Pad e Table, Continued Fraction Expansion and Perfect ... - CiteSeerX
Pade Table, Continued Fraction Expansion and Perfect Reconstruction Filter Banks Masoud R.K. Khansariy University of California Department of EECS 231 Cory Hall Berkeley, CA, 94708 [email protected] y
Eric Dubois University of Quebec INRS-Telecommunications 16 Place du Commerce Verdun, Quebec, H3E 1H6 [email protected]
corresponding author
Abstract We investigate the relationships among the Pade table, continued fraction expansions and perfect reconstruction (PR) lter banks. We show how the Pade table can be utilized to develop a new lattice structure for general two-channel bi-orthogonal perfect reconstruction (PR) lter banks. This is achieved through characterization of all two-channel bi-orthogonal PR lter banks. The parameterization found using this method is unique for each lter bank. Similarly to any other lattice structure, the PR property is achieved structurally and quantization of the parameters of the lattice does not e ect this property. Furthermore, we demonstrate that for a given lter, the set of all complementary lters can be uniquely speci ed by two parameters, namely the end-to-end delay of the system and a scalar quantity. Finally, we investigate the convergence of the successive lters found through the proposed lattice structure and develop a sucient condition for this convergence.
1
1 Introduction The subband decomposition of signals and its application to speech and image coding has been investigated by many researchers in the last decade. At the same time, developments in the existence and the construction of orthonormal wavelet bases have formed a solid theoretical foundation for the theory of perfect reconstruction (PR) lter banks [9] [12]. Factorization of lter banks has proved to be essential in both design and implementation of PR lter banks. Also, they help in understanding the fundamental algebraic structure of PR lter banks. Using factorization, a higher order lter bank is achieved by adding an extra element to a lower order bank. Such a lattice type structure was considered to characterize all M -channel paraunitary lter banks [10]. Partial results on factorization of general PR lter banks have been recently reported where, although the proposed factorization characterizes a large class of PR lter banks, it lacks completeness [11]. Such structures have also been proposed for the bi-orthogonal linear-phase lters [13]. With the use of ladder structures, it can be shown that the factorization of general M -channel bi-orthogonal PR lter banks is possible. This factorization even though is complete lacks uniqueness [4]. In [14] Vetterli and Herley demonstrated the close relation between the continued fraction expansion (CFE) of functions and PR lter banks. They showed that in the case of two-band PR lter banks with lters H (z ) and He (z ), the CFE of the ratios of H (z )=H (?z ) and He (z )=He (?z ) are similar except for the last term. This is expected in light of the fact that the comprimeness of H (z ) and H (?z ) is a necessary condition for PR [14] [5], which also points to the relation between the CFE and the Euclid algorithm as was demonstrated in [14]. Note that the necessary and sucient condition for the pair fH (z ); He (z )g to be a PR pair is
H (z)He (?z) ? He (z)H (?z) = z 2k+1
2 C ? f0g;
k 2 Z:
(1)
The Pade table is a classical method for approximating a power series by a ratio of two polynomials [1][2]. The members of the table approximate a given power series as closely as possible and are distinguished by the respective orders of their numerators and denominators. Clearly, the higher the order of these polynomials, the closer is the approximation to the given power series. In Section 2, we characterize the functions where the numerators of the successive diagonal elements of their Pade table constitute a two-channel PR lter bank. There is also a close relation between the Pade table and certain classes of continued fraction expansion of functions. In Section 3, by demonstrating this relationship, we show that the lter banks found using this approach are modular in structure { the higher order lter banks are based on the lower order banks. Also, the perfect reconstruction property 2
is preserved structurally; the quantization of the lter parameters does not a ect the PR property. This is especially desirable for nite precision implementation of lter banks. In Section 4, we show that given one of the lters of a two-channel lter bank, only two parameters are sucient to characterize all the possible complementary lters (up to a scaling factor). The complementary lters are those lters which together with the original lter constitute a two-channel PR lter bank. Consequently, we nd an n + 2 canonical parameterization for general two-band causal bi-orthogonal PR lter bank of order n. This parameterization besides being unique is complete. Finally, in Section 5 we investigate the convergence of the successive elements of the table to the desired power series (frequency response). We nd a sucient condition which is not too restrictive and under which this convergence is guaranteed.
2 Pade Table and Perfect Reconstruction Filter Bank P
i Let us assume that the function f (z ) has power series expansion f (z ) = 1 i=0 ai z . Then the [m; n] (mth row, nth column) element of the Pade table is the ratio of two polynomials in z , namely Pm;n (z )=Qm;n (z ) where the highest degree of Pm;n (z ) and Qm;n (z ) are at most m and n, respectively [1]. The elements of the Pade table are de ned such that,
f (z) ? QPm;n ((zz)) = O(z m+n+1 )
(2)
m;n
where O(z k ) denotes a function for which its power series expansion has only elements of degree k and higher. The above condition uniquely determines the rational polynomial Pm;n (z)=Qm;n(z) which is called a Pade approximant of f (z). The function f (z) and the corresponding Pade table is called regular or normal if for all m and n, the order of Pm;n (z ) and Qm;n (z ) are exactly m and n, respectively. A sucient condition for the Pade table to be regular is for the Hankel matrix
0 BB am?n+1 am?n+2 am BB am?n+2 am?n+3 am+1 .. .. BB ... . . BB B@ am?1 am am+n am am+1 am+n?1
3
1 CC CC CC CC CA
(3)
to be non-singular for all m 0 and n m + 1. In this paper we are only considering regular functions. A consequence of (2) which we state without giving the proof is [m + 1; n + 1] ? [m; n] =
zm+n+1 Qm+1;n+1 (z)Qm;n (z)
(4)
for some 2 C ? f0g which can be calculated explicitly [1]. Throughout, we take advantage of the following two invariance properties of the Pade table [1]: 1: If P (z)=Q(z) is the [m; n] element of the Pade table for f (z), then Q(z)=P (z) is the [n; m] element of the Pade table for 1=f (z ). 2: If P (z)=Q(z) is the [n; n] element of the Pade table for f (z), then for all scalars and
A + B [P ( y=(1 + y))=Q ( y=(1 + y))] C + D [P ( y=(1 + y))=Q ( y=(1 + y))] is the [n; n] element of the Pade table of
A + Bf ( y=(1 + y)) ; C + Df ( y=(1 + y)) where we have assumed that C + Df (0) 6= 0. We now state the main result of this section.
Theorem 1 If f (z)f (?z) = 1, then the numerators of the consecutive diagonal elements of the Pade table are a PR pair.
Proof:
Since f (z )?1 = f (?z ), from the rst invariance property we have [n; m]f (z) =
1 ; [m; n]f (?z)
and the diagonal elements (m = n) of the Pade tables of f (z ) and f (?z ) are reciprocal of each other. Now if [n; n]f (z) = P (z )=Q(z ) then from the second invariance property [n; n]f (?z) = P (?z )=Q(?z ). Therefore
P (z) = Q(?z) ; Q(z) P (?z) and since P (z ) and Q(z ) are relatively prime, Q(z ) = P (?z ). As a result, the diagonal
4
elements of the Pade table of f (z ) are of the form [n; n]f (z) = HHn((?zz)) : n
and from (2) we have
Hn(z) ? Hn?1 (z) = z 2n?1 Hn (?z) Hn?1 (?z) Hn (?z)Hn?1 (?z) ; or
Hn (z)Hn?1(?z) ? Hn?1 (z)Hn (?z) = z 2n?1 ;
which means that fHn (z ); Hn?1(z )g are a PR pair.
End of Proof
As an example, let us consider f (z ) = g (z )=g (?z ) 1 where we take g (z ) to be the D4 Daubechies lter which is of length 8. Since g (z ) is of order 7, for n; m 7, all [n; m] element of the table are the same and equal to f (z ). Table 1 gives the lter coecients of H7 (z) = g(z) and its rst two approximations where the coecients are normalized so that the leading coecient is one. The approximations are found by taking the numerators of the consecutive diagonal elements of the table. It is straightforward to verify:
H7(z)H6(?z) ? H6(z)H7(?z) = 0:01667z 13 H6(z)H5(?z) ? H5(z)H6(?z) = 0:13348z 11
(5)
and hence both fH7 (z ); H6(z )g and fH6 (z ); H5(z )g are PR lter banks.
3 Continued Fraction Expansion and Lattice Structure The nth approximant of the continued fraction
is de ned as
a1 (z) z) a2 (z) = b (z) + ; b0 (z) + ba(1z()+ 0 b2(z) + 1 b1 (z) + b2a(2z()+z)
(6)
Nn(z) = b (z) + a1(z) a2 (z) an(z) : Dn (z) 0 b1(z)+ b2(z) + bn(z)
(7)
1 Note that f (z ) is a power series in z . In this paper, we denote the delay element by z instead of usual notation of z ?1 used in signal processing.
5
Therefore, it is found by simply removing the tail term an+1 (z )=(bn+1(z ) + ) from the original fraction expansion. It can be shown that both the numerators (Nn (z )) and the denominators (Dn (z )) satisfy the same recurrence equation
8 > > < Nn(z) = bn(z)Nn?1(z) + an (z)Nn?2(z) > > : Dn(z) = bn(z)Dn?1(z) + an(z)Dn?2(z)
n1
(8)
but with di erent initial conditions
8 < N0(z) = b0(z) N?1(z) = 1 : : D0(z) = 1 D?1(z) = 0
(9)
In the literature, the above recurrence equation is known as a three-term recurrence equation [6]. Also, note that the solution space of the above recurrence equation is a two-dimensional linear space and hence any other solution of the above recurrence equation can be found as a linear combinations of these two solutions. There is a close relationship between the Pade table and certain continued fraction expansions of the functions. Every path 2 through the table corresponds to a speci c continued fraction expansion of the function approximated by that Pade table. For example, a path consisting of the diagonal elements of the table corresponds to the consecutive approximants of the P -fraction expansion of the function, where P stands for the principleplus-constant terms. P -fractions have the following form
f (z) = b0(z) + b (z1)+ b (z)1+ 1 2
(10)
where bi (z ) are polynomials in z ?1 [7] [8]. Assuming that the table is normal, the [n; n] element of the table is the same as the nth approximant of the associated P -fraction and can therefore be written as: (11) [n; n] = b0 (z ) + b (z1)+ b (z )1+ b 1(z ) ; 1 2 n P a zi) and the for appropriate bi (z ). If f (z ) does not have principle part (f (z ) = 1 i=0 i corresponding Pade table is normal, then b0(z ) is a constant and bi(z ) (i > 0) is at most of By path we mean to start from upper left corner of the table, and at each step, to move to the right or the down or along the diagonal direction 2
6
order one. Therefore, (7) can be rewritten as: 2 2 [n; n] = a0 + B (zz )+ B (z )z + B z(z ) ; 1 2 n
(12)
where Bi (z ) = zbi (z ), i 1, is a rst order polynomial in z . The three-term recurrence formula for the above continued fraction can be written as
8 > > < H1(z) = B1(z)H0(z) + zH?1(z) > > : Hn(z) = Bn (z)Hn?1(z) + z2Hn?2(z)
n>1
(13)
where H?1 = 1; H0(z ) = a0 for the numerator and H?1 = 0; H0(z ) = 1 for the denominator. We now show that if f (z ) satis es the condition required by Theorem (1), namely f (z)f (?z) = 1, then Bn (z) satis es an additional constraint.
Lemma 1 If f (z) is regular, f (z)f (?z) = 1, and 2 2 f (z) = 1 + B (zz)+ B (z)z + B (zz) + ; 1 2 n
is the P -fraction expansion of f (z ), then 2 z2 ; f (z) = 1 + ? 1 z +z b + b +z b + n 1 2 2
(14)
where bn 2 C .
Proof:
This lemma states that if f (z ) is regular and f (z )f (?z ) = 1 then for n 2, Bn (z ) are constant. Let us de ne fe (z ) and fo (z ) as:
fe (z2) =
pf (z) + pf (?z) 2p
p
fo (z2) = f (z) ?2z f (?z) : Then it is straightforward to show that
s
(z ) = fe (z 2 ) + zfo (z 2) : f (z) = ff(? z) fe(z 2 ) ? zfo (z2)
7
Writing the P -fraction expansion of f (z ) we have 2
o (z ) f (z) = 1 + f (z22zf e ) ? zfo (z 2 ) z = 1+ (fe (z 2) ? zfo (z 2 ))=(2fo(z 2)) = 1 + ?1=2z + b + (f (z 2) z? 2b f (z 2))=(2f (z 2 )) ; 1 e 1o o
where b1 is chosen such that fe (z 2 ) ? 2b1fo (z 2 ) = O(z 2 ). Comparing the above with (12) we have fe (z2) ? 2b1fo (z 2) = z2 z2 z2 : 2fo (z 2) B2(z)+ B3(z) + Bn (z) + Now since the LHS is an even function, B2 (z ); B3(z ); are also even and since they are at most of the rst order, they can only have constant term, as required.
End of Proof
Using the above theorem, we can now rewrite the recurrence equation (13) to arrive at the following recursive relation.
8 > > < H1(z) = (b1 ? 1=2 z)H0(z) + zH?1(z) > > : Hn(z) = bnHn?1(z) + z2Hn?2 (z)
n > 1;
(15)
where H0(z ) = H?1 (z ) = 1. For example, Table 2 shows the value of bi 's for the Daubechies D4 lter. Note that since the above corresponds to the numerators of the diagonal elements of the Pade table of f (z ) with the property that f (z )f (?z ) = 1, for all n, fHn?1 (z ); Hn(z )g is a PR lter bank. In summary, the nth order lter Hn (z ) is found by shifting Hn?2 (z ) by 2 and then adding a proper multiple of Hn?1 (z ), as shown in Figure 2. The PR property is preserved structurally and the quantization of bi parameters does not e ect this property. In general, for large n, the successive lters found through the above procedure have similar frequency response. In most application, however, it is desired that each lter captures di erent frequency portion of the input signal. In the next section we demonstrate that all the possible causal complementary lters of order n can be simply characterized by two parameters. We also show that all the causal complementary lters can be implemented as a linear combination of the lower order lters of the chain fHn (z )g found using recurrence equation (15).
8
4 Parameterization of the Complementary Filter We rst show the following lemma:
Lemma 2 If H (z) and He (z) are of order n and n ? 1 respectively and are a PR pair, then any other lter of order n ? 1 which forms a PR pair with H (z ) is of the form He (z ) for some 2 C ? f0g. Proof:
Since H (z ) and He (z ) are PR pair,
H (z)He (?z) ? He (z)H (?z) = z 2n?1 : Note that k = n ? 1. This is true since the LHS is an odd function and it is not possible for the term corresponding to z 2n?1 to cancel out. Let us assume that there exists a di erent lter of order n ? 1, namely H (z ), such that
H (z)H (?z) ? H (z)H (?z) = z 2n?1 : De ne
G(z) = He (z) ? H (z):
It follows that H (z )G(?z ) ? H (?z )G(z ) = 0, which implies that G(z ) is at most of order n ? 2. A necessary condition for PR is that H (z) and H (?z) be relatively prime (they cannot have common zero) [13] so that the only solutions to the above equation are either G(z) = 0 and G(z) = H (z). The latter is not possible since G(z) is at most of order n ? 2, and thus G(z ) = 0 and H (z ) = He (z ).
End of Proof
The statement of the above lemma is also evident from the Euclid algorithm and the fact that the comprimeness of H (z ) and H (?z ) is a necessary condition for PR. The following two theorems are the direct and converse results that characterizes all the possible causal complementary lter of order n. In summary, Theorems (2) and (3) state that given Hn (z ) (Hn?1 (z ) is unique up to a scaling factor), the complementary lter of H (z ) of order n, He (z ), is uniquely characterized, independently of n, by three parameters k, and 1. The value of k corresponds to the delay between the reconstructed signal at the output of the synthesis lter bank and the input signal at the input of the analysis bank - overall system delay. Therefore, a small value of k corresponds to a low-delay subband decomposition system. Moreover, for a given value of k, the frequency selectivity of He (z ) depends only on the ratio = 1 and not their absolute values. Therefore, the process of nding a complementary lter 9
He (z) with a desired frequency response is a one-dimensional search through di erent values of = 1.
Theorem 2 Let Hn(z) and Hn?1 (z) of order n and n ? 1, respectively, be a PR pair. Then any He (z ), of order n, such that Hn (z ) and He (z ) are a PR pair with Hn (z)He (?z) ? He (z)Hn(?z) = z 2k+1 ; where k is an integer between 0 and n ? 1, can be expressed in the form
0n?k?1 1 X j+1 z ?2j A Hn (z) He (z) = z ?2(n?k?1) Hn?1 (z) + @ j =0
(16)
for some 1; ; n?k ; 2 C ? f0g.
Proof: De ne
G1 (z) = He (z) ? 1 Hn(z)
where 1 is chosen such that g1 (n) = 0. By direct substitution,
Hn(z)G1(?z) ? G1 (z)Hn(?z) = z 2k+1 so that G1(z ) and Hn (z ) form a PR pair. Now if k = n ? 1, g1(n ? 1) cannot be zero, so that by Lemma (2), G1 (z ) = Hn?1 (z ) for some 2 C ? f0g, and He (z ) = Hn?1 (z ) + 1Hn (z ) satisfying (16) and we are done. If k < n ? 1, then de ne
Gi(z) = z 2 Gi?1 (z) ? i Hn(z);
i = 2; 3; ; n ? k
(17)
where i is chosen such that gi(n) = 0. Now,
Hn (z)Gi(?z) ? Gi (z)Hn(?z) = z 2(k+i)?1 which implies that gi (n ? 1) = 0 for i < n ? k and gi(n ? 1) 6= 0 for i = n ? k. By Lemma (2), Gn?k (z ) = Hn?1 (z ), and by repeated substitution of (17) into itself, (16) results.
End of Proof
One has to note that even though negative powers of z appear on the RHS of (16), He (z ) is causal and does not have any negative power of z . We now prove the converse of the above theorem. 10
Theorem 3 Let Hn(z) and Hn?1 (z) of order n and n ? 1, respectively, be a PR pair. Then
any lter of the form
0n?k?1 1 X j+1 z ?2j A Hn (z) He (z) = z ?2(n?k?1) Hn?1 (z) + @ j =0
(18)
for 1; ; n?l ; 2 C ? f0g, k 2 f0; 1; ; n ? 1g forms a PR pair with Hn (z ). Moreover, if k; and 1 are speci ed, only one choice of 2; ; n?k yields a causal lter.
Proof:
By direct substitution
Hn(z)He (?z) ? He (z)Hn (?z) = z ?2(n?k?1) (Hn(z)Hn?1 (?z) ? Hn?1 (z)Hn(?z)) = z 2k+1 for some 2 C ? f0g: Therefore, H (z ) and He (z ) from a PR pair. It remains to show that He (z ) is of the form given by (18). By regrouping di erent terms of (18), we have
0n?k?2 1 X He (z) = (Hn?1 (z) + n?k Hn (z))z ?2(n?k?1) + @ j+1 z?2j A Hn (z); j =0
(19)
and using three-term recurrence equation, it is possible to choose n?k such that
0
1
n?k?2 eH (z) = 1Hn?2(z)z?2(n?k?2) + @ X j+1z?2j A Hn(z): j =0
We can again regroup di erent terms of the above to arrive at
0
1
n?k?3 eH (z) = ( 1Hn?2 (z) + n?k?1 Hn(z))z?2(n?k?2) + @ X j+1z?2j A Hn(z); j =0
(20)
(21)
and again it is possible to choose n?k?1 such that
0n?k?3 1 X j+1 z?2j A Hn (z): (22) He (z) = 2Hn?3 (z)z ?2(n?k?3) + 3Hn?4 (z)z ?2(n?k?4) + @ j =0
By iteratively applying the same procedure, the RHS will only contain positive powers of z and as a result, for this speci c choice of n?k ; n?k?1 ; ; 2, He (z ) is causal. Note that if He (z) is causal, we do not have any freedom in choosing these parameters.
End of Proof
11
As a consequence of the above theorems, a bi-orthogonal causal two-band perfect reconstruction lter bank where the lters are of order n can be parameterized by the following n +2 parameter set: fb1; b2; ; bn; = 1 2 C ? f0g ; k 2 f0; :::; n ? 1gg : (23) Note that this is a canonical (unique) parameterization and there is only one set of parameters corresponding to each lter bank. Moreover, it is a complete parameterization and all lter banks can be parameterized using the above. Also, the end-to-end system delay is given by 2k + 1. Figure 3 shows possible complementary lters to the Daubechies D4 lter. The value of k is xed at 3 which means that the end-to-end delay is 7 as in the case of an orthogonal lter bank. By varying the value of = 1 (the range ?85 to ?95 is shown in the Figure 3) di erent complementary lters can be found. When = 1 = ?90:5, the resultant lter is the power complementary lter of the D4 lter. Note that this search process is only a one-dimensional search which can be readily done. Also note that the the complementary lter is sensitive to the value of = 1 and deviation from ?90:5 can result in a lter which may not have the desired high-pass characteristic. Another example is given in Figure 4, where the possible complementary lters of a linear-phase lter H (z ), tabulated in Table 3, are shown. Again k is set to 3 which translates into an end-to-end delay of 7. The gure shows di erent complementary lters where the value of = 1 is varied between 60 and 100. The value of 87:79 results in the linear-phase lter He (z ) where its coecients are tabulated in Table 3.
4.1 Causal Implementation Implementation of the complementary lter as is proposed by (18) requires non-causal elements. But, since the overall lter is causal, one would expect that causal implementation should be possible. In fact, using similar procedure as was used in the proof of Theorem 3, we can show that He (z ) can be expressed as a linear combination of
fHn (z); Hn?k(z); ; Hn?2k (z)g or
He (z) = 1Hn(z) +
2k X j =k
aj Hn?j (z);
(24)
where aj 2 C for all j . In other words, the complementary lter can be implemented by tapping at the output of the adders of the lattice structure of Figure 2 and adding a proper linear combination of these outputs. We clarify this point using the following example. 12
Example:
8 < H (z) = 1 + z + z2 + z3 (25) : He (z) = 1 + z ? z2 ? z3; where 6= 1. Note that both H (z ) and He (z ) are linear phase lter and fH (z ); He (z )g is a Let
PR pair [13] since
H (z)He (?z) ? H (?z)He (z) = 4(1 ? 2 )z3:
(26)
Using the Pade approximation method, we nd the following family of the lters:
8 2 + z3 > > < H3(z) = 1 + z + z ?1+ 2 z 2 H ( z ) = 1 + z + 2 > > : H1(z) = 1 + z
(27)
Now by applying Theorem (2), we can write He (z ) in terms of H3 (z ) and H2(z ) as:
He (z) = ?2 z ?2 H2 (z) + ?1 + 2 z ?2 H3 (z):
(28)
Therefore, given H (z ), the following two parameter set uniquely speci es He (z ) up to a scaling factor fk = 1 ; = 1 = 2 g : The implementation of He (z ) using the above expression requires non-causal elements. Alternatively, using (24) one can express He (z ) in terms of H3 (z ) and H1(z ) as:
He (z) = 2H1(z) ? H3 (z);
(29)
which can be implemented using only causal elements. Figure 5 shows such a lattice implementation for both H (z ) and He (z ).
5 Convergence of Successive Approximants Similarly to the power series expansion of functions, the continued fraction expansion does not necessarily converge for all z 2 C and one should de ne the associated region of convergence. Clearly, one would want the unit circle to be within this region. In our development, we are primarily interested in the expansion of the function of the form f (z ) = g (z )=g (?z ) where it is assumed that g (z ) and g (?z ) do not have any common zero or are relatively prime. Note that if g (z ) is a polynomial and therefore is of nite order, the convergence of f (z ) is always guaranteed since its continued fraction expansion terminates in a nite 13
number of steps. We now consider the case where g (z ) is not necessarily of nite order. We still however assume that g (z ) and g (?z ) are relatively prime. We nd a set of sucient conditions which guarantees that the P -fraction expansion of f (z ) converges to f (z ) for all z 2 C . Note that since g (z ) and g (?z ) are relatively coprime, this also implies that, up to a scaling factor, the numerators of the successive approximants also approach g (z ). First, we brie y present an important class of continued fractions (C -fractions) and investigate some of its properties which are directly relevant to our mathematical development.
5.1 C -fraction Expansion C -fraction expansion of a function has the following form: 1 (z ) a2(z ) a3 (z ) ; a0 (z) + a1+ 1+ 1 +
(30)
where ai (z ) are monomial polynomials. If f (z ) is a regular function then ai (z ) are of rst order (ai(z ) = i z for all i 1) and a0(z ) = 0 is a constant. Therefore, We can rewrite the above as: 1 z 2 z 3 z : (31) 0 + 1+ 1+ 1 +
An interesting property of the C -fraction expansion is that they correspond to a speci c path in the Pade table as shown in Figure 1. Note that the even approximants of a C -fraction expansion are the same as the approximants of the corresponding P -fraction expansion. The following lemma gives the consequence of the condition f (z )f (?z ) = 1 on the parameters of the C -fraction expansion of f (z ).
Lemma 3 If f (z) has C -fraction expansion of the form (31), then f (z)f (?z) = 1 if and only if in the C-fraction expansion of f (z )
2 = ? 1 1 2 2n?1 = ? 2n
n = 2; 3;
(32)
Proof:
Let us write the continued fraction expansion which corresponds to the even approximants of the C -fractions expansion. It is straightforward to show that it is given by: 2 2 0 + 1 + 1zz? 1 + ( 2 +3 z )z? 1 + ( +4 5 z )z ? : 2 3 4 5 6
(33)
Since the even approximants of the C -fraction expansion of a function are successive ap14
proximants of the P -fraction expansion of the same function, the proof can be completed by comparing the above to (14).
End of Proof
As a corollary of the above lemma, if n ! then = 0. This is the true since n alternates between positive and negative values and as a result if f n g is a convergent sequence, it has to converge to zero.
5.2 Sucient Conditions The following theorem, which we state from [2] without outlining its proof, gives sucient conditions for C -fraction expansion of the function given by (31) to be convergent.
Theorem 4 If n ! as n ! 1, then (31) converges to a function f (z) which is meromorphic 3 in the cut z -plane. The cut is placed in the shadow of ?(1=4 ) from the
origin. The convergence is uniform on any compact set not containing the poles of the limit function, and the limit function has the continued fraction expansion (31).
Clearly, if = 0, then the C -fraction expansion of f (z ) given by (31) converges to f (z ) for all z 2 C . Note that this is exactly the case when f (z )f (?z ) = 1. In summary, if the C -fraction expansion of f (z ), where f (z )f (?z ) = 1, converges, it is convergent for all z 2 C . Therefore, a sucient condition for this convergence is for n to approach to zero as n ! 1. We now show how this sucient condition translates into condition on the parameter of the lter decomposition found in the previous section, namely bi in (15). Comparing (14) and (33) we nd the following relationship between bi and i
8 > > < b1 = 1= 1 bn = ?2= 2n n even > > : bn = ?1=(2 2n) n odd
(34)
Clearly, n ! 0 if and only if bn ! 1. Therefore, starting from the recursive relation (15) and choosing bn such that bn ! 1, we are guaranteed that limn!1 Hn (z )=Hn (?z ) exists and is equal to f (z ) = g (z )=g (?z ) where (35) nlim !1 Hn (z ) = Kg (z ) for some scalar K . 3 A function is said to be meromorphic in a region if it is analytic in that region except at a nite number of poles [15].
15
6 Conclusions
We have investigated relationships among Pade table, continued fraction expansion and two-channel perfect reconstruction lter banks. Through characterization of all two-channel bi-orthogonal PR lter banks, we developed a new lattice structure for this class of lter banks. The parameter space found using this method is unique for each lter bank. Also, it is a complete parameterization and it includes all the possible two-band bi-orthogonal lter banks. Similarly to any other lattice structure, the PR property is achieved structurally and quantization of the parameters of the lattice does not e ect this property. We showed that for a given lter, the set of all complementary lters can be uniquely speci ed by two parameters, namely the end-to-end delay of the system and a scalar quantity. Finally, we developed a sucient condition for the convergence of the successive lters found by the proposed lattice structure.
16
References [1] G.A. Baker, Jr., Essentials of Pade Approximants, Academic Press, 1975. [2] G.A. Baker, Jr., P. Graves-Morris and P.A. Carruthers, Pade Approximants, Part I: Basic Theory, Encyclopedia of mathematics and its applications, Vol. 13, Addison Wesley, 1981. [3] T. Kailath, Linear Systems, Printice-Hall, Englewood Cli s, N. J., 1980. [4] A.C.M. Kalker and I.Shah, \On ladder structures and linear phase condition for multidimensional bi-orthogonal Filter Banks", To appear in IEEE Trans. on Signal Processing, 1995. [5] M. Khansari and A. Leon-Garcia, \Subband decomposition of signals with generalized sampling," Special issue of IEEE Trans. on Signal Processing on Wavelets and Signal Processing, Vol. 41, No. 12, pp. 3365-76, Dec. 1993. [6] L. Lorentzen and H. Waadeland, Continued Fractions with Applications, Studies in computational mathematics 3, North-Holland, 1992. [7] A. Magnus, \ Certain continued fractions associated with the Pade table ", Math. Zeitschr. 78, 1962, pp. 361-374. [8] A. Magnus, \ Expansion of power series into P -fractions ", Math. Zeitschr. 80, 1962, pp. 209-216. [9] S. Mallat, \ A theory of multi-resolution signal decomposition: the wavelet representation ", IEEE Trans. on PAMI, Vol. 11, No. 7, pp. 710-732, July 1989. [10] P.P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993. [11] P.P. Vaidyanathan, \ Multirate digital lters, lter banks, polyphase networks, and applications: A tutorial ", Proceeding of IEEE, pp. 56-93, Jan. 1990. [12] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice-Hall, 1995. [13] M. Vetterli and D. J. LeGall, \ Perfect reconstruction FIR lter banks: Some properties and factorizations ", IEEE Trans. on ASSP, Vol. ASSP-37, No. 7, pp. 1057-1071, July 1989. [14] M. Vetterli and C. Herley, \ Wavelets and lter banks: theory and design ", IEEE Trans. on Signal Processing, No. 9, pp. 2207-2232, Sept. 1992. 17
[15] E.C. Titchmarsh, The theory of Functions, Oxford University Press, Second edition, 1986.
18
0 1 2 3 4 5 6 7
H7(z)
1.0000000000 3.1029314858 2.7384614815 -0.1214690295 -0.8118612163 0.1338739557 0.1427351497 -0.0400009711
H6(z)
1.0000000000 3.1029314858 2.8819998520 0.3239206941 -0.3563540087 0.3101626433 0.2082479716
H5(z)
1.0000000000 3.1029314858 3.1734177538 1.2281704828 0.4564133099 0.3204724809
Table 1: The coecients of D4 lter and its rst two approximants
b1 b2 b3 b4 b5 b6 b7
0.1611379440 2.2342472102 1.6658767160 -2.3332622209 -1.1149202725 3.0777960077 -2.2635601304
Table 2: The value of bi (1 i 7) for D4 lter
0 1 2 3 4 5 6 7
H (z)
-0.0506224053 -0.0481274616 0.1781173490 0.4206325181 0.4206325181 0.1781173490 -0.0481274616 -0.0506224053
He (z)
0.1014472948 0.0964474279 -0.174342018 -0.669342151 0.669342151 0.174342018 -0.0964474279 -0.1014472948
Table 3: The coecients of a bi-orthogonal linear-phase two-band PR lter bank. Note that the low-pass lter is symmetric whereas the high-pass lter is anti-symmetric.
19
0
1
2
3
4
0 1 2 3 4
P-Fraction Expansion C-Fraction Expansion
Figure 1: Paths corresponding to di erent continued fraction expansions of the same function
20
z
z 1/2
H (z)
b 1
1
H (z)
b2
2
z
z2
bn n-1
3
2
z2
H (z)
H (z)
b3
b n+2
b n+1
Hn (z)
H
n+1
(z)
z2
Figure 2: Lattice structure to implement the successive lters Hn (z )
21
-4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 3
-94 2.5
-92
2 1.5 frequency
1
-88 0.5 0
-90 delta/beta_1
-86
Figure 3: Magnitude frequency response of He (z ) (dB) for D4 lter for fk = 3; ?95 = 1 ?85g
22
2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 100
3 2.5
90 2 80 delta/beta_1
frequency 1.5 1
70 0.5 60
Figure 4: Magnitude frequency response of He (z ) (dB) for linear-pahse ler given in Table 3 for fk = 3; 60 = 1 100g
23
z
z 1/2 b1
b3
b2
H1 (z)
H (z)
H (z)
2
z
3
2
~
H (z)
Figure 5: Lattice structure for H (z ) and He (z )
24
Eric Dubois University of Quebec INRS-Telecommunications 16 Place du Commerce Verdun, Quebec, H3E 1H6 [email protected]
corresponding author
Abstract We investigate the relationships among the Pade table, continued fraction expansions and perfect reconstruction (PR) lter banks. We show how the Pade table can be utilized to develop a new lattice structure for general two-channel bi-orthogonal perfect reconstruction (PR) lter banks. This is achieved through characterization of all two-channel bi-orthogonal PR lter banks. The parameterization found using this method is unique for each lter bank. Similarly to any other lattice structure, the PR property is achieved structurally and quantization of the parameters of the lattice does not e ect this property. Furthermore, we demonstrate that for a given lter, the set of all complementary lters can be uniquely speci ed by two parameters, namely the end-to-end delay of the system and a scalar quantity. Finally, we investigate the convergence of the successive lters found through the proposed lattice structure and develop a sucient condition for this convergence.
1
1 Introduction The subband decomposition of signals and its application to speech and image coding has been investigated by many researchers in the last decade. At the same time, developments in the existence and the construction of orthonormal wavelet bases have formed a solid theoretical foundation for the theory of perfect reconstruction (PR) lter banks [9] [12]. Factorization of lter banks has proved to be essential in both design and implementation of PR lter banks. Also, they help in understanding the fundamental algebraic structure of PR lter banks. Using factorization, a higher order lter bank is achieved by adding an extra element to a lower order bank. Such a lattice type structure was considered to characterize all M -channel paraunitary lter banks [10]. Partial results on factorization of general PR lter banks have been recently reported where, although the proposed factorization characterizes a large class of PR lter banks, it lacks completeness [11]. Such structures have also been proposed for the bi-orthogonal linear-phase lters [13]. With the use of ladder structures, it can be shown that the factorization of general M -channel bi-orthogonal PR lter banks is possible. This factorization even though is complete lacks uniqueness [4]. In [14] Vetterli and Herley demonstrated the close relation between the continued fraction expansion (CFE) of functions and PR lter banks. They showed that in the case of two-band PR lter banks with lters H (z ) and He (z ), the CFE of the ratios of H (z )=H (?z ) and He (z )=He (?z ) are similar except for the last term. This is expected in light of the fact that the comprimeness of H (z ) and H (?z ) is a necessary condition for PR [14] [5], which also points to the relation between the CFE and the Euclid algorithm as was demonstrated in [14]. Note that the necessary and sucient condition for the pair fH (z ); He (z )g to be a PR pair is
H (z)He (?z) ? He (z)H (?z) = z 2k+1
2 C ? f0g;
k 2 Z:
(1)
The Pade table is a classical method for approximating a power series by a ratio of two polynomials [1][2]. The members of the table approximate a given power series as closely as possible and are distinguished by the respective orders of their numerators and denominators. Clearly, the higher the order of these polynomials, the closer is the approximation to the given power series. In Section 2, we characterize the functions where the numerators of the successive diagonal elements of their Pade table constitute a two-channel PR lter bank. There is also a close relation between the Pade table and certain classes of continued fraction expansion of functions. In Section 3, by demonstrating this relationship, we show that the lter banks found using this approach are modular in structure { the higher order lter banks are based on the lower order banks. Also, the perfect reconstruction property 2
is preserved structurally; the quantization of the lter parameters does not a ect the PR property. This is especially desirable for nite precision implementation of lter banks. In Section 4, we show that given one of the lters of a two-channel lter bank, only two parameters are sucient to characterize all the possible complementary lters (up to a scaling factor). The complementary lters are those lters which together with the original lter constitute a two-channel PR lter bank. Consequently, we nd an n + 2 canonical parameterization for general two-band causal bi-orthogonal PR lter bank of order n. This parameterization besides being unique is complete. Finally, in Section 5 we investigate the convergence of the successive elements of the table to the desired power series (frequency response). We nd a sucient condition which is not too restrictive and under which this convergence is guaranteed.
2 Pade Table and Perfect Reconstruction Filter Bank P
i Let us assume that the function f (z ) has power series expansion f (z ) = 1 i=0 ai z . Then the [m; n] (mth row, nth column) element of the Pade table is the ratio of two polynomials in z , namely Pm;n (z )=Qm;n (z ) where the highest degree of Pm;n (z ) and Qm;n (z ) are at most m and n, respectively [1]. The elements of the Pade table are de ned such that,
f (z) ? QPm;n ((zz)) = O(z m+n+1 )
(2)
m;n
where O(z k ) denotes a function for which its power series expansion has only elements of degree k and higher. The above condition uniquely determines the rational polynomial Pm;n (z)=Qm;n(z) which is called a Pade approximant of f (z). The function f (z) and the corresponding Pade table is called regular or normal if for all m and n, the order of Pm;n (z ) and Qm;n (z ) are exactly m and n, respectively. A sucient condition for the Pade table to be regular is for the Hankel matrix
0 BB am?n+1 am?n+2 am BB am?n+2 am?n+3 am+1 .. .. BB ... . . BB B@ am?1 am am+n am am+1 am+n?1
3
1 CC CC CC CC CA
(3)
to be non-singular for all m 0 and n m + 1. In this paper we are only considering regular functions. A consequence of (2) which we state without giving the proof is [m + 1; n + 1] ? [m; n] =
zm+n+1 Qm+1;n+1 (z)Qm;n (z)
(4)
for some 2 C ? f0g which can be calculated explicitly [1]. Throughout, we take advantage of the following two invariance properties of the Pade table [1]: 1: If P (z)=Q(z) is the [m; n] element of the Pade table for f (z), then Q(z)=P (z) is the [n; m] element of the Pade table for 1=f (z ). 2: If P (z)=Q(z) is the [n; n] element of the Pade table for f (z), then for all scalars and
A + B [P ( y=(1 + y))=Q ( y=(1 + y))] C + D [P ( y=(1 + y))=Q ( y=(1 + y))] is the [n; n] element of the Pade table of
A + Bf ( y=(1 + y)) ; C + Df ( y=(1 + y)) where we have assumed that C + Df (0) 6= 0. We now state the main result of this section.
Theorem 1 If f (z)f (?z) = 1, then the numerators of the consecutive diagonal elements of the Pade table are a PR pair.
Proof:
Since f (z )?1 = f (?z ), from the rst invariance property we have [n; m]f (z) =
1 ; [m; n]f (?z)
and the diagonal elements (m = n) of the Pade tables of f (z ) and f (?z ) are reciprocal of each other. Now if [n; n]f (z) = P (z )=Q(z ) then from the second invariance property [n; n]f (?z) = P (?z )=Q(?z ). Therefore
P (z) = Q(?z) ; Q(z) P (?z) and since P (z ) and Q(z ) are relatively prime, Q(z ) = P (?z ). As a result, the diagonal
4
elements of the Pade table of f (z ) are of the form [n; n]f (z) = HHn((?zz)) : n
and from (2) we have
Hn(z) ? Hn?1 (z) = z 2n?1 Hn (?z) Hn?1 (?z) Hn (?z)Hn?1 (?z) ; or
Hn (z)Hn?1(?z) ? Hn?1 (z)Hn (?z) = z 2n?1 ;
which means that fHn (z ); Hn?1(z )g are a PR pair.
End of Proof
As an example, let us consider f (z ) = g (z )=g (?z ) 1 where we take g (z ) to be the D4 Daubechies lter which is of length 8. Since g (z ) is of order 7, for n; m 7, all [n; m] element of the table are the same and equal to f (z ). Table 1 gives the lter coecients of H7 (z) = g(z) and its rst two approximations where the coecients are normalized so that the leading coecient is one. The approximations are found by taking the numerators of the consecutive diagonal elements of the table. It is straightforward to verify:
H7(z)H6(?z) ? H6(z)H7(?z) = 0:01667z 13 H6(z)H5(?z) ? H5(z)H6(?z) = 0:13348z 11
(5)
and hence both fH7 (z ); H6(z )g and fH6 (z ); H5(z )g are PR lter banks.
3 Continued Fraction Expansion and Lattice Structure The nth approximant of the continued fraction
is de ned as
a1 (z) z) a2 (z) = b (z) + ; b0 (z) + ba(1z()+ 0 b2(z) + 1 b1 (z) + b2a(2z()+z)
(6)
Nn(z) = b (z) + a1(z) a2 (z) an(z) : Dn (z) 0 b1(z)+ b2(z) + bn(z)
(7)
1 Note that f (z ) is a power series in z . In this paper, we denote the delay element by z instead of usual notation of z ?1 used in signal processing.
5
Therefore, it is found by simply removing the tail term an+1 (z )=(bn+1(z ) + ) from the original fraction expansion. It can be shown that both the numerators (Nn (z )) and the denominators (Dn (z )) satisfy the same recurrence equation
8 > > < Nn(z) = bn(z)Nn?1(z) + an (z)Nn?2(z) > > : Dn(z) = bn(z)Dn?1(z) + an(z)Dn?2(z)
n1
(8)
but with di erent initial conditions
8 < N0(z) = b0(z) N?1(z) = 1 : : D0(z) = 1 D?1(z) = 0
(9)
In the literature, the above recurrence equation is known as a three-term recurrence equation [6]. Also, note that the solution space of the above recurrence equation is a two-dimensional linear space and hence any other solution of the above recurrence equation can be found as a linear combinations of these two solutions. There is a close relationship between the Pade table and certain continued fraction expansions of the functions. Every path 2 through the table corresponds to a speci c continued fraction expansion of the function approximated by that Pade table. For example, a path consisting of the diagonal elements of the table corresponds to the consecutive approximants of the P -fraction expansion of the function, where P stands for the principleplus-constant terms. P -fractions have the following form
f (z) = b0(z) + b (z1)+ b (z)1+ 1 2
(10)
where bi (z ) are polynomials in z ?1 [7] [8]. Assuming that the table is normal, the [n; n] element of the table is the same as the nth approximant of the associated P -fraction and can therefore be written as: (11) [n; n] = b0 (z ) + b (z1)+ b (z )1+ b 1(z ) ; 1 2 n P a zi) and the for appropriate bi (z ). If f (z ) does not have principle part (f (z ) = 1 i=0 i corresponding Pade table is normal, then b0(z ) is a constant and bi(z ) (i > 0) is at most of By path we mean to start from upper left corner of the table, and at each step, to move to the right or the down or along the diagonal direction 2
6
order one. Therefore, (7) can be rewritten as: 2 2 [n; n] = a0 + B (zz )+ B (z )z + B z(z ) ; 1 2 n
(12)
where Bi (z ) = zbi (z ), i 1, is a rst order polynomial in z . The three-term recurrence formula for the above continued fraction can be written as
8 > > < H1(z) = B1(z)H0(z) + zH?1(z) > > : Hn(z) = Bn (z)Hn?1(z) + z2Hn?2(z)
n>1
(13)
where H?1 = 1; H0(z ) = a0 for the numerator and H?1 = 0; H0(z ) = 1 for the denominator. We now show that if f (z ) satis es the condition required by Theorem (1), namely f (z)f (?z) = 1, then Bn (z) satis es an additional constraint.
Lemma 1 If f (z) is regular, f (z)f (?z) = 1, and 2 2 f (z) = 1 + B (zz)+ B (z)z + B (zz) + ; 1 2 n
is the P -fraction expansion of f (z ), then 2 z2 ; f (z) = 1 + ? 1 z +z b + b +z b + n 1 2 2
(14)
where bn 2 C .
Proof:
This lemma states that if f (z ) is regular and f (z )f (?z ) = 1 then for n 2, Bn (z ) are constant. Let us de ne fe (z ) and fo (z ) as:
fe (z2) =
pf (z) + pf (?z) 2p
p
fo (z2) = f (z) ?2z f (?z) : Then it is straightforward to show that
s
(z ) = fe (z 2 ) + zfo (z 2) : f (z) = ff(? z) fe(z 2 ) ? zfo (z2)
7
Writing the P -fraction expansion of f (z ) we have 2
o (z ) f (z) = 1 + f (z22zf e ) ? zfo (z 2 ) z = 1+ (fe (z 2) ? zfo (z 2 ))=(2fo(z 2)) = 1 + ?1=2z + b + (f (z 2) z? 2b f (z 2))=(2f (z 2 )) ; 1 e 1o o
where b1 is chosen such that fe (z 2 ) ? 2b1fo (z 2 ) = O(z 2 ). Comparing the above with (12) we have fe (z2) ? 2b1fo (z 2) = z2 z2 z2 : 2fo (z 2) B2(z)+ B3(z) + Bn (z) + Now since the LHS is an even function, B2 (z ); B3(z ); are also even and since they are at most of the rst order, they can only have constant term, as required.
End of Proof
Using the above theorem, we can now rewrite the recurrence equation (13) to arrive at the following recursive relation.
8 > > < H1(z) = (b1 ? 1=2 z)H0(z) + zH?1(z) > > : Hn(z) = bnHn?1(z) + z2Hn?2 (z)
n > 1;
(15)
where H0(z ) = H?1 (z ) = 1. For example, Table 2 shows the value of bi 's for the Daubechies D4 lter. Note that since the above corresponds to the numerators of the diagonal elements of the Pade table of f (z ) with the property that f (z )f (?z ) = 1, for all n, fHn?1 (z ); Hn(z )g is a PR lter bank. In summary, the nth order lter Hn (z ) is found by shifting Hn?2 (z ) by 2 and then adding a proper multiple of Hn?1 (z ), as shown in Figure 2. The PR property is preserved structurally and the quantization of bi parameters does not e ect this property. In general, for large n, the successive lters found through the above procedure have similar frequency response. In most application, however, it is desired that each lter captures di erent frequency portion of the input signal. In the next section we demonstrate that all the possible causal complementary lters of order n can be simply characterized by two parameters. We also show that all the causal complementary lters can be implemented as a linear combination of the lower order lters of the chain fHn (z )g found using recurrence equation (15).
8
4 Parameterization of the Complementary Filter We rst show the following lemma:
Lemma 2 If H (z) and He (z) are of order n and n ? 1 respectively and are a PR pair, then any other lter of order n ? 1 which forms a PR pair with H (z ) is of the form He (z ) for some 2 C ? f0g. Proof:
Since H (z ) and He (z ) are PR pair,
H (z)He (?z) ? He (z)H (?z) = z 2n?1 : Note that k = n ? 1. This is true since the LHS is an odd function and it is not possible for the term corresponding to z 2n?1 to cancel out. Let us assume that there exists a di erent lter of order n ? 1, namely H (z ), such that
H (z)H (?z) ? H (z)H (?z) = z 2n?1 : De ne
G(z) = He (z) ? H (z):
It follows that H (z )G(?z ) ? H (?z )G(z ) = 0, which implies that G(z ) is at most of order n ? 2. A necessary condition for PR is that H (z) and H (?z) be relatively prime (they cannot have common zero) [13] so that the only solutions to the above equation are either G(z) = 0 and G(z) = H (z). The latter is not possible since G(z) is at most of order n ? 2, and thus G(z ) = 0 and H (z ) = He (z ).
End of Proof
The statement of the above lemma is also evident from the Euclid algorithm and the fact that the comprimeness of H (z ) and H (?z ) is a necessary condition for PR. The following two theorems are the direct and converse results that characterizes all the possible causal complementary lter of order n. In summary, Theorems (2) and (3) state that given Hn (z ) (Hn?1 (z ) is unique up to a scaling factor), the complementary lter of H (z ) of order n, He (z ), is uniquely characterized, independently of n, by three parameters k, and 1. The value of k corresponds to the delay between the reconstructed signal at the output of the synthesis lter bank and the input signal at the input of the analysis bank - overall system delay. Therefore, a small value of k corresponds to a low-delay subband decomposition system. Moreover, for a given value of k, the frequency selectivity of He (z ) depends only on the ratio = 1 and not their absolute values. Therefore, the process of nding a complementary lter 9
He (z) with a desired frequency response is a one-dimensional search through di erent values of = 1.
Theorem 2 Let Hn(z) and Hn?1 (z) of order n and n ? 1, respectively, be a PR pair. Then any He (z ), of order n, such that Hn (z ) and He (z ) are a PR pair with Hn (z)He (?z) ? He (z)Hn(?z) = z 2k+1 ; where k is an integer between 0 and n ? 1, can be expressed in the form
0n?k?1 1 X j+1 z ?2j A Hn (z) He (z) = z ?2(n?k?1) Hn?1 (z) + @ j =0
(16)
for some 1; ; n?k ; 2 C ? f0g.
Proof: De ne
G1 (z) = He (z) ? 1 Hn(z)
where 1 is chosen such that g1 (n) = 0. By direct substitution,
Hn(z)G1(?z) ? G1 (z)Hn(?z) = z 2k+1 so that G1(z ) and Hn (z ) form a PR pair. Now if k = n ? 1, g1(n ? 1) cannot be zero, so that by Lemma (2), G1 (z ) = Hn?1 (z ) for some 2 C ? f0g, and He (z ) = Hn?1 (z ) + 1Hn (z ) satisfying (16) and we are done. If k < n ? 1, then de ne
Gi(z) = z 2 Gi?1 (z) ? i Hn(z);
i = 2; 3; ; n ? k
(17)
where i is chosen such that gi(n) = 0. Now,
Hn (z)Gi(?z) ? Gi (z)Hn(?z) = z 2(k+i)?1 which implies that gi (n ? 1) = 0 for i < n ? k and gi(n ? 1) 6= 0 for i = n ? k. By Lemma (2), Gn?k (z ) = Hn?1 (z ), and by repeated substitution of (17) into itself, (16) results.
End of Proof
One has to note that even though negative powers of z appear on the RHS of (16), He (z ) is causal and does not have any negative power of z . We now prove the converse of the above theorem. 10
Theorem 3 Let Hn(z) and Hn?1 (z) of order n and n ? 1, respectively, be a PR pair. Then
any lter of the form
0n?k?1 1 X j+1 z ?2j A Hn (z) He (z) = z ?2(n?k?1) Hn?1 (z) + @ j =0
(18)
for 1; ; n?l ; 2 C ? f0g, k 2 f0; 1; ; n ? 1g forms a PR pair with Hn (z ). Moreover, if k; and 1 are speci ed, only one choice of 2; ; n?k yields a causal lter.
Proof:
By direct substitution
Hn(z)He (?z) ? He (z)Hn (?z) = z ?2(n?k?1) (Hn(z)Hn?1 (?z) ? Hn?1 (z)Hn(?z)) = z 2k+1 for some 2 C ? f0g: Therefore, H (z ) and He (z ) from a PR pair. It remains to show that He (z ) is of the form given by (18). By regrouping di erent terms of (18), we have
0n?k?2 1 X He (z) = (Hn?1 (z) + n?k Hn (z))z ?2(n?k?1) + @ j+1 z?2j A Hn (z); j =0
(19)
and using three-term recurrence equation, it is possible to choose n?k such that
0
1
n?k?2 eH (z) = 1Hn?2(z)z?2(n?k?2) + @ X j+1z?2j A Hn(z): j =0
We can again regroup di erent terms of the above to arrive at
0
1
n?k?3 eH (z) = ( 1Hn?2 (z) + n?k?1 Hn(z))z?2(n?k?2) + @ X j+1z?2j A Hn(z); j =0
(20)
(21)
and again it is possible to choose n?k?1 such that
0n?k?3 1 X j+1 z?2j A Hn (z): (22) He (z) = 2Hn?3 (z)z ?2(n?k?3) + 3Hn?4 (z)z ?2(n?k?4) + @ j =0
By iteratively applying the same procedure, the RHS will only contain positive powers of z and as a result, for this speci c choice of n?k ; n?k?1 ; ; 2, He (z ) is causal. Note that if He (z) is causal, we do not have any freedom in choosing these parameters.
End of Proof
11
As a consequence of the above theorems, a bi-orthogonal causal two-band perfect reconstruction lter bank where the lters are of order n can be parameterized by the following n +2 parameter set: fb1; b2; ; bn; = 1 2 C ? f0g ; k 2 f0; :::; n ? 1gg : (23) Note that this is a canonical (unique) parameterization and there is only one set of parameters corresponding to each lter bank. Moreover, it is a complete parameterization and all lter banks can be parameterized using the above. Also, the end-to-end system delay is given by 2k + 1. Figure 3 shows possible complementary lters to the Daubechies D4 lter. The value of k is xed at 3 which means that the end-to-end delay is 7 as in the case of an orthogonal lter bank. By varying the value of = 1 (the range ?85 to ?95 is shown in the Figure 3) di erent complementary lters can be found. When = 1 = ?90:5, the resultant lter is the power complementary lter of the D4 lter. Note that this search process is only a one-dimensional search which can be readily done. Also note that the the complementary lter is sensitive to the value of = 1 and deviation from ?90:5 can result in a lter which may not have the desired high-pass characteristic. Another example is given in Figure 4, where the possible complementary lters of a linear-phase lter H (z ), tabulated in Table 3, are shown. Again k is set to 3 which translates into an end-to-end delay of 7. The gure shows di erent complementary lters where the value of = 1 is varied between 60 and 100. The value of 87:79 results in the linear-phase lter He (z ) where its coecients are tabulated in Table 3.
4.1 Causal Implementation Implementation of the complementary lter as is proposed by (18) requires non-causal elements. But, since the overall lter is causal, one would expect that causal implementation should be possible. In fact, using similar procedure as was used in the proof of Theorem 3, we can show that He (z ) can be expressed as a linear combination of
fHn (z); Hn?k(z); ; Hn?2k (z)g or
He (z) = 1Hn(z) +
2k X j =k
aj Hn?j (z);
(24)
where aj 2 C for all j . In other words, the complementary lter can be implemented by tapping at the output of the adders of the lattice structure of Figure 2 and adding a proper linear combination of these outputs. We clarify this point using the following example. 12
Example:
8 < H (z) = 1 + z + z2 + z3 (25) : He (z) = 1 + z ? z2 ? z3; where 6= 1. Note that both H (z ) and He (z ) are linear phase lter and fH (z ); He (z )g is a Let
PR pair [13] since
H (z)He (?z) ? H (?z)He (z) = 4(1 ? 2 )z3:
(26)
Using the Pade approximation method, we nd the following family of the lters:
8 2 + z3 > > < H3(z) = 1 + z + z ?1+ 2 z 2 H ( z ) = 1 + z + 2 > > : H1(z) = 1 + z
(27)
Now by applying Theorem (2), we can write He (z ) in terms of H3 (z ) and H2(z ) as:
He (z) = ?2 z ?2 H2 (z) + ?1 + 2 z ?2 H3 (z):
(28)
Therefore, given H (z ), the following two parameter set uniquely speci es He (z ) up to a scaling factor fk = 1 ; = 1 = 2 g : The implementation of He (z ) using the above expression requires non-causal elements. Alternatively, using (24) one can express He (z ) in terms of H3 (z ) and H1(z ) as:
He (z) = 2H1(z) ? H3 (z);
(29)
which can be implemented using only causal elements. Figure 5 shows such a lattice implementation for both H (z ) and He (z ).
5 Convergence of Successive Approximants Similarly to the power series expansion of functions, the continued fraction expansion does not necessarily converge for all z 2 C and one should de ne the associated region of convergence. Clearly, one would want the unit circle to be within this region. In our development, we are primarily interested in the expansion of the function of the form f (z ) = g (z )=g (?z ) where it is assumed that g (z ) and g (?z ) do not have any common zero or are relatively prime. Note that if g (z ) is a polynomial and therefore is of nite order, the convergence of f (z ) is always guaranteed since its continued fraction expansion terminates in a nite 13
number of steps. We now consider the case where g (z ) is not necessarily of nite order. We still however assume that g (z ) and g (?z ) are relatively prime. We nd a set of sucient conditions which guarantees that the P -fraction expansion of f (z ) converges to f (z ) for all z 2 C . Note that since g (z ) and g (?z ) are relatively coprime, this also implies that, up to a scaling factor, the numerators of the successive approximants also approach g (z ). First, we brie y present an important class of continued fractions (C -fractions) and investigate some of its properties which are directly relevant to our mathematical development.
5.1 C -fraction Expansion C -fraction expansion of a function has the following form: 1 (z ) a2(z ) a3 (z ) ; a0 (z) + a1+ 1+ 1 +
(30)
where ai (z ) are monomial polynomials. If f (z ) is a regular function then ai (z ) are of rst order (ai(z ) = i z for all i 1) and a0(z ) = 0 is a constant. Therefore, We can rewrite the above as: 1 z 2 z 3 z : (31) 0 + 1+ 1+ 1 +
An interesting property of the C -fraction expansion is that they correspond to a speci c path in the Pade table as shown in Figure 1. Note that the even approximants of a C -fraction expansion are the same as the approximants of the corresponding P -fraction expansion. The following lemma gives the consequence of the condition f (z )f (?z ) = 1 on the parameters of the C -fraction expansion of f (z ).
Lemma 3 If f (z) has C -fraction expansion of the form (31), then f (z)f (?z) = 1 if and only if in the C-fraction expansion of f (z )
2 = ? 1 1 2 2n?1 = ? 2n
n = 2; 3;
(32)
Proof:
Let us write the continued fraction expansion which corresponds to the even approximants of the C -fractions expansion. It is straightforward to show that it is given by: 2 2 0 + 1 + 1zz? 1 + ( 2 +3 z )z? 1 + ( +4 5 z )z ? : 2 3 4 5 6
(33)
Since the even approximants of the C -fraction expansion of a function are successive ap14
proximants of the P -fraction expansion of the same function, the proof can be completed by comparing the above to (14).
End of Proof
As a corollary of the above lemma, if n ! then = 0. This is the true since n alternates between positive and negative values and as a result if f n g is a convergent sequence, it has to converge to zero.
5.2 Sucient Conditions The following theorem, which we state from [2] without outlining its proof, gives sucient conditions for C -fraction expansion of the function given by (31) to be convergent.
Theorem 4 If n ! as n ! 1, then (31) converges to a function f (z) which is meromorphic 3 in the cut z -plane. The cut is placed in the shadow of ?(1=4 ) from the
origin. The convergence is uniform on any compact set not containing the poles of the limit function, and the limit function has the continued fraction expansion (31).
Clearly, if = 0, then the C -fraction expansion of f (z ) given by (31) converges to f (z ) for all z 2 C . Note that this is exactly the case when f (z )f (?z ) = 1. In summary, if the C -fraction expansion of f (z ), where f (z )f (?z ) = 1, converges, it is convergent for all z 2 C . Therefore, a sucient condition for this convergence is for n to approach to zero as n ! 1. We now show how this sucient condition translates into condition on the parameter of the lter decomposition found in the previous section, namely bi in (15). Comparing (14) and (33) we nd the following relationship between bi and i
8 > > < b1 = 1= 1 bn = ?2= 2n n even > > : bn = ?1=(2 2n) n odd
(34)
Clearly, n ! 0 if and only if bn ! 1. Therefore, starting from the recursive relation (15) and choosing bn such that bn ! 1, we are guaranteed that limn!1 Hn (z )=Hn (?z ) exists and is equal to f (z ) = g (z )=g (?z ) where (35) nlim !1 Hn (z ) = Kg (z ) for some scalar K . 3 A function is said to be meromorphic in a region if it is analytic in that region except at a nite number of poles [15].
15
6 Conclusions
We have investigated relationships among Pade table, continued fraction expansion and two-channel perfect reconstruction lter banks. Through characterization of all two-channel bi-orthogonal PR lter banks, we developed a new lattice structure for this class of lter banks. The parameter space found using this method is unique for each lter bank. Also, it is a complete parameterization and it includes all the possible two-band bi-orthogonal lter banks. Similarly to any other lattice structure, the PR property is achieved structurally and quantization of the parameters of the lattice does not e ect this property. We showed that for a given lter, the set of all complementary lters can be uniquely speci ed by two parameters, namely the end-to-end delay of the system and a scalar quantity. Finally, we developed a sucient condition for the convergence of the successive lters found by the proposed lattice structure.
16
References [1] G.A. Baker, Jr., Essentials of Pade Approximants, Academic Press, 1975. [2] G.A. Baker, Jr., P. Graves-Morris and P.A. Carruthers, Pade Approximants, Part I: Basic Theory, Encyclopedia of mathematics and its applications, Vol. 13, Addison Wesley, 1981. [3] T. Kailath, Linear Systems, Printice-Hall, Englewood Cli s, N. J., 1980. [4] A.C.M. Kalker and I.Shah, \On ladder structures and linear phase condition for multidimensional bi-orthogonal Filter Banks", To appear in IEEE Trans. on Signal Processing, 1995. [5] M. Khansari and A. Leon-Garcia, \Subband decomposition of signals with generalized sampling," Special issue of IEEE Trans. on Signal Processing on Wavelets and Signal Processing, Vol. 41, No. 12, pp. 3365-76, Dec. 1993. [6] L. Lorentzen and H. Waadeland, Continued Fractions with Applications, Studies in computational mathematics 3, North-Holland, 1992. [7] A. Magnus, \ Certain continued fractions associated with the Pade table ", Math. Zeitschr. 78, 1962, pp. 361-374. [8] A. Magnus, \ Expansion of power series into P -fractions ", Math. Zeitschr. 80, 1962, pp. 209-216. [9] S. Mallat, \ A theory of multi-resolution signal decomposition: the wavelet representation ", IEEE Trans. on PAMI, Vol. 11, No. 7, pp. 710-732, July 1989. [10] P.P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993. [11] P.P. Vaidyanathan, \ Multirate digital lters, lter banks, polyphase networks, and applications: A tutorial ", Proceeding of IEEE, pp. 56-93, Jan. 1990. [12] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice-Hall, 1995. [13] M. Vetterli and D. J. LeGall, \ Perfect reconstruction FIR lter banks: Some properties and factorizations ", IEEE Trans. on ASSP, Vol. ASSP-37, No. 7, pp. 1057-1071, July 1989. [14] M. Vetterli and C. Herley, \ Wavelets and lter banks: theory and design ", IEEE Trans. on Signal Processing, No. 9, pp. 2207-2232, Sept. 1992. 17
[15] E.C. Titchmarsh, The theory of Functions, Oxford University Press, Second edition, 1986.
18
0 1 2 3 4 5 6 7
H7(z)
1.0000000000 3.1029314858 2.7384614815 -0.1214690295 -0.8118612163 0.1338739557 0.1427351497 -0.0400009711
H6(z)
1.0000000000 3.1029314858 2.8819998520 0.3239206941 -0.3563540087 0.3101626433 0.2082479716
H5(z)
1.0000000000 3.1029314858 3.1734177538 1.2281704828 0.4564133099 0.3204724809
Table 1: The coecients of D4 lter and its rst two approximants
b1 b2 b3 b4 b5 b6 b7
0.1611379440 2.2342472102 1.6658767160 -2.3332622209 -1.1149202725 3.0777960077 -2.2635601304
Table 2: The value of bi (1 i 7) for D4 lter
0 1 2 3 4 5 6 7
H (z)
-0.0506224053 -0.0481274616 0.1781173490 0.4206325181 0.4206325181 0.1781173490 -0.0481274616 -0.0506224053
He (z)
0.1014472948 0.0964474279 -0.174342018 -0.669342151 0.669342151 0.174342018 -0.0964474279 -0.1014472948
Table 3: The coecients of a bi-orthogonal linear-phase two-band PR lter bank. Note that the low-pass lter is symmetric whereas the high-pass lter is anti-symmetric.
19
0
1
2
3
4
0 1 2 3 4
P-Fraction Expansion C-Fraction Expansion
Figure 1: Paths corresponding to di erent continued fraction expansions of the same function
20
z
z 1/2
H (z)
b 1
1
H (z)
b2
2
z
z2
bn n-1
3
2
z2
H (z)
H (z)
b3
b n+2
b n+1
Hn (z)
H
n+1
(z)
z2
Figure 2: Lattice structure to implement the successive lters Hn (z )
21
-4 -6 -8 -10 -12 -14 -16 -18 -20 -22 -24 3
-94 2.5
-92
2 1.5 frequency
1
-88 0.5 0
-90 delta/beta_1
-86
Figure 3: Magnitude frequency response of He (z ) (dB) for D4 lter for fk = 3; ?95 = 1 ?85g
22
2 0 -2 -4 -6 -8 -10 -12 -14 -16 -18 100
3 2.5
90 2 80 delta/beta_1
frequency 1.5 1
70 0.5 60
Figure 4: Magnitude frequency response of He (z ) (dB) for linear-pahse ler given in Table 3 for fk = 3; 60 = 1 100g
23
z
z 1/2 b1
b3
b2
H1 (z)
H (z)
H (z)
2
z
3
2
~
H (z)
Figure 5: Lattice structure for H (z ) and He (z )
24
