IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 44, NO. 2, MARCH 1997 259

A Closed Loop ML Algorithm for Phase Aberration Correction in Phased Array

Imaging Systems-Part I: Algorithm Synthesis and Experimental Results

Jos6 Mauro P. Fortes

Abstract-This paper describes a new closed loop algo- rithm, based on the maximum likelihood theory, that cor- rects phase aberrations in phased array coherent imaging systems using phase estimates which are directly obtained from measured data (and not from estimated correlation functions). It has the advantage of estimating phase dif- ferences with respect to the same reference, and not phase differences between signals in adjacent array elements, elim- inating the need to perform an integration to get the phase aberration profile. It allows for a real time implementation, in the sense that corrected data flows out of the circuit at the same rate that input data is fed into it. Being a closed loop algorithm, it is able to track time variations that may occur in the phase aberrations and, as in any phase lock loop circuit, it shows a trade-off between acquisition time and phase jitter.

I. INTRODUCTION

HASE ABERRATIONS induced by inhomogeneities of P the body wall constitute a major problem in ultra- sound medical imaging. Several schemes applicable, in gen- eral, to coherent imaging systems using sampled aperture have been proposed to estimate and correct these phase aberrations. They usually estimate the phase differences between signals reaching pairs of array elements (or chan- nels) and integrate them to obtain the phase profile across the array. The scheme proposed by Flax and O’Donnell [l] , [2], for example, uses information on phase differences between signals reaching adjacent array elements only. The algorithm proposed in [3] (multiple layers algorithm) ad- ditionally uses information on phase differences between nonadjacent array elements. In this algorithm, m addi- tional sets (layers) of correlators are used to estimate phase differences between nonadjacent array elements. It is worth noting that for m = 1 the multiple layers algorithm reduces to the algorithm proposed by Flax and O’Donnell [1],[2]. In both schemes, the phase difference estimates are indi- rectly obtained from estimates of the correlation function associated with pairs of channel signals.

This paper describes a new closed loop algorithm, based on the maximum likelihood theory, that corrects phase

Manuscript received June 27,1995; accepted September 20, 1996. J. M. Fortes is with the Pontificia Universidade Cat6lica do Rio de

Janeiro. This work was developed while he was on sabbatical leave at the General Electric Corporate Research and Development Center in Schenectady, NY (e-mail: jmfortes@cetuc.puc-rio.br).

aberrations in phased array coherent imaging systems, us- ing phase estimates which are directly obtained from mea- sured data (and not from estimated correlation functions). It has the advantage of estimating phase differences with respect to the same reference, and not phase differences be- tween signal reaching adjacent array elements, eliminating the need to perform an integration to get the phase aber- ration profile. It allows for a real time implementation, in the sense that, whenever a sample goes into the circuit an- other comes out, meaning that the input and output data streams have both the same rate. Being a closed loop al- gorithm, it is able to track time variations that may occur in the phase aberrations and, as in any phase lock loop circuit, it shows a trade-off between acquisition time and phase jitter.

A detailed derivation of the 2-dimensional (2-D) closed loop algorithm is presented. Higher dimension circuits can be derived using the same procedure. In this case, equa- tions may become very complex. As opposed to the mul- tiple layers algorithm described in [3 ] , the proposed al- gorithm, in its 2-D form, requires a low degree of inter- connection between the various channels. The degree of interconnection is supposed to increase as the algorithm dimension increases. The proposed closed loop algorithm seems to constitute the natural way to handle the phase aberration problem. As explained in the text, there are two major differences between the proposed algorithm and the generalized cross-correlator method for time delay estima- tion [4],[5]: its closed-loop structure and the modeling of the received signal.

In Section I1 the statistical modeling of the measured signals is presented. The maximum likelihood closed loop circuit for phase aberration correction is developed in Sec- tion 111. Experimental results obtained with the proposed closed loop circuit can be found in Section IV. In Section V, a simpler (1-D) closed loop scheme is presented and finally, in Section VI, comments and concluding remarks are made.

11. THE JOINT PROBABILITY DENSITY FUNCTION OF THE MEASURED SIGNALS

Let wi(k), k = I,. . . , K denote complex samples of the speckle signal reaching array element i (i-th channel) in

0885-3010/97$05.00 @ 1997 IEEE

260 IEEE TRANSACTIONS O N ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL 44, NO. 2 , MARCH 1997

a phased-array imaging system. Assume that these sam- ples are characterized by a complex, wide sense stationary, stochastic process with normalized autocorrelation func- tion 7& ( k - e ) defined by

where

o? = E [ I wi(k) 1 2 ] (2)

Also assume that the stochastic processes, wi(k), i =

I,. . . , N , all have the same autocorrelation function, that is,

Ri(k - e) = R(k - e) ; i = 1,. . . , N ( 3 )

Now, let wi denote the complex random vector con- taining the samples of the speckle signal in channel i, and define w as the NK-dimensional complex random vector containing samples of the signals in all N channels. In this case we have, respectively,

and T

w = (WY, wr . . . W Z ) (5)

When the complex signal reaching any array element consists of a large number of dephased contributions com- ing from the various scatterers, and the conditions for the central l imit theorem are satisfied [8], w can be shown to be well approximated by a complex, zero mean, Gaussian random vector (for the probability density function of a complex Gaussian random vector, see [9]) with probabil- ity density function given by

where A, is the Hermitian-symmetric “covariance” ma- trix, having elements A$) defined by

Now, let then A,, (i = 1,. . . , N , j = I , . . . ,iv) denote the complex correlation of the signals reaching array elements i and j . In this work, we have assumed that the complex “covariance” matrix A, can be expressed in terms of the X i j ’ s and the in-channel normalized autocorrelation func- tion R(k - e), as:

with A being the Toeplitz Hermitian-symmetric matrix having complex elements u k e , given by

1 - k = 1 , . . . , K 2 & = 1 , . . . , K UkE = - R ( k - l); (9)

Note that %!(e - k ) = R*(k - !) and, as a consequence, A*T - - A.

Observe that (8) can alternatively be written as

A, = A @ A (10)

with 8 denoting a Kronecker product, and

A11 A12 ” ’ A 1 N

This means that

det A, = (det A)K (det A)&

and

Substituting (12) and (13) into (6) we finally get

1

( 2 ~ ) ~ ~ ( d e t A)K (det A)N PW(W =

which is the joint probability density function of the com- plex samples of the speckle signals reaching the various array elements.

Now, let r i ( k ) , k = I,. . . , K denote complex samples cor- responding to the total signal received by array element i (i-th channel). Assume that r i ( k ) is equal to a phase- aberrated version of the sum of a desired signal s ( ! ~ ) , and the speckle signal wi(k), that is,

Note that, in (15), Oi(k) represents the phase distortion (aberration) experienced by the signal in channel i , at time k .

In vector notation, we can alternatively write:

where ri is the K-dimensional vector containing samples of the complex signal in channel i, given by

FORTES: CLOSED LOOP ML ALGORITHM: SYNTHESIS AND RESULTS 261

and wi is the K-dimensional vector defined in (4). Also in which can also be written as (16), T, and s are respectively given by

exp { -; (M*R - [A-' @ A-l] (M*R - S)*}

(30) ( 2 ~ ) ~ ~ ( d e t A ) K (det A)N Pr (R) =

T, = ; a = 1 , . . . , N Now, let I' denote the inverse of A, and let yZ3, z = (18) 1, , N, J = 1, . . ,N denote the elements of I?, that is . . .

and T

s = s(l), 4 2 ) . ' . s ( K ) ) (19)

To consider the signals reaching all N array elements,

The probability density function of r can be rewritten, in terms of yij, as (

let us define a NK-dimensional vector r as Pr (R) =

exp { -; E,"=, xgl y i j ( ~ d ~ i - s ) T ~ - l / ~ ~ ~ i - s)*}

( 2 ~ ) ~ " ( d e t A)" (det A)N

(32)

(20) T

r = (r:, r;. . . r;)

The relation in (16) can then be rewritten as

r = M [S + w]

where M is the NK x N K matrix

with Ti given by (18) and S being the

(21) where

R = (RT, R; . . . R;)~ (33)

111. THE MAXIMUM LIKELIHOOD ESTIMATES (22)

The probability density function in ( 3 2 ) can be seen as the joint probability density function of the complex sam- ples of the signals reaching the elements of a phased array when certain parameters are considered to be known. As- sume that these parameters are represented by the entries of a vector x, and that the expression in (32) corresponds

rK-dimensional vector consisting of N repetitions of the vector s, that is

S = (ST, ST.. . qT (23) to a situation in which x has taken the value X. In this case, we can write From (21) we can easily see that the complex random

vector r has mean

m, = M S = ~ J Q ~ s T , e J Q z ~ T . . . ~ J O N T Prlx=x(R) =

T -1

( 2 ~ ) ~ ~ (det A ) K (det A)N

s ) (24) exp { -$ E,"=, E:1 YzJ(T:R, - s) A (T;R, - s ) * }

( and covariance matrix

(25) (34) A, = M*A,M = M* [A @ A] M

Note that (25) allows us to write Note that, in (34), the quantities yZJ (or A), T,, s, and A could all have unknown parameters, in which case they would all be dependent on the vector x of unknown pa- rameters. For the sake of simplicity this dependency is not explicitly shown in the notation.

The maxamum likelihood estimate of x is then given by

X M L = max-'p,ix=x(R) = m a - ' In {prlx=x(R)}

with maxx-'F denoting the value of X that maximizes F. Taking into account the expression in (34), we can write

(26) det A, = det M* det A, det M

and considering that det M* det M = 1, we have

det A, = det A, (27)

(35) x X

Also note that

A-1 - ~ - 1 ~ - 1 ~ * - 1 - - M-1 [A-l @ A-11 M*-I r W

-

(28)

From (2 l ) , (14), (26) and (28) we can finally obtain the probability density function of r, which is given by XML = max-l X f(x) (36)

Pr (R) = with f (X) given by N N exp { -$ (R - Ms)TM* [A-' c3 A-'1 M(R - Ms)'} 1 f(x) = - xx7ij(~;~i - S ) ~ A - ~ ( T ; R ~ - s)*

( 2 ~ ) ~ ~ ( d e t A ) K (det A)N i=l j=1

(29) - Kln(de tA) - Nln(de tA) (37)

The maximum likelihood estimate X M L can then be ob- tained by solving

Note that in the particular case of constant phase aberra- tions (in each channel), that is,

k = 1 , . . . ,K i = 1, . . . , N

Q,(k) = Qi ;

(37) becomes

(39)

- K In (det A) - N In (det A) (40)

Among the parameters that can be considered for esti- mation through (36) and (40), are the phase aberrations Qi, i = 1, . . . , N (assumed to be constant), the desired sig- nal s ( k ) , k = 1,. . . , K and the elements of the covariance matrix A. In this work we have assumed the desired signal s(lc), k = I , . . . , K to be deterministic and known, and we have solved (36) for the elements of A and the phase aber- rations Bi, i = 1,. . . , N. Unfortunately, depending on the value of N , analytically solving (38) for the elements of A may become a very difficult task. However, this is not a big problem when N = 2. In this particular case, assuming that the channels under consideration are channels a and j , the matrix A can be written as

where pij denotes the complex correlation coefficient of the signals reaching array elements i and j , and a?, a; denote, respectively, their variances. Note that pij can be expressed in terms of its magnitude pi? and phase Pi, as

The inverse of A is then given by

Aj(Qj) = K K

k = l !=I

k = l ! = l

with a k e denoting the elements of A-l. The quantities to be estimated are the variances a: and

D:, the magnitude p;j and the phase Q i j of the complex correlation coefficient pi?, and the phase aberrations Bi

and Q j . In this case, the vector x becomes

262 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL 44, NO. 2 , MARCH 1997

k=l e=1

and the function to be maximized becomes

1

where

Ai(&) = K K

with y and Q respectively given by

and

To maximize (44), we have to solve (38 ) , or equivalently solve

The maximization of f ( X ) with respect to X is done in two steps. The first step consists of solving O y f ( X ) = 0 for all possible values of 8, obtaining a solution

y(6) = max-'f(y, Q ) (52) Y

Once ?(e) is determined, we can find 8 , ~ (second step) using

Fortunately, the solution for V y f ( X ) = 0 (first step) can be analytically obtained. After some algebraic manip- ulations, we can get

with

(55)

(57)

FORTES: CLOSED LOOP ML ALGORITHM: SYNTHESIS AND RESULTS 263

In (55) to (58), &(ei) , Aj(0j) and B,j(ei, O j ) are given by (45), (46) and (47), respectively.

By plugging (55) to (58) into (37), we get

f (Y(O) , 8) = -1n [di(Wdj(Qj) - I & ( @ i , @ j ) 1 2 ] (59)

and, as a consequence, ( 5 3 ) becomes

or, equivalently,

e,, = min-l 8 In [ g ( ~ ) ] (61)

with y(8) given by

g ( e ) = ~ ~ ( e ~ ) s l ~ ( e ~ ) - I ~ ~ ~ ( w + ) i 2 (62)

Alternatively, it is possible to write g(0) as

where

K K

K K

k = i e=i

K K

k=i e=i

K K

k=l e=1

K K

k = i e=i

To find the maximum likelihood estimate of 8 , we need to find a solution to

The idea is to obtain this solution through a gradient opti- mization technique, using a closed loop circuit. The com- ponents of the gradient in (70) can be shown to be

9 (6)

and

In (71) and (72), %(x) denotes the imaginary part of x. The circuit used to implement the gradient optimiza-

tion technique is presented in Fig. 1. In designing this cir- cuit, we have considered that the quadratic forms in (64) to (69) can be implemented by correlating adequately fil- tered signals. As indicated in [7], this is possible due to some particular characteristics of the matrix A. The pro- cedure to obtain the filter impulse response h( ) from the in-channel normalized autocorrelation function ’& ( k - l ) , using the Toeplitz matrix inversion algorithm in [lo], is described in [7].

We have individually calculated h( ) using various sets of measurements, each taken on a different imaging object. The results have indicated that the in-channel autocorrela- tion function %(k -e), and consequently the filter impulse response h( ), practically do not vary with the object being imaged. This is a symptom that they basically depend on systems parameters such as, for example, the shape of the transmitted pulse, or the transducer response. This is an important result since it indicates that we can use a fixed filter, as long as the system parameters affecting the in- channel autocorrelation function stay the same. The block labeled PWR performs the power calculation over a given range (or interval) of samples. The block labeled CORR performs the correlation of its inputs over the same in- terval. These blocks can easily be implemented using the structures illustrated in Figs. 2 and 3.

In the closed loop circuit shown in Fig. 1, the phase aberration estimate, based on data within a given obser- vation interval, is used to correct the phase aberration of the first sample after the end of the interval. It seems a better idea to use this estimate to correct the phase aber- ration of the sample in the middle of the interval. This can be accomplished by adding some delays just after the input filters, allowing the circuit to phase correct the de- layed versions of its inputs. Fig. 4 illustrates the closed loop maximum likelihood circuit, as modified to allow the correction of the phase aberration of the sample in the middle of the observation interval. In this case, the block labeled D represents the delay corresponding to half of the interval size.

264 IEEE TRANSACTIONS O N ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL 44, N O . 2, MARCH 1997

I 1

I

I L - 1

filter * 1 1 2 -

1 I I DELAY

I 1 1 I I DELAY

I 2-D PLL I

Fig. 1. Closed loop maximum likelihood estimator.

1 - - - - - r

Fig. 2. Power calculation block.

r - - - 1

CORR L , - - - 1

Fig. 3 . Correlation block.

Note that, instead of minimizing In [g(6)1, we could have chosen to minimize g(0). Although the two approaches lead, in theory, to the same result, there is a difference between them. When we minimize In [ g ( O ) ] , the compo- nents of the gradient are proportional t o the inverse of g(B). In this case, g- ' (B) acts like a controlling gain (see Figs. 1 and 4), making circuit performance less sensible t o the choice of the the gains a, and 0 1 ~ . This is not the case if we try to minimize g(Q) , in which case the choice of the gains a, and 0 1 ~ is more critical.

At this point, it is important to note that some of the major differences between the generalized cross-correlator method for time delay estimation [4],[5] and the algorithm proposed here stem from the modeling of the received sig- nal. In the generalized cross-correlator method, the re- ceived signal corresponds to a delayed version (or phase- aberrated in the frequency domain) of the desired signal corrupted by measurement (sensor) noise, assumed t o be

FORTES: CLOSED L O O P ML ALGORITHM: SYNTHESIS AND RESULTS

I s (k )

r% I

PWR

DELAY

DELAY

265

I I I

I I I

I I I I I I I I

li ,(k)

U k )

- I 2-D PLL

Fig. 4. Closed loop maximum likelihood estimator: correction of delayed data.

additive, white and Gaussian. The variances of the noise components are assumed to be known. Speckle noise is not considered. In our work, as can be seen from (15), the re- ceived signal consists of a phase-aberrated version of the sum of the desired signal and the speckle noise. Note that, in this case, both the desired signal and the speckle noise carry information on the phase aberrations. Additionally note that samples of the speckle noise are assumed to form a non-white gaussian random vector with unknown covari- ance matrix. This covariance matrix and the desired phase aberrations are jointly estimated by the proposed algo- rithm. Another important difference between the proposed algorithm and the generalized cross-correlator method in [4],[5] is the implementation of the ML estimate using a closed-loop scheme which, as explained in the text, have intrinsic properties not available in open loop schemes as that proposed in [4],[5].

IV. EXPERIMENTAL RESULTS

The closed loop m a x i m u m likelihood circuit in Fig. 4 was implemented, in software, in a SUN Sparc2 Station, using the Interactive Data Language (IDL-Research Systems Inc.). The digital data used to produce the results in the paper was obtained at the General Electric Corporate Re- search and Development Center using a research platform. This platform is a digital baseband beamformer, internally known as ADDM (Adaptive Digital Demodulator). The A/D sampling rate was 20 MHz. The data that was used was baseband (demodulated) which was decimated to 5 MHz after demodulation and filtering. It consisted of 128 beams, each of them having 96 channels data (872 sam- ples/channel) .

The proposed algorithm was used to estimate and cor- rect phase aberrations in measured data, taken on a hu- man abdomen. The normalized in-channel autocorrelation

266 I E E E TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL 44, NO. 2, MARCH 1997

-0.5

~ ~ real part imaginary par t

, , , I , , I I I , , I , I , I , I I

Fig. 5. In-channel normalized autocorrelation function.

a, : -1.0-

imaginary part -1 5 t , , , , I I , I , I 1 , 1 I I I I I I i

0 5 10 15 20 sample n u m b e r

Fig. 6. Filter impulse response.

function 7&(k - e ) and the filter impulse response h( ) were determined using the procedure described in [7] and are illustrated in Figs. 5 and 6, respectively.

We have first considered data corresponding to one of the beams (beam number 88) and used the closed loop maximum likelihood circuit to estimate the phase aberra- tions in two of the channels. The reference signal s ( k ) was made proportional to the beam sum. Note that the pro- posed algorithm produces maximum likelihood estimates of the phase aberrations only when the reference signal is exactly equal to the desired signal. This means that the results in this section were obtained using a suboptimal version of the algorithm in which the reference signal is made proportional to the beam sum. As expected, the re- sults have shown the usual trade-off between acquisition time and phase jitter, as illustrated in Fig. 7. As the loop gain (or equivalently the loop bandwidth) increases, phase jitter also increases, while acquisition time is reduced.

The algorithm has shown consistency when it was ap- plied to different pairs of channels having a common chan- nel, as illustrated in Fig. 8. There are two sets of plots in Fig. 8. The first set (solid lines) was obtained using

n m L 0 U ._

E! u

a, 0 u

E .- u m a, c 0 .- u

F a, I) 0

a, m 0

, , , , I , , , , , , , , ,I chonnel 50 __ loop g a , n - - ~ loop gain 5

loop 90," 1 - 1 .0

I Q 0 200 400 600 800 1000

t i m e [sample number ]

Fig. 7. Phase aberration estimate for different values of the loop gain. Two different initial conditions were considered for the phase estimate: starting value equal to 0 and starting value equal to 1. The time interval between samples equals 2 ps.

__ channel pair (50.80) chonnel pair (50.10)

I 1 , / l , , / / , , , I , , I

f a 0 200 400 600 800 1000

t ime [sample number ]

Fig. 8. Phase aberration estimate associated with two pairs of chan- nels having a channel in common. The time interval between samples equals 2 ps.

the proposed 2-D circuit connected to array channels 50 and 80 and corresponds to the time evolution of the phase aberration estimates associated with channels 50 and 80, respectively. The second set (dash-dotted lines) was ob- tained using the proposed 2-D circuit connected to array channels 50 and 10 and corresponds to the time evolution of the phase aberration estimates associated with channels 50 and 10, respectively. The purpose of Fig. 8 is to show that the time evolution of the estimates of the phase aber- ration associated with channel 50 in both cases are very close.

In general, in a N channel imaging system, it would be possible to get N - 1 different estimates for the phase aber- ration in the signal reaching each channel (there are N - 1 possible pairs of channels involving a given channel). Get- ting these N - 1 estimates for each channel would require the use of N ( N - 1)/2 closed-loop circuits. Once these N - 1 estimates are available, the final estimate would be

FORTES: CLOSED LOOP ML ALGORITHM: SYNTHESIS AND RESULTS 267

(u

u?

c 0

c

0.5

0.0

.- I

.~ Y p -0.5 i

0

a,

__ average

......... 95% conf idence interval : -1.5t, , , I , , I I , I , I , , I I , I , i I a 0 20 40 60 80 100

e lement number

Fig. 9. Statistics of the phase aberration estimates as a function of the array element number, for time value corresponding to sample 600.

their average. Fig. 9 shows the statistics of the 95 estimates ( N - 1 = 95), for each channel, all of them corresponding to the same instant of time (sample 600). In Fig. 9, the X % confidence interval ( X = 70,90,95) defines, for each channel, the interval containing X % of the 95 estimates. We can see that the proposed algorithm is indeed highly consistent. This is a great result, since it allows us to use a small number of circuits. We could use, for example, a minimum of $ circuits, without a significant degradation in performance.

The results obtained with the proposed algorithm were also compared to those obtained using other phase aberra- tion correction techniques. Fig. 10 shows estimates of the phase profile for beam 88 (at the instant of time corre- sponding to sample 600), obtained with four different pro- cedures: (i) the closed loop m a x i m u m likelihood algorithm, with 48 circuits; (ii) the closed loop m a x i m u m likelihood algorithm, with 4560 circuits; (iii) the multiple layers al- gorithm in [3], with 20 layers; and (iv) the multiple layers algorithm in [3], with 1 layer, which is equivalent to the Flax and O’Donnell algorithm [1],[2]. These results indi- cate that the first three alternatives perform very close, and that there is no need to increase the number of closed loop circuits to be used, beyond G. The bias that is ob- served in the dotted curve in Fig. 10 is not necessarily a linear component. It depends on the variances of the phase difference estimates. In the case of the phase aber- ration correction method proposed by Flax and O’Doiiiiell [1], [a] only information associated with adjacent array el- ements is used. As explained in [3 ] , phase difference esti- mates based only on measurements taken from adjacent array elements can have reasonably large variances, pro- ducing low reliability estimates. To get rid of this bias, one could either estimate it, or use more measurements so that these variances of the phase difference estimates de- crease. Neither of these possibilities is contemplated in the method proposed by Flax and O’Donnell [1],[2]. A more detailed discussion on this matter can be found in [3].

ML closed loop algorithm 48 c~rcutts) ML closed loop algorithm j4560 circuits) Multiple layers aIgor8thm 20 layers) Multiple layeis algorithm {I layer)

20 40 60 80 100 e lement number

Fig. 10. Phase aberration estimate of the phase profile for beam 88, using four different algorithms.

When $ circuits are used, the structure to be employed is obviously the one in Fig. 11. When more than $ cir- cuits are used, combining circuits (COMB) are necessary to merge the various phase estimates associated with the same array element into a single estimate. If, for exam- ple, N - 1 circuits are used, the structure in Fig. 12 could be employed and, in this case, the diagram of the COMB circuits would be that shown in Fig. 13.

We have also considered the situation in which a silicon rubber plate was used to induce phase aberrations on mea- surements made on a standard graphite-gel AIUM phan- tom. The silicon rubber plate was the same as that iden- tified as the 2X plate in [a] and [3] . The estimated phase profile obtained using the proposed 2-D closed loop circuit is presented in Fig. 14. For comparison purposes, Fig. 14 also includes a plotting of the silicon rubber plate profile. Observe that, as the phase estimates obtained using the proposed circuit are modulo 2n estimates, the solid curve in Fig. 14 does not represent the true values of the esti- mates. In Fig. 14, the phase estimate was unwrapped to be better visually compared with the plate profile. Note how- ever that either possibilities, the solid curve in Fig. 14 or its modulo 27r version, both provide exactly the same phase corrected outputs. Finally observe that the result in Fig. 14 is much better than that obtained using the algorithm in [l],[2], and is equivalent to that obtained with the more complex scheme suggested in [3] (see Fig. 15, which repro- duces results obtained with these two other techniques). In this specific case, the algorithm proposed in [l],[2] was not able to produce a good estimate of the phase aberration profile, and that the algorithm in [3] has produced good phase aberration estimates at the expense of an increase in algorithm complexity (a larger number of correlators had to be used to allow the processing of additional in- formation on phase differences between nonadjacent array elements). In fact, the phase aberration correction method proposed by Flax and O’Donnell [1],[2] uses only informa- tion associated with adjacent array elements. As explained

268 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL 44, NO. 2 , MARCH 1997

2-D PLL

74(k)

.

.

.

Fig. 11. Structure with $! circuits

in [ 3 ] , phase difference estimates based only on measure- ments taken from adjacent array elements can have rea- sonably large variances, being therefore low reliability es- timates. Their reliability can be considerably improved if information on phase differences between signals in nonad- jacent array elements is additionally used (see [ 3 ] ) . Finally, it is worth noting that, in this particular example (2X p la te ) , the estimated phase profile has not experienced big changes after acquisition has been reached (around sam- ple 300). This behavior was indeed expected since we were using a fixed phase aberrating silicon rubber plate.

V. A SIMPLER SCHEME: THE 1-D ML CLOSED LOOP CIRCUIT

COMB F6 (k) - c

2-D PLL _t

. --c .

2-D PLL c

7 COMB FN-l (k) - 2-D PLL

fN(k)

Fig. 12. Structure with N - 1 circuits.

r - - - i

COMB L - - - _ 1 Fig. 13. Combination block.

ration d i associated to channel i, under the condition that the desired signal s ( k ) , k = 1,. . . ,I( and the normalized in- channel autocorrelation function 7&(k - t) are known. In this particular case, the vector x, of unknown parameters, would be

Instead of analyzing the 2-D problem outlined in Sec- tion 111, we could have analyzed the simpler 1-D problem in which we would be searching for the maximum likelihood estimates of the standard deviation cri and the phase aber-

and the function to be maximized would be given by

(74) 1 1

f (X) = -- - [Ai(di)] - 2Kln (ci) 2 a:

FORTES: CLOSED LOOP M L ALGORITHM: SYNTHESIS AND RESULTS 269

I c- 0 5 10 15 20 25

lateral d is tance across t h e plate [ m m ]

Fig. 14. Estimated phase profile using the closed loop ML algorithm, for the standard graphite-get AIUM phantom with the 2X distortion

I I I 9

I d I

y DELAY I 1-D PLL

I L - - - - - - - - - - - - - - - l

Fig. 16. Closed loop maximum likelihood estimator (1-D).

s ( k ) 4 h ( )

CORR

h 0 I * U -

Fig. 17. Maximum likelihood estimator (equivalent circuit).

where

(79) d e , ) = Ai(&)

with Ai(Qi) as defined in (45). Alternatively, g(Qi) can be written as

g(6'i) (pi - Qi(0i ) - Q:(Bi) + P,) (80) L--'-Ai with Ps, Pz, and Qi(0; ) respectively given by (64), (65)

If the idea is to obtain phase aberration estimates 25 and (67). 0 5 10 15 20

lateral d is tance acroSs the p la te [mm]

through a gradient optimization technique, using a closed Fig. 15. Estimated phase profiles obtained using the multiple layers algorithm for m = 1 (equivalent to the Flax O'Donnell algorithm)

loop circuit, we need to consider that

and for m = 25

two steps, by first determining

. ^ t ( ~ z ) = max-'f(o,,e,) 0%

and then finding the maximum likelihood using

= max-if(&z(Qz),Q,) 0,

The solution to (75) can easily be found

shown in Fig. 16. Note that, alternatively, we could have solved (78) an-

(75) alytically, obtaining the solution

estimate of Bi Q i M L = LDi (82)

where Dz)~ is given by

(76) K K

in which case, (76) becomes

diML = ,ax-' {-In [g(ei)l> 0,

to be li=i e = i

This means that an equivalent circuit to estimate the phase aberration in channel i would be the one shown in Fig. 17. This circuit corresponds, except for the input filters, to the algorithm proposed by Rigby in [11]. It is an open loop circuit and, as such, does not have the features available in a closed loop circuit (as, for example, the ability to track

(77)

(78) time phase variations).

270 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL 44, NO. 2, MARCH 1997

At this point it is important to remember that the cir- cuits in Figs. 16 and 17 (as well as those in Figs. 1 and 12) will only implement the maximum likelihood phase aberra- tion estimator when the reference signal s ( k ) , k = 1,. . . ,K is exactly equal to the desired signal. In practice, we have been using, as a reference, a distorted noisy version of the desired signal (the beam sum), which contains speckle noise and phase aberrations. We should also note that the circuit in Fig. 16 corresponds to a particular case of the circuit in Fig. I, in which the cross-channel connections have been taken off. It is then expected that phase esti- mates obtained with the 1-D circuit in Fig. 16 would be more directly affected by the reference signal than the 2- D circuit in Fig. 1, which additionally uses cross channel information. A comparison between the performances of the 2-D and the 1-D closed loop circuits is the subject of another paper (a sequel to this one) [6], which addresses performance analysis and loop design.

VI. CONCLUSION

We have described a new closed loop maximum likeli- hood algorithm that corrects phase aberrations in phased array coherent imaging systems, using phase estimates which are directly obtained from measured data, and not from estimated correlation functions, as in the algorithms proposed in [1],[2] and [3]. It has the advantage of estimat- ing phase differences with respect to the same reference, and not phase differences between signals reaching adja- cent array elements, eliminating the need to perform an integration to get the phase aberration profile. This means that the error variances are not integrated across the phase profile and, as a consequence, the results do not show the undesirable bias which is present in the Flax O’Donnell al- gorithm [1],[2] and, to a less extent, in the multzple layers algorithm in [3].

The proposed algorithm allows for a real time imple- mentation, where corrected data can flow out of the cir- cuit at the same time new data is fed into it. Delays can be added to the circuit to allow for the phase aberration correction of past measurements. The algorithm does not have problems in handling 27r phase slips since the esti- mated phase profile and its mod 27r version both provide exactly the same phase corrected outputs.

Being a closed loop algorithm, the proposed scheme is able to track time variations that may occur in the phase aberrations and, as in any phase lock loop circuit, it shows a trade-off between its tracking and acquisition perfor- mances. This fact has to be considered when designing the circuit.

The proposed algorithm also has a built-in deconvolu- tion of the time correlation, which is performed by the filters h( ). As opposed to the multzple layers algorithm described in [ 3 ] , the proposed algorithm, in its 2-D form, requires a low degree of interconnection between the vari- ous channels.

Finally, we think that these closed loop circuits consti- tute a more natural way to handle the phase aberration problem than those in [1],[2] and [3], which base their phase estimates on estimated correlation functions.

REFERENCES

S. W. Flax and M. O’Donnell “Phase-aberration correction using signals from point reflectors and diffuse scatterers: basic princi- ples,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 35, November 1988. pp. 758-767. -, “Phase-aberration correction using signals from point re- flectors and diffuse scatterers: measurements,” IEEE Trans. Ul- trason., Ferroelect., Freq. Contr., vol. 35, November 1988. pp.

J . M. Fortes, “Improving on phase aberration correction in phased-array ultrasound imaging systems,” GE Research and Development Center, Technical Information Series no. 92CRD224, General Electric Company, Corporate Research and Development Center, Schenectady, NY, November 1992. J. C. Hassab and R. E. Boucher, “Optimum estimation of time delay by a generalized correlator,” IEEE Trans. Acoust., Speech, Sig. Proc., vol. ASSP-27, August 1979. pp. 373-380. W. R. Kahn and S. A. Tretter, “Optimum processing for delay- vector estimation in passive signal arrays,” IEEE Trans. Inform. Theory, vol. IT-19, September 1973. pp. 608-614. J . M. Fortes, “A closed loop ML algorithm for phase aberra- tion correction in phased array imaging systems-Part 11: Per- formance analysis,” IEEE Trans. Ultrason., Ferroelect., B e y . Contr., vol. 44, March 1997. pp. 271-286. J . M. Fortes, “On the maximum likelihood estimation of the cor- relation coefficients of pairs of speckle signals in phased-array imaging systems,’’ Ultrasound Memo # 92-33, General Electric Company, Corporate Research and Development Center, Sch- enectady, NY, September 1992. 3. W. Goodman, “Statistical properties of laser speckle pat- terns,” in Topics in Applied Physics, vol. 9, New York: Springer- Verlag, 1975, ch. 2. I. S. Reed, “On a moment theorem for complex gaussian pro- cesses,” I R E Trans. Inform. Theory, April 1961. pp. 194-195. J. H. Justice, “An algorithm for inverting positive definite toeplitz matrices,” SIAM J . App l . Math., vol. 23, November 1972. W. Rigby, “Method of phase estimation for phase aberration correction,” private communication.

768-774.

Jose Mauro P. Fortes was born in Rio de Janeiro, Brazil on May 25, 1950. He received the Diploma de Engenheiro and the M.Sc. degree, both in electrical engineering, from Pontificia Universidade Cat6lica do Rio de Janeiro (PUC-Rio) in 1973 and 1976, respec- tively. He received a Ph.D. degree in electrical engineering from Stanford University in 1980, with fellowships from Conselho Nacional de Pesquisas (CNPq/Brazil) and PUC-Rio. From 1980 to 1986 he was an Assistant Professor at PUC-Rio, and since 1987 he has been an As-

sociate Professor at PUC-Rio, teaching in the Electrical Engineering Department and doing research in communications within the Center for Studies in Telecommunications (CETUC). During 1992 he was on leave from PUC-Rio, working as part of the Ultrasound Team, at the General Electric Research and Development Center in Schenectady, NY. Since 1987 he has been Vice Chairman of Study Group 4 of the International Telecommunication Union (ITU) Radiocommunication Sector (Fixed-Satellite Service). His research interests are mainly on satellite communications and digital signal processing.