satisfy the following nonlinear di erence equation 8] nan = +1X k= 1 kck an k;. (2) which is a basis .... the e cient method of Levinson and Rybicki 10] to solve. (8).
On Time{Domain Deconvolution and the Computation of the Cepstrum Eduard KRAJNIK Abstract| A novel time{domain computation method of the sequences this equation can be rearranged into a recursive complex cepstrum is proposed. We approximate the com- formula to compute cn . Recently, a sophisticated method plex cepstrum by the solution of a di erence equation sys-
tem and employ a projection method by Gohberg and Feldman to solve the system. It is proved that under fairly general assumptions the approximations converge to the complex cepstrum. The projection method, together with the proposed time{domain algorithm for the calculation of the linear phase component of a sequence, can also be used as a time{domain deconvolution method.
how to employ (2) also for general mixed phase sequences has been developed [4]. The in nite{dimensional system of equations (2) has been approximated by the nite{dimensional one
nan =
I. Introduction
m X
k= m
kck an k ; n = m; : : : ; m
(3)
Homomorphic systems are, for the most part, based on and solved by means of generalized inverse matrices. The the complex cepstrum. Though computation of the com- approximation invokes several questions that have not been plex cepstrum has received considerable attention in the answered in full in [4]: literature since the cepstrum potential was recognized in When do the solutions of (2) and (3) exist and are [1] no absolutely reliable method has yet been developed. they unique? The known methods split into two categories depending on How well do the solutions of (3) approximate the cepwhether the computation is performed in the time domain strum; in particular, do the solutions of (3) converge or frequency domain. The frequency domain methods are to the cepstrum as m ! 1? based on the de nition of the complex cepstrum [2], [3] Are there any other methods to solve (3) for ck , possibly without using generalized matrix inverses? n o As indicated by (1), the sequence a must be approc = F 1 log FfTk ag (1) prietly shifted before (2) or (3) can be applied. A P def time{domain algorithm to nd the shift would be in j! jn! where Ffan g = A(e ) = an e is the Fourier transorder. form operator, Tmfan g = fan m g is the shift operator, complex logarithm function, k = ind A(ej! ) def = It is an aim of this paper to give complete answers to all logargisAthe 2 (ej! ) 0 is the index of the origin with respect to the stated questions and thus introduce a novel time{domain for the computation of the cepstrum. curve A(ej! ); ! 2 [0; 2] which is equivalent to the linear method If we denote dk = kck ; d = fdk g; bk = kak ; b = fbk g component of the phase of A(ej! ) and c is the complex then (2) becomes cepstrum of the sequence a. b=da The limitations of the frequency{domain methods are caused by the necessity to discretize the Fourier transform Since a is assumed to be invertible in l1 (otherwise its cepwhich in turn makes calculation of the unwrapped phase strum does not exist [3]) we have (needed for the logarithm) dicult. If the Fourier transform A(ej! ) is di erentiable and d=ba 1 (4) j! ind A(e ) = 0 then the complex cepstrum coecients cn and the problem is essentially reduced to nding the consatisfy the following nonlinear di erence equation [8] volution inverse of a. We will investigate the problem of +1 X time{domain deconvolution in Section II. nan = kck an k ; (2) A similar approach was used in [7]; the calculation, howk= 1 ever, was performed in the frequency domain. Note also which is a basis for the time{domain computation of the that d is equivalent to the di erential cepstrum as introcomplex cepstrum. For minimum phase or maximum phase duced in [6] and is related to the complex cepstrum by This research is supported by the Grant No. 8188 of the Czech Technical University and the Grant No. 201/93/0932 of the Grant Agency of Czech Republic. E. Krajník is with the Faculty of Electrical Engineering, Czech Technical University, 166 27 Prague 6, Czech Republic.
ck = dkk ; k 6= 0
(5)
The value c0 can be found in the same way as in [8] or [4].
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II. Time{Domain Deconvolution
Finding the convolution inverse of a sequence a is equivalent to solving the equation
ax=
(6)
where is the unit impuse sequence:
k =
1
for k = 0 0 for k 6= 0
The equation (6) is in fact an in nite{dimensional system +1 X
k= 1
an k xk = n ;
n 2 Z:
(7)
A reliable general method of solving (7) is based on the following theorem : Theorem 1. [5] Let a = fak g+11 be an absolutely summable sequence and A(ej! ) its Fourier transform. Let A(ej! ) 6= 0 for all ! 2 R and i = ind A(ej! ). Then there exists a positive integer m0 such that for each m > m0 the system of equations m X
k= m
an k xk = yn ; n = m + i; : : : ; m + i
(8)
where B (z ) def = a0 z n + a1 z n 1 + + an (overbar denotes the compex conjugate) and choose any z0 such that jz0 j < 1 and jw(z0 )j 6= 1. Now, for k = 0; 1; : : : we set (9) wk+1 (z ) = 1z zkzz wk (zk ) wk (z ) k 1 wk (zk )wk (z ) where zk is again chosen such that jzk j < 1 and jwk (zk )j 6= 1. Each wk (z ) is a rational function of degree one less than wk 1 (z ) and hence jwr (z )j 1 for some r n. The di erence n r is equal to the number of common zeros of A and B . From the de nition of B it follows that such zeros are symmetric with respect to the unit circle and since we assume that A has no zeros on the unit circle, half of them must be inside. From Theorem 20 of [9] it then follows Theorem 2. Let A(z ) = a0 + a1 z + + an zn and A(ej! ) 6= 0 for ! 2 R. Let w0 (z ); : : : ; wr (z ) be given by (9) and z0 ; ; zr 1 are such that jwk (zk )j 6= 1 for k = 0; : : : ; r 1. Let i0 ; : : : ; im 1 denote the values of k for which jwk (zk )j > 1. Then, the number of zeros of A(z ) inside the unit circle is 1 2 (n
r) + (i1 i0 ) + (i3 i2 ) + + (im 1 im 2 )
if m is even or 1 2 (n + r)
i0 (i2 i2 ) (im 1 im 2 )
if m is odd. The points zk need not be necessarily distinct and major simli cation of (9) results when all zk can be selected 0. If A(ej! ) 6= 0 the assertion of Theorem 2 can be exx(m) = f: : : ; 0; 0; x(mm); : : : ; xm(m) ; 0; 0; : : :g pressed by the following algorithm. The integer array out ( m ) saves values of i0 ; ; im 1 . then, for any absolutely summable sequence y, x converge in the l1 {norm to the solution x of the equation Input and initialization: a x = y. n n X X The matrix of the system (8) is of the form A(z ) = ak z k ; B (z ) = ak z n k ;
has a unique solution fx(km) gm k= m for any absolutely summable sequence y = fyk g+1 . 1 If we denote
0 a i B ai+1 B ai+2 A=B B B @ ...
ai 1 ai ai+1 .. .
ai 2 ai 1 ai .. .
: : : ai 2 m : : : ai 2m+1 : : : ai 2m+2 ...
ai+2m ai+2m 1 ai+2m 2
.. .
ai
1 CC CC CA
k=0
k=0
z0 = 0; k = 0; m = 0; out[0] = 0 Main loop: repeat
f (z ) = BA((zz)) ; w = f (z0) if jwj = 1 then nd z0 ; jz0 j < 1 such that jf (z0 )j 6= 1 if z0 not found then go to evaluation (Remark: It is sucient to check n k + 1 di erent values z0 ) if jwj > 1 then out[m] = k; m = m + 1 (z ) B (z ) wA(z ) B (z ) A(z ) A(z )z wB 1 z0 z 0 z
which is a nonsymmetric Toeplitz matrix. In most cases all the principal minors of A are nonzero and we can use the ecient method of Levinson and Rybicki [10] to solve (8). The index i, however, must be known beforehand. We recommend the following method to calculate i = ind A(ej! ). Since ind A(ej! ) = m + ind ejm! ind A(ej! ) we may assume without loss of generality that A is a polynomial in nonnegative powers of ej! . Then the index is (Remark: Both fractions can be cancelled; the cancelequal to the number of zeros of A(z ) inside the unit circle lation is not automatic and must be included in the and we follow [9] to calculate it. program.) We rst de ne the function z0 = 0; k = k + 1 until k=n w0 (z ) = BA((zz)) ; 1909
Evaluation: index = (n k) div 2 if Odd(m) then (mX 1)=2 out[2i] out[2i 1] index = index + k out[0] else
index = index +
i=1
m= X2 i=1
out[2i 1] out[2i 2]
We remark that the index is unde ned when A(ej! ) = 0 for some ! 2 R. The algorithm, however, does not check this condition. III. Computation of the Cepstrum
Combining Theorem 1, (4) and (5) we get a conceptually simple approximation to the cepstrum. The following theorem gives details and guarantees that the approximations converge to the cepstrum. Theorem 3. Let a = fak g+11 be an absolutely summable sequence. Let
A(z ) =
+1 X
k= 1
ak z k
c0 ) and therefore coincides with that obtained in [4] by a
di erent, and in our opinion more involved method. If the sequence a is minimum phase (i.e. unilateral and ind A(ej! ) = 0) then the matrix A is lower triangular and the method is equivalent to computing ck recursively from (3) with the initial value c0 = log a0 . Moreover, c(km) are already exact values of cepstral coecients, i.e. c(km) = ck for any m. Interestingly, for the minimum phase sequences our method coincides also with the approach via the { transform [11] and via the one{dimensional isomorphic operator [12]. Both [11] and [12] give, however, di erent results from our method for mixed phase sequences. Even though the assumption of analycity imposed by Theorem 3 on the Fourier transform of a is not too restrictive in practice (we usually work with sequences of nite length), it is not known if Theorem 3 holds without this assumption. Another open problem is whether Theorem 3 can be extended to multidimensional sequences. Obviously, (4) holds irrespective of dimension and (5) can easily be generalized to higher dimensions. Theorem 1 has been proved, however, only for the one{dimensional case and though [5] conjectures its validity for higher dimensions, to the author's knowledge no proof has been given as yet. IV. Numerical Examples
be analytic for all z , jz j = 1, let A(ej! ) 6= 0 for all ! 2 R and ind A(ej! ) = 0. Then there exists a positive integer To illustrate the performance of the method we provide two examples and compare the obtained results with the m0 such that for each m > m0 the system of equations theoretical values. In both examples sequences with known m X zero{locations of their z {transforms have been taken to an k xk = n ; n = m; : : : ; m (10) enable calculating the exact cepstrum fck g [8]. The tables k= m show the approximations c(km) for some m and the exact values ck . has a unique solution fxk(m) gm k= m . If we denote TABLE I { CEPSTRUM APPROXIMATIONS x(m) = f: : : ; 0; 0; x(mm); : : : ; x(mm) ; 0; 0; : : :g A1 (z) = (1 0:7z)(1 0:5z)(1 0:8z 1 )(1 + 0:8z 1 )(1 0:7z 1 ) ( m ) b = f: : : ; 0; 0; ma m; : : : ; mam; 0; 0; : : :g k ak m = 4 m = 10 m = 20 ck ( m ) d = b(m) x(m) -10 0 { -0.0120 -0.0244 -0.0243
c(0m) =
c(m) =
Z
log A(ej! ) d!; c(km) =
n (m)o ck
n d(km) o k
; k 6= 0
then c(m) converge in the l1 {norm to the cepstrum c of a. Let bk = kak ; b = fbk g. Since A(z ) is analytic on the unit circle and ind A(z ) = 0, log A(z ) is also analytic on the unit circle [3] and, consequently, both b and d are absolutely summable. Then, using (4) and Theorem 1 we get (the norm is taken in l1 ): Proof:
kd d(m) k = kb a 1 b(m) x(m) k = kb a 1 b x(m) + b x(m) b(m) x(m) k kb a 1 b x(m) k + kb x(m) b(m) x(m) k kbk ka 1 x(m) k + kx(m) k kb b(m) k ! 0 as m ! 1. Hence c(m) ! c in the l1 norm. 2
We remark that since the solution of (10) is uniquely determined, the solution of (3) is also unique (except for 1910
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
0 0 0 0 0 0 0.4480 -1.1776 0.2248 1.6160 -1.4450 0.3500 0 0 0 0 0 0 0 0
{ { { { { -0.1676 -0.3979 -0.9593 -1.0737 0 -0.9974 -0.7812 -0.1196 -0.0218 { { { { { {
-0.0445 -0.0761 -0.0313 -0.1207 -0.0456 -0.2722 -0.1241 -0.8868 -0.7173 0 -1.1866 -0.3859 -0.1515 -0.0895 -0.0367 -0.0376 -0.0103 -0.0268 -0.0028 -0.0004
-0.0046 -0.0492 -0.0118 -0.1070 -0.0336 -0.2648 -0.1144 -0.8850 -0.7001 0 -1.2000 -0.3700 -0.1560 -0.0757 -0.0398 -0.0223 -0.0129 -0.0077 -0.0047 -0.0030
-0.0045 -0.0492 -0.0118 -0.1070 -0.0336 -0.2648 -0.1143 -0.8850 -0.7000 0 -1.2000 -0.3700 -0.1560 -0.0757 -0.0399 -0.0222 -0.0129 -0.0077 -0.0047 -0.0029
It is to be expected that the performance of the method V. Conclusions depends on the zero locations of A(z ). The next example We have presented a time{domain method for the compushows that the rate of convergence of c(m) slows down as tation of the complex cepstrum suitable for mixed phase zeros of A(z ) approach the unit circle. sequences with analytic Fourier transform. Though our method may be computationally less ecient than frequenTABLE II cy domain methods it provides an alternative in the cases CEPSTRUM APPROXIMATIONS when phase unwrapping problems render frequency domain A2 (z) = (1 + 0:8z)(1 + 0:6z 1 )(1 + 0:9z 1 )(1 + 0:95z 1 ) methods unreliable. An asset in comparison with the time{ domain cepstral transformation by Sokolov and Rogers [4] k ak m = 10 m = 20 m = 30 ck is that our method requires neither generalized inverses of -10 0 0.8161 -0.1107 -0.0969 -0.0953 matrices nor repeated matrix multiplications. -9 0 0.2862 0.1278 0.1155 0.1142 As self-contained parts we have described a time-domain -8 0 -0.2922 -0.1510 -0.1400 -0.1388 deconvolution method together with an algorithm for the -7 0 0.3105 0.1831 0.1732 0.1721 time{domain calculation of the linear phase component of -6 0 -0.3459 -0.2290 -0.2199 -0.2189 a sequence. -5 0 0.4075 0.2980 0.2894 0.2884 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
0 0.5130 2.3754 4.0220 2.9600 0.8000 0 0 0 0 0 0 0 0 0
-0.5133 0.7182 -1.1683 2.6422 0 0.7159 -0.2985 0.1681 -0.1082 0.0759 -0.0569 0.0453 -0.3840 0.0347 -0.0084
-0.4095 0.6106 -1.0475 2.4671 0 0.7910 -0.3170 0.1695 -0.1020 0.0656 -0.0440 0.0304 -0.0215 0.0155 -0.0114
-0.4010 0.6018 -1.0374 2.4518 0 0.7990 -0.3197 0.1705 -0.1023 0.0655 -0.0437 0.0300 -0.0210 0.0150 -0.0108
-0.4001 0.6008 -1.0363 2.4500 0 0.8000 -0.3200 0.1707 -0.1024 0.0655 -0.0437 0.0300 -0.0210 0.0149 -0.0107
References
In both examples the sequences are already approprietly shifted so that ind A1 (ej! ) = ind A2 (ej! ) = 0. Originally we started from mixed phase sequences; e.g. in the second example we took A~2 (z ) = (z + 1:25)(z + 0:6)(z + 0:9)(z + 0:95) for which it was found that ind A~2 (ej! ) = 3. Hence
A~2 (z )=1:25z 3(1+0:8z )(1+0:6z 1)(1+0:9z 1)(1+0:95z 1) = 1:25z 3(0:513z 3 + 2:3754z 2 + 4:022z 1 + 2:96 + 0:8z ). The scaling factor 1.25 in uences only c0 and was dropped in A2 (z ).
[1] A. V. Oppenheim, R. W. Schafer and T. G. Stockham, \Nonlinear ltering of multiplied and convolved signals," Proc. IEEE, vol. 65, pp. 1264{1291, 1968. [2] E. Krajník and B. Pondìlíèek, \Cepstrum as logarithm in a Banach algebra," Proc. Circuit Theory and Design Seventh Europ. Conf., Prague 1985, pp. 414{416. [3] D. M. Goodman, \Some properties of the multidimensional complex cepstrum and their relationships to the stability of multidimensional systems," Circuit Systems Signal Process., vol. 6. pp. 3{30, 1987. [4] R. T. Sokolov and J. C. Rogers, \Time{domain cepstral transformations," IEEE Trans. Signal Processing, vol. 41, pp. 1161{ 1169, March 1993. [5] I. C. Gohberg and I. A. Feldman, Convolution Equations and Projection Methods for their Solution. AMS, Providence, RI 1974, Sec. 6.2. [6] A. Polydoros and A. T. Fam, \The di erential cepstrum: de nition and properties," 1981 IEEE Int. Symp. on Circuits and Systems, pp. 77{80, April 1981. [7] J. B. Bednar and T. L. Watt, \Calculating the complex cepstrum without phase unwrapping or integration," IEEE Trans. Acoust. Speech, Signal Processing, vol. ASSP-33, pp. 1014{ 1017, Aug. 1985. [8] A. V. Oppenheim and R. W. Scha er, Discrete{Time Signal Processing. Prentice{Hall, Englewood Cli s, N.J., 1989. [9] N. N. Meiman, \Some aspects of polynomial root locations" (in Russian), Uspekhi matem. nauk, vol. 4 (1949), No.6, pp. 154{ 188. [10] W. H. Press et al., Numerical Recipes in Pascal. Cambridge Univ. Press, Cambridge, N.Y. 1990. [11] E. Krajník, \{transform: a new tool for homomorphic signal processing," in H. Dedieu (ed.), Circuit Theory and Design 93 { Proc. ECCTD'93, Elsvier, Amsterdam 1993, pp. 341{346. [12] I. Yamada, K. Sakaniwa and S. Tsujii, \A multidimensional isomorphic operator and its properties { a proposal of nite{extent multidimensional cepstrum," IEEE Trans. Signal Processing, vol. 42, pp. 1766{1785, July 1994.
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