IEEE TRANSACTIONS O N ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 44, NO. 2, MARCH 1997 271

A Closed Loop ML Algorithm for Aberration Correction in Phased

Imaging Systems-Part 11: Performance Analysis

Josi! Mauro P. Fortes

Abstract-This paper analyses the performance of the maximum likelihood closed loop circuits proposed in [4] for phase aberration correction in phased array imaging sys- tems. This analysis is helpful in designing and also in deter- mining the tracking mode phase error variance performance of both closed loop circuits (1-D and 2-D). It is an approx- imate analysis, suitable under small errors conditions, and considered accurate for high values of the signal to noise ratio.

I. INTRODUCTION

HASE ABERRATIONS induced by inhomogeneities of P the body wall constitute a major problem in ultra- sound medical imaging and is usually cited as the main cause of poor image quality. Several schemes have been proposed to solve this problem. Some of them require the use of a beacon signal, i.e., a reflection from an ideal point or surface, to appropriately evaluate the phase aberration corrections. Other techniques, in which there is no need for a beacon signal, are more general and usually applica- ble to any coherent imaging system using sampled aper- ture. They estimate and correct phase aberration based on observations of the measured signal itself [1]-[4]. Among these general techniques, are the closed loop algorithms proposed in [4], which were developed based on the max- imum likelihood theory. These closed loop techniques are able to correct phase aberrations using phase estimates directly obtained from measured data, and have the ad- vantage of allowing for real t ime implementation and for the tracking of time variations that may occur in the phase aberrations. It is important to clarify that the term real t ime is used here to mean that the circuit input and out- put data streams both have the same rate. This does not mean, however, that there is no delay between the input and output signals. As indicated in [4], the application of the 2-D closed loop technique to measured data has led to results considered better than those obtained using the algorithms proposed in [1],[2] and comparable to those ob- tained with the multiple layers algorithm in [3].

The objective of this work is to analyze the performance

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

Rio de Janeiro. This work was performed under a General Elec- tric (GE-CRD) Fellowship in Ultrasound Signal Processing (e-mail: jmfortes@cetuc.puc-rio.br).

of the 1-D and 2-D closed loop

Phase Array

circuits proposed in [4]. This analysis is helpful in designing these circuits and also in determining their performance in terms of the tracking mode phase error variance. The complexity of the problem has forced us to work with simplified versions of the closed loop schemes which produce, in tracking mode, the same phase estimates as the original closed loop circuits (both versions implement the m a x i m u m likelihood estimates).

Section I1 of this paper presents the ultrasound sig- nal modeling used throughout this work. An approximate analysis, suitable under small errors conditions, is devel- oped in Section 111. It contemplates both the 1-D and the 2-D closed loop circuits and can be considered accurate for high values of the signal to noise ratio. Finally some concluding remarks are presented in Section IV.

11. SIGNAL MODELING

Let r i ( k ) , k = 1,. . . , K denote complex samples of the 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 ( k ) and a speckle signal wi(k), that is,

In (l), Oi(k) represents the phase distortion (aberration) experienced by the signal in channel i , at time k . In vector notation, we can alternatively write

(2) r i = T i [ s + w i ] ; i = l , . . . , N

where ri, wi and s are K-dimensional vectors respectively defined as

T 7-i(1),7-i(2).. .ri(~)) (3)

T

wz = ( w2(1),w2(2)-'w2(K)) (4)

and

T s = s(l), 4 2 ) . . . s ( K ) ) ( (5)

0885-3010/97$05.00 @ 1997 IEEE

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

Now, let r be the NK-dimensional vector containing complex samples of the signals reaching all N array el- ements, defined by

T T T T r = (rl , r2 . . . rN)

(7)

As explained in [4], the complex vector r can, under cer- tain assumptions, be adequately modeled as a complex gaussian random vector with probability density function

P 4 R ) =

exp {-$ (R-Ms)*M* [Ap1 @A-l] M(R-MS)*}

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

(8)

where

(9)

The closed loop circuits presented in 141 have both a whitening filter h( ) at their front end. It is then useful to determine the statistics of the signals after this filter. Let then F denote the NK-dimensional vector containing samples of the filtered signals. According to [SI, r can be written as

- r = P r (14)

with

where IN is the N x N identity matrix and P is an upper triangular matrix satisfying the condition

Clearly, F is also a complex Gaussian random vector, with mean vector

(17) where

and covariance matrix

with Ti given by ( 6 ) , and

with s given by (5). In (8), A is the N x N matrix whose entries { A i j ,

i = I,. . . , N , j = I,. . . , N } are complex correlations associ- ated with signals reaching pairs of array elements (for ex- ample, A,, is the complex correlation of the signals reach- ing array elements i and j ) , and A is a Toeplitz Hermitian- symmetric matrix having complex elements ake, given by

where R ( k - k ) is the normalized time-autocorrelation func- tion of the speckle signal wi(k), defined by

with

o? = E [ I wi(k) 12] (13)

Observe that R(k ~ k ) is assumed to be the same for all channels ( i = 1,. . . , N ) , and also that %(i? - k ) = R*(k - e ) , implying that A*T = A.

i = l , . . . ,N A, = P*M* [A @I A] PTMT = {Rij}; . 3 = 1 , . . . ,N (19)

with Rij being a K x K matrix given by

From (18) and (19) we have, after some algebraic manip- ulations,

E [ F ; ( k ) F j ( k ) ] = 2X-.ej[03(e)-e,(k)l 23 S ( k - e ) + m f ( k ) m j ( k ) (21)

where S( ) denotes the delta function, defined by

Now, note that the elements of mi in (18) can be written in terms of the elements of Ti and s , and the elements pke of the upper triangular matrix P, as

k

m i ( k + 1) = Cpk:ee '""(+l)s(! + 1) ( 2 3 ) !=O

As pointed out in [5], in the particular case under con- sideration, P is a Toeplitz matrix, meaning that pke is a function of the difference k - i?. In this case, we can write

FORTES: CLOSED LOOP M L ALGORITHM: PERFORMANCE ANALYSIS 273

I I I " I I I

Fig. 1. 1-D closed loop circuit.

and ( 2 3 ) becomes

n

mi(n) = h~-le'oi(n-e+l)~(~ - l + 1) (25) e=i

Assuming that Oi( ) is a slow varying quantity, so that it stays approximately constant over an interval equal to the duration of the filter impulse response {hj , j = 0,1, . . . }, we can approximate (25) by

.,o - 1 u I I - ,

DELAY PT.1.

I c--- - - - - - - - - - - - - - - 1

Fig. 2. Simplified 1-D closed loop circuit

where S ( n ) represents the filtered version of s(n) . Under this approximation, (21) can be rewritten as

E [F;(k)Fj( l )] = [2Xij6(k - l ) + s*(k)S(!)] ej[Qj(e)-Q'(k)I

(27)

with S(k - l ) given by (22).

111. PERFORMANCE ANALYSIS

In this section an approximate linear analysis is applied to the 1-D and the 2-D closed loop maximum likelihood circuits described in 141. This analysis allows us to quantify the gain in performance, in terms of phase error variance, when we go from the 1-D to the 2-D circuit. The linear analysis is simpler for the 1-D closed loop circuit, which will be considered first, in Section 111-A. The analysis for the 2-D closed loop circuit will be conducted in Section 111-B.

A. The I -D Closed Loop Circuit

Fig. 1 shows the 1-D closed loop maximum likelihood circuit, as presented in [4]. It has an automatic gain control feature which is performed by the part of the circuit indi- cated in thick lines. Analyzing a closed loop circuit with such an automatic gain feature can be a complex task.

n-20-i L L 1-1 0 200 400 600 800 1000

Lime [samp le number]

Fig. 3. Results obtained using the original and the simplified versions of the 1-D closed loop circuit.

A simpler scheme that leads to the same tracking mode performance is presented in Fig. 2. It works with a fixed feedback gain that has to be carefully designed. As an il- lustration, Fig. 3 shows the behavior of the original and the simplified circuits when they were used to estimate the phase aberrations in a particular sequence of ultrasound image data. The ultrasound data used to generate these curves corresponds to measurements made on a human abdomen. The true value of the phase aberrations is not known. For this reason, two initial conditions were con- sidered: starting from 0 rad (solid line curve) and from -1.5 rad (dash-dotted curve). The true value of the phase aberrations are expected to be somewhere near the region between these two curves. Note that, although the acqui- sition time is much larger when a fixed gain is used, both circuits seems to have the same performance in tracking mode (after sample number 400). As shown in Fig. 4, in the original closed loop circuit the gain control parameter g varies with time. In working with the simplified circuit, the gain control parameter was set to a fixed average value (g = 5.67 x lo7). Note that it took some time for the phase estimate resulting from the simplified circuit to move from its starting value. This was indeed expected since, during

274 TEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. 44, NO. 2, MARCH 1997

update the phase aberration estimate 8, (n) through the difference equation

ii

= 2 I s(n - k + 1) l 2 sin (+i(n - k + 1)) (38) k=l

8,(n + 1) = + oliei(n) (28) Note that, in this case, the random component of e i (n) will also be a function of the phase error vector a,", and is expressed as

which can alternatively be written as

+i(n + 1) = +i(n) - aiei(n) + AQi(n)

h(.) = &(n) - &(n)

(29) = ei(n) - S a, ( "1

K

where &(n) denotes the phase error

(30) = 2 - y { E ( n - k + 1 ) - I S ( n - k + l ) 12 x and k=l

Ad,(?%) = f&(n + 1) - &(n) (39)

- K

where Now, let 0; and 0, denote, respectively, K-dimensional vectors containing the last K past values of the phase e,( ) and the phase estimates e,( ), that is E,(n - k + 1) =

9 e-3e,(n-k+l) s -* (n - k + 1) F,(n- k -t I)}

(40) { "

(32) Q,(n), Q,(n - 1) . . . Q,(n - K + I)

and As a consequence, (35) becomes - K T 0, = (e&), 8,(n - 1). . - Ei + I) ) (33) +i(n f 1) = +i(..) - ai (s (a,") + wi (n; a,")} + AQ,(n)

141) \ ,

- K Assuming that 05 and 0, are known, the random se- quence e,(n) can be written as the sum of a deterministic with S (+:) given by (38) and W, (n; +:) given by (39).

cess ~ , ( n ) is wide sense stationary and to make a theoret- ical analysis feasible, let us assume that ~(n), and conse- quently s(n), are constant. This means that

- K e,(n) 1 Of , 0, ] and a random component At this point, to guarantee that the input random pro-

K e,(n) = E [e,(n) 1 OF, eii] + 6, (n; OF, 6,)

(34) S ( n - k + l ) % C ; k = l , . . . , K (42)

FORTES: CLOSED LOOP ML ALGORITHM: PERFORMANCE ANALYSIS 275

This assumption is justified when the reference signal s ( t ) has small variations over a time window of size K . It corre- sponds to neglecting the variations of the reference signal around its mean value over this time window. Using the approximation in (42), (41) becomes

or, equivalently,

where

and wi n; an is the normalized loop noise, defined by: ( "> To proceed with the linear (small phase errors) analysis, let us assume that the loop operates in its linear region and, as a consequence,

Under this assumption, equation (44) becomes:

Taking the z-Transform of both sides of (48), we obtain:

z @ ~ ( x ) = ai(.)- G F ( z ) @ ~ ( z ) + GK [-W?(Z)] + ( Z - l)O,(.) (49)

with F ( z ) given by:

K

F ( z ) =

k=I

Note that (49) permits us to write:

ai(.) = H ( z ) [-W?(Z,] + H"O&) (51)

where H ( z ) is the normalized, linearized, digital closed loop transfer function, given by:

GK z - 1 + G F ( z )

H ( z ) =

and

2 - 1 z - 1 + G F ( z )

H " ( Z ) = ( 5 3 )

Usually, it is interesting to define H ( z ) in such a way that it incorporates the low pass filter in the correlator block of the closed loop circuit. Although this is a straight forward thing to do for the 1-D closed loop circuit, it is not feasible for the 2-D circuit due to the nonlinear dependence of the control sequence on the input signals. Since one of the objectives of this work is to compare the performance of the two circuits (1-D and 2-D), it is adequate to analyze both circuits on the same basis, and for this reason the low pass filter in the correlator block will not be incorporated into H ( z ) .

Considering (51), and assuming that e,(.) is a deter- ministic sequence, it is possible to determine the linear phase error variance, in steady state, which is given by:

where S6: ( f ) is defined as the power spectrum density of

the normalized loop noise sequence n 'Pn , parame- ( ; "> terized by a fixed value 'P of the phase error vector +:, that is

where

To evaluate R,%(m, 'P), observe from (39) that the auto- correlation function Ret (m, +) of the non-normalized loop noise is given by:

R,,(m, 'P) = E .Wi(n).Wi(n + m) I 'P: = 'PI [

(57)

sin (&(n - k + 1)) sin (4i(n + m - e + 1))

with &( ) given by (40). Now, observing that

1 qx) S(y) = $R((Zy* - q/)

276

we can write I ,

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

10-1 3 e,(n - k + l)e,(n + m - e + 1) j = a] =

iR {,-jet ( n - k + l ) (n+m-e+l) X

1 { e -3e , (n -k+l )e -3~2(n+m-e+l ) X

E [ F z ( n - k + l)Fz(n + m - e + l)] }

2 ~*(n - k + I)s(n + m - + 1) x

(

E [F,(n - k + l);i;,*(n+m -Q+ l)] -

2 s* (n - k + l )S* (n + m - e + 1) x

(59)

Moreover, considering (27) and (as), and also noting [7] that

E [Fz(n - k + l)Fz(n + m - Q + l)] =

m,(n - k + 1) m,(n + m - Q + 1) (60)

we have, after some algebraic manipulations,

E e , ( n - k + l ) e , ( n + m - e + 1 ) la:=+] =

s*(n - k + 1)s(n + m - + I)&, S(m + k - e) + I s(n - k + 1) 1 2 1 ~ ( n + m - Q + 1) l 2 x

sin (&(n - k + 1)) sin (&(n + m - ! + 1))

[

(61)

Substituting (61 ) into (57) we finally obtain

K K

R~~ (m, +) = 4 A,, >: F: [s*(n - IC + 1)x k = l e=1

3(n + m - e + 1) S(m + k - e ) ] (62)

which, considering the approximation in (42), becomes

K K

Finally, performing the double summation in (63), we get:

R G ~ (m, a) = 4 I c 1’ A,, g(m) (64)

with ij(m) being the triangle function defined by

The autocorrelation function of the normalized loop noise is then obtained from (63) and (46), and is

given by

where u is the signal to noise ratio defined by

I I2

I 10-2

10-6 t/ , , I , , , I I I , 1 0.0001 0.001 0 0.01 00 0 1000

digital loop noise bandwidth BP

Fig. 5. Parameter G as a function of the digital loop nozse bandwidth, for different values of K.

Note that u is the ratio of the signal power E[l S ( k ) 1’1 = I c 1’ t o the variance of the received data Fi(k) in channel i (see equation (21)). Both, S ( k ) and F i ( k ) , correspond to quantities after the whitening filter h( ). Note further that the variance of the received signal is closely related to the speckle noise power E[I wi(k) 1’1 in channel i. Some authors refer to u as the beam ratio [8]. Observe that this signal to speckle-noise ratio is defined at the output of the whitening filter. This is indeed convenient since in this way our final results will not be dependent on the whitening filter impulse response h( ). Note that, as pointed out in [5], h( ) depends basically on systems parameters such as the shape of the transmitted pulse and the ultrasound probe response.

The linear phase error variance is finally obtained from (54), (55) and (66), resulting

with G(f) given by

00

G(f) = g(m) e P g a n f m d f m=--00

An important parameter usually considered when design- ing (choosing the value of G) a closed loop circuit is the digital loop noise bandwidth E:, defined as

It is well known that decreasing the value of E: would decrease the value of the steady state phase error variance at the expense of an increase in the acquisition time. To help in selecting an adequate value of the parameter G, Fig. 5 presents curves of G as a function of the digital loop noise bandwidth, for different values of K . Note from Fig. 5 that a given digital loop noise bandwidth can be

FORTES: CLOSED LOOP ML ALGORITHM: PERFORMANCE ANALYSIS 277

a U L1

I

0

> a, U

0

0

U L 0 005

0 001 0 0005

6 10.00

._

Y 100

i a,

m v7

5 10 15 20 25 2 0.01 m

--IO -5 0 ra t i o v = (c];/2h,, [dB]

Fig. 6. Phase error standard deviation for the simplified 1-D closed loop circuit, for K = 50.

Y U U L 72 v1 Y

L e a, L 4 a, E I 1 2 O O I L . L L , , , I , , , , I , , 1 1 I . , , I , , , , I CL

-10 -5 0 5 10 15 20 2 5 rotio v = lc172h,, [dB]

Fig. 7. Phase error standard deviation for the simplified 1-D closed loop circuit, for K = 200.

achieved by choosing different combinations of G and K . It is worth noting that in Fig. 3 the fixed gain curve was obtained with a simplified 1-D closed loop circuit having a digital loop noise bandwidth approximately equal to 0.001.

Curves of the phase error standard deviation associated with the 1-D closed loop circuit, versus the signal to noise ratio v, for different values of the digital loop noise band- width BE, are presented in Figs. 6 and 7, respectively, for K = 50 and K = 200.

B. The 2-0 Closed Loop Circuit

Fig. 8 shows the 2-D closed loop maximum likelihood circuit, as presented in [4]. As in the I-D circuit, it has an automatic gain control feature which is performed by the part of the circuit indicated in thick lines. Here again, the analysis is carried out for an equivalent simpler scheme, that leads to the same steady state performance as the original scheme. It works with a fixed feedback (that has to be carefully designed) and is presented in Fig. 9. According

to Fig. 9, control sequences e , (n) and e 3 ( n ) are used to respectively update the phase aberration estimates 8, (n) and i, (n) through the difference equations

8,(n + 1) = 8,(n) + Q,e,(n) { O, (n + 1) = 4, (n) + a,e3 (n) (71)

which can alternatively be written as

(72) 4,(n + 1) = 4,(n) - a,e,(n) + AQ,(n) { 4, (n + 1) = 4J (n) - %e, (n) + A@, (n)

{ 4, (4 = 0 , ( 4 - 8 , ( 4

{ AQ,(n) = Q,(n + 1) - Q,(n)

where &(n) and 4,(n) denote the phase errors

(73) $,(.) = 4(n) -

and

(74) AQ,(n) = Q,(n + 1) - &(n)

Note that (72) can be written, in matrix form, as:

@(n + I) = @(n) - Ye(n) + A@(n) (75)

where

and

(78) A 0 ( n ) = 0 ( n + 1) - 0 ( n )

with T

(79)

In (75), Y denotes the diagonal matrix given by

2K Now, let etK and 0, denote, respectively, 2K-

dimensional vectors containing the last K past values of the phases Qi( ) and Qj( ), and the last K past values of the phase estimates e,( ) and 8 j ( ), that is:

@ i K =

Q,(n), ej(n- 1) . . . O j ( n - K + 1))

0, =

(Qi (n), Qi(n - 1) . . .Qi (n - K + l),

(81) T

arid 2K

(8i(n), 8,(n- 1). . .8i(n- K + l),

e,(n),e,(n- l ) . . . 8 j ( n - K + l ) ) T (82)

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

Fig. 8. 2-D closed loop circuit

I I I I L - - - PLL

DELAY I / '

Fig. 9. Simplified 2-D closed loop circuit

FORTES: CLOSED LOOP ML ALGORITHM: PERFORMANCE ANALYSIS 279

E [ ~ ~ ( ~ 1 I (89) 2K

Assuming that OiK and 0, are known, the random vec- tor sequence e(n) can be written as the sum of a determin-

istic component E e(.) I OiK, 6tK] and a random com-

ponent W n; @ E K , This means that we can write E [ F t ( k ) f j ( l ) I 6:K] [2Aijd(k - e)+ e(n) as:

and [ g* (k)g(!)l ej[',(')-'z(k)l

( (90) e(n) = E [e(.) I aiK, 6, + i~ ( n; 6, 2K>

(83) The deterministic part of e i (n) can then be shown to be a function Si ( ) of the phase error vector aiK = OiK - 0, , given by:

2K Equation (75) can then be rewritten as:

si (aiK) = E [ei(n) 1 a(n + 1) = +(n) - T E e(.) I O ~ K , etK] + { [

Note that E e(.) I OiK, 6:K] is a 2-dimensional vec- [ E [e(,,) I o : K , ~ : ~ ] =

.Gir (n; OiK , 6 tK)} + a q n ) (84) K K

= 4 y x { l s ( n - k + 1 ) I ~ [ 1 - d ( k - l ) ] x k = l e=1

[A j jd i (n - k + 1) - %{A,,} 4j(n - k + l)]} tor, given by

(91) T

( E [ei(n) I OiK, , E [ e j ( n ) 1 (85) or K

To evaluate (85) note, from Fig. 9, that si (a:K) = 4(K - 1) I s(n - k + 1) 12

[Ajj$i(n - k + 1) - % { A j i } $ j ( . - k + l)] k = l

e i (n ) =

(92) K K

2 3 { { ,-&(n-l+l) s*(n ~ - e + l)ri(n - e + 1)

Note that in obtaining (91), and consequently (92), the small phase error approximation was used. This means that we have considered that:

k = l e=1

x [ I F j ( . - k + 1) 12 -

,8,(n-k+l) s ( n - - k + 1 ) q n - k + 1)]

+,-~ , (~-e+i) ,e , (n-e+i) ri(n - - e + 1 ) q n - e + 1) sin(a + x) sin(u) + II: cos(u) (93)

x [ I s ( n - k + l ) 12-

e-',(n-k++1)%*(n- k + ~ ) ~ j ( n - k + l ) l }

where 2 represents small values of phase errors (or sums of phase errors).

In analogy, we would get for the second element of (85)

(86) Sj (aiK) = E [ e j ( n ) I @zK,6)n K

= 4(K - 1) I s(n - k + 1) 12

[Azzdj(n - k + 1) - %{(x i j }d i (n - k + 1)]

1 and

e j ( n ) = k = l

K K

23{ ~ { e - ~ ~ ( n - ~ + 1 ) s * ( n - e + l ) F j ( n - e + 1)

,-8,(n-'i+l)s*(n- IC + 1)Fi(n-k+1)]}} 0

(94) k = l e=1

The results in (92) and (94) can be written in matrix form as

S (aiK) = 4(K - 1) B

x [ I Fz(n -k+l ) 12 -

(n-k+l) - s(n - k + l)Fz*(n - k + l)] K

+,-8,(n-e+l),ei(,-e+l) r j (n - - l + l ) ~ ; ( n - l + 1) I s(n - k + 1) l 2 (a(n - k + 1) x[I S ( n - k + l ) 12- k=l (95)

where S aiK denotes the 2-dimensional vector

s (dy) =

T (87)

Also note, from (26) and (27), that E [e(.) I @ E K , 6:K] = (Si (a:") , Sj (a iK))

ri(n) I @iK, (88) (96) = ,jei(,) - 4 n )

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

and At this point, to guarantee that the input random pro- cesses rz (n) and rJ (n) are wide sense stationary, let us as- sume that s(n), and consequently s(n), are constant. This

(97) means that

S ( n - k + 1 ) R 5 c ; k = l , . . . , K (103) i 4, -%{&I

-%{&} A,,

Note that in this case the random component of e(n) will also be a function of the phase error vector aiK, and is expressed as

w (n; @i~) = e(.) - s (aiK)

In this case, (95) becomes

K

S (aiK) = 4(K- 1) 1 c l 2 B x @ ( n - k + 1) k=l (104)

K K and consequently - - 2 y x q n - k + l , n - c + I) -

k=l t=1 K K +(n + 1) = @(n) - G @(n - k + 1) -

4 ( K - 1) BE I S ( n - k + l ) l 2 @(n- IC+ 1) k=l

K G W (n; + E K ) + A@(n) (105) k=l

(98) where where G(n - IC + 1, n - ! + 1) is the 2-dimensional vector

(106) G = 4(K - 1) I c l 2 YB G ( n - k + l , n - l + l ) =

and n; @zK is the normalized loop noise, given by: 0 T ( q n - k + 1 , n - c + l),E,(n - k + l , n - e + I)) (99)

with elements

Ez(n - IC + l , n - Q+ 1) =

( n - ! + l ) F i ( n - ! + l ) x

[ I F j ( . - k + 1) l 2 - ee, (n-k+l) s ( n -

- k + 1) F p 2 - k + 1)] +

Taking the z-Transform of both sides of (105), we obtain

z g ( z ) = 6 ( z ) - F ( z ) G 6 ( z ) + K G [-W"(z)] + (z - 1) Q ( z ) (108)

,&n-e+l) e e, (n-!+l) r,(n - - l+ 1) q n - Q + 1) x

[ I s(n - k + 1) 12 -

with F ( z ) given by (50). Note that (108) permits us to write:

( n - k + l ) -* &(z) = H(z) [-W"(z)] + H"(z) 6 ( z ) (109)

s (n - k + 1) .i;,(n - k + l)]} (100)

and where H(z) is the normalized, linearized, digital closed loop transfer function, given by e, (72 - IC + I, n - e + 1) =

29 ,--B,(n-t+l) s -* ( n - l + 1 ) F 1 ( n - t + 1 ) x

[ I F,(n - IC + 1) l 2 -

,-e, (n-t+1) ,B,(n-e+l) - (111)

H(z) = K [ ( z - 1)1 + F(z )G]- 'G (110)

and { e

,B,(n-k+l) s ( n - - k + 1) F ; ( n - k + 1)] + H"(z) = ( z - 1) [ ( z - 1)I+ F ( z ) G ] - l

r J ( n - e + 1) F ; ( n - Q+ 1) x

As a consequence, (84) becomes:

+ 1) = a(n) - { s ( a ; K ) + w (.; a")} + a@(n) At this point, it is worth defining the signal to noise ratio (10.4 I/, which is assumed to be the same in both channels ( i

and j ) , as

with S ( @ E K ) given by (95) and W n; aiK given by 0 (98).

FORTES: CLOSED LOOP ML ALGORITHM: PERFORMANCE ANALYSIS 281

Also, consider the complex correlation coefficient defined bY

(114) A,, .7qt3

d m =

and having magnitude p,, and phase Q Z J .

and A,, as functions of u, p,,, and Q Z J . We then have Note that (113) and (114) permit us to write A,,, A,,

We also assume that the loop gain parameters ai and aj are the same in both channels, that is

or

Y = a I (119)

Considering (115), (116), (117), and (119) we can write B and G, respectively given by (97) and (106), as:

1 -pij C O S ( 9 i j ) B = - ( I c l 2

-p i j COS(Q2j) 1 2 u

and

G = G ( 1

-pij COS(Q2j)

with G being the scalar quantity

G = 4(K - 1) I c 1' 2 1 c I a - 2 u

Substituting (121) into (110) we get, after some algebraic manipulations,

& ( z ) = H J z ) =

KG [ z - 1 + F(z)G (1 - p;.7 cos2(q,,))]

( z - 1)2 + 2 ( z - 1) F ( z ) G + F2(z)G2 (1 - p:, cos2(QZ3))

(123)

and

Considering (108), and assuming that @(n) is a deter- ministic vector sequence, it is possible to determine the linearized phase error variance, in steady state. We then have, for channel i

and for channel j

In (125) and (126) Sui (f, 'P) and Sui ( f , @) are respec- tively defined as the power spectrum densities of the com- ponents i and j of the normalized loop noise vector se- quence w n, @, , parameterized by a fixed value @ of

the phase error vector @.,",", that is: ( "1

00

and

where

and

Also in (125) and (126) SG,w, (f, @) and S w 3 a , ( f , @) are defined as cross spectrum densities of the compo- nents i and j of the normalized loop noise vector sequence

, parameterized by a fixed value of the phase

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

-6 10

error vector aiK, that is:

K=50 *,,= 45" ~ average curve

I

-00

and

where

and

To compare the performance of the 2-D closed loop cir- cuit with that of the l-D circuit, it is convenient to ex- press both results in terms of the same digital loop noise bandwidth Bf. Note that for p i j = 0 the in-channel filters Hii(z) and H j j (2) in (123) become equal to the closed loop filter H ( z ) of the l-D circuit given by (52). This suggests that the performance results associated with the 2-D cir- cuit should be expressed in terms of the digital loop noise bandwidth of the in-channel filters, which is given by

Note that this digital loop noise bandwidth depends not only on the parameters G and K , as that in the l-D closed loop circuit, but also on the magnitude pij and phase Q i j

of the complex correlation coefficient in (114). Note from equations (123) and (135) that, for given values of K and p i j and for F ( z ) as in (50), it is possible to evaluate the digital loop noise bandwidth BE as a function of G. Also note that this function depends on cos2(Qij), being there- fore periodic in 9 i j . Curves relating the parameter G to the digital loop noise bandwidth Bf are presented in Fig. 10 for fixed values of K and p i j , and for Qi j varying from 0" to 360" in 15" steps. Note from Fig. 10 that for K = 50 and pi j = .95 as Q i j varies from 0" to 360", these curves touch an upper bound for three values of Q i j which are respectively close to 0" , 180" , and 360" and a lower bound for two values of Qfij which are respectively close to 90" and 270". The average curve occurs around Q i j = 45", which would be a convenient value to use when Qi j is not known. Similar curves, for other values of K and p i j have shown that, as dictated by (123), the dependence of these curves on Q\~rij decreases as pi j decreases. This suggests the use of Q i j = 45". For this particular value of Q i j and set- ting K to a fixed value, it is possible to produce curves of

Fig. 10. Parameter G as a function of the digztal loop noise bandwidth, for K = 50 and p z j = 0.95, and for Qtj varying from 0' to 360' in 15" steps.

G versus BE, which will, of course, depend on p i j . Exam- ples of such curves are presented in Fig. 11, in which p i j

varies from 0. to .99 and K = 50. Curves as those in Fig. 11 should be used in design of the 2-D closed loop circuit in the following way: once a value for BF is chosen, these curves will supply the value of G, which will provide a value for the circuit parameters ai and c i j through (122). Remember that we have assumed both ai and aj to be equal to a. Clearly, the dotted curves must be used when pij is known or can be estimated from measured data. If this is not the case, the average curves appearing in Fig. 11 must be used. This average curve is also shown in Fig. 12, which includes average curves of G versus BE for K = 10 and K = 200. Note that, as in the l-D closed loop circuit, a given digital loop noise bandwidth can be achieved by choosing different combinations of G and K .

At this point, observe that, to evaluate the phase error variances in (125) and (126), the autocorrelation functions

FORTES: CLOSED LOOP ML ALGORITHM: PERFORMANCE ANALYSIS 283

0.000 1 0.00 10 0.0100 0.1000 dig i ta l loop noise bandwidth 6:

Fig. 12. Average value, with respect to p i j , of the parameter G, as a function of the digital loop noise bandwidth Bf.

RGt (m, @) and RG3 (m, @) and the cross-correlation func- tion Rfitm3 (m, a), of the components of the normalized loop noise, must be determined. This is a nontrivial, time consuming, and tedious task (even with the help of the re- sults in [7]) that requires expressing several moments (up to 6th order) of a nonzero mean, complex, Gaussian ran- dom vector in terms of its 1st and 2nd order moments. Full expressions for the autocorrelation functions of the compo- nents 'Lui ( ) and t i j j ( ) of the non-normalized loop noise and of their cross-correlation function can be found in [6]. They are very long expressions, each having 85 nonzero terms. For a linear (small phase errors) analysis, it is usual to con- sider the particular case where @ = 0. In this particular case, these expressions become more tractable. As shown in [6], they are reduced to:

Rej (m, 0) = E [Gj(n)Gj(n + m) I @iK = 0, 2K = 01

= 16 I c l 2 X i i [ X i i X j j - X i j X j i ] x

[K3g(m) - 2K2g(m) - Kg(m) + 2K2g2(m)]

(137)

and

Considering (107), (120), and (115) through (117), we have, for the normalized loop noise

and

X ( K + l ) 1 (1 - P?j)

K ( K - 1) 2y (1 - p$ COS2(Qij)) RGiGj (m, 0) =

P i j COS(Qi j ) f ( m ) (141)

where

K3g(m) - 2K2g(m) - Kg(m) + 2K2g2(m) K ( K + 1)(K - 1)

(142) f ( m ) =

Plugging (139), (140), and (141), respectively, into (127), (l28), and (131), and substituting the resulting ex- pressions into (125), we have for the phase error variance associated with channel i

a& =

(K+1) 1 (1 - P?j) K ( K - 1) G (1 - pzj COS2(@ij ) )

X

where

m=-00

Obviously, by symmetry, channel j will have the same phase error variance, which will be expressed by:

2 a$,? =

X ( K + 1 ) 1 (1 - P j i )

K ( K - 1) G (1 - pji C O S 2 ( 9 j i ) )

J - - l

H*. 3% ( e 9 9 1 M ( f ) d f where g(m) is given by (65).

(145)

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

7

1000 - ? a 1000 OI

Y IT ru U 72 Y

i 0 0 I

? 1 0 0 9 100

ar Y

LI

+

> a, u

W U

e Y

: 0 10 ;; 0 1 0

0 U i

0 D L 0

i L

0 P - L a, W

W in a 2-D c rcu t 2-D circuit (II

," 001 2 0 0 1 a 20 10 20 -10 0 10 Q

-10 0 ratio u = IclY2h [dB] ratio v = JcJY2h [dB]

Fig 13 Phase error standard devlatlon for the simplified closed loop circuits for K = 10 and BE = 0 001 The dotted curves correspond respectively to pt3 equal to 0 99,O 9 , 0 8,0 ?,0.6,0 5,0 4,0 3,0 2,O 1, and 0 and 0

Fig 14 Phase error standard deviation for the simplified closed loop circuits for K = 50 and BE = 0 001 The dotted curves correspond respectively to pZ3 equal to 0 99,0.9,0 8,O.?, O.6 ,O 5,0 4,O 3,0 2,0 1,

- At this point it would be of interest to particularize

(143) and (145) for pz3 = 0. Note that, in this case, the normalized cross-channel loop filters ) and GJz( ) in (124) are equal to zero and the normalized in-channel loop 2 filters gtz( ) and RJJ( ) in (123) tend to the normalized loop filter H ( ) of the 1-D closed loop czrcuzt, in (52). Also note that for K sufficiently large (K > 10 is enough), f ( m ) in (142) tends to g(m) in (65) and consequently M ( f ) in (144) will tend to G ( f ) in (69). In this case, the phase t; i

e

O0

O0

0 10

error variances in (143) and (145) become

D c i rcu i t D circuit

I , ,___u 1 , , ' ' ' ' ' '

(146)

given by (52) and

Fig. 15. Phase error standard deviation for the simplified closed loop circuits for K = 200 and BE = 0 001 The dotted curves correspond respectlvely to pZ3 equal to 0 99,O 9,0 8,O ? , O 6,O 5,0 4,0 3,0 2,0 1, where &( ) and G(f) are

(69). and 0

Curves of the phase error standard deviation associated with the 2-D closed loop circuit, versus the signal to noise 7 10 00 2 ratio v, for different values of the dzgztal loop nozse band- wzdth and K are presented in Figs. 13-16. The first three ;I3,

figures refer to a dzgztal loop notse bandwzdth Bf = 0.001 h and correspond respectively to K = 10, K = 50, and 5 K = 200. Observe, for example, that in Fig. 15 the per- 2 formance of the 2-D closed loop czrcuzt is 4 dB better (in y terms of the signal to noise ratio v ) than that of the 1-D cir- cuit when pa3 = 0.8. This gain increases to approximately + 6.5 dB when pz3 = 0.9. Also note that, since this perfor- mance gain increases with the correlation coefficient pz3, it is attractive to apply the 2-D circuit to adjacent channels, which are supposed to have higher correlation coefficients. i"

of phase error standard deviation, that we have when we

to 0.00015. closed loop circuits for K = 200 and BF = 0 00015 Also note from Figs. 13-15 that, as dictated by (146)

and (68), the 1-D closed loop czrcuzt performs a little better

I 00

l o

0 0 1 20

Q Additionally, Fig. 16 illustrates the kind of gain, in terms -10 0 10 ratio u = /clYZh [dB]

the dzgztal loop bandwzdth from o'Ool Fig 16 Phase error standard deviation for the simplified

The dotted curves correspond respectlvely to pt3 equal to 0 99,o 9,* 8,O 7,O 63 0 5,O 4 ,o 3 ,o 2 , o 1) and 0

FORTES: CLOSED L O O P ML ALGORITHM: PERFORMANCE ANALYSIS 285

I

I

I I I I I I

I I I DELAY I

Fig. 17. 2-D closed loop circuit for p23 = 0.

than the 2-D closed loop circuit for p i j = 0. This difference in performance becomes negligible as K increases. This is indeed expected since the 2-D closed loop circuit for pij = 0 is not exactly equal to the 1-D closed loop circuit. It ideally tends to the circuit shown in Fig. 17. Note from Fig. 17 that the signal in channel j produces a sequence fi,? (n) that works as an uncorrelated noise to the closed loop circuit associated with channel i. As K increases, the filter inside the channel j correlator becomes narrower and narrower and as a consequence its output becomes nearly constant. In this situation, the circuit in Fig. 17 approaches the 1-D closed loop circuit.

IV. CONCLUSION

We have analyzed the performance of the closed loop circuits proposed in [4] to estimate and correct phase aber- rations in coherent imaging systems that utilize sampled aperture. These closed loop circuits have the advantage of allowing for real time phase correction and for the tracking of time variations that may occur in the phase aberrations.

The complexity of the circuits has forced us to work with simplified versions of the closed loop schemes which produce, in steady state, the same phase estimates as the original closed loop circuits (both versions implement, in steady state, the m a x i m u m likelihood estimates). An ap- proximate analysis, suitable under small errors conditions, and considered accurate for high values of the signal to noise ratio, was performed on both closed loop circuits.

Analysis results have indicated that, when compared with the I-D circuit, the 2-D closed loop circuit has a supe- rior performance. For a digital loop noise bandwidth equal to 0.001 and K = 200, for example, the 2-D circuit per- forms 4 dB better (in terms of the signal to noise ratio 71)

than the 1-D circuit, when pij = 0.8. This gain increases to approximately 6.5 dB when pij = 0.9.

As an additional result, the analysis work has also pro- vided us with curves that are very helpful when designing the closed loop circuits. Once we have decided upon a value for the digital loop noise bandwidth, these curves allow us to obtain the value of the feedback gains ai = a3 = Q to be used.

Finally, we have noted that the 1-D closed loop circuit performs a little better than the 2-D closed loop circuit for p i j = 0. This difference in performance becomes negligible as K increases. We have concluded that this behavior is indeed expected since the 2-D circuit for p i j = 0 is not exactly the same as the 1-D circuit but it tends to it for large values of K .

ACKNOWLEDGEMENTS

The author would like to thank Dr. Raimundo Sampaio- Net0 for the fruitful discussions during the development of this work.

REFERENCES

[l] S. W. Flax and M. O’Donnell “Phase-aberration correction using signals from point reflectors and diffuse scatterers: Basic princi-

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

ples,” I E E E 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 T+-uns. U1- truson., 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 . M. Fortes, “A closed loop ML algorithm for phase aberra- tion correction in phased array imaging systems-Part I: Algo- rithm synthesis and experimental results,” I E E E Trans. Ultra- son., Ferroelect., Freq. Contr. , vol. 44, March 1997, pp. 259-270. J. M. Fortes, “On the maximum likelihood estimation of the cor- relation coefficients of pairs of speckle signals in phased-array imaging systems,” GE Research and Development Center, U1- trasound Memo #92-33, September 1992. J . M. Fortes, “Performance analysis of the 1-D and 2-D closed loop circuits for phase aberration correction in phased-array imaging systems,” Technical Report D-ST-01/94, Pontificia Uni- versidade Cat6lica do Rio de Janeiro, January 1994. I. S. Reed, “On a moment theorem for complex gaussian pro- cesses,” I R E D u n s . Infomn. T h e o q , April 1961, pp. 194-195. J . W. Goodman, “Statistical properties of laser speckle pat- terns,” in Topics in Applied Physics, vol. 9, New York: Springer- Verlag, 1975, ch. 2.

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.