RATIONAL CANONICAL FORM OF POLYPHASE MATRICES WITH APPLICATIONS TO DESIGNING PARAUNITARY FILTER BANKS. Peter Vouras, Trac Tran, Michael Ching

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1 RATIONAL ANONIAL ORM O POLYPASE MATRIES WIT APPLIATIONS TO ESIGNING PARAUNITARY ILTER BANKS Peter Vouras Trac Tran Michael hing Naval Research Laboratory Johns opkins University University of Georgia ABSTRAT In this paper we consider the rational canonical form of arbitrary polyphase matrices and use it to derive a simple implementation of paraunitary filter banks (PUBs) based on a cascade of elementary building blocks. urthermore this decomposition is shown to be easily extendable to include a large class of perfect reconstruction filter banks (PRBs) and can be especially useful for deriving the initial condition of PUB design algorithms. Index Terms rational canonical form polyphase matrices paraunitary filter banks. INTROUTION The polyphase matrix of an M-channel causal finite impulse response (IR) filter bank may be written as K () = k = k k where = re j and k is a scalar M-by-M matrix. The matrix () resides within the ring of M-by-M matrices with Laurent polynomial entries denoted by M(N [ - ]) with the field of complex numbers. The rational canonical form of this matrix is not guaranteed to exist within the Laurent polynomial ring [ - ] since this ring is not a field. In this paper we will show that the rational canonical form of () does indeed exist as another matrix in the ring M(N [ - ]). If () corresponds to the polyphase matrix of a paraunitary filter bank (PUB) then the algorithm for computing the rational canonical form illustrated in this paper may be used to implement () as a particularly simple product of elementary building blocks. urthermore this new decomposition of () may be used to represent a large class of perfect reconstruction filter banks (PRBs) as well and can be used to provide initial conditions for PUB design algorithms.. EXISTENE O RATIONAL ANONIAL ORM The rational canonical form of a polyphase matrix () over the ring [ - ] exists as a matrix over the field K. K denotes the field of rational functions p()/q() q() with coefficients in. The field K is the field of fractions of the ring [ - ]. This field is also denoted (). Any matrix () in M(N [ - ]) is similar to a matrix in M(N K) in rational canonical form. In other words there exists an invertible M-by-M matrix P() such that P() - ()P() is in rational canonical form. A matrix R() in rational canonical form is a matrix such as R M where q() q p p i M( N K). pn qn() The matrix i () is the companion matrix corresponding to the invariant factor a i (x) of (). The invariant factors a i (x) may be made unique by requiring that p i () and q i () are monic polynomials. The characteristic polynomial of () is a polynomial in the indeterminant x with coefficients in the Laurent polynomial ring [ - ] namely c(x) = det[xi ()]. The invariant factors of R() are the monic factors of c(x) []. The invariant factors of () and R() are the same namely the monic polynomials a i (x). The important contribution of this section is the following claim. laim: The rational canonical form of the polyphase matrix () is a matrix with entries in the ring [ - ]. In other words it is an element in the ring M(N [ - ]). Proof: Since is a field the ring [ - ] is a unique factoriation domain (U). The field K = () is the field of fractions of [ - ]. The characteristic polynomial c(x) is in the ring [ - ][x]. By Gauss Lemma since c(x) can be factored in ()[x] then it is reducible in [ - ][x]. onsequently the invariant factors of a matrix () in M(N [ - ]) are polynomials in the ring [ - ] and therefore the rational canonical form R() is a matrix in the ring M(N [ - ]). 3. AN ALTERNATIVE ORM OR R(Z) laim: The rational canonical form R() of a polyphase matrix () can be written as a Laurent polynomial with matrix coefficients. In other words R () N = R + R + R + + RN. Proof: The claim follows once one proves that the ring of matrices with Laurent polynomial entries is isomorphic to the ring of Laurent polynomials with matrix coefficients. It is sufficient to prove that [] M ( N [] x ) M( N )[] x which implies that M ( N [ ]) M( N )[ ]. The proof is omitted for brevity. M /9/$5. 9 IEEE 3 IASSP 9

2 Report ocumentation Page orm Approved OMB No Public reporting burden for the collection of information is estimated to average hour per response including the time for reviewing instructions searching existing data sources gathering and maintaining the data needed and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information including suggestions for reducing this burden to Washington eadquarters Services irectorate for Information Operations and Reports 5 Jefferson avis ighway Suite 4 Arlington VA -43. Respondents should be aware that notwithstanding any other provision of law no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.. REPORT ATE MAY. REPORT TYPE N/A 3. ATES OVERE - 4. TITLE AN SUBTITLE Rational anonical orm Of Polyphase Matrices With Applications To esigning Paraunitary ilter Banks 5a. ONTRAT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTOR(S) 5d. PROJET NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERORMING ORGANIZATION NAME(S) AN ARESS(ES) Naval Research Laboratory 8. PERORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENY NAME(S) AN ARESS(ES). SPONSOR/MONITOR S ARONYM(S). ISTRIBUTION/AVAILABILITY STATEMENT Approved for public release distribution unlimited. SPONSOR/MONITOR S REPORT NUMBER(S) 3. SUPPLEMENTARY NOTES See also AM34. IEEE International onference on Acoustics Speech and Signal Processing (34th) held in Taipei Taiwan on 9-4 April 9. U.S. Government or ederal Purpose Rights License 4. ABSTRAT In this paper we consider the rational canonical form of arbitrary polyphase matrices and use it to derive a simple implementation of paraunitary filter banks (PUBs) based on a cascade of elementary building blocks. urthermore this decomposition is shown to be easily extendable to include a large class of perfect reconstruction filter banks (PRBs) and can be especially useful for deriving the initial condition of PUB design algorithms. 5. SUBJET TERMS 6. SEURITY LASSIIATION O: 7. LIMITATION O ABSTRAT SAR a. REPORT b. ABSTRAT c. TIS PAGE 8. NUMBER O PAGES 4 9a. NAME O RESPONSIBLE PERSON Standard orm 98 (Rev. 8-98) Prescribed by ANSI Std Z39-8

3 4. OMPUTATION To compute the rational canonical form R() of the M-by-M matrix () one approach is to diagonalie the matrix xi () []. The invariant factors of () will then appear on the diagonal. The invariant factors should be monic and satisfy the divisibility condition a i (x) divides a i+ (x). The direct sum of the companion matrices associated with each invariant factor yields the matrix R() in rational canonical form. By keeping track of the row operations used to diagonalie xi () one can also construct the matrix P() such that P() - ()P() = R(). The following three elementary row and column operations can be used to diagonalie xi ():. interchange two rows or columns. add a multiple in K[x] of one row or column to another e.g. add p(x) times the jth row to the ith row 3. multiply any row or column by a unit in K[x] i.e. by a nonero element in K. The matrix P() can also be computed systematically []. irst let d d m denote the degrees of the monic nonconstant polynomials a (x) a m (x) appearing on the diagonal. Begin with the matrix S = I the identity matrix. or each row operation used to diagonalie xi () change the matrix S as follows.. If the ith and jth rows were interchanged then interchange the ith and jth columns of S. if Row i + p(x)row j Row i then subtract the product of the matrix p(()) times the ith column of S from the jth column of S i.e. ol j p(())ol i ol j 3. if the ith row is multiplied by a unit u then divide the ith column of S by u. Once the matrix xi () has been diagonalied the first M m columns of S will be ero. Then for each i = m multiply the ith nonero column of S successively by () = I () () () di- where d i is the degree of a i (x). Use the resulting column vectors in this order as the next d i columns of a matrix P(). Then P() - ()P() = R(). 5. APPLIATION TO PUBS In this section we consider the example of paraunitary filter banks. The analysis (or synthesis) polyphase matrix of a normalied PUB satisfies the condition jω = I for = e. urthermore M-channel PUBs may be decomposed into a product of elementary building blocks as in [3] ( ω) = GL( ω) G( ω)qj () where L is the Smith-McMillan degree of () G k and Q are M-by-M orthogonal matrices () = diag(i e -j I) for M even and J = diag(± ±). Each matrix G k and Q can be written as the product of ½M(M ) Givens rotation matrices in the sequence Gk = { BM M } { B M B}{ B M B}. () The matrix B ij corresponds to a Givens rotation matrix with cos( n ) in the ith row and ith column sin( n ) in position (i j) -sin( n ) in position (j i) and cos( n ) in position (j j) with n ½M(M ). 5.. ase L = M = or a simple example let M = and L = in () above. Without loss of generality set QJ = I. Then the polyphase matrix is cos sin = G() = ( ) ( ). (3) sin θ cos θ Table lists the steps to compute the rational canonical form of () and the change of basis matrix P(). In this table R i denotes the ith row and j denotes the jth column. Table. Sample Rational anonical orm omputations Steps to compute R() Steps to compute P() Step : orm the matrix Step : orm the matrix xi () S = I Step : sin()r R Step : -(/sin()) (x cos())r R (() cos()i) + Step 3: -R + R R Step 3: sin() Step 4: (/sin())r R Step 4: (/sin()) - Step 5: - sin()r R Step 5: (() (x xcos()(+ - ) + - )R cos()()( + - ) + - I) R Step 6: -R R Step 6: (() cos()i) sin() - Step 7: R + R R Step 7: (() cos()()( + Step 8: (/(x cos())r R (/( - sin())r R Step 9: (/(x xcos()( + - ) + - )R R The result of these computations is R() = = cos θ + - ) + - I) Step 8: Now ( ) S = sin θ Also m = d =. Step 9: orm the matrix P = I () [ () ] ( )( ) cos cos sin cos + P = sin() θ (4) cot P = csc. csc urthermore P() - ()P() = R(). Using the rational canonical form of () one can write () = () + - () where () and () are rank one matrices defined as = cos cos ( ) ( ) θ cot θ cos cos [( ) tan] () ( ) [ csc ]. = Now for M = the decomposition of PUBs given in () can be rewritten as L = ( i + ). i= 6. SIMPLE IMPLEMENTATION O PUBS Using the decomposition P i ()R i ()P i () - = G i () for each building block of a -by- polyphase matrix in () a two channel PUB may be constructed as a cascade of the lattice sections shown in ig.. ere v v are the input signals and u u are the output signals. This lattice structure lends itself easily to an implementation in hardware. i 3

4 using two parameters for every building block includes the set of PUBs as a proper subset. ig.. Lattice Implementation of PUB Similar lattice structures can be derived for PUBs with greater than M = channels using () the rational canonical form for (3) and the rational canonical form for a scalar Givens rotation matrix. Using the same computational algorithm described in Section 4 a scalar Givens rotation matrix can be written in rational canonical form with PRP - = G as cos sin sin R = ( ) G = cos sin( ) cos( ) P = cos( ) θ θ θ θ (5) cot P =. csc( ) θ 6.. ase M = 3 Using () a 3-by-3 paraunitary polyphase matrix may be written as = B( θ) B( θ ) B( θ3) () with () = diag( - - ). A lattice implementation can be constructed from the following matrices derived using (4) and (5) B B B ( θ) = sin( θ) cos cos sin( θ ) ( θ) = cos( θ ) cos( θ ) sin( θ3) ( ) = cos θ csc cot( θ) ( θ) cot csc ( ) cos( + ) cot csc 3 ( θ3) Similar matrices and lattice structures may be derived using () (4) and (5) for paraunitary polyphase matrices with M = 4 or larger. Those examples will not be presented here to conserve space. 7. PERET REONSTRUTION ILTER BANKS onsider the case M = again. A perfect reconstruction filter bank satisfies the property that ()E() = c -m I where () is the analysis polyphase matrix and E() is the synthesis polyphase matrix c and m is an integer. A PUB is the special case where E() = (). A perfect reconstruction filter bank may be constructed by parameteriing the rational canonical form R() and the conjugation matrix P() independently using one angle for each. or example the system with sin( θ) cos( θ) sin( θ) = csc( θ) cos( θ) cos( θ ) sin ( θ) cos cos sin E = csc cos (6) is PR since E() = () -. Note that for the special case when = () is paraunitary. Therefore this set of PRBs constructed 8. APPLIATION TO PUB ESIGN Since the space of PRBs is larger than and includes the space of PUBs one can design a PUB by searching over the space of PRBs for an optimal solution and then choosing the paraunitary filter bank closest to it for an initial condition in a strictly PUB search. The advantage of this approach is that if the initial condition of a strictly PUB search is closer to the optimal solution then the more likely it is that a nonlinear optimiation program will settle on the global solution instead of a local minimum. onsider the following PUB design algorithm which minimies the Mean Square Error (MSE) between a desired polyphase matrix () and an approximation (). 8.. PUB esign Algorithm Step. orm the MSE objective function MSE ( ( ω) P( ω) ) = ζ = W ( ω) ( ω) P( ω) where. denotes the robenius norm W() is a scalar weighting function set equal to one P() is an M-by-M perfect reconstruction polyphase matrix as described in Section 7 and () is the desired or ideal polyphase matrix. Step. With the angles i as free parameters minimie over all P() using a nonlinear optimiation program such as the Broyden- letcher-goldfarb-shanno (BGS) algorithm. all the optimal solution P(). Step 3. ompute the paraunitary polyphase matrix '() parameteried as in () and () which minimies MSE(P() ()) over all (). Step 4. Use '() as the initial starting point in a nonlinear optimiation program which minimies MSE(() ()) over all (). all this optimal solution (). enote MSE(() ()) by MSE. The advantage of using the proposed PUB design algorithm is that it satisfies the following property. laim: efine MSE = MSE(() '()). Using the above filter bank design algorithm MSE MSE + ε MSE( ( ω) ( ω) ) MSE + ε where is a small nonnegative constant. Proof: Since the MSE is a norm on the space of polyphase matrices by the triangle inequality P ( ω) ( ω) P ( ω) ( ω). + Since P() is the optimal solution over a larger set of matrices than () As a result ( ω) P ( ω) ( ω) ( ω). 33

5 ( ω) ( ω) ( ω) P ( ω) or in other words MSE MSE +. The proof that MSE('() ()) MSE + is similar ase M = 4 or M = 4 a PR polyphase matrix P() which can be used to design a PUB using the previously described algorithm is constructed as P = G( θ θ ) () = T3( θ θ ) T3( θ3 θ4 ) T( θ5 θ6 ) T3( θ7 θ8) T( θ9 θ ) T( θ θ ) ( ). The inverse of P() is given by P = B θ θ B θ9 θ B3 θ7 θ8 B θ5 θ6 B3 θ3 θ4 B3 θ θ with the matrices B ij as in (5) and () except with two angle parameters as in (6). The system so described is PR since P()P() - = I. The matrix T 3 is () () ( ) ( ) ( ) ( ) ( ) ( ) T 3 =. sin cos( θ ) cot( θ) cos( θ) csc( θ) The matrices T 3 T T 3 T T are constructed similarly using two angle parameters for each. Note that if i = i+ for i odd then P() is paraunitary. 9. RESULTS The PUB design algorithm was used to design a finite impulse response PU approximation to an ideal principal component filter bank (PB). PBs are described in detail in [4] [5] and [6]. A PB is the solution to the problem of finding optimal Q-by-M and M-by-Q analysis and synthesis polyphase matrices with Q < M such that the time-averaged mean squared error between the vector input to the filter bank and the vector output is minimied. An ideal PB corresponding to an infinite order PU filter bank has channel filters with brick-wall responses. This ideal filter bank can be approximated using a IR PU filter bank by minimiing the mean squared robenius norm error between the desired polyphase matrix of the ideal PB () and the IR PU synthesis polyphase matrix (). PBs are an interesting design example because they are optimal for maximiing coding gain and minimiing mean-squared error in the presence of quantiation noise. It has been proven that they are also optimal for any concave function of the subband variance vector [6]. Table lists the performance of the PUB design algorithm and compares it to another elegant design algorithm described by Tkacenko in [4]. Tkacenko generously made available the MATLAB code to duplicate his results on the Internet. The PR polyphase matrix was populated with incrementally more twoparameter building blocks. As the table shows the MSE of the optimal PR solution decreased monotonically with greater degrees of freedom until the number of free parameters was too great for the algorithm to converge. A PU approximation to the optimal PR solution with the largest number of free parameters was then used as the initial condition for a search over PU matrices to arrive at the final filter bank solution. The length of the channel filters in the final PUB was eight taps. Table. Performance of PUB esign Algorithm Algorithm MSE(() ()) Tkacenko PU solution.74 PU search with random start value.5747 PR matrix with 4 angles.5578 PR matrix with 6 angles.5549 PR matrix with 8 angles.5539 PR matrix with angles.545 PR matrix with angles.5395 PR matrix with 4 angles (algorithm did not converge) PU search with PU initial condition.574 As Table shows the lowest MSE for a PUB was attained with the PU initial condition. igure illustrates the frequency response of one of the optimal channel filters (red) and overlays the ideal filter response (blue dotted line) with the Tkacenko solution (black). ig. hannel ilter One As the curves show the derived filter is an excellent approximation to the ideal brick-wall filter with high frequency selectivity in the narrow bandpass regions.. ONLUSIONS In this paper we showed that the rational canonical form of a polyphase matrix exists in the ring M(N [ - ]) and can be written as a matrix in the ring M(N )[ - ]. The factoriation of paraunitary polyphase matrices into a similarity transformation of the rational canonical form makes possible a very simple lattice implementation of the filter bank. Also this decomposition lends itself easily to a representation of a large class of perfect reconstruction filter banks and can be used to find an initial condition for PUB design algorithms that ultimately yields better MSE performance.. REERENES []. S. ummit and R. M. oote Abstract Algebra John Wiley and Sons oboken NJ 4. []. E. Rose Linear Algebra: A Pure Mathematical Approach Birkhauser Basel Switerland. [3] P. P. Vaidyanathan Multirate Systems and ilter Banks Prentice all NJ 993. [4] A. Tkacenko and P. P. Vaidyanathan Iterative Greedy Algorithm for Solving the IR Paraunitary Approximation Problem IEEE Transactions on Signal Processing pp Jan. 6. [5] M. K. Tsatsanis and G. B. Giannakis Principal omponent ilter Banks for Optimal Multiresolution Analysis IEEE Transactions on Signal Processing pp Aug [6] S. Akkarakaran and P. P. Vaidyanathan ilterbank Optimiation with onvex Objectives and the Optimality of Principal omponent orms IEEE Transactions on Signal Processing pp. 4 Jan.. 34