M-CHANNEL LIFTING STRUCTURE FOR UNIMODULAR ... - CiteSeerX

M -CHANNEL LIFTING STRUCTURE FOR UNIMODULAR FILTER BANK Rohit Kumar∗ , Ying-Jui Chen† , Soontorn Oraintara∗ , and Kevin Amaratunga† ∗

Department of Electrical Engineering, University of Texas at Arlington, 416 Yates St., Arlington, TX, 76019-0016 USA Email:[email protected], [email protected] † Intelligent Engineering Systems Lab, Massachusetts Inst. Technology Cambridge, MA 02139, USA, http://wavelets.mit.edu Email: {yrchen, kevina}@mit.edu ABSTRACT

We extend the method of lifting from M -channel paraunitary filter banks (PUFB) to M -channel unimodular filter banks. In particular, the lifting factorizations of Type-I and Type-II building blocks for the first-order unimodular matrices are presented, where the McMillan minimality is preserved. The proposed factorizations continue to have unity diagonal scaling, and perfect reconstruction (PR) is thus guaranteed under finite precision. Using degreeone unimodular building blocks, we present the design of firstorder unimodular transforms (a.k.a. lapped unimodular transform, or LUT) and their lifting factorizations. Multiplierless (while reversible) implementations of the resulting LUT are obtained by dyadic approximation of the lifting coefficients. 1. INTRODUCTION Multirate filter banks (FBs) have found several applications in the field of signal processing [1, 2]. Signals can be represented more compactly with subband samples than their time-domain representations without losing information. Fig. 1 shows the polyphase representation of a PRFB, where E(z) and R(z) are the analysis and the synthesis polyphase matrices with R(z)E(z) = I. As FIR FBs are usually preferred, it is necessary that det{E(z)} = c z −` for some c 6= 0 and some integer `, so that FIR analysis filters Hk (z) are jointly inverted by FIR synthesis filters Fk (z). Such FIR PRFBs assume the so-called lifting factorization, which is suitable for fast, reversible, and possibly multiplierless implementations of the FBs, and is also efficient by allowing for in-place computation [3, 4]. Reversibility is structurally guaranteed even when the lifting multipliers are quantized. Lifting factorization has also been used in filter bank design, including PUFBs and a class of biorthogonal filter banks [4, 5, 6]. Unimodular filter banks are a special class of FIR PRFBs, where the polyphase matrix E(z) is unimodular, i.e. det{E(z)} = c for some c 6= 0. Some important properties of unimodular FBs are summarized below: • The inverse of a unimodular FB is not only FIR but also causal unlike the paraunitary FB which has an anticausal inverse [2, 7, 8]. • The system delay of a unimodular FB is (M − 1), where M is the number of channels. It is dependent on M but is independent on the filter length. This system delay of unimodular FB is found to be less than other (causal) FBs.

This property is of utmost use in the applications where system delay is important, for e.g. speech coding and adaptive filtering, etc. [7]. • The coding gain is a measure of energy compaction capability of a FB. Some examples have shown that a first-order unimodular FB can be optimized so as to have coding gain for highly correlated input signals (correlation coefficient close to 1) greater than the lapped orthogonal transform (LOT) and the lapped biorthogonal transform (BOLT) [7]. Hence it is possible that it can be used for image coding application. • According to Smith McMillan decomposition, any general polyphase matrix E(z) of size p × r for an FIR FB can be decomposed into a product of unimodular matrices and a diagonal matrix as: E(z) = U(z)D(z)W(z)

(1)

where U(z) and W(z) are unimodular matrices of size p× p and r × r, respectively, and D(z) is a p × r diagonal matrix [2, 8]. From (1), it is clear that parameterization of unimodular matrix would be useful to parameterize any general E(z). • It is shown in [9] that any causal FIR polyphase matrix E(z) with det{E(z)} = delay is a product of a paraunitary matrix and a unimodular matrix. Hence, in order to parameterize an FIR biorthogonal FB, it suffices to independently parameterize paraunitary and unimodular FBs. Unimodular FBs of an arbitrary order do not necessarily have a degree-one factorization as pointed out in [7]. However, when the order is limited to one, it has been proved that a unimodular matrix can be parameterized as a product of degree-one building blocks [7, 8]. Such factorizations are complete, and are minimal in terms of the number of delays. In this paper we consider exclusively first-order unimodular matrices, and present novel lifting parameterizations of the underlying building blocks. In Section 2, unimodular FB construction using Type-I and Type-II building blocks is reviewed. Then Section 3 presents their M -channel lifting factorizations, based on which multiplierless implementations of the FB are discussed and obtained in Section 4. Concluding remarks are found in Section 5. Notations: Uppercase boldfaced alphabets indicate matrices while lowercase boldfaced alphabets denote column vectors. Superscript † indicates the conjugate transpose of matrices or vectors.

Processing Block

M E(z)

z

-1

-1

R(z)

M

M

-1

-1

z

z

-1

M z

M

-1

z

x(n)

M

y(n)

z

Figure 1: The polyphase representation of an M -channel maximally decimated filter bank.

Di (−z), which is also causal and has degree one. One notices that the factor E0 in (6) contains the first M coefficients for the filters. It also has an interesting interpretation as pointed out in [11]. In particular, one can view this as a concatenation between a nonsingular matrix (E0 in this case) and a vector differential pulse code modulation (DPCM). The role of the non-singular matrix is to decorrelate the subband signals while the vector DPCM is used for linear prediction. The advantage of the vector DPCM constructed by a unimodular FB is that its inverse is also causal and FIR, and thus eliminates the stability problem. 3. LIFTING FACTORIZATION OF THE DEGREE-ONE UNIMODULAR BUILDING BLOCKS

2. UNIMODULAR FILTER BANK The M × M polyphase matrix E(z) for a causal unimodular FB of order N can be written as: E(z) = E0 + E1 z −1 + . . . + EN z −N

(2)

where E0 is a non-singular matrix, EN 6= 0 and det{E(z)} = c, for some c 6= 0. It has been proved in [8, 7] that there do not exist any finite-degree structures that can be used as a general building block for unimodular matrices of any order. However if we restrict ourselves to N = 1 E(z) = E0 + E1 z −1 ,

(3)

factorization into a product of degree-one building blocks is possible. This class of FBs is referred to as lapped unimodular transform (LUT). The degree of E(z) in (3) is determined by the rank of E1 [2]. The polyphase matrix E(z) in (3) with degree ρ can be factorized into degree-one building blocks as: ˆ 0 (z)D ˆ 1 (z) . . . D ˆ ρ−1 (z), E(z) = E(1)D

(Type I)

(4)

ˆ i (z) = I − where D + with = 0 is the degree-one unimodular building block of Type I. It is clear that, ˆ −1 (z) = I + u ˆ i† u ˆ i = 0, D ˆ iv ˆ i† − u ˆ iv ˆ i† z −1 . Hence the since v i inverse of E(z), R(z), will also be causal and have degree ρ. One ˆ i (1) = I and thus E(z)|z=1 = E(1) = should also notice that D E0 + E1 . It is known that, in order for the FB to possess one degree of regularity (vanishing moment), ˆ iv ˆ i† u

ˆ iv ˆ i† z −1 u

ˆ i† u ˆi v

E(1)1M = αe0

(5)

where 1M is the vector with all elements equal to one, e0 is the first basis vector, and α is a non-zero constant [10]. Therefore the ˆ i (z), condition in (5) is dependent on the degree-one matrices D ˆ i and v ˆ i . Hence, the unimodular FB has regularity of and thus u degree one if and only if E(1) does. For example, if E(1) is chosen to be the discrete cosine transform (DCT), for any given choice ˆ i (z), all the filters except for the first one of the corresponding of D FB will have exactly zero response at DC. This type of factorization can be useful in design and optimization since regularity is structurally imposed. The polyphase matrix E(z) in (3) can also be factorized into a product of different degree-one building blocks as: E(z) = E0 D0 (z)D1 (z) . . . Dρ−1 (z),

(Type II)

(6)

where Di (z) = I + ui vi† z −1 with vi† ui = 0 is the degree-one unimodular building block of Type II. Since vi† ui = 0, D−1 i (z) =

It is clear that, in order to obtain lifting parameterizations for the products in (4) and (6), it suffices to parameterize the non-singular ˆ i (z) and matrices (E(1) and E0 ) and the degree-one factors D Di (z) separately. When the determinant of E(1) or E0 is equal to one, one can always factorize it as a product of upper and lower triangular matrices [12]. In special cases where they are some popular transforms, for instance DCT and DFT, the structures proposed in [13, 14] can be employed. In this section, the lifting factorˆ i (z) and Di (z) are presented. The subscript i will izations for D be omitted from the building blocks to simplify the notation and x ¯ , −x. ˆ The lifting factorization for Type-I building block D(z) used in (4) is presented in (7), where αi = u∗i /u∗r and βi = u∗r vi∗ for any r ∈ {1, 2, . . . , M } with ur 6= 0 and βr = 1. It is clear that ˆ and v ˆ are re-parameterized by the free parameters ui and vi of u a new set of free parameters αi and βi . The number of degrees of ˆ freedom of each D(z) is 2(M − 1) for real-valued and 4(M − 1) for complex-valued filters, which is consistent with the condition ˆ†v ˆ = 0. The parameterization in (7) has 3(M − 1) lifting steps, u which correspond to 3(M − 1) multipliers. Fig. 2(a) shows a realˆ ization of D(z) in (7). As illustrated in Fig. 2(a), though there are M − 1 delay elements z −1 in (7), they can be jointly implemented using just one delay, which is consistent with its (McMillan) degree of unity. Similarly, Type-II building block D(z) employed in (6) can also be factorized as given in (8), where αi = u∗i /u∗r and βi = u∗r vi∗ for any r ∈ {1, 2, . . . , M } with ur 6= 0 and βr = 1. The structure has 3(M − 1) lifting steps (3(M − 1) multipliers) with 2(M − 1) or 4(M − 1) degrees of freedom, depending on whether it is real- or complex-valued. Fig. 2(b) shows a realization of D(z), in which again only one delay is used. Hence the new parameterization of D(z) is also minimal in the McMillan sense. 4. APPROXIMATION WITH DYADIC COEFFICIENTS There is unity diagonal scaling in both factorizations and hence the proposed factorizations allow for the reversible and multiplierless implementation of the FB even under finite precision conditions and/or with non-linear operations at the lifting steps. With the proposed factorizations, the FB can be optimized directly in the lifting domain. Since the expressions in (7) and (8) can be implemented for any value of r ∈ {1, · · · , M } as long as ur 6= 0, the factorization is not unique. However, one can choose r so that |ur | is the maximum among all |ui |, i.e. the parameters αi = u∗i /u∗r can be limited within the unit circle.

2 6 6 6 6 6 ˆ D(z) =6 6 6 6 6 4

2 6 6 6 6 6 D(z) = 6 6 6 6 6 4

1 ..

α1 .. . 1 αr−1 1 αr+1 1 .. . αM

.

1 ..

.

α1 .. . 1 αr−1 1 αr+1 1 .. . αM

32

..

76 76 76 76 76 76 76 76 76 76 54

. 1

32

..

76 76 76 76 76 76 76 76 76 76 54

.

3T 2

(z −1 −1)β1 .. .. . . −1 1 (z −1)βr−1 1 (z −1 −1)βr+1 1 .. . (z −1 −1)βM

1

1 ..

.

1

β1 z −1 .. . 1 βr−1 z −1 1 βr+1 z −1 1 .. . βM z −1

..

7 7 7 7 7 7 7 7 7 7 5

.

6 6 6 6 6 6 6 6 6 6 4

1 ..

.

1 3T 2

..

7 7 7 7 7 7 7 7 7 7 5

.

6 6 6 6 6 6 6 6 6 6 4

1 ..

.

1

α ¯1 .. . 1 α ¯ r−1 1 α ¯ r+1 1 .. . α ¯M α ¯1 .. . 1 α ¯ r−1 1 α ¯ r+1 1 .. . α ¯M

3

..

7 7 7 7 7 7 7 7 7 7 5

.

(7)

1 3

..

7 7 7 7 7 7 7 7 7 7 5

.

(8)

1

is greater than 7.57dB of the DCT and is comparable to 7.93dB of LOT [5]. The dyadic lifting parameters are given in Table 1. A good dynamic range is guaranteed as the magnitudes of all αi and βi are less than 1. Notice that since the DCT has one degree of regularity, with this Type I parameterization, the resulting FB also has one vanishing moment on the analysis side. (a)

Table 1: The design examples: Types I and II lifting parameters with r = 4 and E(1) being the integer-approximated DCT [4]. The frequency responses are shown in Figs. 3 and 4. Type

α1

α2

α3

β1

β2

β3

I

45 − 256 41 256

− 11 64 109 256

5 16 185 256

9 − 256 25 − 128

− 18 3 − 32

3 − 32

II

1 − 256

(b) ˆ i (z) in Type-I factorFigure 2: Lifting parameterizations of: (a) D ization and (b) Di (z) in Type-II factorization. Both structures are drawn for M = 4 and r = 1. The floating point parameters αi and βi can be approximated with VLSI-friendly coefficients of the form k/2n , which can be implemented by multiplication by an integer k and then right shifting by n bits. Consequently, the multiplications are reduced to mere add and shift operations, by which the complexity of the transforms can be reduced, resulting in faster calculation. 4.1. Design Examples—Multiplierless LUT Using the Type-I structure (4), we present below the design of a four-channel real-valued LUT with degree ρ = 1, namely, E(z) = ˆ 0 (z). The integer-approximated DCT in [4] is employed as E(1)D the non-singular matrix E(1) and thus only six free parameters are left to be determined. The FB is optimized for coding gain and stopband attenuation [1]. Figs. 3(a) and (b) show the frequency responses of the resulting design, with coding gain 7.90dB which

The second example is a Type II parameterization in (6) with E0 chosen to be the integer-approximated DCT [4]. The FB is optimized for coding gain and the corresponding dyadic lifting coefficients are presented in Table 1. Fig. 4 shows the frequency and impulse responses of the resulting FB which show that the first four samples of the analysis and synthesis filters are the same as the DCT. The corresponding coding gain is 7.81 dB.

5. CONCLUSION Two novel lifting structures for first-order unimodular FBs are presented in this paper. A first-order unimodular filter bank is designed and then the lifting parameters are calculated. The parameter ur is properly chosen so that all the lifting parameters are less than 1 in magnitude. Consequently the input range is preserved. The resulting lifting coefficients are quantized for multiplierless implementation. The FB can also be designed by optimizing lifting coefficients in the lifting domain. Examples of unimodular with dyadic coefficients in Types I and II parameterizations are also presented.

Type−I Analysis: 4 channels, 8 taps, coding gain=7.8978dB 5 h0

−5 −10

h1

−15 −20 h2

−25 −30

h3

−35 0.1

0.2

ω/2π

0.3

0.4

h0

0

Magnitude Responses (dB)

Magnitude Responses (dB)

0

−40 0

Type−II Analysis: 4 channels, 8 taps, coding gain=7.8078dB 5

−5 −10

h1

−15 −20 h2

−25 −30

h3

−35 −40 0

0.5

0.1

0.2

ω/2π

0.3

(a)

0

−5 −10

f1

−15 −20 f2

−25 −30

f3

−35 0.2

ω/2π

0.3

0.4

0.5

f

0

0

Magnitude Responses (dB)

Magnitude Responses (dB)

Type−II Synthesis: 4 channels, 8 taps, coding gain=7.8078dB 5

f

0

0.1

0.5

(a)

Type−I Synthesis: 4 channels, 8 taps, coding gain=7.8978dB 5

−40 0

0.4

−5 −10

f1

−15 −20 f2

−25 −30

f3

−35 −40 0

0.1

0.2

ω/2π

0.3

0.4

0.5

(b)

(b)

Figure 3: Frequency and impulse responses of a four-channel order-one unimodular FB of degree one (Type I): (a) analysis bank and (b) synthesis bank.

Figure 4: Frequency and impulse responses of a four-channel order-one unimodular FB of degree one (Type II): (a) analysis bank and (b) synthesis bank.

6. REFERENCES [1] G. Strang and T. Nguyen. Wavelets and Filter Banks. Wellesley-Cambridge Press, 2 edition, 1997. [2] P. P. Vaidyanathan. Prentice Hall, 1992.

Multirate Systems and Filter Banks.

[3] I. Daubechies and W. Sweldens. Factoring wavelet transforms into lifting steps. J. Fourier Anal. Appl., 4(3):245–267, 1998. [4] Ying-Jui Chen and Kevin Amaratunga. M -channel lifting factorization of perfect reconstruction filter banks and reversible M -band wavelet transforms. IEEE Trans. Circuits Syst. II, Dec. 2003 (to appear). [5] Ying-Jui Chen, Soontorn Oraintara, and Kevin Amaratunga. M -channel lifting-based design of paraunitary and biorthogonal filter banks with structural regularity. In Proc. IEEE Int’l Symp. Circuits Syst., May 2003. [6] Ying-Jui Chen, Soontorn Oraintara, and Kevin Amaratunga. Dyadic-based factorizations for regular paraunitary filter banks and M -band orthogonal wavelets with structural vanishing moments. IEEE Trans. Signal Processing, submitted, 2003.

[7] S.-M. Phoong and Y.-P. Lin. Lapped unimodular transform and its factorization. IEEE Trans. Signal Processing, 50(11):2695–2701, Nov. 2002. [8] P. P. Vaidyanathan and Tsuhan Chen. Role of anticausal inverses in multirate filter-banks—Part II: The FIR case, factorizations, and biorthogonal lapped transforms. IEEE Trans. Signal Processing, 43(5):1103–1115, May 1995. [9] P. P. Vaidyanathan. How to capture all FIR perfect reconstruction qmf banks with unimodular matrices? Proc. IEEE Int’l Symp. Circuits Syst., 3:2030–2033, 1990. [10] S. Oraintara, T.D. Tran, and T.Q. Nguyen. A class of regular biorthogonal filter banks. IEEE Trans. Signal Processing, Dec. 2003 (to appear). [11] S.-M. Phoong and Y.-P. Lin. Application of unimodular matrices to signal compression. Proc. IEEE Int’l Conf. Acoust., Speech, Signal Processing, 2002. [12] G. Strang. Every unit matrix is a LULU. Linear Algebra and Its Applications, 265:165–172, November 1997. [13] J. Liang and T.D. Tran. Approximating the DCT with the lifting scheme: Systematic design and applications. Signals Systems and Computers, 1:192–196, 2000. [14] S. Oraintara, Ying-Jui Chen, and T. Q. Nguyen. Integer fast Fourier transform. IEEE Trans. Signal Processing, 50:607– 618, March 2002.