Bispectrum

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40. NO. 3 . MARCH 1992 60 I

Signal Reconstruction from the Phase of the Bispectrum

Athina P . Petropulu, Member, IEEE, and Chrysostomos L. Nikias, Fellow, IEEE

Abstract-In this paper, we first present a simple procedure to recover the Fourier phase of a signal from the phase of its bispectrum; namely the bispectrum signal reconstruction (BSR) algorithm. By simple analogy, a procedure that recovers the Fourier magnitude of a signal from the magnitude of its bispectrum is also presented. Jn addition, we also propose in this paper an iterative scheme, the bicepstrum iterative reconstruction algorithm (BIRA), for the reconstruction of a FIR sequence from only the phase of its hispectrum, and we demon- strate how some apriori information on the energy of the cepstra coefficients can improve significantly the convergence rate of the algorithm. Both schemes are based on the key observation that the differences of the bicepstrum coefficients contain all the information concerning the Fourier phase of the signal, whereas their sums contain the Fourier-magnitude information.

I. INTRODUCTION

has been well established in the signal processing lit- r erature that, under certain conditions, a signal can be reconstructed based on partial Fourier domain information. Several schemes have been proposed to reconstruct a signal from some samples of its Fourier phase, or its Fourier magnitude, or both. Gerchberg and Saxton [I], [2] introduced an iterative procedure (G-S) for the restoration of a signal from the magnitude of its Fourier transform. Later, Papoulis [3] used the G-S algorithm to recover a sequence based on the knowledge of a part of its time or frequency representation. Fienup [4] also used similar ideas to come up with a procedure for the reconstruction of a sequence from only its magnitude. Refer- ences [5]-[lo] pointed out the importance of the phase in the signal recovery, and developed iterative schemes to reconstruct a signal from only its Fourier phase, as well as the conditions under which such a reconstruction is possible. Youla and Webb [ l l ] , as well as Sezan and Stark [12], employed the method of convex projections to reconstruct a sequence from partial information.

The importance of the bispectrum/trispectrum in signal processing has been well established over the last five

Manuscript received July 19, 1990; revised January 25, 1991, This work was supported by the Office of Naval Research under Contract ONR- N00014-91-J-1124. Portions of this paper were presented at ICASSP’90, Albuquerque, NM.

A. P. Petropulu is with the Communication and Digital Signal Process- ing (CDSP) Center for Research and Graduate Studies, Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 021 15.

C. L. Nikias is with the Department of Electrical Engineering-Systems, Signal and Image Processing Institute, University of Southern California, Los Angeles, CA 90089-2564.

IEEE Log Number 9105669.

years [ 131. Many different algorithms (parametric and nonparametric) have been developed to reconstruct a signal from both the magnitude and phase of its higher order spectra. However, there are applications in signal processing where the magnitude bispectrum of the source signal is distorted, while the phase remains “clean. ” Such situations occur approximately in long-term exposure to atmospheric turbulence or when images are blurred by de- focused lenses with circular aperture stops [ 141. Further- more, being able to reconstruct a sequence from its bispectral phase only, we may reduce the dimensionality of certain problems. Petropulu and Nikias [15] utilized the fact that under certain conditions a sequence can be reconstructed from its bispectral phase only, to blindly de- convolve an unknown signal that propagates in a multi- path environment with unknown characteristic function.

The problem of a signal reconstruction from partial information in the bispectrum domain has been investigated by Cetin [16] who proposed an iterative algorithm based on the method of projections onto convex sets.

The latter problem can be transformed into a reconstruction problem of a sequence based on Fourier domain information provided there is a way to convert the knowledge we have in the bispectrum domain into Fourier domain information. Several algorithms have been introduced in the literature to reconstruct the phase or magnitude of a sequence from the phase or magnitude of its bispectrum. The phase estimation problem from the bispectrum (or trispectrum) has been addressed by Brillin- ger, Lii-Rosenblat, Matsuoka-Ulrych (see [ 171 and references therein), Pan-Nikias [ 181, Alshbeili-Cetin [ 191, and Rangoussi-Giannakis [20]. The Fourier-magnitude estimation problem has been addressed by Sundaramoor- thy et al. [21].

The purpose of this paper is twofold: First, to present a procedure to reconstruct the phase of a sequence from the phase of its bispectrum. Second, to introduce an iterative scheme that will reconstruct a signal from only the phase of its bispectrum, bypassing calculation of the phase of the sequence first, and to show how some a priori information on the energy of the cepstra coefficients can improve the convergence rate of the algorithm.

11. PRELIMINARIES

Consider a deterministic ARMA energy sequence { ~ ( n ) } . Its Z transform is generally nonminimum phase

1053-587>(/92$03 00 1992 IEEE

_ ~ _ - I 1TT

602 IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 40. NO 3. MARCH 1992

and can be written as

Z(z) = A,z- ' I (z- ' )O(z)

with A , being a constant, r integer, and LI

1 - , = I II (1 - a , z - ' )

II (1 - c , z - 9 4 z - ) - L1

( = I

LZ

O(z) = II (1 - b,z) I = I

the minimum and the maximum phase components, respectively, with Ja , ) < 1 , Ib,J < 1, Ic,I < 1.

Let X ( w ) = X(eJW) be the Fourier transform of x ( n ) . Then the following functions are defined:

Power spectrum: P,(w) L x(w)x*(w) (4)

Bispectrum: B,(w l , w2) X ( o l ) X ( w 2 ) X * ( w ~ + 02)

(5) Bicepstrum: c,(m, n ) F-'[ln B,(wl, 4 1 (6)

m = O , n = O

m = 0, n > 0 - - A ( n )

where B,(wl , w 2 ) is the bispectrum of x(n) and P,(w) is the corresponding power spectrum.

The complex cepstrum of the bicoherence, cb,(m, n ) , is defined as the inverse Fourier transform ( F - ' [ -1) of the complex logarithm of the bicoherence index; i.e.,

( ' )

c,,(m, n) F-'[ln b,(ol , w2)1. (1 1 )

Rewriting (10) as (2)

where aB(wI, 02) is the phase of the bispectrum, we get ( 3 )

where

1 - B(-n) n

- -B(n )

m = 0, n < 0

1 n

m = n > 0

otherwise L o

LZ

B(k) = bf i = I

(9)

contain the minimum and maximum phase information, respectively.

The bicoherence index or normalized bispectrum of a sequence x ( n ) is defined as

from ( 1 1 ) that

ch,(m, n> = j F - I { @ d w I , U ? ) } . (13)

In other words, the cepstrum of the bicoherence is the inverse Fourier transform of the phase of the bispectrum.

It was shown in [22] that

1 2m cbr(m, 0) = -- [A(m) - B(m)], m > 0 (14)

where A(m) and B(m) are the cepstral coefficients.

of the power spectrum and it was found to be [22] The power cepstrum cp,(m) is defined as the cepstrum

- ( l / m ) [ A ( m ) + B(m)] m > 0 cp,(m) = In IAI l 2 m = O

( I / m ) [ A ( - m ) + B(-m)] m < 0. (15)

From (14) and (15), it is obvious that the differences of cepstral coeficients contain the phase information while their sums contain the magnitude information. The proposed reconstruction algorithms will be based on the aforementioned key observation.

i

111. BISPECTRUM SIGNAL RECONSTRUCTION (BSR) METHOD

A . Phase Estimation

be the phase of the bispectrum of x(n) .

given by

Define O,(w) to be the phase of x ( n ) and a B ( w I , 02) to

By definition, the complex cepstrum of x(n) , c,(m), is

c,(m) A F-'{log X(w) )

= F-I{log IX(w)I} + jF-I{O.,(w)}. (16) In terms of A(m) and B(m), the complex cepstrum is given by

- ( l /m)A(m) m > 0 i ( l / m ) B ( - m ) m < 0.

c,(m) = ln (AI I m = O (17)

Also, from the definition of the power cepstrum cp,(m) we have that

cp,(m) = 2F-I {log IX(w)I}. (18)

PETROPULU A N D NIKIAS: S I G N A L RECONSTRUCTION 603

Define In other words, there is a duality between the following quantities:

(28) j ( l / 2 m ) [ A ( Iml) - B( I4 )I m + 0

(19) +B(wl, w2) + In IB(wl, U,)(' - In ( X ( w , + wZ)l4 m = 0. d(m) A

Considering (13) and (14), d(m) can also be written as Consequently, we can utilize (20), (21) to calculate

d(m) = d&n, 0) (20) In I X ( ~ > ( as follows:

In (X(o)I* = F { d h ( m , 0 ) } (30) where

where

From (15)-(19), it follows that

O,(w) = F { d ( m ) ) (22) where d(m) can be evaluated either from (20) and (2 l ) , or from (19). Since the bispectrum suppresses all linear phase information, the reconstructed phase will correspond to a shifted version of the true sequence x(n).

Depending on how we calculate d(m) , (i.e., using (19) or (21)) O,(w) can be calculated by either a parametric or a nonparametric approach, utilizing the phase of the bispectrum. For the nonparametric BSR approach, to obtain the correct phase of the bispectrum, we may need a two- dimensional phase unwrapping algorithm. This will occur when the given bispectrum phase needs to be unwrapped first. Besides an existing phase unwrapping scheme [23], we suggest the following procedure for two-dimensional phase unwrapping.

1) 2 -0 Phase Unwrapping: The unwrapped phase of

d a m , n ) 4

Defining

d l ( m , n) = 2F- ' {In ( X ( w , + U,)('} (32)

we can show that

with cp,(0) being the power cepstrum at the origin. Since d,(rn, 0) is a constant, it will only affect the gain of I X ( W ) / ~ . As such, In I X ( W ) ( ~ can be calculated from the formula

(34)

the bispectrum can be calculated in the following way. Let

where Q! is a positive constant and dM(m, n ) is given by

F-'{ln ( ~ ( w , , w,>l') m # o i o m = 0.

d d m , n ) = ~ ( w ~ , w2) = I B ( ~ , , (23)

If c(m, n ) is the bicepstrum, then (35)

The advantage of computing the magnitude of the Fou- rier transform of the sequence from the bispectrum

1 m

c(m, n) = F - ' {In B(wl, U,)} = - F-I

m # O (24) through (30) lies on the fact that we are utilizing all the available information from the bispectrum.

The BSR procedures that reconstruct a sequence from its bispectrum, in a parametric or in a nonparametric way, are summarized in Table I .

- where R(m, n ) = F - ' { B ( w l , U,)} . When m = 0, c(0, n ) = c(n, 0) . Since we do not know the value of c(0, 0) we set c(0, 0) = 0. Then, the unwrapped phase +B(oI, U,)

of B ( w , , U,) is computed from

(25) C. Treating Zeros or Poles On or Close to the Unit Circle

By setting '('7 = we are loosing a linear phase in The parametric BSR phase and magnitude reconstruction schemes do not allow zeros or poles on the unit cir- the computation of the bispectral phase through (25).

cle, since they are based on the cepstra coefficients which are not defined in such a case. In the case where there are zeros or poles close to the unit circle, the lengths of the cepstra coefficients need to be very long, which implies that their computation has to be based on long two-dimensional Fourier transforms.

In the noise free case, if the observed sequence x ( n ) has zeros or poles on the unit circle, then by multiplying x(n) by a", we force the zeros to move towards the interior (a < l ) , or towards the exterior (a > 1 ) of the unit circle.

B. Magnitude Estimation

that From the definition of the bispectrum, (see ( 5 ) ) , we see

ln I B ( ~ , , w 2 ) ~ 2 - I X ( ~ , + w 2 ) 1 4 = In I x ( w , ) ( ' + In (x(w,)('

-In ( ~ ( w , + (26)

(27) iPB(wI, 0,) = O(w,) + O(w,) - O(w, + U?).

604 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 40. NO. 3, MARCH 1992

TABLE 1 PHASE A N D MAGNITUDE ESTIMATION FROM T H E BISPECTRUM, USING THE PARAMETRIC A N D NONPARAMETRIC BSR METHODS

~

Phase Estimation Magnitude Estimation

BSR

Consider now the case where the observed sequence is corrupted by additive noise, i.e.,

y(n) = x(n) + w(n) (36) where w(n) is a Gaussian, zero-mean stationary process, and x ( n ) and y (n ) are the transmitted and observed sequences, respectively. Multiplication of y(n) by an will have the undesired effect of turning the noise into a non- stationary process. On the other hand, in the third-order moment domain, in theory, the Gaussian noise is already suppressed; i.e.,

R,(m, n) = RAM, n) (37) where R,(m, n ) and R,(m, n) are the third-order moments of y (n ) and x ( n ) , respectively. Consider

RZ(m, n) = ami"R,.(m, n) . (38)

~ , ( z ~ , z2) = Z{RZ(m, n ) } = ~ , ( a - ' z ~ , a- l zz ) (39)

where B,(zl, z 2 ) is the 2 transform of R,(m, n) . From the definition of the bispectrum we have that

Then,

B,(a-'zI, a&) = B,(a-'zI, a- l zz )

= X(a -Iz,)X(a -Izz)X(a2z~Iz~I). (40)

From (39), (40), and assuming that z(n) is the sequence that corresponds to the bispectrum B ( w l , w2) , we get that

(41) where no is an integer, denoting the time shift that is lost in the bispectrum domain, and c is a real constant, denoting the scale that is also lost in the bispectrum domain.

In this way, the zeros or poles that were on, or close to the unit circle, will be pushed away from it, towards the center of the unit circle if a < 1, or outside the unit circle if a > 1. Of course, there may be a case where a zero that does not reside close to the unit circle, after the windowing, will come closer to the unit circle, creating a problem. Therefore, a suitable value for a must be se- lected.

If z(n) is a real signal, the zeros and poles of its Fourier transform are either real or appear in complex conjugate

a-"Z(n) = cx(n - no)

pairs. Hence, its cepstra coefficients can be written as

R I II

A,@) = ,z & + r = l ,c 2Ja,,;lk cos 4,; 1 = 1

12 R ,

R3 h

r = l r = l B,(k) = c b;,r + c 2(b,,, I k cos (43)

where c , , ~ are the zeros and poles of the minimum phase part of z(n); bZ,I are the zeros of the maximum phase part; R I , R2 are the number of real zeros and real poles of the minimum phase part; ZI , Z2 are the number of complex conjugate zero, and pole pairs of the minimum phase part; R3 is the number of the real zeros in the maximum phase part: and Z3 is the number of complex conjugate zero pairs. Hence, from (41) we obtain that

A,(k) = A,(k)ak (44)

B,(k) = B,(k)a" (45)

where A&), B,(k) and A,(k) , BZ(k) are the minimum and maximum phase cepstra coefficients of x(n) and z(n), respectively. Hence, the two-dimensional third-order moment window will introduce a one-dimensional window on the cepstra coefficients of the signal.

Note also that in the case where

y(n) = x ( n ) * h(n) + w(n) (46)

the application of the window on the third-order moment sequence of y (n ) will be equivalent to applying the same window on both the third-order moment sequences of h(n) and x ( n ) . This can be simply drawn by observing that

akA,(k) = ak(A,(k) + A&))

a 'B, ( k ) = a'B, ( k ) + Bh(k))

(47)

(48)

where A , , B , , are the cepstra coefficients of y ( n ) ; A , , B,, are the cepstra coefficients of x ( n ) ; A h , B,, are the cepstra coefficients of h(n); and a"A(k) , a"B(k) are the cepstra coefficients that correspond to the windowed third-order moments.

PETROPULU A N D NIKIAS: SIGNAL RECONSTRUCTION 605

IV. ITERATIVE SIGNAL RECONSTRUCTION FROM THE

PHASE OF THE BISPECTRUM

Hayes er al . presented in [ 5 ] the assumptions under which a FIR sequence can be reconstructed from its Fou- rier phase only. Those assumptions are repeated here in the following theorem, and applied in the bispectrum reconstruction problem.

Theorem 1: Let x(n) and y(n) be two finite length sequences, which are zero outside the interval 0 < n I N - 1, and their 2 transforms have no zeros on the unit circle nor in reciprocal pairs. Define O,(w) and O,(w) to be the phases of x ( n ) and y (n ) , respectively. If O.,(w) = O,,(w) at N - 1 distinct frequencies in the interval 0 < w < 7r, then x ( n ) = ay@), for some positive constant a.

Proof: The proof is shown in [ 5 ] . The assumptions under which a FIR sequence can be

reconstructed from its bispectral phase only are stated in the following lemma.

Lemma 1: Let x(n) and y (n ) be two sequences that sat- isfy the requirements of Theorem 1, and let Bx(wl, w2) and Oy(wl, w2) be the phases of the bispectrum of x(n) and y (n ) , respectively. Suppose we sample the bispectral phases at L > 2N - 1 equispaced points. If Ox(wlr w2) = O,(wl, w2) at the discrete frequencies within the region described by 0 < wI + w2 I R, w2 5 wI, and wI 1 0, thenx(n) and ay(n - no) for some positive constant a and some integer no.

Proof: Given that O,(wl, w2) = O,(wl, w z ) , by utilizing (21) and (22) , or (19) and (22), we end up with the phase O'(w) that corresponds to the phase response of both x(n - n l ) and y(n - nl ) , for some integers n l and n 2 . Then, as a result of theorem 1, x ( n - n l ) = ay(n - n,), o r x ( n ) = ay(n - no) with no = n2 - n I .

V . THE BICEPSTRUM ITERATIVE RECONSTRUCTION ALGORITHM (BIRA)

Using the formulas already derived in Section I11 to compute the phase or the magnitude of the Fourier transform of a sequence from the phase or the magnitude of its bispectrum, existing algorithms can be applied to reconstruct the sequence from its phase or magnitude [5]. How- ever, in this section, an iterative algorithm is introduced that will reconstruct a sequence from only the phase of its bispectrum, without calculating the phase of its Fourier transform first. Eventually, we will show that the BIRA algorithm is equivalent to the algorithm in [SI, in the sense that both algorithms reconstruct the same sequence at each iteration, but BIRA gives us the flexibility to exploit available U priori information and therefore speed up the algorithm's convergence rate.

Based on the knowledge of the phase of the bispectrum, the signal itself is recovered by the BIRA algorithm as follows.

, N - 1 that has no zeros on the unit circle or in reciprocal pairs. From the definition of { A @ ) ) , { B ( m ) ) , it follows that we

Consider an FIR signal { ~ ( n ) } , n = 0, 1, . -

can truncate A's and B's up to some integers p and q , respectively. Let r = max { p , q } .

Step 1. Initialize the power cepstrum cp,(m) to some arbitrary value, i.e.,

7 r. (49)

Since we know the phase of the bispectrum of x ( n ) , from (13), (14) it follows that the differences of cepstrum coefficients are also known, i.e.,

(0) p,(m) = 1, m = 0, 1,

A(m) - B(m) = D(m) , m = 1, * * 3 r. (50)

Step 2. Solve (14) and (15) for A(m) and B(m) , m > 0, and let the solution A(')(m), and B(i)(m), rn > 0, be

(51) -mc$l(m) + D(m)

2 A("(m) =

Step 3. Reconstruct x:'(n) from the following equa- tions:

X = p { e F k ( : t n ) ) } , n = 0, . . . , L - 1 (53)

where

-(I / m ) A ("(m) m > o c!)(m) = O m = 0 (54) r ( l / m ) B ' " ( - m ) m < 0.

Step 4 . Define y'"(n) = x'"(n)R,(n) where

Rdn) = 0, i f N - n o 5 n s L - n o - 1

(55)

where L is the length of the Fourier transform used in (53), and no is the time shift that is introduced due to the reconstruction of x(')(n) from its bicepstrum (or equivalently from its cepstra coefficients). Due to that shift, x"'(n) will appear for n in the interval [-no, N - no - 13. When x("(n) is reconstructed from (53), the sample x ( ' ) ( - n ) will appear at position L - n , as long as L > 2N. This is demonstrated in the following equation:

i 1, otherwise

X(')(k) = eF{cvtLl}, k = 0, . . . , L - 1. (57)

Step 5. Calculate the power spectrum of y("(n) and up-

(58 )

date c,,(m); i.e.,

c$,+"(m) = F - - ' {log Y( ' ) (w)12) .

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Repeat steps 2 to 5 until the reconstructed signal ( x ' (n ) } at each step remains unchanged. In other words, after defining

L - I

E, = C {x(I)(n) - x('- "(n)>2 (59) n = O

the algorithm will stop at iteration i , if E, < 6 (very small). At the ith iteration, x"'(n) will be equal to x(n) , within a scale factor.

Steps 1 through 4 correspond to the BIRA(no) algorithm. The parameter no is the time shift required in step 4. Since we don't know its value, we may guess any value in the interval [0, N - 11. The algorithm will converge only for the correct value of no. For an incorrect no, the algorithm may soon reach a constant E, for i > I . In order to check if this corresponds to the algorithm convergence, we can compare the differences of the ceptra coefficients that are initially given to BIRA, i .e . , D(m) = A(m) - B(m) , m = 1, * * * , r, with the one of the coefficients that c?rresponds to the reconstructed seqtence, y'"(n), i.e., D(m) , m = 1 , , r. If D(m) f D(m) , m = 1, . . . , r , then the guess for no was wrong, and we should repeat the same procedure for a new no value. Otherwise, x'"(n) corresponds to a convergent solution. The control for no can be implemented as an external loop connected to BIRA(no). The whole structure then corresponds to the BIRA algorithm.

A. Convergence Just@cation From steps 4 and 5 of the algorithm, it can be seen that

'I(m) = c!&m) (60) which is equivalent to

IX(' + y k ) I = I Y"'(k) I. (61)

That is, the magnitude of the Fourier transform of the sequence y(n) at the end of iteration ( i ) is taken to be the magnitude of x(n) at the beginning of iteration ( i + 1 ) .

The differences of the cepstral coefficients are constant in all iterations. From (13), (14), (19), (ZZ), this can be viewed as keeping the phase of the bispectrum or, equivalently, the phase of the Fourier transform constant and equal to the true phase of x(n) .

After these observations, the algorithm turns out to be equivalent with the one proposed by Hayes et al. in [5], in the sense that both algorithms produce the same sequence at the end of each iteration. Therefore, the same proof of convergence applies here too [lo]. We mention here that the proof was based on the theory of nonexpansive maps with respect to the Euclidean distance.

B. Convergence Improvement by Imposing Energy Constraints on Cepstral Coeficients

By setting an upper bound E to the energy of the cepstrum, or equivalently to the energy of ( A ( i ' ( m ) / m ) and (B" ' (m) /m) of the signal x( i l (n) at each iteration i , the

convergence rate of the algorithm can be substantially improved. At each iteration of the algorithm, we want

To guarantee (62), (see [12]), we include the following step 2.a immediately after step 2 of the algorithm:

Step 2.a Let

Then,

(A( ' ) (m), E(i) 5 E

and

( B 'i'(m), E(i) 5 E

Even further reduction in the number of iterations can be achieved if we separate energy bounds in the energies of the A's and B's , i.e.,

To guarantee this we have to modify the algorithm at each iteration (i) as follows:

fA" ' (m) , E,(i) 5 E ,

Remark: The algorithm that includes the energy constraints is equivalent to successive applications of the following three operators: time limiting, phase substitution, and cepstrum scaling. The latter does not correspond to a nonexpansive mapping with respect to the Euclidean distance. However, with respect to the distance

d 2 ( x , y ) = C llog X(k) - log Y(k)I2 (70)

where X ( k ) and Y(k) are the Fourier transforms of the sequences x(n) and y(n) , respectively, the phase substitution as well as the cepstrum scaling are proved to be nonexpansive maps (see Appendix A). The time limiting oper-

PETROPULU AND NIKIAS: SIGNAL RECONSTRUCTION

-

607

ator, however, is not nonexpansive with respect to this distance. Although we have not been able to show this theoretically, the combination of all three operators, i.e., the phase substitution, the cepstrum scaling, and the time limiting, always leads to a convergent solution.

VI. SIMULATIONS A . BSR for Phase Estimation

The model

z(n) = h(n) + w(n) (71)

of length N, = 128 was considered, where w(n) is zero- mean Gaussian white noise of variance 0 2 . In order to reduce the variance of the bispectrum estimate, it was assumed that L = 50 sensors were available to record L versions of z(n). The third-order moment sequence of z(n), i.e., RZ(m, n ) , was taken to be the average over the third- order moment sequences recorded at all sensors. The bicepstrum of z(n) was computed as

and from there, based on (7), its ceptra coefficients were computed, and given as input to BSR for phase estimation.

The following signal h(n) was considered:

h(n) = (0.7, l . , 0.8, -0.7, 0.6, -0.5, 0.4,

-0.3, 0.2, -0.1)

of length N = 10. By applying (22), the reconstructed phase for different noise levels is shown in Fig. 1. For all simulation examples, the signal-to-noise ratio (SNR), is taken to be

1 - c h2(n) N n

SNR = (73) o2

where o 2 is the noise variance.

B. Image Reconstruction from the Bispectrul Phase Only (No Observation Noise)

The BIRA was applied for the recovery of an image from the phase of its bispectrum. It was assumed that the image was transmitted line by line through the linear phase channel h(n) = (0.8, 1.0, 0.8}, and as a result, its magnitude was distorted. In Fig. 2, the originally transmitted image (14 X 14) is shown, as well as the received image (14 x 16).

A typical Fourier-magnitude plot for a line of the "NU" image is shown in Fig. 3 . The deep nulls in this plot correspond to zeros in the vicinity of the unit circle. Therefore, in order to apply cepstra-based operations, a windowing on the third-order moment sequence of every

0 1 2 3 4 5 6 7

frequency

Fig. 1. The true and reconstructed (through BSR) phase for different signal-to-noise ratios.

Fig. 2. Recovery of an image convolved with a linear phase channel, with BIRA. (a) Original image. (b) Image convolved with h(n ) = (0.8, I ., 0.8). (c) Recovery after 1 iteration. (d) Recovery after 10 iterations (e) Recovery after 30 iterations.

line of the received image was necessary (for the window description see section 111-C). For this particular example the window was taken to be

w(m, n) = (0.9)'""'.

BIRA was applied on the exponentially weighted cepstra coefficients, calculated based on the windowed third-or-

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1- r

1 2 3 4 5 6

frequency

Fig. 3. Typical Fourier-magnitude plot of an image line.

der moment sequence, and the reconstructed sequence had to be multiplied by the inverse exponential window. The reconstructed image obtained after I , 10, and 30 iterations is also shown in the same figure.

C. Image Reconstruction from the Bispectral Phase Only (With Observation Noise)

The same algorithm was applied for the recovery of an image embedded in additive, zero-mean, white, Gaussian noise, from only the phase of its bispectrum.

The original transmitted image (60 x 60) is shown in Fig. 4. The processing was made on half line each. It was assumed that 50 received sequences were available from different sensors, and the third-order moment sequence was taken to be the average over the third-order moment sequences from all receivers.

In this case too, a third-order moment window (a! = 0.82) was applied on the third-order moment sequence of every segment. The recovered images for 30 and 25 dB SNR are shown in Fig. 4 against the received images at one receiver, under the same SNR. For the recovery, the BIRA was applied for 50 iterations.

D. Signal Reconstruction This example employs the FIR sequence x ( n ) = (0.3,

1.0, -0.4, 0.7, -0.1). The BIRA algorithm (with and without energy constraints) was applied to reconstruct x ( n ) from only the phase of its bispectrum. We pointed out earlier that BIRA without energy constraints, reconstructs the same sequence as the algorithm proposed by Hayes et al. in [5] . However, Hayes et al. employed the method of adaptive relaxation to accelerate the convergence of their algorithm [8]. In Fig. 5 , the mean-squared error between the original sequence and the reconstructed one versus the number of iterations, is shown for BIRA, the adaptive relaxation technique, and BIRA with energy constraints. For the calculation of the phase of x ( n ) , (22) was utilized. From these figures, it is apparent that the

(d) (e)

Fig 4 Recovery of an image corrupted by additive white Gaussian noise, with BIRA (a) Origlnal image (b) Noisy image at 25 dB (c) Recovery at 25 dB (d) Noisy image at 30 dB (e) Recovery at 30 dB

10-7 I 1 0 10 20 30 40 50 60 70 80 90 100

number of iterations

Fig. 5 . The mean-squared error between the original sequence and the reconstructed one versus the number of iterations, for BIRA, the adaptive relaxation technique, and BIRA with energy constraints, for different energy bounds.

tighter the energy bounds are, the faster is the convergence of BIRA. However, the goal of this comparison is not to show that one scheme is better than the other, since the adaptive relaxation scheme utilizes no a priori information about the sequence. Fig. 5 demonstrates how much

PETROPULU AND NIKIAS: SIGNAL RECONSTRUCTION 609


(a) 1 0 ’

adapave relaxation

1 0 5 0 10 20 30 40 50 60 70 80 90 100


(b) Fig. 6 . Mean-square error versus the number of iterations for the adaptive relaxation technique, the BIRA algorithm with energy constraints and the “hybrid” scheme for (a) SNR = 30 dB and (b) SNR = 25 dB.

improvement can be achieved in the BIRA convergence rate when a priori information about x ( n ) is available.

In Fig. 6(a), the experiments are repeated for the case of additive white Gaussian noise. The rate of convergence of BIRA with energy constraint is very high in the first 5 iterations and then slows down. On the other hand, the adaptive relaxation scheme has uniform rate of convergence, except for some isolated points. Note that those isolated points are due to the fact that the relaxation parameter is not restricted to lie within the interval (0 , 2). This result indicates that we can run BIRA with energy constraint for a few iterations and then switch to the adaptive relaxation algorithm. The mean-square error versus the number of iterations for the “hybrid” scheme is also shown in the Fig. 6(a). Clearly, the convergence rate is greatly improved.

The same experiments were repeated for a different noise level, and the results are shown in Fig. 6(b).

VII. CONCLUSIONS In this paper, we proposed a computationally simple

formula to reconstruct the Fourier phase of a signal from

the phase of its bispectrum (BSR). Also, we presented an iterative scheme to reconstruct a FIR sequence from the phase of its bispectrum only (BIRA), bypassing the computation of its Fourier phase first, and we showed how we can incorporate some a priori information about the signal, in order to accelerate the convergence of the reconstruction algorithm. Finally, we proposed a “hybrid” scheme, consisting of BIRA and the adaptive relaxation algorithm, that succeeds an even faster convergence rate.

APPENDIX A A map F is said to be nonexpansive with respect to a

distance d(. , .) if

Let P denote the phase substitution operation, i.e., PX = Y implies that Y = I Xle ie for a given 8.

Considering the distance defined in (70),

d 2 ( x , y ) = c /log X(k) - log Y(k)I2 k

=

= c [log PX(k) - log PY(k)I2

= d2(Px, Py ) . (75)

liog ~x(k)[ + j e - log [Y(~)I - j e t 2 k

k

So, P is a nonexpansive mapping. Let S denote the scaling operation, i.e., SX = Y im-

plies that log X = a log Y with 0 I a < 1. Then

d(x , y ) = c [log X(k) - log Y(k)I2

I c la(2 llog X(k) - log Y(k)(? k

= d(Sx, Sy). (76) So, S is also a nonexpansive map. From (75) and (76) the composition of P and S, i.e.,

the mapping F = PS is also nonexpansive.

REFERENCES

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[3] A. Papoulis. “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trrms. Circuirs S w . , vol. CAS-22, no. 9, pp. 735-742, Sept. 1975.

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869-891, July 1987.

Athina P. Petropulu (S’86-M’91) was born in Kalamata, Greece. She received the Diploma degree in electrical engineering from the National Technical University of Athens, Greece, in 1986, the M.Sc. degree in electrical and computer engineering in 1988, and the Ph.D. degree in electrical and computer engineering in 1990, both from Northeastern University, Boston, MA.

From September 1986 through December 1990 she was a Research Assistant at Northeastern Uni- versity. She is currently a Visiting Professor at

Northeastern University. Her research interests are higher order spectra with applications in signal recovery, image processing, and signal reconstruction from partial information.

Chrysostomos L. Nikias (S’79-M’81-SM’87- F’91) received the Diploma degree in electrical and mechanical engineering from the National Technical University of Athens, Greece, in 1977 and the M.S. and Ph.D. degrees in electrical engineering from the State University of New York at Buffalo, in 1980 and 1982, respectively.

From 1982 to 1985 he was on the Faculty of the Department of Electrical and Systems Engineer- ing, University of Connecticut, Storrs. From 1985 to 1991 he was with Northeastern University in

Boston where he also served as the Director of the Communications and Digital Signal Processing (CDSP) Center for Research and Graduate Stud- ies of Northeastem University. He is currently a Professor of Electrical Engineering-Systems at the University of Southern California in Los An- geles. His research interests span the fields of higher order spectral analysis with applications, array processing, bioengineering, and detection and estimation theory. He has made contributions to these areas and has written over 60 published journal papers and 4 patents. Two of his publications have been reprinted in the IEEE Press book Modern Spectrum Analysis II . He is a contributor to the books Advances in Geophysical Data Processing, vol. 2 (JAI Press, Inc., 1985), Trauma Care (J. B. Lippincott Company, 1987) and Advances in Spectrum Estimation (Prentice-Hall, Inc., 1991). He has organized and extensively taught short courses in signal processing devoted to continuing engineering education. He is the designer and in- structor of a 15-hour videotape seminar entitled Modern Spectrum Esti- mation and Array Processing (Network Northeastern, Boston, MA, 1987).

Dr. Nikias is Chairman of the IEEE Signal Processing (SP) Technical Committee on Statistical Signal and Array Processing. He has previously served as Associate Editor of the IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, A N D SIGNAL PROCESSING (1985-1987), as a member of the Con- ference Board of the IEEE Signal Processing Society (1986-1989) and as organizer and Cochairman of the following workshops: Third ASSP on Spectrum Estimation and Modeling (Boston, MA, 1986), Higher Order Spectral Analysis (Vail, CO, 1989), and Higher Order Statistics (France, 1991). He was a Coeditor of the July 1990 IEEE TRANSACTIONSON Acous- TICS, SPEECH. A N D SIGNAL PROCESSING special section on higher order spectral analysis.

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Bispectrum