A Design of Paraunitary Polyphase Matrices of Rational Filter Banks Based on Shift-Invariant Systems
In this paper we present a method to design paraunitary polyphase matrices of critically sampled rational filter banks. The method is based on shift-invariant systems, and so any kind of rational splitting of the frequency spectrum can be achieved using this method. Ideal shift-invariant system with smallest and that map of a band of input spectrum to the output spectrum are obtained. A new set of filters is obtained that characterize a shift-invariant system. Ideal frequency spectrum of these filters are obtained using ideal shift-invariant systems. Actual paraunitary polyphase matrices are then obtained by minimizing the stopband energies of these filters against the parameters of the paraunitary polyphase matrices.
Keywords : Rational filter bank, filter design, optimization, paraunitary matrix.
Rational Filter Banks (RFB) are now well known tools for nonuniform subband decomposition of the signal. Given rational numbers , where , the task of an channel RFB is to split the frequency spectrum of the signal into nonuniform bands, where -th band is given by .
An RFB uses a sample rate changer, shown in Fig.(1) in each channel.
Such RFB are treated in detail in  and it is shown that all kind of rational band splitting can not be achieved by the RFB based on the above sample rate changer, even if the ideal real coefficients filters are used.
If and are coprime, the structure shown in Fig.(1) can be converted into a structure shown in Fig.(2). In this figure is a polyphase matrix that consists of polyphase components of . It is shown in  that this structure is more general than the previous structure, and if and are allowed to have common factors, then any kind of rational splitting of the input frequency spectrum can be achieved.
Such structures are called shift-invariant systems, since shifting the input by samples results in shifting the output by samples. Many properties of shift-invariant systems are studied in . It is shown that such structures can be represented by a LTI polyphase matrix . In other words, let input and output be written in polyphase form as and respectively, and let
Let represent the frequency spectrum of the signal between frequencies . The shift-invariant systems described above can be used to obtain the ideal mapping of an input subband on the output band or . Let
then it can be shown that
where , and will be given by
is called alias component matrix, or modulation matrix. It can be seen that (5) gives a relationship between input subbands and output subbands. By proper choice of the dimensions of the modulation matrix and by proper selection of the elements of from any required mapping can be obtained. It is shown in  that mapping can be obtained by using a shift-invariant system, or a modulation matrix, where and . A modulation matrix consisting of elements from will be referred as Ideal Modulation Matrix (IMM).
In this paper we consider the problem of obtaining the paraunitary polyphase matrices of the filter banks that can be realized by shift-invariant systems. In the next subsection we define the problem more precisely and highlight the contributions of the paper.
I-B Problem definition and contributions of the paper
Let us say that we are given any rational numbers . It is required to split the input frequency band into bands, such that -th channel gives the band . Our objective is to design a paraunitary polyphase matrix of minimum dimensions that achieves this splitting. In order to achieve this objective, we address the following issues:
If an input subband is to be mapped to output band , or , what would be the smallest dimensions of the ideal modulation matrix ? We show in section II that though the above mapping can be obtained by a ideal modulation matrix, where and , as given in , these are not the smallest dimensions in all the cases. In the same section we obtain smallest dimension ideal modulation modulation matrices for a given mapping.
We prove that a shift-invariant system can be characterized by filters. This result as such is not new. Many structures that realize shift-invariant systems using filters are given in . However we could not find a method to use these structures to design an RFB based on these structures. In section III we obtain a set of filters that is different from the filters given in . We show that using the ideal modulation matrices, we can obtain ideal frequency spectrums of these filters. These ideal frequency spectrums are then used to form an objective function. By minimizing this objective function we obtain the required paraunitary polyphase matrix.
The notations that we use in this paper are as follows. The frequency spectrum of a signal between and will be denoted by . All vectors will be represented by boldface letters and matrices will be represented by upper boldface letters. For two integers and , modulo is denoted by . For a matrix , the -th element is denoted by . A set of integers from to will be denoted by . For a polynomial matrix , , where the subscript indicates taking complex conjugate of polynomial coefficients. denote a frequency spectrum that is for and zero elsewhere.
Ii Structure of the Ideal Modulation Matrices
We recall that the modulation domain equation is given by (5). Putting in this equation, we get
It can be noted that for , the elements of cover disjoint segments of and the union of these segments is equal to . Consider a subset of . For this subset we define an ideal modulation matrix to be a matrix that maps some segments of on the respective segments of according to given desired mapping when . We say that a mapping is possible if we can obtain disjoint subsets of such that their union is and for each subset, we can obtain an ideal modulation matrix.
Example : Consider the following mapping
For this mapping we have and are given by
It can be seen that where and where together give the required mapping.
We now consider the following problem - given two rational numbers and , where and , find out the the smallest modulation matrices that achieve one of the following mappings:
Let , and . Let , , , . The following lemma gives the smallest ideal modulation matrices.
The ideal modulation matrices will have dimensions if and only if either of the or is even. If both and are odd, then the ideal modulation matrices will be of dimensions . Moreover the ideal modulation matrices are given as follows-
is even: In this case we will have two modulation matrices and . These matrices are given as follows - for , if is given by
else . is obtained from as follows
is even: In this case also we will have two modulation matrices and . These matrices are given as follows -
for , if is given by
else . is obtained by using (13).
Proof: We first prove that the number of columns in the modulation matrix must be a multiple of . Let the dimension of the modulation matrix be and . The modulation equation will then be . We first prove that needs to be a multiple of , i.e. . The -th term in and -th term in will be and respectively. The term covers the input spectrum for . If , then this input spectrum maps to the output spectrum . Consider now the mapping given in (10). In order to map the input frequency to the output frequency , we require , and such that
Substituting from (15) into (16), we obtain . Since , this condition will be fulfilled only if is a multiple of . By a similar argument it can be shown that needs to be a multiple of in order to map the input frequency to the output frequency . Thus needs to be a multiple of both and , or in other words, needs to be a multiple of , i.e. .
If , then the condition reduces to . Thus if , and the mapping is given by (10) then we require to be even. By a similar argument, it can be shown that if , and the mapping is given by (11) then we require to be even. If both and are odd, then it is clear from this condition that is required to be equal to .
We now obtain when is even. Consider the frequency in . For , this frequency will be in . The frequency in that should map to this frequency is
Let be defined as
Then if , it would map to .
For , the frequency in will be equal to . The corresponding frequency is given by
which gives as
It can be seen that and together achieve the mapping given in (10).
The structure of the modulation matrices for rest of the two cases can be obtained in the similar manner.
Remark: Two results close to this lemma appear in , proposition 3.2, and in , theorem 1. However the result of this lemma is more general than the above two results and also unifies these results. Proposition 3.2 of  considers only the case when is even and . Theorem 1 of  always obtains ideal modulation matrices of dimensions , that are not the smallest ideal modulation matrices if or is even.
Example 2: Consider the following mapping
Since both and are odd, the ideal modulation matrix dimensions will be , and the ideal modulation matrices can be given either from (12) or from (14). Considering (14) for ideal modulation matrices, we obtain
If is even (i.e. both and are even, or both and are odd) then both the kinds of mappings given in (10) and (11) are possible. We now obtain the relationship of ideal modulation matrices for one mapping with those of the other mapping. Let the dimensions of ideal modulation matrices be . Then the modulation equation will be
where . Let be the permutation matrix that rotates upward by . Thus if then
Proof: Consider the modulation domain equation
changing to , we obtain
In the following section, we assume without loss of generality that either or is even, since if both of them are odd then we can double the dimensions of the shift-invariant system as shown in the above lemma.
Iii Filters Characterizing the Modulation Matrix
We recall from (6) that the modulation matrix is given by
Let , and . Then . Let be a pair of integers such that . An element of is given by
We now have the following lemma.
The modulation matrix can be characterized by filters. In perticular we assert the following
For a given integer there are pairs such that , where and .
If and are two pairs of indices such that then
for some integer such that .
Proof: In order to prove the first part, for a given we count all the that give . Such are given by . Thus for a given we have values of that give . Since takes values from to , total number of pairs satisfying will be .
In order to prove the second part, we note that
Let and , for some integers and , then it can be shown that
where . Using this in (30) we get
This proves the second part.
To prove the third part, we note that if then and we have . Now if , then for some integer . This gives . Thus is a multiple of . But since , the only possibllity is , and this gives . In order to prove , we note that with we have . Now since is not a multiple of , should be a multiple of q. Let , then , giving , and .
To show that the modulation matrix can be characterized by filters, for a choose any index such that . Define . Then from the above results, elements of the modulation matrix, namely such that can be written as
In this manner all the elements of can be written in the form of for some and some .
By using the lemma(1) and the lemma(3) we can obtain ideal frequency spectrum of the filters that characterize the modulation matrix. The following lemma gives ideal frequency spectrum of these filters:
Only maximum two filters out of filters will have non-zero ideal frequency spectrum. Moreover by appropriate choice of the filters, it can be shown that the filters having non-zero ideal frequency spectrum will have frequency spectrum given by and for some integer .
Proof: In order to prove that only two filters out of filters have non-zero frequency spectrum, we note that when is even, then it can be seen that if , then either or . Let and be two filters out of filters such that when and when , then it can be seen that the ideal frequency spectrum of only and will be non-zero.
Similarly if is even, then and will have nonzero ideal frequency spectrum if when and when . Rest of the filters will have zero ideal frequency spectrum.
In order to prove the second part, we consider two cases
is even: For this case if we choose and such that
then it can be shown that
If then we have and for such a case
is even: For this case if we choose and such that
then it can be shown that
Again if then we have and for such a case
The ideal frequency spectra of these filters are used to form an objective function, minimizing which would give us desired paraunitary polyphase matrix.
Iv Design of the Paraunitary Polyphase Matrix
Iv-a Formation of an objective function
Given rational numbers , our objective is to design a paraunitary polyphase matrix that gives the required splitting of the input frequency spectrum into bands.
Consider -th channel of such a filter bank. The part of the input spectrum corresponding to this band is , where