- Email: [email protected]
Minimum-phase parts of zero-phase sequences
ARTICLE IN PRESS Signal Processing 89 (2009) 1032–1037
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Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Minimum-phase parts of zero-phase sequences Corneliu Rusu a,, Jaakko Astola b a
Faculty of Electronics, Telecommunications and Information Technology, Technical University of Cluj-Napoca, Cluj-Napoca, Str. Baritiu 26-28, RO-400027, Romania b Tampere International Center for Signal Processing, Tampere University of Technology, Tampere, Finland
a r t i c l e i n f o
abstract
Article history: Received 26 May 2008 Received in revised form 8 September 2008 Accepted 4 December 2008 Available online 24 December 2008
For more than a decade it has been empirically known that the causal portion of the inverse Fourier transform of the magnitude spectrum of the speech signal behaves like a minimum-phase signal. Later on, this statement has been shown for an all-pole model. In this paper, we consider related results for both discrete-time Fourier and discrete Fourier transforms of arbitrary sequences. We indicate how the presence of aliasing in circular autocorrelation might be detected. The energy concentration property of zerophase sequences is discussed. & 2008 Elsevier B.V. All rights reserved.
Keywords: Signal reconstruction Discrete-time Fourier transform Discrete Fourier transform Nonuniform discrete Fourier transform Minimum-phase function
1. Introduction In many cases where it is necessary to avoid any phase distortion, one solution is to make the frequency response real and nonnegative, i.e., to have a filter with a zerophase characteristic [1]. Besides filter design, zero-phase signals appear in applications like phase retrieval or phase-only reconstruction [2] and in speech analysis [3]. In spectral factorization theorems, the zero-phase condition represents a key assumption. Perhaps the most known result on zero-phase sequences is the celebrated Feje´r and Riesz theorem [4]: P n Theorem 1 (Feje´r and Riesz). If XðzÞ ¼ M n¼M xðnÞz PM jo and Xðe ÞX0, then there is YðzÞ ¼ n¼0 yðnÞzn such that Xðejo Þ ¼ jYðejo Þj2 and YðzÞ unique if minimum-phase. In this paper, we discuss some properties of minimumphase and zero-phase sequences. As the properties of such sequences are related to the properties of rational Corresponding author. Tel.: +40 2 64 401804; fax: +40 2 64 591340.
E-mail addresses: [email protected] (C. Rusu), Jaakko.Astola@tut.fi (J. Astola). 0165-1684/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.12.006
functions and analytic functions in general, as well as those of Fourier series, results that are closely connected appear implicitly in circuits and systems literature. About two decades ago it was shown that if a positive real function belongs to the class of rational functions, then it must be minimum-phase [5]. To best of our knowledge, there is no published result indicating that the Fourier transform of a causal part of an arbitrary zero-phase sequences is a positive real function belonging to the class of rational functions. Recently, it was noticed that the causal portion of the inverse Fourier transform of the magnitude spectrum of the speech signal behaves like a minimum-phase signal [6]. This property has been verified for an all-pole model [7,8]. Here we present related properties that may be of interest from a digital signal processing point of view, but do not appear explicitly in the literature. In the following we recall few definitions (Section 2). We consider the causal part of an arbitrary zero-phase sequence and show that it is a minimum-phase sequence (Section 3). We also give a direct proof of the fact that the inverse DFT transform of a strictly positive sequence is minimum-phase (Section 4). We also extend this to NDFT (nonuniform discrete Fourier transform) in Section 5.
ARTICLE IN PRESS C. Rusu, J. Astola / Signal Processing 89 (2009) 1032–1037
1033
The presence of aliasing in circular autocorrelation and the energy concentration property of zero-phase sequences are discussed in Section 6.
If jzj41, then the sequence ejon zn is absolute summable. In such situation we can interchange the order of summation and integration in (1). We get
2. Definitions
X þ ðzÞ ¼
Definition 2. A sequence is causal if xðnÞ ¼ 0 for no0. A sequence is anti-causal if xðnÞ ¼ 0 for n40. Definition 3. A sequence is minimum-phase if all zeros of its z-transform are inside the unit open disk. A sequence is maximum-phase if all zeros of its z-transform are outside the unit closed disk.
1 2p
Z p "X 1 p
Re
jXðejo Þj 1 r 1 ejðoyÞ
Thus
Definition 5. A complex valued function XðzÞ is positive real if
RefX þ ðzÞgX
Definition 6. A sequence is zero-phase sequence if its Fourier transform is a nonnegative function. Sometimes a zero-phase sequence is said to be a positive sequence [9]; in this paper, we shall refer to this as zerophase sequence, to distinguish from positive real function. 3. Zeros location of causal part of zero-phase aperiodic sequences
n¼0
jo
Xðe Þ ¼
1 X
jon
xðnÞe
Z p
1 r 1 2pð1 þ r 1 Þ2
jXðejo Þj40;
arg Xðejo Þ ¼ 0.
In this case by using inverse DTFT, the sequence can be computed as follows: Z p Z p 1 1 Xðejo Þejon do ¼ jXðejo Þjejon do. xðnÞ ¼ 2p p 2p p The z-transform of causal portion of xðnÞ is the one-sided z-transform: Z p 1 1 X X 1 xðnÞzn ¼ jXðejo Þjejon do zn . (1) X þ ðzÞ ¼ 2p p n¼0 n¼0 Our goal is to show that the causal portion of xðnÞ is a minimum-phase sequence. Theorem 7. Let xðnÞ be a stable sequence. If jXðejo Þj40 and arg Xðejo Þ ¼ 0 for all o 2 R, then its causal portion is a minimum-phase sequence. Proof. First we establish that for any jzj41, we have RefX þ ðzÞg40.
jXðejo Þj do. 1 ejo z1
jXðejo Þj do40.
p
It follows that for any jzj41, we have X þ ðzÞa0. To show that X þ ðejo0 Þa0, first notice that Z p Z p 1 1 Xðejo Þ do ¼ jXðejo Þj do40. xð0Þ ¼ 2p p 2p p
(2)
Assume now that X þ ðejo0 Þ ¼ 0. Then Xðejo0 Þ ¼ X þ ðejo0 Þ þ
1 X
xðnÞejo0 n ¼ X þ ðejo0 Þ xð0Þ
n¼1
þ
0 X
xðnÞejo0 n
n¼1 1 X
xðnÞejo0 n ¼ X þ ðejo0 Þ xð0Þ
n¼0
þ
1 X
jo 0 n
xðnÞ e
n¼0 þ
jo0
¼ X ðe
" Þ xð0Þ þ
1 X
# jo0 n
xðnÞe
¼ X þ ðejo0 Þ xð0Þ þ ½X ðe
satisfies
p
jXðejo Þj½1 r 1 cosðo yÞ j1 r 1 ejðoyÞ j2 jo jXðe Þjð1 r 1 Þ X . ð1 þ r 1 Þ2
n¼0 þ jo0
n¼1
Z p
¼
¼ X þ ðejo0 Þ xð0Þ þ Let xðnÞ be zero-phase sequence with no zeros or poles on the unit circle. Then its DTFT (discrete-time Fourier transform)
1 2p
Now, for z ¼ r ejy, r41 Z p 1 jXðejo Þj do X þ ðzÞ ¼ , 2p p 1 r 1 ejðoyÞ
Definition 4. A sequence is absolute summable or stable sequence if the region of convergence of its z-transform includes the unit circle.
(1) z 2 R ) XðzÞ 2 R; (2) jzjX1 ) RefXðzÞgX0.
# ejon zn jXðejo Þj do ¼
Þ ¼ xð0Þ.
This is a contradiction since Xðejo Þ is positive for all o 2 ½p; p. & Example 8. Let XðzÞ ¼ z þ 52 þ z1 , then Xðejo Þ ¼ 12 þ 4 cos2 o=240. The causal portion of XðzÞ is X þ ðzÞ ¼ 52 þ z1 and its zero is z1 ¼ 25. According to Feje´r–Riesz theorem, there is YðzÞ such that pffiffiffi Xðejo Þ ¼ jYðejo Þj2 . Indeed, for Y 1 ðzÞ ¼ 2ð1 þ 1=2z1 Þ and pffiffiffi Y 2 ðzÞ ¼ 2=2ð1 þ 2z1 Þ we have Xðejo Þ ¼ jY 1 ðejo Þj2 ¼ jY 2 ðejo Þj2 . Moreover, Y 1 ðzÞ is minimum-phase function, and Y 2 ðzÞ is maximum-phase function. Corollary 9. For any sequence having no zeros or poles on unit circle, the causal portion of autocorrelation is minimumphase sequence. Example 10. The causal portion of a symmetric sequence may be minimum-phase, but the sequence is not always zero-phase, e.g., XðzÞ ¼ 0:7z þ 1 þ 0:7z1 .
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4. Minimum-phase property of DFT magnitude One can find a counterexample that the causal part of circular autocorrelation is not always a minimum-phase sequence (Section 6.1). However, it can be shown that for any finite sequence of length at most N, having no zeros on unit circle, the N-point circular autocorrelation is a minimum-phase sequence. We start with the following result: Proposition 11. If m0 ; m1 ; . . . mN1 are positive numbers, then the polynomial RðzÞ ¼
N1 Y
1 ej2pk=N z1
1 NX
k¼0
k¼0
1
mk ej2pk=N z1
Proof. Clearly mk ¼ DFTfan g. Since mk 40, then mk ¼ jDFTfan gj. It remains to show that an is a minimum-phase sequence. To obtain this result, a little manipulation is needed. " # N 1 N 1 1 X X 1 NX n j2pkn=N AðzÞ ¼ an z ¼ m e zn N k¼0 k n¼0 n¼0 " # N 1 N 1 1 X X 1 zN NX mk ¼ mk ej2pkn=N zn ¼ . j2pk=N z1 N 1 e n¼0 k¼0 k¼0 Consequently AðzÞ ¼ RðzÞ=N and using Proposition 11, AðzÞ has all zeros inside the unit disk. &
,
has all zeros inside the unit disk.
Example 13. Consider N ¼ 3 and
Proof. Consider z ¼ rejy where rX0 and y 2 R. We have
XðkÞ ¼ 52 þ ej2pk=3 þ ej4pk=3 ,
( Re
N1 X
1 k¼0
mk ej2pk=N z1
)
1 mk Re 1 jðð2 p k=NÞ y Þ 1r e k¼0
9 8 2 pk > > 1 > <1 r cos N y > = N1 X ¼ mk Re 2 > > 1 jðð2 p k=NÞ y Þ > > k¼0 : 1r e ;
9 8 pk y > > >1 r1 þ 2r1 sin2 > < N 1 X 2 = N . ¼ mk Re 2 > > 1 jðð2 p k=NÞ y Þ > > 1r e k¼0 : ; ¼
N1 X
(3) First we note that Rðej2pk=N Þ ¼ mk 40. Then for r41 or r ¼ 1 and ya2pk=N we get ( ) N 1 X mk Re 1 ej2pk=N z1 k¼0
9 8 y > 2 pk > 1 1 > > 1 r þ 2r sin < N 1 X N 2 = 40. mk Re ¼ 2 1 jðð2 p k=NÞ y Þ > > j1 r e j > > k¼0 ; : It is easy to see that RðzÞa0, when jzjX1.
&
Let xðnÞ be a discrete signal of length N and let XðkÞ be its N-point DFT XðkÞ ¼
N 1 X
xðnÞej2pkn=N ;
k ¼ 0; 1; . . . ; N 1.
(4)
n¼0
Its inverse DFT is computed via xðnÞ ¼
1 1 NX XðkÞej2pkn=N . N k¼0
for k ¼ 0; 1; 2. Then XðkÞ ¼
1 pk þ 4 cos2 40. 2 3
The inverse DFT of XðkÞ is ( 5 ; n ¼ 0; xðnÞ ¼ 2 1; n ¼ 1 and n ¼ 2; and its z-transform XðzÞ ¼ 52 þ z1 þ z2 has zeros z1;2 ¼ ð1 3jÞ=5 which are inside the unit circle. Note that the symmetric sequence 85 > < 2 ; n ¼ 0; x1 ðnÞ ¼ 1; n ¼ 1 and n ¼ 1; > : 0 otherwise; has zeros z1 ¼ 12 and z2 ¼ 2. By aliasing x1 ðnÞ over the interval 0pnp2, we get xðnÞ. Corollary 14. For any finite sequence of length at most N, having no zeros on unit circle, the N-point circular autocorrelation is minimum-phase sequence. We note that autocorrelation and N-point circular autocorrelation of a sequence differ as time-aliasing appears. Moreover, the succession of samples in autocorrelation and circular autocorrelation is not the same. Thus, in general, their z-transforms do not have common zeros. 5. An extension to NDFT
Now we shall proceed with the main result of this section. Theorem 12. Let fmk gk¼0;1;...;N1 a strict positive sequence and consider the complex sequence fan gn¼0;1;...;N1 given by an ¼
1 1 NX m ej2pkn=N N k¼0 k
Then (1) The sequence an is a minimum-phase sequence. (2) The magnitude of an is the given sequence mk .
If the restriction of equally spaced points on unit circle is dropped, i.e., instead of DFT magnitude spectrum, the NDFT magnitude spectrum is considered, then Theorem 12 is not valid. This is easily seen by a counterexample. However, we can establish a situation when for certain sampling points wk and sample values Xðwk Þ of XðzÞ, the resulting interpolation polynomial is minimum-phase. Nevertheless, when jwk j ¼ 1 we retrieve the NDFT case. Before proving the main result of this section, we refine some results from the geometry of zeros.
ARTICLE IN PRESS C. Rusu, J. Astola / Signal Processing 89 (2009) 1032–1037
complex sampling points. If # " #" N 1 X Y xðnÞwnk ð1 wl w1 Þ 40 k
Lemma 15. Let P be a polynomial of degree N PðzÞ ¼
N 1 Y
ð1 wk z1 Þ,
k¼0
n¼0
where w0 ; w1 ; . . . ; wN1 are the zeros of P. If m0 ; m1 ; . . . mN1 are positive numbers, then any closed half-plane that contains all zeros of polynomial P also contains all zeros of polynomial Q ðzÞ ¼
N 1 Y
ðz wk Þ
k¼0
N 1 X
mk . z wk k¼0
(5)
Proof of Lemma 15 is presented in Appendix A. The zeros of Q ðzÞ can be interpreted in various ways from the standpoint of physics, geometry and function theory [10]. Lemma 15 implies that the zeros of Q are contained in the intersection of all closed half-planes containing the zeros of P. This intersection is known as the closed convex hull of the set of zeros and may be described as the smallest convex polygon containing all zeros. Consider a sequence xðnÞ; n ¼ 0; 1; . . . ; N 1 and its ztransform XðzÞ ¼
N 1 X n¼0
Now let fwk jk ¼ 0; 1; . . . ; N 1g be a set of complex sampling points and we denote by N 1 X
xðnÞwnk .
n¼0
The z-transform can be reconstructed from its values at wk ; k ¼ 0; 1; . . . ; N 1 by Lagrange interpolation N 1 X
Lk ðzÞ X , XðzÞ ¼ L ðwk Þ k k¼0 k
Y ð1 wl z1 Þ ¼ lak
PðzÞ . 1 wk z1
Now we shall expand XðzÞ using appropriate partial fractions XðzÞ ¼
Q N1 X ð1 wl z1 Þ X k PðzÞ Xk ¼ Q Q lak 1 1 1 Þ ð1 w w Þ ð1 w l l wk Þð1 wk z lak lak k k¼0 k¼0
N1 X
¼ PðzÞ
N1 X
mk , 1 wk z1 k¼0
where mk ¼ Q
lak ð1
then the roots of XðzÞ are in the convex hull of w0 ; w1 ; . . . ; wN1 . Corollary 17. Let xðnÞ; n ¼ 0; 1; . . . ; N 1, be a complex valued sequence, fok jk ¼ 0; 1; . . . ; N 1g a set of distinct frequencies and X k their corresponding NDFT. If Y ½1 ejðol ok Þ 40 for all k ¼ 0; 1; . . . ; N 1, Xk lak
then XðzÞ is a minimum-phase function. Notice that the term
Q
lak ð1
wl w1 k Þ can be evaluated
as Y 0 ð1 wl w1 k Þ ¼ wk P ðwk Þ.
(6)
lak
Thus, for PðzÞ ¼ 1 zN , coefficients from (6) turn out to be N, as we have
and we obtain the uniform case. It should be also noted that the convex hull of ej2pk=N , k ¼ 0; 1; . . . ; N 1 is inside the unit disk, but this is not valid anymore for the convex hull of ej2pk=N , k ¼ 0; 1; . . . ; N=2 1. This explains why the circular autocorrelation is a minimum-phase sequence, although its causal part is not minimum-phase. 6. Additional aspects 6.1. Aliasing in circular autocorrelation
where Lk ðzÞ are the fundamental polynomials for Lagrange interpolation Lk ðzÞ ¼
lak
for all k ¼ 0; 1; . . . ; N 1,
ej2pk=N ðNzN1 Þjz¼ej2pk=N ¼ Nej2pk=NN ¼ N,
xðnÞzn .
X k XðzÞjz¼wk ¼
1035
Xk . wl w1 Þ k
It means that if mk is positive for all k ¼ 0; 1; . . . ; N 1, Lemma 15 implies that the roots of XðzÞ are in the convex hull of w0 ; w1 ; . . . ; wN1 . Thus, we obtain: Theorem 16. Let xðnÞ; n ¼ 0; 1; . . . ; N 1, be a complex valued sequence and fwk jk ¼ 0; 1; . . . ; N 1g a set of distinct
Aliasing may appear when one tries to generate autocorrelation as inverse DFT of set of arbitrary magnitudes. If the first part of the circular autocorrelation is not a minimum-phase sequence, then according to Section 3, this is not a causal part of a valid autocorrelation. Clearly, autocorrelation cannot be recovered from circular autocorrelation. This can be helpful in applications where the support or some bound of the support is not known, and the sampling requirements of Fourier transform magnitude cannot be specified [11]. In this case aliasing of autocorrelation may appear after sampling. Using Theorem 7 the presence of aliasing can be detected directly from circular autocorrelation, without no other supplementary information. Example 18. Consider XðoÞ the Fourier transform of a sequence of a certain length. Let X 1 ðkÞ and X 2 ðkÞ be the samples of XðoÞ in 5, respectively 15 points. If 8 > < 0:0651; k ¼ 0; X 1 ðkÞ ¼ 0:4031; k ¼ 1; 4; > : 0:0069; k ¼ 2; 3;
ARTICLE IN PRESS 1036
C. Rusu, J. Astola / Signal Processing 89 (2009) 1032–1037
(Appendix B):
and 8 0:0651; > > > < 0:4031; X 2 ðkÞ ¼ > 0:0069; > > : 1;
k ¼ 0; k ¼ 3; 12; k ¼ 6; 9; k ¼ 1; 2; 4; 5; 7; 8; 10; 11; 13; 14:
Then the corresponding circular autocorrelations are given by 8 n ¼ 0; > < 0:1770; n ¼ 1; r˜ x1 x1 ðnÞ ¼ 0:0606; > : 0:1166; n ¼ 2; and 8 0:7257; > > > > > 0:0202; > > > > > 0:0389; > > > > < 0:0389; r˜ x2 x2 ðnÞ ¼ > 0:0202; > > > > > 0:2743; > > > > > 0:0202; > > > : 0:0389;
n ¼ 0; n ¼ 1; n ¼ 2; n ¼ 3; n ¼ 4; n ¼ 5; n ¼ 6; n ¼ 7:
The causal part of r˜ x2 x2 ðnÞ is minimum-phase, but the causal part of r˜ x1 x1 ðnÞ is not minimum-phase. Thus, one can conclude that in r˜ x1 x1 ðnÞ the aliasing is present.
6.2. The energy concentration property for zero-phase sequences The energy concentration theorem [4] states that any minimum-phase sequence possess the attribute of minimum energy delay. In other words, they tend to have their energy concentrated toward the beginning. Conversely, a maximum-phase sequence tends to concentrate their energy toward the end of the sequence. The anti-causal part of zero-phase sequence is a maximum-phase sequence. This comes from the symmetry of a zero-phase sequence. It follows that both causal part (minimumphase part) and the anti-causal part (maximum-phase part) of a zero-phase sequence tend to preserve their energy toward to zero. Let us denote by EL ðxÞ the partial energy of both causal and anti-causal parts of a sequence xðnÞ EL ½x ¼
0 X n¼L
jxðnÞj2 þ
EL ½x ¼
L X
jxðnÞj2 .
n¼0
Note that EL ðxÞ contains twice the energy of the sample in origin. Previous remarks suggest that of all sequences xðnÞ having the same magnitude, the quantity EL ½x is maximum when xðnÞ is zero-phase sequence. We have verified the related conjecture: of all finite sequences xðnÞ having the same DFT magnitude, the quantity EL ½x is maximum when xðnÞ is zerophase sequence. We have the following relationship
1 N 1 X 2 NX
jXðkÞjjXðlÞj cos½arg XðkÞ arg XðlÞ N 2 k¼0 l¼0 " # L X 2pnðl kÞ cos . N n¼0
(7)
When L ¼ 0 or 1, then L X n¼0
cos
2pnðl kÞ X0, N
and it can be easily shown that EL ½x is maximum when xðnÞ is zero-phase sequence. This is not anymore valid for L ¼ 2. Example 19. The sequences x1 ðnÞ and x2 ðnÞ are given by 8 0:2767; n ¼ 0; > > > > > < 0:0034; n ¼ 1; x1 ðnÞ ¼ > 0:1464; n ¼ 2; > > > > : 0:1867; n ¼ 3; 8 0:0377; > > > > > > 0:2682; > > > > > > > 0:0489; > < x2 ðnÞ ¼ 0:1593; > > > 0:1532; > > > > > > > 0:2192; > > > > : 0:1284;
n ¼ 3; n ¼ 2; n ¼ 1; n ¼ 0; n ¼ 1; n ¼ 2; n ¼ 3:
Their magnitude is equal 8 0:3641; > > > < 0:0096; jX 1 ðkÞj ¼ jX 2 ðkÞj ¼ > 0:7717; > > : 0:0050;
k ¼ 0; k ¼ 1; k ¼ 2; k ¼ 3:
It can be verified that x1 ðnÞ is zero-phase E0 ðx1 Þ4E0 ðx2 Þ, E1 ðx1 Þ4E1 ðx2 Þ, but E2 ðx1 ÞoE2 ðx2 Þ.
and
One can conclude that energy concentration property for zero-phase sequences is valid for L ¼ 0 and 1. We suggest that from practical point of view energy concentration property of zero-phase sequences is almost valid. We have performed simulations and we have found that the energy is not concentrated for zero-phase sequences in less than 0.01% cases, and only for length of sequences between 7 and 17. 7. Conclusions In this paper, we have proved for the general case that the causal portion of the inverse discrete-time Fourier transform of the magnitude spectrum is a minimumphase signal. Then a similar statement has been shown for DFT. The NDFT case has been discussed. The energy concentration property of zero-phase sequences has been analyzed. We have also shown how the presence of aliasing can be detected in circular autocorrelation. We have assumed that the sequences involved do not have
ARTICLE IN PRESS C. Rusu, J. Astola / Signal Processing 89 (2009) 1032–1037
zeros or poles on the unit circle. In the last case, the Fourier transform does not exist. In the first case, the minimum-phase property cannot be anymore considered.
Then, we have EL ½x ¼
0 X
jxðnÞj2 þ
n¼L
Acknowledgment
¼
The first author would like to thank Prof. Zoran Cvetkovic for pointing out Ref. [5]. His research was supported by CNCSIS project number 162.
Given any closed half-plane H, there exist a complex number ca0 and a real number a such that z 2 H if and only if ReðczÞXa [12, p. 463]. Therefore, if all zeros are in H, then Reðcwk ÞXa;
k ¼ 0; 1; . . . ; N 1.
Let z not be in H. We wish to show that Q ðzÞa0. Because ReðczÞoa, we have Re½cðz wk Þo0;
k ¼ 0; 1; . . . ; N 1.
Hence Re
c o0; z wk
k ¼ 0; 1; . . . ; N 1,
and if mk 40 we get
mk o0; k ¼ 0; 1; . . . ; N 1. Re c z wk There follows: N1 X k¼0
mk a0. z wk
Consequently Q ðzÞa0. Appendix B. Proof of Eq. (7) For the beginning we just note that jxðnÞj2 ¼
1 N 1 X 1 NX
N2
k¼0 l¼0
XðkÞX ðlÞej2pnðlkÞ=N .
L X n¼0
( L N1 X N1 X 1 X
jxðnÞj2 ¼
L X ðjxðnÞj2 þ jxðnÞj2 Þ n¼0
h i XðkÞX ðlÞ ej2pnðlkÞ=N þ ej2pnðlkÞ=N
)
N 2 n¼0 k¼0 l¼0 ( ) 1 N 1 L X X 2 NX 2pnðl kÞ XðkÞX ðlÞ cos ¼ 2 N N n¼0 k¼0 l¼0
¼ Appendix A. Proof of Lemma 15
1037
1 N 1 X 2 NX
jXðkÞjjXðlÞj cos½arg XðkÞ arg XðlÞ N 2 k¼0 l¼0 " # L X 2pnðl kÞ cos . N n¼0
References [1] S.K. Mitra, Digital Signal Processing: A Computer Based Approach, McGraw-Hill, New York, 1998. [2] M.H. Hayes, J.S. Lim, A.V. Oppenheim, Signal reconstruction from phase or magnitude, IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-28 (6) (1980) 672–680. [3] L.R. Rabiner, R.W. Schafer, Digital Processing of Speech Signals, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1978. [4] A. Papoulis, Signal Analysis, McGraw-Hill Book Company, New York, 1977. [5] J.O. Smith, Techniques for digital filter design and system identification with application to the violin, Report no. STAN-M-14, available on-line at hhttp://ccrma.stanford.edu/STANM/stanms/stanm14 (1983)i. [6] H.A. Murthy, B. Yegnanarayana, Formant extraction from minimum phase group delay function, Speech Communication 10 (1991) 209–221. [7] T. Nagarajan, V.K. Prasad, H.A. Murthy, Minimum phase signal derived from root cepstrum, Electronics Letters 39 (12) (2003) 941–942. [8] T. Nagarajan, H.A. Murthy, Subband-based group delay segmentation of spontaneous speech into syllable-like units, EURASIP JASP 17 (2004) 2614–2625. [9] J.A. Cadzow, Y. Sun, Sequences with positive semidefinite Fourier transforms, IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-34 (6) (1986) 1502–1510. [10] M. Marden, Geometry of Polynomials, American Mathematical Society, Providence, RI, 1989. [11] D. Izraelevitz, J.S. Lim, A new direct algorithm for image reconstruction from Fourier transform magnitude, IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP-35 (4) (1987) 511–518. [12] P. Henrici, Applied and computational complex analysis, Power Series—Integration—Conformal Mapping—Location of Zeros, vol. I, Wiley, New York, 1974.
Contents lists available at ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Minimum-phase parts of zero-phase sequences Corneliu Rusu a,, Jaakko Astola b a
Faculty of Electronics, Telecommunications and Information Technology, Technical University of Cluj-Napoca, Cluj-Napoca, Str. Baritiu 26-28, RO-400027, Romania b Tampere International Center for Signal Processing, Tampere University of Technology, Tampere, Finland
a r t i c l e i n f o
abstract
Article history: Received 26 May 2008 Received in revised form 8 September 2008 Accepted 4 December 2008 Available online 24 December 2008
For more than a decade it has been empirically known that the causal portion of the inverse Fourier transform of the magnitude spectrum of the speech signal behaves like a minimum-phase signal. Later on, this statement has been shown for an all-pole model. In this paper, we consider related results for both discrete-time Fourier and discrete Fourier transforms of arbitrary sequences. We indicate how the presence of aliasing in circular autocorrelation might be detected. The energy concentration property of zerophase sequences is discussed. & 2008 Elsevier B.V. All rights reserved.
Keywords: Signal reconstruction Discrete-time Fourier transform Discrete Fourier transform Nonuniform discrete Fourier transform Minimum-phase function
1. Introduction In many cases where it is necessary to avoid any phase distortion, one solution is to make the frequency response real and nonnegative, i.e., to have a filter with a zerophase characteristic [1]. Besides filter design, zero-phase signals appear in applications like phase retrieval or phase-only reconstruction [2] and in speech analysis [3]. In spectral factorization theorems, the zero-phase condition represents a key assumption. Perhaps the most known result on zero-phase sequences is the celebrated Feje´r and Riesz theorem [4]: P n Theorem 1 (Feje´r and Riesz). If XðzÞ ¼ M n¼M xðnÞz PM jo and Xðe ÞX0, then there is YðzÞ ¼ n¼0 yðnÞzn such that Xðejo Þ ¼ jYðejo Þj2 and YðzÞ unique if minimum-phase. In this paper, we discuss some properties of minimumphase and zero-phase sequences. As the properties of such sequences are related to the properties of rational Corresponding author. Tel.: +40 2 64 401804; fax: +40 2 64 591340.
E-mail addresses: [email protected] (C. Rusu), Jaakko.Astola@tut.fi (J. Astola). 0165-1684/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.12.006
functions and analytic functions in general, as well as those of Fourier series, results that are closely connected appear implicitly in circuits and systems literature. About two decades ago it was shown that if a positive real function belongs to the class of rational functions, then it must be minimum-phase [5]. To best of our knowledge, there is no published result indicating that the Fourier transform of a causal part of an arbitrary zero-phase sequences is a positive real function belonging to the class of rational functions. Recently, it was noticed that the causal portion of the inverse Fourier transform of the magnitude spectrum of the speech signal behaves like a minimum-phase signal [6]. This property has been verified for an all-pole model [7,8]. Here we present related properties that may be of interest from a digital signal processing point of view, but do not appear explicitly in the literature. In the following we recall few definitions (Section 2). We consider the causal part of an arbitrary zero-phase sequence and show that it is a minimum-phase sequence (Section 3). We also give a direct proof of the fact that the inverse DFT transform of a strictly positive sequence is minimum-phase (Section 4). We also extend this to NDFT (nonuniform discrete Fourier transform) in Section 5.
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1033
The presence of aliasing in circular autocorrelation and the energy concentration property of zero-phase sequences are discussed in Section 6.
If jzj41, then the sequence ejon zn is absolute summable. In such situation we can interchange the order of summation and integration in (1). We get
2. Definitions
X þ ðzÞ ¼
Definition 2. A sequence is causal if xðnÞ ¼ 0 for no0. A sequence is anti-causal if xðnÞ ¼ 0 for n40. Definition 3. A sequence is minimum-phase if all zeros of its z-transform are inside the unit open disk. A sequence is maximum-phase if all zeros of its z-transform are outside the unit closed disk.
1 2p
Z p "X 1 p
Re
jXðejo Þj 1 r 1 ejðoyÞ
Thus
Definition 5. A complex valued function XðzÞ is positive real if
RefX þ ðzÞgX
Definition 6. A sequence is zero-phase sequence if its Fourier transform is a nonnegative function. Sometimes a zero-phase sequence is said to be a positive sequence [9]; in this paper, we shall refer to this as zerophase sequence, to distinguish from positive real function. 3. Zeros location of causal part of zero-phase aperiodic sequences
n¼0
jo
Xðe Þ ¼
1 X
jon
xðnÞe
Z p
1 r 1 2pð1 þ r 1 Þ2
jXðejo Þj40;
arg Xðejo Þ ¼ 0.
In this case by using inverse DTFT, the sequence can be computed as follows: Z p Z p 1 1 Xðejo Þejon do ¼ jXðejo Þjejon do. xðnÞ ¼ 2p p 2p p The z-transform of causal portion of xðnÞ is the one-sided z-transform: Z p 1 1 X X 1 xðnÞzn ¼ jXðejo Þjejon do zn . (1) X þ ðzÞ ¼ 2p p n¼0 n¼0 Our goal is to show that the causal portion of xðnÞ is a minimum-phase sequence. Theorem 7. Let xðnÞ be a stable sequence. If jXðejo Þj40 and arg Xðejo Þ ¼ 0 for all o 2 R, then its causal portion is a minimum-phase sequence. Proof. First we establish that for any jzj41, we have RefX þ ðzÞg40.
jXðejo Þj do. 1 ejo z1
jXðejo Þj do40.
p
It follows that for any jzj41, we have X þ ðzÞa0. To show that X þ ðejo0 Þa0, first notice that Z p Z p 1 1 Xðejo Þ do ¼ jXðejo Þj do40. xð0Þ ¼ 2p p 2p p
(2)
Assume now that X þ ðejo0 Þ ¼ 0. Then Xðejo0 Þ ¼ X þ ðejo0 Þ þ
1 X
xðnÞejo0 n ¼ X þ ðejo0 Þ xð0Þ
n¼1
þ
0 X
xðnÞejo0 n
n¼1 1 X
xðnÞejo0 n ¼ X þ ðejo0 Þ xð0Þ
n¼0
þ
1 X
jo 0 n
xðnÞ e
n¼0 þ
jo0
¼ X ðe
" Þ xð0Þ þ
1 X
# jo0 n
xðnÞe
¼ X þ ðejo0 Þ xð0Þ þ ½X ðe
satisfies
p
jXðejo Þj½1 r 1 cosðo yÞ j1 r 1 ejðoyÞ j2 jo jXðe Þjð1 r 1 Þ X . ð1 þ r 1 Þ2
n¼0 þ jo0
n¼1
Z p
¼
¼ X þ ðejo0 Þ xð0Þ þ Let xðnÞ be zero-phase sequence with no zeros or poles on the unit circle. Then its DTFT (discrete-time Fourier transform)
1 2p
Now, for z ¼ r ejy, r41 Z p 1 jXðejo Þj do X þ ðzÞ ¼ , 2p p 1 r 1 ejðoyÞ
Definition 4. A sequence is absolute summable or stable sequence if the region of convergence of its z-transform includes the unit circle.
(1) z 2 R ) XðzÞ 2 R; (2) jzjX1 ) RefXðzÞgX0.
# ejon zn jXðejo Þj do ¼
Þ ¼ xð0Þ.
This is a contradiction since Xðejo Þ is positive for all o 2 ½p; p. & Example 8. Let XðzÞ ¼ z þ 52 þ z1 , then Xðejo Þ ¼ 12 þ 4 cos2 o=240. The causal portion of XðzÞ is X þ ðzÞ ¼ 52 þ z1 and its zero is z1 ¼ 25. According to Feje´r–Riesz theorem, there is YðzÞ such that pffiffiffi Xðejo Þ ¼ jYðejo Þj2 . Indeed, for Y 1 ðzÞ ¼ 2ð1 þ 1=2z1 Þ and pffiffiffi Y 2 ðzÞ ¼ 2=2ð1 þ 2z1 Þ we have Xðejo Þ ¼ jY 1 ðejo Þj2 ¼ jY 2 ðejo Þj2 . Moreover, Y 1 ðzÞ is minimum-phase function, and Y 2 ðzÞ is maximum-phase function. Corollary 9. For any sequence having no zeros or poles on unit circle, the causal portion of autocorrelation is minimumphase sequence. Example 10. The causal portion of a symmetric sequence may be minimum-phase, but the sequence is not always zero-phase, e.g., XðzÞ ¼ 0:7z þ 1 þ 0:7z1 .
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4. Minimum-phase property of DFT magnitude One can find a counterexample that the causal part of circular autocorrelation is not always a minimum-phase sequence (Section 6.1). However, it can be shown that for any finite sequence of length at most N, having no zeros on unit circle, the N-point circular autocorrelation is a minimum-phase sequence. We start with the following result: Proposition 11. If m0 ; m1 ; . . . mN1 are positive numbers, then the polynomial RðzÞ ¼
N1 Y
1 ej2pk=N z1
1 NX
k¼0
k¼0
1
mk ej2pk=N z1
Proof. Clearly mk ¼ DFTfan g. Since mk 40, then mk ¼ jDFTfan gj. It remains to show that an is a minimum-phase sequence. To obtain this result, a little manipulation is needed. " # N 1 N 1 1 X X 1 NX n j2pkn=N AðzÞ ¼ an z ¼ m e zn N k¼0 k n¼0 n¼0 " # N 1 N 1 1 X X 1 zN NX mk ¼ mk ej2pkn=N zn ¼ . j2pk=N z1 N 1 e n¼0 k¼0 k¼0 Consequently AðzÞ ¼ RðzÞ=N and using Proposition 11, AðzÞ has all zeros inside the unit disk. &
,
has all zeros inside the unit disk.
Example 13. Consider N ¼ 3 and
Proof. Consider z ¼ rejy where rX0 and y 2 R. We have
XðkÞ ¼ 52 þ ej2pk=3 þ ej4pk=3 ,
( Re
N1 X
1 k¼0
mk ej2pk=N z1
)
1 mk Re 1 jðð2 p k=NÞ y Þ 1r e k¼0
9 8 2 pk > > 1 > <1 r cos N y > = N1 X ¼ mk Re 2 > > 1 jðð2 p k=NÞ y Þ > > k¼0 : 1r e ;
9 8 pk y > > >1 r1 þ 2r1 sin2 > < N 1 X 2 = N . ¼ mk Re 2 > > 1 jðð2 p k=NÞ y Þ > > 1r e k¼0 : ; ¼
N1 X
(3) First we note that Rðej2pk=N Þ ¼ mk 40. Then for r41 or r ¼ 1 and ya2pk=N we get ( ) N 1 X mk Re 1 ej2pk=N z1 k¼0
9 8 y > 2 pk > 1 1 > > 1 r þ 2r sin < N 1 X N 2 = 40. mk Re ¼ 2 1 jðð2 p k=NÞ y Þ > > j1 r e j > > k¼0 ; : It is easy to see that RðzÞa0, when jzjX1.
&
Let xðnÞ be a discrete signal of length N and let XðkÞ be its N-point DFT XðkÞ ¼
N 1 X
xðnÞej2pkn=N ;
k ¼ 0; 1; . . . ; N 1.
(4)
n¼0
Its inverse DFT is computed via xðnÞ ¼
1 1 NX XðkÞej2pkn=N . N k¼0
for k ¼ 0; 1; 2. Then XðkÞ ¼
1 pk þ 4 cos2 40. 2 3
The inverse DFT of XðkÞ is ( 5 ; n ¼ 0; xðnÞ ¼ 2 1; n ¼ 1 and n ¼ 2; and its z-transform XðzÞ ¼ 52 þ z1 þ z2 has zeros z1;2 ¼ ð1 3jÞ=5 which are inside the unit circle. Note that the symmetric sequence 85 > < 2 ; n ¼ 0; x1 ðnÞ ¼ 1; n ¼ 1 and n ¼ 1; > : 0 otherwise; has zeros z1 ¼ 12 and z2 ¼ 2. By aliasing x1 ðnÞ over the interval 0pnp2, we get xðnÞ. Corollary 14. For any finite sequence of length at most N, having no zeros on unit circle, the N-point circular autocorrelation is minimum-phase sequence. We note that autocorrelation and N-point circular autocorrelation of a sequence differ as time-aliasing appears. Moreover, the succession of samples in autocorrelation and circular autocorrelation is not the same. Thus, in general, their z-transforms do not have common zeros. 5. An extension to NDFT
Now we shall proceed with the main result of this section. Theorem 12. Let fmk gk¼0;1;...;N1 a strict positive sequence and consider the complex sequence fan gn¼0;1;...;N1 given by an ¼
1 1 NX m ej2pkn=N N k¼0 k
Then (1) The sequence an is a minimum-phase sequence. (2) The magnitude of an is the given sequence mk .
If the restriction of equally spaced points on unit circle is dropped, i.e., instead of DFT magnitude spectrum, the NDFT magnitude spectrum is considered, then Theorem 12 is not valid. This is easily seen by a counterexample. However, we can establish a situation when for certain sampling points wk and sample values Xðwk Þ of XðzÞ, the resulting interpolation polynomial is minimum-phase. Nevertheless, when jwk j ¼ 1 we retrieve the NDFT case. Before proving the main result of this section, we refine some results from the geometry of zeros.
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complex sampling points. If # " #" N 1 X Y xðnÞwnk ð1 wl w1 Þ 40 k
Lemma 15. Let P be a polynomial of degree N PðzÞ ¼
N 1 Y
ð1 wk z1 Þ,
k¼0
n¼0
where w0 ; w1 ; . . . ; wN1 are the zeros of P. If m0 ; m1 ; . . . mN1 are positive numbers, then any closed half-plane that contains all zeros of polynomial P also contains all zeros of polynomial Q ðzÞ ¼
N 1 Y
ðz wk Þ
k¼0
N 1 X
mk . z wk k¼0
(5)
Proof of Lemma 15 is presented in Appendix A. The zeros of Q ðzÞ can be interpreted in various ways from the standpoint of physics, geometry and function theory [10]. Lemma 15 implies that the zeros of Q are contained in the intersection of all closed half-planes containing the zeros of P. This intersection is known as the closed convex hull of the set of zeros and may be described as the smallest convex polygon containing all zeros. Consider a sequence xðnÞ; n ¼ 0; 1; . . . ; N 1 and its ztransform XðzÞ ¼
N 1 X n¼0
Now let fwk jk ¼ 0; 1; . . . ; N 1g be a set of complex sampling points and we denote by N 1 X
xðnÞwnk .
n¼0
The z-transform can be reconstructed from its values at wk ; k ¼ 0; 1; . . . ; N 1 by Lagrange interpolation N 1 X
Lk ðzÞ X , XðzÞ ¼ L ðwk Þ k k¼0 k
Y ð1 wl z1 Þ ¼ lak
PðzÞ . 1 wk z1
Now we shall expand XðzÞ using appropriate partial fractions XðzÞ ¼
Q N1 X ð1 wl z1 Þ X k PðzÞ Xk ¼ Q Q lak 1 1 1 Þ ð1 w w Þ ð1 w l l wk Þð1 wk z lak lak k k¼0 k¼0
N1 X
¼ PðzÞ
N1 X
mk , 1 wk z1 k¼0
where mk ¼ Q
lak ð1
then the roots of XðzÞ are in the convex hull of w0 ; w1 ; . . . ; wN1 . Corollary 17. Let xðnÞ; n ¼ 0; 1; . . . ; N 1, be a complex valued sequence, fok jk ¼ 0; 1; . . . ; N 1g a set of distinct frequencies and X k their corresponding NDFT. If Y ½1 ejðol ok Þ 40 for all k ¼ 0; 1; . . . ; N 1, Xk lak
then XðzÞ is a minimum-phase function. Notice that the term
Q
lak ð1
wl w1 k Þ can be evaluated
as Y 0 ð1 wl w1 k Þ ¼ wk P ðwk Þ.
(6)
lak
Thus, for PðzÞ ¼ 1 zN , coefficients from (6) turn out to be N, as we have
and we obtain the uniform case. It should be also noted that the convex hull of ej2pk=N , k ¼ 0; 1; . . . ; N 1 is inside the unit disk, but this is not valid anymore for the convex hull of ej2pk=N , k ¼ 0; 1; . . . ; N=2 1. This explains why the circular autocorrelation is a minimum-phase sequence, although its causal part is not minimum-phase. 6. Additional aspects 6.1. Aliasing in circular autocorrelation
where Lk ðzÞ are the fundamental polynomials for Lagrange interpolation Lk ðzÞ ¼
lak
for all k ¼ 0; 1; . . . ; N 1,
ej2pk=N ðNzN1 Þjz¼ej2pk=N ¼ Nej2pk=NN ¼ N,
xðnÞzn .
X k XðzÞjz¼wk ¼
1035
Xk . wl w1 Þ k
It means that if mk is positive for all k ¼ 0; 1; . . . ; N 1, Lemma 15 implies that the roots of XðzÞ are in the convex hull of w0 ; w1 ; . . . ; wN1 . Thus, we obtain: Theorem 16. Let xðnÞ; n ¼ 0; 1; . . . ; N 1, be a complex valued sequence and fwk jk ¼ 0; 1; . . . ; N 1g a set of distinct
Aliasing may appear when one tries to generate autocorrelation as inverse DFT of set of arbitrary magnitudes. If the first part of the circular autocorrelation is not a minimum-phase sequence, then according to Section 3, this is not a causal part of a valid autocorrelation. Clearly, autocorrelation cannot be recovered from circular autocorrelation. This can be helpful in applications where the support or some bound of the support is not known, and the sampling requirements of Fourier transform magnitude cannot be specified [11]. In this case aliasing of autocorrelation may appear after sampling. Using Theorem 7 the presence of aliasing can be detected directly from circular autocorrelation, without no other supplementary information. Example 18. Consider XðoÞ the Fourier transform of a sequence of a certain length. Let X 1 ðkÞ and X 2 ðkÞ be the samples of XðoÞ in 5, respectively 15 points. If 8 > < 0:0651; k ¼ 0; X 1 ðkÞ ¼ 0:4031; k ¼ 1; 4; > : 0:0069; k ¼ 2; 3;
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C. Rusu, J. Astola / Signal Processing 89 (2009) 1032–1037
(Appendix B):
and 8 0:0651; > > > < 0:4031; X 2 ðkÞ ¼ > 0:0069; > > : 1;
k ¼ 0; k ¼ 3; 12; k ¼ 6; 9; k ¼ 1; 2; 4; 5; 7; 8; 10; 11; 13; 14:
Then the corresponding circular autocorrelations are given by 8 n ¼ 0; > < 0:1770; n ¼ 1; r˜ x1 x1 ðnÞ ¼ 0:0606; > : 0:1166; n ¼ 2; and 8 0:7257; > > > > > 0:0202; > > > > > 0:0389; > > > > < 0:0389; r˜ x2 x2 ðnÞ ¼ > 0:0202; > > > > > 0:2743; > > > > > 0:0202; > > > : 0:0389;
n ¼ 0; n ¼ 1; n ¼ 2; n ¼ 3; n ¼ 4; n ¼ 5; n ¼ 6; n ¼ 7:
The causal part of r˜ x2 x2 ðnÞ is minimum-phase, but the causal part of r˜ x1 x1 ðnÞ is not minimum-phase. Thus, one can conclude that in r˜ x1 x1 ðnÞ the aliasing is present.
6.2. The energy concentration property for zero-phase sequences The energy concentration theorem [4] states that any minimum-phase sequence possess the attribute of minimum energy delay. In other words, they tend to have their energy concentrated toward the beginning. Conversely, a maximum-phase sequence tends to concentrate their energy toward the end of the sequence. The anti-causal part of zero-phase sequence is a maximum-phase sequence. This comes from the symmetry of a zero-phase sequence. It follows that both causal part (minimumphase part) and the anti-causal part (maximum-phase part) of a zero-phase sequence tend to preserve their energy toward to zero. Let us denote by EL ðxÞ the partial energy of both causal and anti-causal parts of a sequence xðnÞ EL ½x ¼
0 X n¼L
jxðnÞj2 þ
EL ½x ¼
L X
jxðnÞj2 .
n¼0
Note that EL ðxÞ contains twice the energy of the sample in origin. Previous remarks suggest that of all sequences xðnÞ having the same magnitude, the quantity EL ½x is maximum when xðnÞ is zero-phase sequence. We have verified the related conjecture: of all finite sequences xðnÞ having the same DFT magnitude, the quantity EL ½x is maximum when xðnÞ is zerophase sequence. We have the following relationship
1 N 1 X 2 NX
jXðkÞjjXðlÞj cos½arg XðkÞ arg XðlÞ N 2 k¼0 l¼0 " # L X 2pnðl kÞ cos . N n¼0
(7)
When L ¼ 0 or 1, then L X n¼0
cos
2pnðl kÞ X0, N
and it can be easily shown that EL ½x is maximum when xðnÞ is zero-phase sequence. This is not anymore valid for L ¼ 2. Example 19. The sequences x1 ðnÞ and x2 ðnÞ are given by 8 0:2767; n ¼ 0; > > > > > < 0:0034; n ¼ 1; x1 ðnÞ ¼ > 0:1464; n ¼ 2; > > > > : 0:1867; n ¼ 3; 8 0:0377; > > > > > > 0:2682; > > > > > > > 0:0489; > < x2 ðnÞ ¼ 0:1593; > > > 0:1532; > > > > > > > 0:2192; > > > > : 0:1284;
n ¼ 3; n ¼ 2; n ¼ 1; n ¼ 0; n ¼ 1; n ¼ 2; n ¼ 3:
Their magnitude is equal 8 0:3641; > > > < 0:0096; jX 1 ðkÞj ¼ jX 2 ðkÞj ¼ > 0:7717; > > : 0:0050;
k ¼ 0; k ¼ 1; k ¼ 2; k ¼ 3:
It can be verified that x1 ðnÞ is zero-phase E0 ðx1 Þ4E0 ðx2 Þ, E1 ðx1 Þ4E1 ðx2 Þ, but E2 ðx1 ÞoE2 ðx2 Þ.
and
One can conclude that energy concentration property for zero-phase sequences is valid for L ¼ 0 and 1. We suggest that from practical point of view energy concentration property of zero-phase sequences is almost valid. We have performed simulations and we have found that the energy is not concentrated for zero-phase sequences in less than 0.01% cases, and only for length of sequences between 7 and 17. 7. Conclusions In this paper, we have proved for the general case that the causal portion of the inverse discrete-time Fourier transform of the magnitude spectrum is a minimumphase signal. Then a similar statement has been shown for DFT. The NDFT case has been discussed. The energy concentration property of zero-phase sequences has been analyzed. We have also shown how the presence of aliasing can be detected in circular autocorrelation. We have assumed that the sequences involved do not have
ARTICLE IN PRESS C. Rusu, J. Astola / Signal Processing 89 (2009) 1032–1037
zeros or poles on the unit circle. In the last case, the Fourier transform does not exist. In the first case, the minimum-phase property cannot be anymore considered.
Then, we have EL ½x ¼
0 X
jxðnÞj2 þ
n¼L
Acknowledgment
¼
The first author would like to thank Prof. Zoran Cvetkovic for pointing out Ref. [5]. His research was supported by CNCSIS project number 162.
Given any closed half-plane H, there exist a complex number ca0 and a real number a such that z 2 H if and only if ReðczÞXa [12, p. 463]. Therefore, if all zeros are in H, then Reðcwk ÞXa;
k ¼ 0; 1; . . . ; N 1.
Let z not be in H. We wish to show that Q ðzÞa0. Because ReðczÞoa, we have Re½cðz wk Þo0;
k ¼ 0; 1; . . . ; N 1.
Hence Re
c o0; z wk
k ¼ 0; 1; . . . ; N 1,
and if mk 40 we get
mk o0; k ¼ 0; 1; . . . ; N 1. Re c z wk There follows: N1 X k¼0
mk a0. z wk
Consequently Q ðzÞa0. Appendix B. Proof of Eq. (7) For the beginning we just note that jxðnÞj2 ¼
1 N 1 X 1 NX
N2
k¼0 l¼0
XðkÞX ðlÞej2pnðlkÞ=N .
L X n¼0
( L N1 X N1 X 1 X
jxðnÞj2 ¼
L X ðjxðnÞj2 þ jxðnÞj2 Þ n¼0
h i XðkÞX ðlÞ ej2pnðlkÞ=N þ ej2pnðlkÞ=N
)
N 2 n¼0 k¼0 l¼0 ( ) 1 N 1 L X X 2 NX 2pnðl kÞ XðkÞX ðlÞ cos ¼ 2 N N n¼0 k¼0 l¼0
¼ Appendix A. Proof of Lemma 15
1037
1 N 1 X 2 NX
jXðkÞjjXðlÞj cos½arg XðkÞ arg XðlÞ N 2 k¼0 l¼0 " # L X 2pnðl kÞ cos . N n¼0
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