = a(t) middot exp j2π f0 minus B

2 t +

2B

Tt 0 le t le T

(3)

The complex envelope of the linear FM signal described by (1) when the real envelope is rectangular is

t 2jπ(BT )tmicro(t) = rect e (4) T

The matched filter output when the returned signal is not frequency shifted by attenuation is the autocorrelation of the signal which can be expressed as a function of the modulation function

infin infin j2πf0 τRψψ(τ) = ψ(t)ψ lowast(t + τ)dt = e micro(t)micro lowast(t + τ)dt

(5) minusinfin minusinfin

Substituting (4) into (5) one can obtain an analytical exshypression of the matched filter response for the linear FM signal The derivation can be found in any book from the radar literature [2] and the result is

1 minus |τ |sin πD τ T T j2πf0 τRψψ(τ) = T e (6) πD T

τ

where D = TB is the time-bandwidth product Matched filtering removes any frequency modulation and the outshyput is an amplitude-modulated signal at the carrier freshyquency f0 with an approximately sinc shape The apshyproximation to the sinc function improves as the timeshybandwidth product D increases

The first zero of (6) is often taken as a measure of the time resolution It is given by [3]

T 4 τr = 1 minus 1 minus

2 BT(7)

T 2 1 asymp 1 minus 1 minus = 2 BT B

Fig 1 Resolution for pulsed and linear FM excitation The pulse shown here (gray line) is the envelope of an apodized sinusoid of the carrier frequency with a Hanning apodization The length is 27 cycles and is chosen to match the bandwidth of the chirp for direct comparisons The black thin line is the compressed envelope of a linear FM signal with D = TB = 36

in which the binomial expansion has been used retaining only first-order terms Imaging with a short pulse of length T will yield axial resolution of T = 1B Thus axial resshyolution for a short (conventional) excitation pulse and for an FM-modulated (coded) excitation will be roughly the same when the signals use the same bandwidth This is shown in Fig 1

The side effect of pulse compression on the linear FM is the resulting sinc sidelobes not present in conventional pulse excitation Range sidelobes represent an inherent part of the pulse compression mechanism In an imagshying situation the effects of the time (or range) sidelobes extending on either side of the compressed pulse will be self-noise along the axial direction and masking of weaker echoes For the linear FM signal the highest of these sideshylobes are the first ones only minus132 dB below the peak of the compressed pulse The near sidelobes fall off at apshyproximately 4 dB per sidelobe interval and the sidelobe null points are spaced approximately 1B apart

III Mismatched Filtering and the Ultrasound Transducer

The temporal sidelobes after compression are thus high and this will significantly degrade the image conshytrast because a dynamic range of 60 dB or more is typically used in ultrasound imaging This section studies methods for reducing the temporal sidelobes to an acceptable level

A Weighting in Frequency and Time Domain

The usual approach for sidelobe reduction is to apply a window function on the matched filter Due to symmetry properties which will be discussed in detail in Section IV weighting can be performed either in the time or in the frequency domain


The response for a matched filter given by (5) can be expressed as a function of the signal spectrum

infin infin

γ (t) = Ψ(f ) middot H (f )ej2πftdf = |Ψ(f )|2 ej2πftdf

0 0 (8)

where H (f ) = Ψlowast(f ) Frequency-domain filtering is the shaping of the amplitude spectrum of the signal with a real window function W (f ) The output of the so called mismatched filter then is

infin

j2πftdfγw(t) = Ψ(f ) middot Hw(f )e0 (9) infin

2= W (f ) middot |Ψ(f )| ej2πftdf

0

where Hw(f ) = W (f ) middot Ψlowast(f ) Using the rectangular FM spectrum approximation the function W (f ) can be deshysigned as the Fourier transform of a desired output time function γw(t)

Hw(f ) = γw(t)e minusj2πftdt (10)

With this technique the mismatched condition applies only on the amplitude spectrum and the receiver transfer phase remains conjugate function of the spectrum phase The sinc function of the matched filter response correshysponds to the rectangular window

Time-domain processing is based on amplitude weightshying of the envelope of the transmitted FM signal or of the matched filter impulse response The former case usually is avoided because it is desirable to transmit as much enshyergy as possible In the latter case the transfer function of a mismatched filter using time-weighting is the complex conjugate of an amplitude-weighted version of the excitashytion signal Time weighting filter design is based on the following relationships

FFM signal ψ (t) larrrarr Ψ(f ) FMismatched filter ψ (t) middot w(t) larrrarr H (f ) rarr H lowast(f ) F lowast(f ) (11) Compression output γ (t) larrrarr Ψ(f ) middot H

Sidelobe reduction in matched filter receivers is reduced to the choice of an appropriate window function There is a plethora of window functions reported and Harris [4] gives a thorough review of the most common windows and their properties Windows such as Hanning Kaiser Blackman Hamming etc are discussed A more systematic window design optimization technique is reported in an excellent paper by Adams [5] It is based on a trade-off between the window design parameters which are the mainlobe width the total sidelobe energy and the peak sidelobe level The energy contained in the sidelobe region is associated with the contrast in the image High sidelobe energy will cause leakage of energy from bright areas into dark areas in the

Fig 2 Compression outputs for two mismatched filters based on time weighting with Hamming (upper graph) and Dolph-Chebyshev windowing (lower graph)

image Such a case is imaging of cysts in which an hyshypoechoic dark region is surrounded by a bright contour The window function that minimizes the sidelobe energy is the prolate-spheroidal window However the resulting peak sidelobe level can be very high

On the other extreme stands the Dolph-Chebyshev window which minimizes the peak sidelobe level The Dolph-Chebyshev window exhibits the minimum mainlobe width for a specified constant sidelobe level and has been used extensively in radar systems Adams [5] describes a method for optimal windows that lie between these two extremes

Fig 2 gives two examples of mismatched time-weighting with a Hamming and a Dolph-Chebyshev window apshypliedThe matched filter response also is plotted for comshyparison A FM signal with a time-bandwidth product of 62 has been used For the Hamming window the first sidelobes close the mainlobe have been reduced to about minus40 dB The deviation from the theoretical minus428 dB [4] as well as the lack of symmetry are an indication that the limitedshyenergy relatively short FM signal is not band-limited and therefore not equal to the analytical signal Further reshyduction of the first sidelobes is achieved with the Dolph-Chebyshev window designed to yield minus60 dB constant sidelobe level The lower sidelobe levels achieved with misshymatched filtering are accompanied with a broadening of the mainlobe and associated loss in axial resolution and a loss in SNR The minus20 dB mainlobe width (axial resolution) of the mismatched filter output using Hamming weightshying is 18λ and the mainlobe width for Dolph-Chebyshev weighting is 22λ For comparison the mainlobe width for the matched filter case is 11λ and for a conventional 4shycycle pulse with Hanning apodization is 12λ


Fig 3 Compression outputs in the absence (top graph) and presence (bottom graph) of the transducer showing the effect of the transshyducer on pulse compression The faint lines are the matched-filter outputs and the bold lines are the outputs when a Dolph-Chebyshev window has been applied to the compression filter The specified sidelobe level for the window was minus90 dB

B The Effect of the Ultrasonic Transducer on Pulse Compression

In ultrasound bandpass filtering of the excitation signal

that although suboptimal is beneficial to sidelobe reducshytion Fig 3(bottom) shows that matched filtering of linear FM signals transmitted by an ultrasound transducer yields compression with sidelobe levels of minus37 dB without any weighting on the receiver Transmitting a weighted pulse (which occurs by the transducer) is desirable because a weighted transmitted signal or spectrum is less sensitive to mismatches in frequency shifts [6] In high power radars weighting of the transmitted pulse is generally not used beshycause the final amplifier stages operate in class C which have no amplitude control [2]

Fig 2 shows that weighting reduces the sidelobes that are close to the mainlobe but not the distant sidelobes that are in the plusmnT2 region The distant sidelobes increase the mainlobe-to-sidelobe energy ratio (MSR) and thus degrade the image quality

IV Fresnel Ripples and Paired-Echoes Sidelobes

The efficiency of windowing is based on the concept of shaping a rectangular spectrum in order to smooth out the sidelobes of the sinc function in the time domain in accordance with (10) However the rectangular spectrum assumption is only an approximation As will become clear from the spectrum analysis that follows the distant sideshylobes are attributed to the spectral distortions and thus are not controlled by the weighting function

For a linear FM signal with rectangular envelope (a(t) = 1) its Fourier transform gives ⎡

T2

⎛ ⎢⎢ ⎜ 1 B Ψ( ) =

exp ⎢ 2 ⎜⎜ + 2f j π f0t t

⎞⎤

T2

⎟⎣ ⎝ 2 Tminus

k

⎟⎥⎟⎠ ⎥

( j2πft) dt

⎦⎥middot exp minus (12) T2

= exp

k 2j2π (f0 minus f)t + t

T

dt

2minus 2

Completing the square in the brackets

2k

( 2 k f f0 (f f 0)2

f0 minus f)t + t = t 2 2

minus minus minus

k minus

2k

the integral becomes

T2 π

Ψ( e f) = xp minusj (f minus 2f0)

k

minusT

2

f fexp

2

π 0j 2k

t

minus 2

minus k

dt (13)

and by introducing the new variable y =radic

2k t minus fminusf0k

we get

occurs inherently from the transducer This has two conseshyquences the gain in SNR is significantly lower than what radic is achieved in radar systems (ie lower than TB) and the sidelobes from pulse compression is less of a problem The transducer effect on pulse compression is illustrated in Fig 3 The FM signal is convolved with the measured pulse-echo impulse response of an ultrasonic transducer and the resulting signal is used as the input in the matched filter The transducer used is a 4 MHz single-element transshyducer (Model 8534 B-K Medical AS Gentofte Denmark) and the calculated minus6 dB fractional bandwidth was 65

When the bandwidth of the chirp matches the transshyducerrsquos bandwidth the presence of the transducer reduces the near range sidelobes from minus2 to minus32 dB below the mainlobe The linear FM signal used for Fig 3 had 12 higher bandwidth and that reduces the sidelobes further down to minus42 dB When mismatched filtering is applied (bold lines of Fig 3) near range sidelobes can be furshyther reduced at the expense of an additional broadenshying in the main lobe Further reduction of the near sideshylobes is possible when the compression filter is matched not to the excitation signal but to the signal after that has passed through a simulated transducer However the latter approach shapes the filter according to the transshyducerrsquos bandwidth and is less stable to deviations of the simulated impulse response to the actual one The effect of the transducer is equivalent to an additional weighting


Fig 4 The Fresnel integrals (left) and the spectrum amplitude of the linear FM signal (right)

Fig 5 The phase distortion term ϑ2(f) of the spectrum of a linear FM signal with a time-bandwidth product of 120

radic 1 π(f minus f0)2Ψ(f) = exp minusj

2k k The phase ϑ(f) of the frequency spectrum from (18) ⎤⎡

Y2 Y2 can be written as a sum of two terms ϑ1(f) and ϑ2(f) where

π π2 2 (14) dy + j sin dy⎦cos⎣middot y y2 2

minusY1 minusY1 2π f minus f0

ϑ1(f) = minus (f minus f0)2 = minusπDThe new limits of the integral are k B

⎫ ⎪⎪⎪⎬

⎪⎪⎪⎭

(19) S(Y1) + S(Y2)

C(Y1) + C(Y2) radic radicT f minus f0 T f minus f0 ϑ2(f) = tanminus1

2k + Y2 = 2k minus Y1 = 2 2k k (15)

The limits can be written in an alternative form as a function of the time-bandwidth product D where D = TB

D f minus f0Y1 = 1 + 2

2 B (16)

The term ϑ2(f) is shown in Fig 5 It is approximately constant and equal to π4 within the signal bandwidth The higher the time-bandwidth product the better the approximation to a constant value Thus for high values of D the phase of the signal is only the quadratic function of frequency given by ϑ1(f) plus the constant π4

The amplitude term

[C(Y1) + C(Y2)]2

radic +

S(Y1) 2 12D f minus f0 + S(Y2)1 minus 2 is approximately equal to 2 for f = f0Y2 =

2 The Fresnel ripples oscillate around this value and for B

The two integrals in (14) can be written as sum of the Fresnel integrals given by

z z

2 2C(z) = cos π

y dy S(z) = sin π

y dy 2 2

0 0 (17)

The final form of the frequency spectrum of the linear FM signal then is

Ψ(f) = radic 1 exp minusj

π(f minus f0)2

2k k

middot C(Y1) + C(Y2) + j [S(Y1) + S(Y2)] (18)

The Fresnel integrals defined in (17) are shown in Fig 4(a) They approach the value 12 when the argushyment is much larger than 1 The spectrum amplitude of the linear FM signal is plotted in Fig 4(b) Because the argushyments of the Fresnel integrals are a function of the timeshybandwidth product the spectrum amplitude approaches the rectangular for high values of D In particular the amshyplitude of the oscillation decreases with D and the number of the ripples increases with D

high TB the amplitude can be considered rectangular in radic the passband with a value of 2 Rihaczek considers that the rectangular approximation is valid when D ge 10 [7] The approximate phase of the spectrum is ϑ1(f) + π4 Therefore the high TB approximation of (18) for the specshytrum of the linear FM signal is

1 f minus f0 π πΨ(f) = radic rect exp minusj (f minus f0)2 +

B k 4k (20)

Rihaczek [7] derived the approximate spectrum of the genshyeral linear FM signal with an arbitrary real envelope a(t) given in (2) as

1 f minus f0 π πΨ(f) = radic a exp minusj (f minus f0)2 +

k kk 4 (21)

Therefore the spectrum amplitude has the same funcshytion as the envelope of the signal in the time domain simshyply being obtained by substituting t with (f minus f0)k and scaling the amplitude Additionally the phase functions φ(t) and ϑ(f) in the time and frequency domains respecshytively are both quadratic functions or alternatively the


instantaneous frequency and group delay are linear funcshytions There is therefore a symmetry in the time and freshyquency functions which is a unique property of the linshyear FM signal However this symmetry is only a high time-bandwidth product approximation A rectangular enshyvelope in the time domain actually yields a rectangular spectrum amplitude with Fresnel ripples as well as phase distortion

For high-quality imaging the Fresnel distortions have to be taken into account in the design of the transmitted signal and the appropriate matched filter The efficiency of windowing as described in Section III is based on the concept of shaping a rectangular spectrum which does not take into account the Fresnel ripples of the spectrum amshyplitude and phase described by (18) and (19) The distant range sidelobes are paired-echo distortions due to devishyation from the ideal rectangular spectrum amplitude and quadratic phase The ripples given by the Fresnel integrals of (18) can be approximated with a combination of sinushysoids [2] With such an approximation calculations with the nonclosed form of the Fresnel integrals are avoided and useful results can be obtained The effect of amplishytude error in the compression output can be assessed by the following Fourier pairs [8]

G(f) G(f) 1 + an cos 2πn f B

(22) 1 1 ang(t) t + n + g(t) + an t minus n 2 g B 2 g B

The presence of n spectrum amplitude ripples of amplishytude an over the passband B of a signal spectrum G(f) create symmetrical paired echoes in the time domain deshylayed and advanced from the main signal by nB and scaled in amplitude by an2 Similar paired-echoes result from the ripples in the spectrum phase [8] Fig 6 shows an application of (22) for a spectrum having 20 amplitude ripples (n = 20) which oscillate 20 about the constant value of the spectrum amplitude (an = 02) In this case the resulting distant sidelobes have a relative amplitude of 20 log(an2) = 20 dB below the mainlobe peak The numshyber of ripples and their oscillating amplitude is a function of the time-bandwidth product Therefore the amplitude of the resulting sidelobes and their time displacement also will be a function of TB For an FM signal with TB = 100 the approximate paired-echo sidelobes are at about plusmnT2 relative to the center of the compressed pulse with an amshyplitude of about minus35 to minus40 dB [9] This is higher than the displayed dynamic range of ultrasound images therefore methods that compensate for the distant axial sidelobes have to be used This is an important consideration in the design of efficient coded excitation systems with low sidelobes

V Linear FM Signal Design

The origin of the distant sidelobes around the plusmnT2 region was discussed in detail in the Section IV Based on

Fig 6 The presence of n spectrum amplitude ripples of amplitude an

over the passband B of a signal spectrum G(f) create symmetrical paired echoes in the time domain delayed and advanced from the main signal by nB and scaled in amplitude by an2

these findings FM signals with reduced Fresnel distortions in the spectrum amplitude will be discussed in this section

A Amplitude and Phase Shaping

In order to eliminate the paired-echo distortions freshyquency weighting functions could be designed and applied on the received signal to cancel the spectrum ripples of the linear FM signal The ripple structure of the weighting function to be designed is a function of the time-bandwidth product and thus different weighting functions should be designed for various TB products There are two drawshybacks of such attempt the transmitted FM signal passes through a transducer with an unknown impulse response before it starts propagating through the tissues This will cause distortions and the ripple cancellation of the weightshying function might not be exact The second drawback is that the sidelobe structure will become sensitive to freshyquency shifts of the returned signal

A more practical and robust approach is to search for methods to reduce the ripples of the FM spectrum then apply a smooth weighting function on the receiver [2] Three different approaches for ripple reduction have been tested by our group [10] envelope modulation phase preshydistortion and time weighting of the edges

B Envelope Modulation

This approach is based on the symmetry property of the linear FM signal in time and frequency A linear FM signal with a constant amplitude envelope in the time doshymain yields a rectangular amplitude spectrum with ripple


Fig 7 FM signal with Fresnel distortion in amplitude and phase (up) and its spectrum amplitude (down) The spectrum of a linear FM signal with constant amplitude envelope is shown for comparison in gray in the bottom graph

distortion The reciprocal signal with a constant amplitude spectrum and strictly quadratic phase will have an amplishytude and phase in the time domain given by (18) if T is reshyplaced with B and t with f [7] Using this concept an FM signal with a duration of T = 20 micros and a total bandwidth of 57 MHz (TB = 114) has been designed The time signal and its spectrum amplitude are shown in Fig 7 Although there is some improvement there are still substantial ripshyples in both the amplitude and the phase of the spectrum (not shown here) The technique however is very efficient for small fractional bandwidths when the exponential sigshynal is analytic and (18) is a good approximation for the signal spectrum The disadvantages of this technique are the time signal is complex such amplitude-modulation reshyquires amplifiers with fast response and the efficiency is low for time-bandwidth products greater than 30

C Phase Predistortion

An alternative approach with similar performance is to use a phase predistortion of the transmitted signal because amplitude and phase distortions have functional similarity and produce similar effects [9] Both quadratic and cubic predistortion functions were reported previously [9] [11] In both papers the quadratic phase function of the signal was modified at the beginning and at the end approxishymating essentially a nonlinear FM instantaneous frequency function using linear [9] or quadratic [11] segments The design parameters are the length of the two end segments and the slopes of their instantaneous frequency curves Similar to nonlinear FM signals phase predistorted chirps have the advantage of sending out more energy but they

Fig 8 FM signal with amplitude tapering of the edges (up) and its spectrum amplitude (down) showing substantial ripple reduction The spectrum of a linear FM signal with constant amplitude envelope is shown for comparison in gray in the bottom graph

are more vulnerable to phase distortions added from acousshytic propagation and frequency shifts

D Amplitude Tapering

The amplitude ripples of the spectrum can be attributed to the sharp rise-time of the time envelope because a linshyear FM signal with infinite duration has no ripples Data correlating pulse rise-time and spectrum ripple were reshyported previously [9] Based on this analysis a modified chirp with amplitude tapering of the transmitted signal has been generated The attainable ripple reduction is a function of the signalrsquos bandwidth the choice of the tashypering function and its duration Fig 8 shows the effect of amplitude tapering on spectrum ripple reduction The tapered function used in Fig 8 is a Tukey window with a duration of 015 middot T

Amplitude tapering is the most efficient way to reduce the Fresnel ripples of the spectrum if the power amplifier allows control of the transmitted pulse rise time For a given duration T the design parameters are

bull the frequency band that is swept relative to the transshyducerrsquos bandwidth

bull the choice of the tapering function and its duration bull the choice of the weighting function

The transducerrsquos bandwidth can be modeled but the phase of its transfer function introduces an unknown phase factor in the signal after convolution which cannot be compensated for The effort here is to minimize the efshyfect of the convolution between the transducer impulse response and the excitation signal on the design of the


Fig 9 Frequency contents of the excitation FM signal of the acshytual transmitted FM signal (convolved with the one-way transducer impulse response) and of the received and compressed signal

Fig 10 Optimized compression outputs for FM signals with amplishytude tapering The left graph uses a weighted filter matched to the tapered signal In the right graph the filter is matched to the signal convolved with a simulated transducer impulse response The filters are applied to a received signal that takes into account the transducer pulse-echo impulse response

waveforms and the compression filter This is done in the excitation signal by sweeping a bandwidth slightly larger than the transducerrsquos frequency passband [12] This has an additional advantage the effect of the two main rise and fall overshoots of the spectrum is minimized (see Figs 8 and 9) The increased bandwidth of the excitation waveshyform also yields a higher gain The frequency contents of the signals are shown in Fig 9 With this design the range sidelobes can be as low as minus882 dB [Fig 10(left)] Fig 10(right) shows the compression output when the filshyter is matched to the tapered signal convolved with a simshyulated transducer impulse response This approach broadshyens the mainlobe but makes the sidelobe performance comshypletely independent on the actual impulse response of the transducer Fig 11 shows optimized compression outputs for different frequencies and signal duration Higher timeshybandwidth product tapered FM signals have nearly ideal rectangular spectrum allowing efficient sidelobe reduction well below 100 dB by the weighting function If the length of the excitation signal is doubled the sidelobes drop down to minus1055 dB with a good resolution of 145λ (Fig 11) For comparisons the minus20 dB axial resolution with a typishycal pulse excitation is 146λ The design gives even better results for higher frequencies

Fig 11 Optimized compression outputs for amplitude tapered FM signals with double the signal duration (left graph) and double the carrier frequency (right graph) compared to the tapered FM signal with the output shown in Fig 10

Fig 12 Trade-off between sidelobe level and axial resolution for a number of Dolph-Chebyshev mismatched compression filters For most applications the appropriate choice is at points in the knees of the curve as the one indicated by the arrow

The appropriate choice of the weighting function is a tradeoff between axial resolution and sidelobe levels This is illustrated in Fig 12 in which the compression resolution and peak sidelobe level of 15 compression filshyters with different Dolph-Chebyshev window functions are shown An implementation of the new scheme easily can give the flexibility to switch between finer resolution or lower range sidelobes depending on the application needs Dolph-Chebyshev windows although optimal in sidelobe level performance exhibit spikes at their leading and trailshying edges The use of other optimal windows described in [5] might result in better performance

The compression for the design shown in Fig 10(right) retains its characteristics regardless of the actual transshyducer impulse response For the more sensitive design of Fig 10(left) the effect of the transducer is shown in Fig 13 The degradation in the compression properties still is not significant for this design

The design presented in this section combines low sideshylobes immunity against the effect of the transducer and resolution comparable to that achieved with pulsed excishytation Additionally as was shown in the first paper of the series [1] the linear FM will retain its good compression properties in the presence of attenuation in tissues Evalshy


Fig 13 The effect of the transducer on the new scheme The black line is the compression output of Fig 10(left) The dotted line is the compressed output for the same filter when the actual transducer impulse response is used on the received signal It is shaped by the envelope of the measured impulse response shown by the gray line

Fig 15 Pulsed versus FM-coded excitation imaging in a medium with attenuation of 07 dB[MHz times cm]

Fig 14 Pulsed versus FM-coded excitation imaging in a medium with no attenuation

uation of the above statements will be presented in the following sections through simulations and clinical images obtained using this coded excitation scheme

VI Imaging with Linear FM SignalsmdashSimulation Results

Figs 14 and 15 show simulated images using convenshytional pulsed and linear FM excitation Simulations have been performed with the simulation program Field II [13] The simulation parameters correspond to the singleshyelement mechanically rotating transducer with a nominal frequency of 4 MHz and a 65 minus6 dB bandwidth that has been used in the experiments Eight point scattershyers along the axial direction are imaged The pulsed exshycitation is a sinusoidal signal of four cycles with Hanshyning apodization which matches the transducer impulse

Fig 16 The minimal effect of attenuation in sidelobe levels using tapered linear FM excitation with mismatched filtering The graphs show the central rf-lines of the coded images in the absence (left) and presence (right) of attenuation in the medium

response The FM-coded excitation is the one whose misshymatched filter response is shown in Fig 10(right) The simulated images shown in Figs 14 and 15 correspond to a nonattenuating medium and a medium with attenuation of 07 dB[MHz times cm] respectively The visual appearance of pulsed and coded images is very similar The sidelobe performance of pulse compression can be seen in Fig 16 in which the compressed central rf-line is plotted With no attenuation the sidelobe levels are very close to what is expected from Fig 10(right) ie the sidelobes are close to minus88 dB for all depths In the attenuating medium (Fig 15 and the right graph of Fig 16) the sidelobes increase with depth but remain below minus60 dB for all depths Compresshysion is robust to attenuation and this is also the case when the actual transducer impulse response has been used in the simulations


Fig 17 The ultrasound scanner (B-K Medical Model 3535) used in the experiments

Fig 18 The single-element transducer (B-K Medical) that makes sector images by mechanical rotation

VII Clinical Evaluation

A Experimental Setup

The experimental system used was based on a commershycial scanner (B-K Medical 3535) shown in Fig 17 with a single-element mechanically rotating transducer The transducer is motor driven and scans a sector as shown in Fig 18 The scan angle is fixed at 75 degrees and the pulse repetition frequency can be set in six modes from 1 to 34 kHz Depending on the settings for imaging depth the machine can acquire from 34 to 212 lines The machine was set for most experiments to a mode for acquisition of 106 lines scanning a depth from 0 to 180 mm with a pulse repetition frequency of 3 kHz

The transducer crystal has a nominal frequency of 4 MHz and a fixed focus at 60 cm The impulse response of the transducer was measured and the minus6 dB bandwidth was found to be 65 The single-element module on the scanner was modified in order to operate with external excitation The transmitter was inactivated and was intershyfaced to an external transmitter board A power radio freshyquency (RF) amplifier (RITEC 5000 Ritec Inc Warwick

Fig 19 The experimental system using boards from the Center for Fast Ultrasound Imagingrsquos newly constructed RASMUS experimenshytal system [14]

RI) specifically designed to drive ultrasound transducers was used subsequently for amplification Sampling of the rf-data was done through a receiver board The transmitshyter and receiver boards that were used are from the Censhyter for Fast Ultrasound Imagingrsquos newly constructed RAS-MUS experimental system [14] A picture of this system is shown in Fig 19 The software control of both boards is implemented as a MATLABTM (The Mathworks Natick MA) toolbox of high-level commands written by Dr Sveshytoslav Nikolov at the Center for Fast Ultrasound Imaging The transmitter board is capable of transmitting differshyent complex arbitrary waveforms for each line of an imshyage with a few lines of MATLABTM code The board was programmed to allow alternating excitation on every secshyond frame That allowed direct comparison of the same set of image pairs one with conventional and one with enshycoded excitation or image pairs with complementary seshyquences The analog RF data of the scanner were sampled at 40 MHz The receiver sampling unit has 12-bit ADCs and 2 GBytes storage SDRAM That allowed storage of 140 consecutive ultrasound images corresponding to about 25 s of scan data After sampling the stored RF data were read from the memory and all post processing and display was done on the computer This includes high-pass filtershying pulse compression interpolation and scan conversion logarithmic compression envelope detection and display The whole system was running under Linux and could be operated from any computer connected to the network

The peak excitation voltages are 32 V p-p for the conshyventional pulse and 20 V for the chirp that yield the same Isptp The applied excitations are very low compared to what is commonly used in ultrasound scanners Therefore the noise level in the conventional images shown here is higher that that of routinely used images of the BK-3535 scanner However the purpose of this study is to show a relative comparison of noise and resolution between pulsed and coded excitation images


Fig 20 Detail of images of a wire phantom (right) and central RF lines (left) for coded and pulsed excitation Matched filtering has been applied to both images The dynamic range of the images is 45 dB From the graphs on the left an improvement in SNR of about 10 dB can be seen Axial resolution is also higher for the coded image

B Phantom Images

Images of a wire phantom are presented in Fig 20 The phantom that was used (Model 525 Dansk Fantom Sershyvice Jyllinge Denmark) has an ultrasonic attenuation of 07 dB[MHz times cm] and consists of wires of 02 mm in in diameter positioned every 1 cm axially Additional wires were placed at a 15 degree angle with decreasing distance down to 1 mm In the pulsed image of Fig 20 a twoshycycle pulse of the carrier frequency was used followed by a matched filter In the coded image the excitation was a 20 micros tapered FM signal with the theoretical compression output shown in Fig 10 The minus6 dB axial resolution of the coded image measured at the wires in depth 12 and 14 cm was 131λ The resolution of the pulsed image was 149λ In conventional imaging short broadband pulses have to be used in order to achieve good axial resolution and to use all available system bandwidth with the drawshyback of degrading the SNR If higher SNR is needed longer pulses have to be transmitted which in turn use less of the

available bandwidth and result in worse axial resolution Coding releases this constraint and allows high SNR usshying the available bandwidth independently of the duration of the signal The images of Fig 19 show a comparison under equal conditions in terms of bandwidth and SNR optimization through matched filtering Under these conshyditions coded images using FM modulated signals result in better SNR with the incentive of slightly better axial resolution Fig 20 shows that there is an effective gain in SNR of 10 dB or more that corresponds to an additional penetration of 3ndash4 cm with the 4 MHz probe

C Clinical Images

Clinical images of the abdomen using the proposed scheme are shown in Fig 21 The interested reader can find more clinical images in [12] The images show an imshyproved performance of the encoded excitation in terms of noise reduction at large depths and resolution The autoshycovariance matrix on the image gives an indication of the speckle size The lateral resolution of speckle data taken from the rectangular areas shown in the images of Fig 21 is very similar for both images while axially the coded imshyage has slightly better performance (Fig 22) These results clearly demonstrate that abdominal ultrasound imaging can benefit from coded excitation yielding a higher SNR and therefore penetration while maintaining both axial and lateral resolution The higher SNR can be exchanged with resolution by increasing the center frequency (ie for GSNR = 10 dB) going from 4 to 5 MHz without comproshymising SNR Longer codes can make this frequency step even bigger

VIII Imaging with Complementary Golay Codes

In the previous section a new FM-coded scheme was presented with distinct features that make it attractive to the implementation of high-performance coded ultrasound systems For comparison phase (or binary) codes will be discussed in this section

The considerations on the applicability of binary codes in ultrasound imaging of attenuating media was previously discussed by the authors [1] [15] using the tool of the amshybiguity function An additional undesired property in comshyparison with the linear FM signal is their poorer sidelobe performance [12] The sidelobes of the linear FM signal are relatively independent of the time-bandwidth product once the spectrum shape is fixed Practically FM signals with a TB greater than 25 have similar sidelobe behavior and thus the same sidelobe reduction techniques are apshyplicable and equally effective regardless of the TB In conshytrast the level of the range sidelobes in phase-coded signals is a function of the time-bandwidth product ie the code length In radar systems in which phase codes have been used successfully code lengths as high as 1000 are possishyble which result in pulse compression systems with range sidelobes down to minus45 to minus50 dB without any weighting


Fig 21 Clinical images of the right kidney for coded and pulsed excitation The portal vein and the inferior vena cava are at the right side of the images and liver tissue is left from the kidney The dynamic range of the images is 45 dB Improvement in resolution and noise reduction at large depths are visible

Fig 22 Evaluation of the lateral (left) and axial (right) resolution in speckle using autocovariance matrix analysis Speckle data are taken from the images of Fig 21 The gray lines correspond to the pulsed image

Fig 23 Imaging with complementary Golay codes in a medium with no attenuation (left two images) and with attenuation of 07 dB[MHz times cm] (right two images)

In ultrasound the length of the code cannot exceed 80ndash 100 at most which in conjunction with the band-limiting effect of the transducer will result in very poor sidelobe performance of about minus25 dB below the autocorrelation peak

A possible exception in terms of sidelobe behavior can be the complementary codes due to their property that the sidelobes of the autocorrelation functions from two codes have opposite signs and can be theoretically canshycelled by addition

Simulated images using complementary Golay code exshycitation are shown in Fig 23 Golay-coded imaging reshyquires two transmit events for every line which decreases the frame rate by half Additionally motion artifacts are expected to degrade the sidelobe cancellation The simshyulation results of Figs 23 and 24 show that even in the case of imaging of stationary tissues the compleshymentary property severely degrades in an attenuating medium Perfect cancellation occurs when the medium has no attenuation Fig 24(left) while with an attenuation of 07 dB[MHz times cm] the range sidelobes increase up to minus25 dB at a depth of 16 cm in contrast to the tapered


Fig 24 The effect of attenuation in sidelobe levels using complemenshytary Golay codes Simulation results with Field II show the echoes from eight point scatterers in the absence (left) and presence (right) of attenuation in the medium

Fig 25 Images with Golay pair excitation of a wire phantom with attenuation of 07 dB[MHz times cm] On the left is the image with one of the Golay codes and on the right is the sum of the two comshyplementary images The dynamic range is 45 dB

FM signal (Fig 16) whose compression is very robust to attenuation When the actual transducer impulse response is used in the simulations the compression performance of Golay codes in the presence of attenuation is rather poor The results of Fig 24 are in agreement with the discussion given in [1] Images of the same phantom using a compleshymentary pair of Golay codes with length 40 are shown in Fig 25 The matched filter used is the phase code conshyvolved with the simulated transducer impulse response A single Golay code has high axial sidelobes that are visible on Fig 25(left) When the echoes from two Golay codes are added coherently there is a degree of cancellation on the axial sidelobes However confirming the simulation reshysults the cancellation of the sidelobes is not perfect due to attenuation even for the stationery phantom imaging and shadows still are visible along the wires

This result indicates the sidelobe degradation for direct phase coding in ultrasound imaging of attenuating media The results shown here involve phase coding of a sinusoidal burst but the same effect applies in a cheaper implementashytion of a bipolar square code An additional coding might be needed in order to shield the coded waveform from the negative effect of attenuation Such an approach is taken in [16] in which the Golay-coded waveform is embedded in another carrier code The simulation results presented in [16] using this approach show a higher robustness to a

Fig 26 Response from a point scatterer positioned at a depth of 160 mm for pulsed linear FM and Golay code excitation waveforms in a medium without (left) and with (right) attenuation

simple model of amplitude-only attenuation that does not include a phase term

IX Discussion and Outlook

A Evaluation of Compression and Resolution

Fig 26 shows the compressed rf-data response from a point scatterer for the coded signals considered in this pashyper The tapered linear FM signal is the one with the comshypression output of Fig 10(right) The mainlobe width is a measure of the axial resolution In the nonattenuating medium the minus20 dB mainlobe width from pulsed excitashytion is 149λ for the tapered linear FM it is 192λ for a nonlinear FM (not shown here) it is 177λ and for the sum of Golay codes it is 217λ That is the mainlobe width of the tapered linear FM is 22 wider than that of the conshyventional pulsed excitation The widening for the nonlinear FM and Golay coded excitation are 16 and 31 respecshytively Imaging with the linear FM with the compression output shown in Fig 10(left) results in axial resolution of 152λ which is only 2 wider than that of the pulsed excishytation In this case the sidelobes are only slightly higher

Fig 26(right) shows the effect of attenuation on the exshycitation signals The superiority of the tapered linear FM signal over pulsed and Golay code excitation is apparent The minus20 dB mainlobe width of the pulse is 09λ of the linear FM it is 213λ of the nonlinear FM (not shown here) it is 241λ and the mainlobe of the Golay code is 3λ severely distorted and split into three lobes The sideshylobes for the linear FM signal are at minus67 dB The energy outside the mainlobe for the linear FM signal is less than that of the pulsed excitation ie image contrast will be improved using linear FM-coded excitation

The rf-lines of Fig 20 contain speckle data and thus do not reveal any information on the range sidelobe level beshycause the scanned phantom contains densely packed scatshyterers similar to soft tissues In further experimental work from our group in synthetic aperture ultrasound imaging Gammelmark and Jensen [17] have used the technique preshysented in this paper to design an FM signal and the correshysponding compression filter in order to obtain theoretical


temporal sidelobe levels of minus60 dB Their measurements from a wire phantom in water has shown temporal sideshylobes below minus55 dB

B Array Imaging

Coded excitation systems described in the literature use a single correlator at the output of the beamformer [18] [19] This approach although advantageous in terms of imshyplementation poses requirements on the code length and arises issues about the effect of time-gain compensation (TGC) and dynamic focusing on pulse compression Dyshynamic focusing is a pixel-based nonlinear operation that will distort the phases of a long coded waveforms In order to minimize this effect OrsquoDonnell [19] suggested that the code should have a duration less than 64f0 which for the typical bandwidth of the ultrasound transducer restricts the time bandwidth of the code to less than 60 Although this time-bandwidth product theoretically should give a SNR gain of more than 10 dB the simulations presented in [1] showed that the transducer weighting and the atshytenuation significantly reduce the expected SNR gain Adshyditional mismatched or inverse filtering required for the high sidelobes of such time-bandwidth product waveforms will shrink the SNR gain further Such a coded excitation system might thus have some limitations in terms of SNR improvement

In a more comprehensive study on the effect of dyshynamic focusing in pulse compression systems Bjerngaard and Jensen [20] have calculated the residual error that reshysults from dynamic focusing when compression is done after the beamformation process They showed the comshyplex dependence of this error from the depth of the transshymit focus and the distance between dynamic receive focal points An important conclusion from their experimental data is that the error decreases with depth and for typical imaging parameters is below minus40 dB at a depth higher than 12 cm and their simulated data show smaller errors at about minus60 dB

In a fully digital system if implementation allows pulse compression should precede and beamformation should follow In this way there are no requirements for the code length in terms of focusing and time-gain compensation Long codes can be used which will result in good compresshysion and SNR gain The disadvantage of this is that comshypression has to be done in all transducer channels which requires one compression filter on every receive channel If there are implementation restrictions and beamformshying has to be done before pulse compression the imaging depth can be divided into several zones and code lengths can be used that increase with depth [21] For instance no coding is used in the nearfield a relatively short code is used in the midfield and a relatively long code is used in the farfield in which SNR gain is most needed SNR improvement is higher [1] and the postcompression errors are negligible [20] This approach also solves the problem in coded excitation systems of a nearfield deadzone equal to half the length of the coded pulse The size of the virshy

tual point source also can increase from shallow to deep zones for further SNR gain without compromising quality of the point source [21]

C Nonlinear Propagation and Harmonic Imaging

The transmitted FM signals do not contain any higher harmonics This has the advantage in harmonic imaging that all higher harmonics are solely due to nonlinear propshyagation The compression filter does not contain any higher harmonics either because it has essentially the same amshyplitude spectrum as the transmitted signal except some weighting This eliminates any nonlinear distortion when imaging in the fundamental frequency Bi-phase codes and their corresponding matched filter however have higher harmonics which will not be filtered out

For second harmonic imaging FM coding combined with the concept of matched filtering can be used with no need of pulse inversion The harmonic field of an FM signal is also an FM signal with double the bandwidth therefore it can provide good axial resolution A small freshyquency band can be swept at the fundamental frequency thereby eliminating bandwidth overlap between harmonshyics while at the same time axial resolution is restored However axial sidelobes are introduced and an asymmeshytry in the harmonic bandwidth These issues along with the design and implementation of FM-coded harmonics are work in progress from our group and they will be reshyported in the near future Implementing coded harmonics with bi-phase codes is more of a challenge because the code phases are not maintained in the harmonic domain In this case different transmit sequences have to be generated for the fundamental and for the second harmonic A quadrashyture encoding has been suggested by Chiao and Hao [16] in order to combine first and second harmonic phase coding with the cancellation of the fundamental frequency

D Flow Estimation

The cross-correlation approach for flow estimation calshyculates time shifts between successive received signals and generally has better performance (no bias and lower varishyance) than the conventional phase-shift systems currently using the autocorrelation flow estimators as long as SNRs are higher than 20 dB [22] The performance of the timeshyshift estimator improves when the transmitted pulse is shorter and more broadband That can increase the frame rate because the same short pulses can be used for B-mode images and flow estimation In a coded excitation system the compressed rf-data entering the flow estimator canshynot be distinguished from conventional rf-data obtained from broadband pulsed excitation apart from a decrease in the noise floor as it can be seen from the data shown in Fig 20 Coded excitation therefore can improve velocity estimates providing the necessary SNR for high probabilshyity of correct estimation in the time-shift estimator Flow estimation using linear FM excitation cross-correlation esshytimator and synthetic aperture imaging techniques has


been studied in our lab by Nikolov and Jensen [23] Apshyplying a phase-shift flow estimator on the uncompressed data also could be theoretically possible but the effect of the varying frequency of the signal on the estimator has still to be investigated

X Conclusions

Inspired by techniques from radar this paper describes how properly designed FM-modulated signals can be emitshyted by conventional ultrasound transducers and subseshyquently compressed to yield the same axial resolution and improved contrast and at the same time to increase the SNR by more than 10 dB It has been shown how the higher demands on resolution in medical ultrasound can be met by amplitude tapering of the emitted signal and by mismatched filter design during receive processing to keep temporal sidelobes below 60 to 100 dB which is well below the limits of the typical dynamic range of an ultrashysound image The range resolution that can be achieved is comparable to that of a conventional system The enshyergy within the sidelobe region is reduced by eliminating the distant sidelobes caused by the ripples of the specshytrum amplitude yielding improved image contrast Other coded signals such as the complementary Golay codes also have been considered and characterized in terms of axial resolution temporal sidelobes and attenuation effects

One of the main results is the conclusion that linear FM signals have the best and most robust performance for ultrasound imaging These results are demonstrated both through computer simulations and phantom and in-vivo measurements

Acknowledgments

The authors would like to thank Dr B Tomov for his extensive help with the experimental system Dr S Nikolov for his software support and Dr P Munk for valushyable discussions

References

[1] T Misaridis and J A Jensen ldquoUse of modulated excitation sigshynals in medical ultrasound Part I Basic concepts and expected benefitsrdquo IEEE Trans Ultrason Ferroelect Freq Contr vol 52 no 2 pp 176ndash190 2005

[2] C E Cook and M Bernfeld Radar Signals Norwood MA Artech House 1993

[3] C J Oliver Synthetic Aperture Radar Norwood MA Artech House 1998

[4] F J Harris ldquoOn the use of windows for harmonic analysis with the discrete Fourier transformrdquo Proc IEEE vol 66 pp 51ndash83 1978

[5] J W Adams ldquoA new optimal windowrdquo IEEE Trans Signal Processing vol 39 pp 1753ndash1769 1991

[6] A W Rihaczek and S J Hershkowitz Theory and Practice of Radar Target Identification Boston Artech House 2000

[7] A W Rihaczek Principles of High-Resolution Radar New York McGraw-Hill 1969

[8] M I Skolnik Radar Handbook 2nd ed New York McGraw-Hill 1990

[9] C E Cook and J Paolillo ldquoA pulse compression predistorshytion function for efficient sidelobe reduction in a high-power radarrdquo Proc IEEE vol 52 pp 377ndash389 1964

[10] T X Misaridis and J A Jensen ldquoAn effective coded excitation scheme based on a predistorted FM signal and an optimized digital filterrdquo in Proc IEEE Ultrason Symp 1999 pp 1589ndash 1593

[11] M Kowatsch and H R Stocker ldquoEffect of Fresnel ripples on sidelobe suppression in low time-bandwidth product linear FM pulse compressionrdquo Proc IEEE vol 129 pp 41ndash44 1982

[12] T Misaridis ldquoUltrasound imaging using coded signalsrdquo PhD dissertation OslashrstedbullDTU Technical University of Denmark Lyngby Denmark 2001

[13] J A Jensen ldquoField A program for simulating ultrasound sysshytemsrdquo Med Biol Eng Comput vol 34 Suppl 1 pt 1 pp 351ndash353 1996

[14] J A Jensen O Holm L J Jensen H Bendsen H M Pedshyersen K Salomonsen J Hansen and S Nikolov ldquoExperimenshytal ultrasound system for real-time synthetic imagingrdquo in Proc IEEE Ultrason Symp 1999 pp 1595ndash1599

[15] T X Misaridis M H Pedersen and J A Jensen ldquoClinical use and evaluation of coded excitation in B-mode imagesrdquo in Proc IEEE Ultrason Symp 2000 pp 1689ndash1693

[16] R Y Chiao and X Hao ldquoCoded excitation for diagnostic ultrashysound A system developerrsquos perspectiverdquo in Proc IEEE Ultrashyson Symp 2003 pp 437ndash448

[17] K Gammelmark and J A Jensen ldquoMulti-element synthetic transmit aperture imaging using temporal encodingrdquo in Proc SPIE Progress in Biomedical Optics and Imaging 2002 pp 25ndash36

[18] Y Takeuchi ldquoAn investigation of a spread energy method for medical ultrasound systems Part One Theory and investigashytionrdquo Ultrasonics pp 175ndash182 1979

[19] M OrsquoDonnell ldquoCoded excitation system for improving the penetration of real-time phased-array imaging systemsrdquo IEEE Trans Ultrason Ferroelect Freq Contr vol 39 pp 341ndash351 1992

[20] R T Bjerngaard and J A Jensen ldquoShould compression of coded waveforms be done before or after focusingrdquo in Proc SPIE Progress in Biomedical Optics and Imaging 2002 pp 47ndash58

[21] R Y Chiao and L J Thomas ldquoMethod and apparatus for ulshytrasonic synthetic transmit aperture imaging using orthogonal complementary codesrdquo US Patent 6048315 2000

[22] J A Jensen Estimation of Blood Velocities Using Ultrasound A Signal Processing Approach New York Cambridge Univ Press 1996

[23] S I Nikolov and J A Jensen ldquoVelocity estimation using synshythetic aperture imagingrdquo in Proc IEEE Ultrason Symp 2001 pp 1409ndash1412

Thanassis Misaridis received the BE and MS degrees in 1992 from the National Techshynical University of Athens in electrical engishyneering He received the MS degree in 1997 from the Pennsylvania State University in bioengineering and the PhD degree in 2001 from the Technical University of Denmark Lyngby Denmark

He was a research scientist at the Laboshyratoire Ondes et Acoustique in Paris France until 2003 He is currently a collaborator of Kretz GE Medical Systems in conjunction

with the National Technical University of Athens (NTUA) Greece He teaches ultrasound imaging at the University of Patras Patras Greece and at NTUA

Dr Misaridisrsquo current research interests include coded excitation in medical ultrasound array processing synthetic aperture and nonshylinear imaging


Joslashrgen Arendt Jensen (Mrsquo93ndashSrsquo02)earned his MS degree in electrical engineering in 1985 and the PhD degree in 1989 both from the Technical University of Denmark Lyngby Denmark He received the DrTechn degree from the Technical University of Denmark in 1996

Dr Jensen has published a number of pashypers on signal processing and medical ultrashysound and the book Estimation of Blood Veshylocities Using Ultrasound Cambridge Univershysity Press in 1996 He is also the developer of

the Field II simulation program He has been a visiting scientist at Duke University Durham NC Stanford University Stanford CA and the University of Illinois at Urbana-Champaign He is currently

full professor of Biomedical Signal Processing at the Technical Unishyversity of Denmark at OslashrstedbullDTU and head of the Center for Fast Ultrasound Imaging at the Technical University of Denmark He has given courses on blood velocity estimation at both Duke University and the University of Illinois and teaches biomedical signal processshying and medical imaging at the Technical University of Denmark He has given several short courses on simulation synthetic aperture imaging and flow estimation at international scientific conferences He is also the co-organizer of a new biomedical engineering education offered by the Technical University of Denmark and the University of Copenhagen

His research is centered around simulation of ultrasound imaging synthetic aperture imaging and blood flow estimation and constructshying systems for such imaging


derivative of the phase is defined as the instantaneous freshyquency fi and is given by

1 dΦ(t) d f0t + B t2

B fi = = 2T = f0 + t (2)

2π dt dt T

The linear FM signal has a quadratic phase modulation function and therefore a linear instantaneous frequency function of time Although the term instantaneous freshyquency contradicts with Fourier theorymdashin which a signal has to be infinite in time in order to be band-limitedmdashin practical terms fi indicates the spectral band in which the signal energy is concentrated at the time instant t The pashyrameter k = BT is referred to as the FM slope or the rate of FM sweep The signal sweeps linearly the frequencies in the interval [f0 minus B

2 f0 + B ]2 For digital generation of the FM signal an alternative

expression that has no negative times has the form 2ψ(t) = a(t) middot exp j2π f0 minus B

2 t +

2B

Tt 0 le t le T

(3)

The complex envelope of the linear FM signal described by (1) when the real envelope is rectangular is

t 2jπ(BT )tmicro(t) = rect e (4) T

The matched filter output when the returned signal is not frequency shifted by attenuation is the autocorrelation of the signal which can be expressed as a function of the modulation function

infin infin j2πf0 τRψψ(τ) = ψ(t)ψ lowast(t + τ)dt = e micro(t)micro lowast(t + τ)dt

(5) minusinfin minusinfin

Substituting (4) into (5) one can obtain an analytical exshypression of the matched filter response for the linear FM signal The derivation can be found in any book from the radar literature [2] and the result is

1 minus |τ |sin πD τ T T j2πf0 τRψψ(τ) = T e (6) πD T

τ

where D = TB is the time-bandwidth product Matched filtering removes any frequency modulation and the outshyput is an amplitude-modulated signal at the carrier freshyquency f0 with an approximately sinc shape The apshyproximation to the sinc function improves as the timeshybandwidth product D increases

The first zero of (6) is often taken as a measure of the time resolution It is given by [3]

T 4 τr = 1 minus 1 minus

2 BT(7)

T 2 1 asymp 1 minus 1 minus = 2 BT B

Fig 1 Resolution for pulsed and linear FM excitation The pulse shown here (gray line) is the envelope of an apodized sinusoid of the carrier frequency with a Hanning apodization The length is 27 cycles and is chosen to match the bandwidth of the chirp for direct comparisons The black thin line is the compressed envelope of a linear FM signal with D = TB = 36

in which the binomial expansion has been used retaining only first-order terms Imaging with a short pulse of length T will yield axial resolution of T = 1B Thus axial resshyolution for a short (conventional) excitation pulse and for an FM-modulated (coded) excitation will be roughly the same when the signals use the same bandwidth This is shown in Fig 1

The side effect of pulse compression on the linear FM is the resulting sinc sidelobes not present in conventional pulse excitation Range sidelobes represent an inherent part of the pulse compression mechanism In an imagshying situation the effects of the time (or range) sidelobes extending on either side of the compressed pulse will be self-noise along the axial direction and masking of weaker echoes For the linear FM signal the highest of these sideshylobes are the first ones only minus132 dB below the peak of the compressed pulse The near sidelobes fall off at apshyproximately 4 dB per sidelobe interval and the sidelobe null points are spaced approximately 1B apart

III Mismatched Filtering and the Ultrasound Transducer

The temporal sidelobes after compression are thus high and this will significantly degrade the image conshytrast because a dynamic range of 60 dB or more is typically used in ultrasound imaging This section studies methods for reducing the temporal sidelobes to an acceptable level

A Weighting in Frequency and Time Domain

The usual approach for sidelobe reduction is to apply a window function on the matched filter Due to symmetry properties which will be discussed in detail in Section IV weighting can be performed either in the time or in the frequency domain



infin infin


0 0 (8)


infin



0




















T2

⎛ ⎢⎢ ⎜ 1 B Ψ( ) =

exp ⎢ 2 ⎜⎜ + 2f j π f0t t

⎞⎤

T2

⎟⎣ ⎝ 2 Tminus

k

⎟⎥⎟⎠ ⎥

( j2πft) dt


= exp


T

dt

2minus 2


2k

( 2 k f f0 (f f 0)2


minus minus minus

k minus

2k


T2 π


k

minusT

2

f fexp

2

π 0j 2k

t

minus 2

minus k

dt (13)



we get












⎫ ⎪⎪⎪⎬

⎪⎪⎪⎭

(19) S(Y1) + S(Y2)


2k + Y2 = 2k minus Y1 = 2 2k k (15)



2 B (16)


The amplitude term

[C(Y1) + C(Y2)]2

radic +




z z

2 2C(z) = cos π

y dy S(z) = sin π

y dy 2 2

0 0 (17)



π(f minus f0)2

2k k





B k 4k (20)



k kk 4 (21)





























































B Phantom Images



C Clinical Images


























B Array Imaging








D Flow Estimation




X Conclusions



Acknowledgments


References




































infin infin


0 0 (8)


infin



0




















T2

⎛ ⎢⎢ ⎜ 1 B Ψ( ) =

exp ⎢ 2 ⎜⎜ + 2f j π f0t t

⎞⎤

T2

⎟⎣ ⎝ 2 Tminus

k

⎟⎥⎟⎠ ⎥

( j2πft) dt


= exp


T

dt

2minus 2


2k

( 2 k f f0 (f f 0)2


minus minus minus

k minus

2k


T2 π


k

minusT

2

f fexp

2

π 0j 2k

t

minus 2

minus k

dt (13)



we get












⎫ ⎪⎪⎪⎬

⎪⎪⎪⎭

(19) S(Y1) + S(Y2)


2k + Y2 = 2k minus Y1 = 2 2k k (15)



2 B (16)


The amplitude term

[C(Y1) + C(Y2)]2

radic +




z z

2 2C(z) = cos π

y dy S(z) = sin π

y dy 2 2

0 0 (17)



π(f minus f0)2

2k k





B k 4k (20)



k kk 4 (21)





























































B Phantom Images



C Clinical Images


























B Array Imaging








D Flow Estimation




X Conclusions



Acknowledgments


References











































T2

⎛ ⎢⎢ ⎜ 1 B Ψ( ) =

exp ⎢ 2 ⎜⎜ + 2f j π f0t t

⎞⎤

T2

⎟⎣ ⎝ 2 Tminus

k

⎟⎥⎟⎠ ⎥

( j2πft) dt


= exp


T

dt

2minus 2


2k

( 2 k f f0 (f f 0)2


minus minus minus

k minus

2k


T2 π


k

minusT

2

f fexp

2

π 0j 2k

t

minus 2

minus k

dt (13)



we get












⎫ ⎪⎪⎪⎬

⎪⎪⎪⎭

(19) S(Y1) + S(Y2)


2k + Y2 = 2k minus Y1 = 2 2k k (15)



2 B (16)


The amplitude term

[C(Y1) + C(Y2)]2

radic +




z z

2 2C(z) = cos π

y dy S(z) = sin π

y dy 2 2

0 0 (17)



π(f minus f0)2

2k k





B k 4k (20)



k kk 4 (21)





























































B Phantom Images



C Clinical Images


























B Array Imaging








D Flow Estimation




X Conclusions



Acknowledgments


References











































⎫ ⎪⎪⎪⎬

⎪⎪⎪⎭

(19) S(Y1) + S(Y2)


2k + Y2 = 2k minus Y1 = 2 2k k (15)



2 B (16)


The amplitude term

[C(Y1) + C(Y2)]2

radic +




z z

2 2C(z) = cos π

y dy S(z) = sin π

y dy 2 2

0 0 (17)



π(f minus f0)2

2k k





B k 4k (20)



k kk 4 (21)





























































B Phantom Images



C Clinical Images


























B Array Imaging








D Flow Estimation




X Conclusions



Acknowledgments


References





























































































B Phantom Images



C Clinical Images


























B Array Imaging








D Flow Estimation




X Conclusions



Acknowledgments


References























































B Array Imaging








D Flow Estimation




X Conclusions



Acknowledgments


References




































X Conclusions



Acknowledgments


References


































use of modulated excitation signals in ultrasound....

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