Reconstruction of signals from sign information alone

Reconstruction Information

of Signals from

Sign

Alone

by ARUNA BAYYA U.S. West Advanced CO 80111, U.S.A.

Technologies,

6200 South

Quebec

Street,

Englewood,

of Pittsburgh,

Pittsburgh,

and L.F.CHAPARRCI Department PA 15261,

of Electrical U.S.A.

Engineering,

University

ABSTRACT: A new representation of signals by sign information alone is derived. This representation is based on the fact that the si~qn information in the frequency domain is directly related to the zero distribution of the z-transform of the signal. Algorithms to reconstruct signals from sign information alone are also proposed.

I. Introduction

The problem of reconstruction from partial Fourier domain information has been studied by many physicists and engineers for the past few years (14). The partial information assumes many forms depending on the application. Among these, phase and magnitude have received considerable attention because of the need for magnitude recovery and phase retrieval in many fields. In this paper, we investigate the problem of reconstructing signals from sign information of the Fourier transform (FT) real part of the signal which is also referred to as l-bit FT phase. In Ref. (5), it has been shown that a two-dimensional (2-D) signal can be uniquely specified by the sign of its FT real part under certain conditions. One of the constraints is the non-factorability of the z-transform of the signal. This representation is possible because of the lack of a fundamental theorem of algebra in two dimensions. However, since a one-dimensional (1 -D) polynomial can always be factored in the complex domain, the 1-D counterpart of this result does not exist. Our representation of one-dimensional signals requires two sign functions for uniqueness. The rest of the paper is organized in the following way. After some preliminaries, uniqueness conditions to represent 1-D signals by its sign information are discussed in Section II. Algorithms for reconstructing these signals from sign information alone are derived in Section III. Examples illustrating the performance of the proposed reconstruction methods are presented in Section IV and the concluding remarks follow in Section V.

m,* The Frankhnlnstilute 001~0032,'90$3.CQ+000

993

A. Buyyu and L. F. Chuppmro II. Representation

by Sign Information

As a complex valued function can be expressed as

X(e”‘), the Fourier

X(e”‘) = (X(o)le”(w)

transform

of a sequence

x(n)

(1)

= X,(w) +jX,(o)

where IX(w)] is the magnitude and Q(w) is the phase of the Fourier transform. X,(w) is the real part and X,(w) is the imaginary part of the Fourier transform. The ;roblem of reconstruction from sign is to obtain the signal -Y(M)given Sgn [X,(o)], where the sign function is defined as Sgn [u] = i-

1

ifa

1

ifa<

(2)

Since the sign information is an indication of the location, with respect to the unit circle, of the zeros of the z-transform, we examine the following three categories (based on the location of zeros) of signals. The minimum phase sequence is a sequence with z-transform that has all zeros inside the unit circle. The z-transform of maximum phase signals on the other hand, has all zeros outside the unit circle. We refer to a sequence with z-transform that has zeros both inside and outside of the unit circle, as a mixed phase sequence. The information content of the sign of the FT real part depends on the type of signal. It is known that the phase of a maximum phase sequence of length Nf 1 varies monotonically, and changes exactly by Nrc as the frequency varies from 0 to rc. Therefore, the real part of the FT of the maximum phase sequence has exactly N sign changes for frequencies between 0 and rr. This implies that this type of sequences can be uniquely specified by its sign information or equivalently by lbit phase. However, the maximum phase condition is too stringent and hence the above uniqueness condition is not useful in practice. The mixed phase signals, on the other hand, represent a broader class of signals, but information in the sign of FT real part is not sufficient to represent them uniquely. In particular, signals that belong to the specific class of minimum phase signals carry very little information in the sign of FT real part. Additional information is required to complement the information content of the l-bit phase in order to represent these signals uniquely. Because of the relation between zero locations with respect to unit circle and the sign information. under some mild conditions this additional information turns out to be another sign function. These conditions are formally stated in the next theorem. Before stating the theorem, a result used in the theorem is presented in the following lemma. Lemma

Let x(n) be a sequence of length N+ 1 such that its z-transform X(z) has no zeros on the unit circle and no zeros at frequencies nk/N and 2nk/N. Denote x(N--n) by x,(n). If wi, i = 1, 2,. . . , K are the frequencies at which Re (X(o)} changes sign, and cG,, j = 1, 2,. . . , L are the frequencies at which Re {X,(o)} changes sign, the u), # 6, for any i, ,j defined above. 994

Journal

of the Franklin Pergamo”

Institue Press plc

Reconstruction

of Signals from Sign Information

Alone

Pro@‘: By definition, Re X(0,) = 2

x(n) cos (qn)

= 0,

i= 1,2,...,K

fl=O

ReX(G,)

= 1 x(N-n)cos(&jn)

=O,

j=

1,2 ,...,

L.

(4)

n=O

if we assume (4) we get :

Expanding

that w, = ~5~for some i and j, then adding

and subtracting

(3) and

j.

x(n) cos (qn) + Eio x(n) cos (o,(N-n))

= 0

(5)

j.

x(n) cos (wn) - n$o x(n) cos (0, (N-n))

= 0.

(6)

cos o, (N-

n) using trigonometric

identities

we get from (5) and (6),

-sin (w;N) cot [&w,)] = ~~ 1+ cos (WiN)

(7)

sin (o,N) cos [$J(wJ] = -~ 1 -cos (w,N) Multiplying

(7) and (8) and using trigonometric cot2 [C$(f&)]= - I

which is impossible 2,...,L.

for real &o,),

therefore

identities,

we get (9)

oi # 07, i = 1, 2,. . . , K and j = 1,

Theorem I Let x(n) and y(n) be two sequences of length N+ 1 satisfying the conditions stated in the Lemma. Defining S,(o) = Sgn {X,(o)} and S,(o) = Sgn { YR(w)}, if S,(w) = S,,(o) and S,,(o) = S,,(w) for all w, then x(n) = q(n) for some real constant C. Proof: In the absence of zeros on the unit circle, if X(z) has A4 zeros outside the unit circle, then X,(Z) has N -A4 zeros outside the unit circle and A4 zeros inside the unit circle. In Ref. (6), we have shown that if X(z) has A4 zeros outside the unit circle, then Re(X(o)j changes sign at least A4 times as cc) increases from 0 to rc. Likewise, Re {X,(o)} changes sign at least N-M times as w increases from 0 to rr. Thus, adding the number of sign changes of Re{X(w)} and number of sign changes of Re{X,(w)), the total number of sign changes is bigger than or equal to N for 0 6 o < TC.Let us define the sequences u(n) and a,(n) as : u(n) = x(n)*y(

-n)

al(n) = x,(n)*y,(-n). These sequences A(z) = X(z)Y(l/z) Vol. 321, No 6. pp. 993-1001. Pnntcd in Great Bntain

(10) (11)

a(n) and a,(n) are zero outside the interval -N Q n d N and and A,(z) = X(l/z)Y(z) = A(l/z). Therefore, a,(-n) = u(n). 1990

995

A. Bayya and L. F. Chaparro Given that Re {X(o)} = Re { Y(w)) = 0, i = 1, 2,. . . , K, $Joi) = &(o,) - 4,,(q) = kz for i = 1, 2,. , K where k is an integer and thus Im [A(cuJ] = Similarly,

Re{X,(G,))

f a(n) sin (w,n) = 0, II= .Y

1,2 ,...,

K.

(12)

= Re{ Y,(G,)} = 0, ,j = I, 2,. . , L which in turn gives .Y *TNa,(n)sin(ril,n)

(a(n) - a( -n))

j=

=O,

Equati -ns (12) and (13) can be rearranged .i,

i=

I,2 ,...,

L.

(13)

i= 1,2,...,K

(14)

to yield

sin (o,n) = 0,

and ,,i, (a,(n)-a,(--))sin(G,n)

=O,

j=

1,2 ,...,

Since a,(n) = a( - n) and w, # (;j, for any i, j (as established and (1.5) can be combined to form one equation as i,

(a(n)- a( - n)) sin (w,,n) =

0,

m = I,2 ,...,

L.

(15)

in the Lemma),

(14)

K+L.

(16)

As K+ L 2 N, the above equation is true for at least N distinct w,,,. Because {sin (o,,n)} for n = 1, 2,. . . , N form a Chebyshev set, it follows that a(n) -a( -n) = 0 for n = 1, 2,. . , N, that is a(n) is symmetric about n = 0. Then, 6(o)

= $r(w)+kn

(17)

where k is either 0 or k 1. According to (2), if X(z) and Y(Z) do not have reciprocal zero pairs or unit circle zeros, (17) implies that x(n) = cy(n), where c is a real constant. QED For a maximum phase sequence x(n), the sign information of the reversed signal .x,(n) is redundant as the sign information of x(n) is sufficient to represent the sequence uniquely. Thus the following Corollary. Corollary. Let x(n) and y(n) be two finite length maximum phase sequences with S,(o) = S,.(w) for all w. Then x(n) = cy(n), where c is a real constant.

I/I. Reconstruction

Algorithms

We now focus our attention on developing algorithms to reconstruct the signals from the sign of the FT real part. The problem of reconstruction is formulated, as in the case of signal reconstruction from phase (2), by the following set of linear equations :

Reconstruction

Re X(w) = 2 x(n) cos (qn) n=O .Y ReX,(cuJ = c .xn( ) cos(w,(N-n))=O, ,i= 0

of Si.ynals,from = 0,

i=

Sign Information

Alone

1,2,.,.,/C

i=K+l,K+2

(18) ,...,

K+L.

(19)

Since all w,, wj are distinct and K+ L 3 N, we can choose any N equations from the above K+ L equations. By choosing K equations from (18) and the rest of N-K equations from (19), these equations can be represented in matrix form as cos(w,)

...

cos(No,)

41)

cos (~0~)

...

cos (NoI)

-w ... wo

..

..

cos (wK)

cos ((N-

..

l)OK+ ,)

...

.. cos ((N-

...

1)~~)

.. cos (NoJ

1

x(K+ 1)

...

..

1

-I

-x(O) -

=

-40) ..

-x(O) - 40) ~0s WOK+I) . -x(O)

cos (NW/J

( )> Since the problem has a unique solution, the coefficient matrix is non-singular. The solution [x( 1) x(2) . . . x(N)ITis obtained in terms of x(0) which is not known, and thus the reconstruction is correct up to a constant. Our second reconstruction algorithm, is a modification of the iterative algorithm proposed by Van Hove ef al. (7) to reconstruct a signal from its signed FT magnitude. Since in our case only sign information (of the sequence and the reversed sequence), instead of sign and magnitude functions, is available an additional pair of transformations is required in each iteration. The modified algorithm is depicted by a block diagram of Fig. 1. Theorem I is valid only when the sign function S,(o) = S,(w) and ,S,, (w) = S,.,(o) for all frequencies. For computational purposes the sign function is taken as a discrete function of o with the restriction that the frequency o is very finely sampled.

ZV. Examples

The following two examples illustrate the reconstruction of a minimum phase sequence and a mixed phase sequence using the non-iterative reconstruction algorithm as well as by solving linear equations. A minimum phase sequence of length 25 is shown in Fig. 2. The real part of the Fourier transform of this sequence is positive for all frequencies (Fig. 3) and therefore the sign function has no information about the sequence. Therefore we consider the sign information of the reversed sequence for reconstruction. The reconstructed sequence using linear equations formulation is indistinguishable from the original and therefore is not shown. As a second example, we consider a mixed phase sequence of length 20 which is shown in Fig. 4. Even though the real part of the FT of this sequence changes sign Vol 327. No. 6. pp YY3-I(X Pr1ntcd I” Grea Brllaln

IYYO

997

A. Buyyn and L. F. Chupurro

Initial

Condition i =O

-Q(n)

= IDFT[S,(k)]

r

x.(n) = 0

for

n2N

x,(n) = x,(N-n)

X,(k) =DE”k(n)l

Xi+,(k)=

Ix,(k)l%(k)+JX,,(k)

x, (n ) = IDFT K, +l(k

v I--

i = i+l

)I

1

FIG. I. Iterative method of reconstruction

from sign information.

11 times between 0 and rc (Fig. S), this does not provide sufficient information to reconstruct the signal. As a result, the l-bit phase of the FT of the reversed sequence is also used for the reconstruction of this sequence. The result of reconstruction using the iterative method is presented in Fig. 6 which is almost identical to the original sequence. In the simulation all these sequences required a Discrete Fourier Transform (DFT) of size 2048 for the sign functions to obtain the results shown. This is to be expected since the reconstruction depends on the location of zero crossings. Journal

998

of the Frankhn Pergamon

Institute Press plc

Reconstruction

of Signals from Sign Information

Time

FIG. 2. Reconstruction

of minimum

Alone

(n)

phase sequence

from sign : original.

60 ‘; ;;

40

_; d

20 0 -20 1

A

1

-HO

2

Frequency FIG. 3.

Reconstruction

of minimum

-0

5

10

Reconstruction

Vol.327.Na h. pp.993-1001. Prmted tn Greal Br,ta,n

4

(WI

from sign : real part of FT.

phase sequence

Time FIG. 4.

3

15

20

;5

(n)

of mixed phase sequence

from sign : original.

,990

999

A. Buyya and L. F. Chuparro

4 Frequency FIG.

5. Reconstruction

of mixed phase sequence

-6 I 0

5

10 Time

FIG. 6. Reconstruction

(WI

15

from sign : real part of FT.

20

J

;!I

(n)

of mixed phase sequence

from sign : reconstructed

V. Conclusions In this paper we have shown that a 1-D signal can be represented by the sign information alone in the frequency domain. The result can be viewed as the counterpart of the result for 2-D signals due to Curtis it ~1. (5). Theoretically, the sign information is considered to be available as a continuous function of frequency. However, as illustrated by our examples, if the frequency is sampled at a high sampling rate, then the reconstruction is possible using sign information as a function of discrete frequency Finally, the extension of this representation to twodimensional signals (not presented in this paper) also yielded promising results. References (1) J. R. Fienup, “Reconstruction of an object from modulus of its Fourier Transform”, 0ptic.v Lctt.. Vol. 3. pp. 21-29. July, 1978. “Signal reconstruction from phase or (2) M. H. Hayes, J. S. Lim and A. V. Oppenheim, magnitude”, IEEE Truns. Acoust. Speech Sipnl Process., Vol. 28, pp. 672-680, December, 1980. 1000

Reconstruction

of Signals from Sign Information

Alone

(3) A. V. Oppenheim (4)

(5)

(6) (7)

and J. S. Lim, “The importance of phase in signals”, Proc. IEEE, Vol. 69, pp. 329-341, May, 1981. A. V. Oppenheim, J. S. Lim and S. R. Curtis, “Signal synthesis and reconstruction from partial Fourier domain information”, J. Opt. Sot. Am., Vol. 73, pp. 141331420, November, 1983. S. R. Curtis, J. S. Lim and A. V. Oppenheim, “Signal reconstruction from Fourier transform sign information”, IEEE Trans. Acoust. Speech Signal Process., Vol. 33, pp. 643-657, June, 1985. L. F. Chaparro and Aruna Bayya, “Non-iterative reconstruction of signals from signed Fourier transform magnitude”, J. Franklin Inst., Vol. 324, pp. 27742, October, 1987. P. L. Van Hove, M. H. Hayes, J. S. Lim and A. V. Oppenheim, “Signal reconstruction from signed Fourier transform magnitude”, IEEE Trans. Acoust. Speech Signul Process., Vol. 31, pp. 128661293, October, 1983.

Vol 327. No h. pp YYX-1001. Printed m Great Bntm

IYYI)

1001