IEEE TRANSACTIONS ON ACOUSTICS, SPEECH,AND SIGNAL PROCESSING, VOL. ASSP-29, NO. 6, DECEMBER 1981 1187 Iterative Techniques for Minimum Phase Signal Reconstruction from Phase or Magnitude Absstract-In this paper, wedevelop iterative algorithms for recon- quency domain. Another iteration in this same style recovers structing a minimum phase sequence from phasethe or magnitude ofits a finite length mixed phase signal from the phase of its Fourier Fourier transform. These iterative solutions involve repeatedly impos- transform by imposing a finite length constraint in the time ing a causality constraint in the time domain and incorporating the known phase or magnitude function in the frequency domain. This ap- domain and the known phase in the frequency domain [4]. proach is the basis of a new means of computing the Hilbert transform In this paper, we begin in Section I1 with a discussion of a of the logmagnitude or phase of the Fourier transform of aminimum number of equivalent conditions for a sequence to be mini- phase sequence which does not require phase unwrapping. Finally, we mum phase. In Sections I11 and lV, we use these conditions in discuss the potential use of this iterative computation in determining developing two iterative reconstruction algorithms for mini- samples of the unwrapped phaseof a mixed phasesequence. mum phase signals, one for reconstruction when the phase is known and the other for reconstruction when themagnitude is I. INTRODUCTION known. NDER certain conditions a signal can be reconstructed In Section V, wediscuss the discrete Fourier transform U from a partial specification in the time domain, in the (DFT) realizations of the algorithms and illustrate the recon- frequency domain, or in both domains. A minimum or maxi- struction process with examples. mum phase signal, in particular, can be recovered from the In Section VI, we propose the use of the algorithms of Sec- phase or magnitude of its Fourier transform [l] . The conven- tions I11 and IV in implementing the Hilbert transform. Of tional reconstruction algorithm involves applying the Hilbert particular importance is reconstruction of the log-magnitude transform to the log-magnitude or phase of the Fourier trans- from phase since the proposed iterative approach requires only form to obtain the unknown component. the principal value of the phase, while the direct DFT imple- In this paper, we take an alternative approach by developing mentation of the Hilbert transform requires the unwrapped iterative algorithms for reconstructing a minimum (or maxi- phase [5]. The proposed technique,therefore, avoids prob- mum) phase signal from the phase or magnitude of its Fourier lems typical of phase unwrapping such as detection of the dis- transform. Specifically, we develop algorithms which impose continuities in the principal value of the phase [l], [6] . Also, causality in the time domain and the given phase or magnitude in Section VI, we suggest the use of this new approach to im- in the frequency domain, in an iterative fashion. plementing the Hilbert transform as the basis for a phase un- Iterative algorithms similar to those we discuss here have wrapping algorithm. been useful in a number of areas where partial information in thetwo domains is available. Inparticular, the algorithms 11. THE MINIMUMPHASE CONDITION presented in this paper are similar in style to the Gerchberg- In general, a signal cannot be uniquely specified by only the Saxton algorithm [2] and an iterative algorithm by Fienup phase or magnitude of its Fourier transform. However, one [3] , in alternately incorporating partial information in the condition under which the magnitude and phase are related is time and frequency domains. The Gerchberg-Saxton algo- the minimum phase condition and under this condition a signal rithm recovers a two-dimensional complex signal by iteratively can be uniquely recovered from the magnitude of its Fourier imposing the finite extent of the signal in the space domain transform or to within a scale factor, from the phase of its and its magnitude in both the space and frequency domains. Fourier transform. In this section, we discuss anumber of Similarly, Fienup's algorithm recovers a real two-dimensional equivalent conditions for a signal to be minimum phase. These signal by iteratively imposing the finite extent and positivity conditions will be of particular importance in Section 111 in de- of the signal in the space domain and its magnitude in the fre- veloping the iterative algorithms. In the following discussion we restrict the z transform of Manuscript received June 13,1980; revised June 30, 1981. Thiswork the sequence h(n) to be a rational function, which we express was supported in part by the Department of the Air Force under Con- tractF19628-80-C-0002, in part by theNational Aeronautics and in the form Space Administration under Grant NAGS-2, and in part by the Ad- vanced Research Projects Agency, monitoredby the Office of Naval Re- Mi MO search under Contract NOOO14-75-C-0951-NR 049-328. The views and conclusions contained in this document are thoseof the contractor and n (1 - ~1)rI (1 - bkz) k=1 should not be interpreted as necessarily representing the official poli- H(Z) = AZno = cies, either expressedor implied, of the US.Government. Pi PO (1) T. F. Quatieri, Jr. is with Lincoln Laboratory,Massachusetts Institute (1 - Ckz-l) (1 - dkZ) of Technology, Lexington,MA 02173. n n k= 1 k=l A. V. Oppenheim is with the Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cam- bridge, MA 02139. where la,/, Ibkl, Ickl, and ldkl areless thanor equal to 0096-3518/81/1200-1187$00.75 0 1981 IEEE 1188 IEEETRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING,VOL. ASSP-29, NO. 6, DECEMBER 1981 unity, zno is a linear phase factor, and A is a scale factor. Again, from (2) it follows that these conditions are necessary When, in addition, h(n) is stable, i.e., En Ih(n)l < 00, Ick 1 and since (2) has no poles or zeros outside the unit circle or at in- Idk\ are strictly less than one. finity, guaranteeing causality, and from the initial value theo- A complex function H(z) of a complex variable z is defined rem h(0) = lim H(z) =A. To demonstrate that these con- z+m to be minimum phase if it and its reciprocal H-’ (z)are both ditions are sufficient, we note that again causality of h(n) will analytic for Iz I > 1. A minimum phase sequence is then de- eliminate factors of the form (1 - dkz) in the denominator of fined as a sequence whose z transform is minimum phase. For (1). Furthermore, since the conditions require that h(n) be H(z) rational, as in (2), the minimum phase condition excludes causal, the initial value theorem can be applied with the result poles or zeros on or outside the unit circle in the z plane or at that infinity. As a consequence, the factors of the form (1 - bkz) corresponding to zeros on or outside the unit circle and the factors of the form (1 - dkz) corresponding to poles on or out- h(0) = lim H(z) = lim AZno (1 - Z’m 2’- n side the unit circle will not be present. Furthermore, in (l), k=l no = 0 to exclude poles or zeros at infinity. Thus, for H(z) Since h(0) = A, minimum phase, (1) reduces to M, Mi n (1 - akz-’) k=l H(z) =A (2) and since Ibk 1 this requires that no = 0 and the bk’s be Pi 1. < equal to zero. Thus, again (1) reduces to (2). rJ (1 - ckz-l) k=l Another condition which can be shown to be equivalent to minimum phase condition A or B or our original definition of where lak] and I ck I are both strictly less than unity. a minimum phase sequence is that the log-magnitude and un- From (2) other conditions can be formulated for a signal to wrapped phase of H(a) are related through the Hilbert trans- be minimum phase. Two conditions in particular which we form [l] . The Hilbert transform relation guarantees that a discuss below are particularly useful in the context of the iter- minimum phase sequence can be uniquely specified from the ative algorithms to be discussed in Sections I11 and IV. Fourier transform magnitude and, to within a scale factor, Minimum Phase Condition A from the Fourier transform phase. One technique for minimum phase signal reconstruction Consider h(n) stable and H(z) rational in the form of (1) from phase or magnitude relies on a DFT implementation of with no zeros on the unit circle. A necessary and sufficient the Hilbert transform [5]. In the next two sections, we take condition for h(n) to be minimum phase is that h(n) be causal, an alternate approach which invokes an iterative computation. i.e., h(n)= 0, n < 0, and no in (1) be zero. Motivated by the minimum phase condition A, when the phase From (2), it follows that these conditions are necessary. To is given weimpose, in an iterative fashion, causality in the time show that they are sufficient, we want to show that they force domain and the known phase in the frequency domain. When (1) to reduce to (2). Clearly, factors of the form (1 - dkz), the resulting sequence satisfies minimum phase condition A ldkl < 1 in the denominator introduce poles outside the unit and has the given phase, it must equal h(n) to within a scale circle which would violate the causality condition since h(n) is factor. Likewise, motivated by the minimum phase condition restricted to be stable.