Design and Multiplierless Implementations of 9 th order linear-Phase Bireciprocal Lattice Wave Digital Wavelet Filter Banks

In this paper, a filter bank structure for the implementation of infinite impulse response (IIR) discrete wavelet transform (DWT) is proposed. Bireciprocal lattice wave filters (BLWDFs) are utilized in a linear-phase design of 9 th order IIR wavelet filter bank (FB). Each of the two branches in the structure of the BLWDF bank realizes an all-pass filter. Filters of this bank belong to the intermediate design group, maintaining linear-phase property with best roll-off characteristics in their frequency responses. The design is first simulated using Matlab7.10 programming in order to investigate the resulting wavelet filter properties and to find the suitable wordlength to represent the BLWDF’s coefficients in quantized forms for best selection of some prescribed performance measures. All adopted measures show an excellent closeness to some typical cases. Each coefficient in the resulting structure is realized in a multiplierless manner after representing it as sum-of-powers-of-two (SPT). Multiplications are then achieved by only shift and add. Multiplierless FPGA implementations of the proposed IIR wavelet filter banks are also achieved with less complexity and high operating frequency.


I. Introudaction
Discrete Wavelet Transform (DWT) is a powerful tool for signal processing and analysis [1]. The success of DWT stems from its ease of computation and its inherent decomposition of an image into non-overlapping subbands that enables the design of efficient algorithms and allows for incorporation of the human visual system [2]. The extension of DWT to efficient implementation of filter banks, has provided a way to efficiently utilize wavelets for signal processing [1]. Filter banks can be realized using finite impulse response (FIR) or infinite impulse response (IIR) filters. FIR filters have large transition bands. If both sharp transition band and reasonable stopband attenuation are required, one must use very high order FIR filters. In VLSI and other implementations, high order means that a large amount of multipliers are required. Lower order IIR filters can be achieved with sharp transition bands and good stopband attenuations [3]. However, linear-phase and orthonormality are not mutually exclusive properties for IIR filter banks, as they are in an FIR case [4].
In this paper, IIR wavelet filter, with perfect reconstruction and linear-phase properties is designed. The structure of this filteris based on the bireciprocal lattice wave filter (BLWDF) to simulate a two channel wavelet filter bank. BLWDF is employed in a 9 th order IIR filter to form an intermediate filter bank. The wave lattice is composed of two parallel real all-pass sections. 2 nd order to 1 st order all-pass filter section reduction method is applied via downsampling position alteration in the designed structure.Consequently, the resulting bireciprocal lattice wave discrete wavelet filter banks(BLW-DWFBs) structure possess efficient computations. Linear-phase is accomplished by laying the impulse responses of the synthesis bank filters as time-reversed versions of the analysis filters.The generations of scaling and wavelet functions pertaining to these structures are also conducted in this paper. The proposed designs are implemented on Spartan-3E FPGA kit for one and three wavelet levels. To accelerate the hardware implementations and make them less-complex on an FPGA, the filter coefficients are implemented in a multiplierless manner after representing them as sum-and-difference-of-powers-of-two (SPT). Multiplications are then achieved only by shift and add. This paper is organized as follows: Section II discusses the theory of bireciprocal lattice wave digital filters and reviews the design and realization of IIR filter banks. In section III, the proposed design method for 9 th order linear-phase IIR wavelet filter bank with perfect reconstruction property is described and the corresponding scaling and wavelet functions after five-level analysis banks are generated. The comparison between biorthogonal FIR and the designed orthogonal IIR filters are also included. Matlab programming is used in section IV to verify the results and to estimate a suitable coefficient's wordlength to work with. In section V, the descriptions of the FPGA implementations are given and the obtained results are displayed. A comparative study is illustrated in section VI. Finally, section VII concludes this paper.

II. Background Theory
The wave digital filter is one of the structures of the IIR digital filters [5]. It possess a number of notable advantages in comparison to other IIR filter structures. Their low sensitivity, good round-off noise characteristics,optimal scaling and wide dynamic range lead to short wordlengths implementations [6]. In other words, it is possible to represent filter coefficients utilizing only few bits. This could allow for decreasing the size of applied multipliers or even replacing them by shift and add operations [2]. Therefore this kind of digital filter becomes more popular to be implemented into many varieties of integrated circuits. A bireciprocal Lattice Wave digital filter (BLWDF), is a special form of lattice wave digital filters (LWDFs), that reduces the number of adaptors under certain conditions and can also be used as a base for decimation and interpolation filters. In BLWDF transfer functions, numerators are polynomials in z -2 .This reduces their implementation complexity. It also increases the throughput, compared to the same order LWDF implementations [7]. So, BLWDFs conducted in this paper, represent the most efficient family of IIR filters and are therefore of great interest. As shown in Fig. 1, the low pass transfer function of BLWDF can be written as where ( ) and ( ) are all-pass functions and can be expressed respectively, as [8] ( ) ∏ ( ) ( ) where α i 's are the all-pass sections coefficients and N is the ( ) filter order. The two-channel quadrature mirrorfilter (QMF) bank structure is a parallel connection of two all-pass subfilters (PCAS) [9]. The aliasing error and amplitude distortion can be eliminated by representing the QMF bank with polyphase structure [10]. Figure 1 shows a two channel all-pass filter based IIR QMF banks with polyphase realization [11]. Because the all-pass filters cannot have exact linear-phase, the QMF banks may have phase distortion [12].
To remove this phase distortion and to achieve a perfect reconstruction condition [3], a branch-interchange method is applied in the analysis side of the bank and the two all-pass sections in the synthesis side are made as function of instead of [13]. Furthermore, in Fig.1, the filtering by 2 nd order all-pass filters followed by downsampling is equivalent to having downsampling first, followed by the 1 st order all-pass filters [14]. The resulting stable and efficient linear-phase filter bank (FB) structure is shown in Fig. 2. The transfer function of ( ) and ( ) in Fig. 2 can be expressed as follows: Let H 0 (z) and H 1 (z) denote the transfer functions of the low-pass and high-pass filter of the analysis part of the two-channel QMF bank. Also let G 0 (z) and G 1 (z) denote the transfer functions of the low-pass and high-pass filters of the synthesis part. By choosing transfer functions to satisfy thefollowing conditions: ) then the filter bank will possess both perfect reconstruction and orthonormality properties [15]. Analysis filters H 0 (z) and H 1 (z) can respectively, be written thenfor the low-pass side as

III. The Pproposed Design For Linear-Phase IIR PR Wavelet FBS
The design procedure presented here is based on placing poles and zeros judiciously in the z-plane, to obtain IIR filter coefficients such that the resulting filter has the desired frequency response which is whose characteristics are between IIR Butterworth and Daubechies filters (intermediate filters).The half band filter's poles are placed on an imaginary axis of the complex z-plane, where one of them is placed at the origin and the remaining conjugate complex pole pairs are located between and . All zeros of such filters are located on the unit circle, where seven of them are placed at z = -1 for 9 th order filter to obtain maximally flat filter which is generating regular wavelets with maximum number of vanishing moments, the remaining two zeros are required to locate on the unit circle in order to obtain the best frequency selectivity [16]. However, for a given order filter, regularity and frequency selectivity some what contradict each other [17]. For this reason, the design of the intermediate filter that has the best possible frequency selectivity for a given flatness condition is considered.
Linear-phase and orthonormality are not mutually exclusive properties for IIR filter banks. To accomplish both properties, the impulse responses of the synthesis bank filters must be the time-reversed versions of the corresponding analysis filters. If the IIR analysis filter bank uses causal filters, then the synthesis has anticausal filters [18].

A. 9 th order linear-phase intermediate IIR filter design
The proposed filter is shown in Fig. 3. It has nine zeros and nine poles. The values of these poles and zeros can be found out depending on the desired frequency response that is shown in Fig.  4.
The transfer function of such IIR filter depends on the positions of its poles and zeros as in Fig. 3. Such transfer function is given by where a, b, α , β ,σ and γ are constants less than 1, as shown in Fig. 3, with k as amagnitude scaling factor. Equation (8) can be factorized and reordered as It can be seen that (9) is expressed as a function of . In order to be used in terms of , both numerator and denominator of (9) can be multiplied by , yielding Finding the transfer function of such 9 th order filter means that the design of an intermediate IIR filter has been accomplished. The corresponding IIR filter can be implemented on BLWDF bases as a parallel connection of two all-pass IIR sections. By Gazsi method [19], that uses an alternative pole technique as illustrated in Fig. 5, the transfer functions of the two all-pass sections ( ) and ( )in (1) can then be derived. In Fig. 5, the poles of the region R 1 can formulate the transfer function of the all-pass section ( ) and the poles of the region R 2 can formulate the transfer function of the all-pass section ( ). The pole at the origin represents the delay element z -1 in (1). Substituting (11) and (12) in (1) to form the low-pass transfer function of BLWDF as ( ) The design ( )of and ( ) is illustrated in the Appendix. Using the equations (A14) and (A15) in such Appendix, the resulting intermediate IIR order BLWDF lowpass transfer function depending on (A1) can be written as The corresponding transfer function of the high-pass filter can then be given by

( ) ( )
As can be seen from Fig. 4, the main drawback of IIR filter is the non-linear-phase response. To obtain linear-phase IIR wavelet filters, the structure in Fig. 2

B. Generations of scaling and wavelet functions
The basis vectors; mother wavelet and scaling functions of orthogonal wavelet transform are obtained from the iteration of the filter bank on its low-pass branch [20]. A scaling function is used to approximate an input signal at different levels of approximation. Each approximation is differed by a factor of two from the approximation at the nearest neighboring level. It should be noted that wavelet function is a type of scaling function [8], [21]. Iteration of filter bank generates equivalent band-pass filters of the form [20].
Letting gives the "mother wavelet" ( ): where [ ] is the impulse response of ( ) . The impulse response [ ] of the equivalent low-pass filters of the form is referred to as scaling function after i iterations.
In this section, a five-level orthogonal IIR wavelet filter bank of the type shown in Fig.7. is proposed. It is realized by intermediate IIR filter pairs for the approximate generation of the corresponding wavelet (mother wavelet) and scaling functions.The mother wavelet and scaling functions can be generated after five iterations of the analysis filter banks on its lowpass branch. Iterations of the bank generate equivalent band-pass filter of the form The impulse response of (21) is referred to as mother wavelet ψ(n), while the impulse response of the eqivalent low-pass filter given by the form

is referred to as scaling function ø(n).
When all-pass IIR filter sections ( )and ( ) in Fig. 7 are realized as 9 th orders BLWDFs and the Dirac impulse signal is employed as input signal to the system, the output signals will represent the corresponding wavelet and scaling functions. These functions are shown in Fig. 8. They are generated after five iterations of the analysis filter bank on its lowpass branch according to (21) and (22), respectively. Figure 7 with the aid of wavelet and scaling functions of Fig. 8 show how the DWT represents the input signal as shifted and scaled versions of the wavelet and scaling functions. Since an impulse function representation is used, the wavelet function appears in the"High wave" signals, while the scaling function appears in the''Low wave" signals.

C. Frequency response of 9 th order intermediate IIR versus Biorthogonal 9/7 DWT filters
The frequency response of the designed filter is compared with those of the lifting scheme implementations of Bio. 9/7 wavelet filter bank in Refs. [22] & [23]. Half-band filters in wavelet transform using 9 th order BLWDF possess superior band discriminations and perfect (a) roll-off frequency characteristics as compared to their Bio. 9/7 wavelet FIR counterparts (see Fig. 9). In addition, the biorthogonal wavelets with different number of coefficients in the low-pass and high-pass filters increase the number of operations and the complexity of the design [24].

IV. Quantizations of Filter Coefficients
Filter coefficients are quantized before implementation in fast hardware. This may change the frequency response of the filter. The filter coefficients are quantized such that the filter bank properties are preserved. This is accomplished by the right selection of wordlengths for these coefficients. In order to select a sufficient wordlength, the proposed filter should be designed according to the application (Gray scale image). Matlab programs are used to find the minimum wordlength of the BLWDF coefficients for acceptable PSNR and SNR values which are considered to be about 30 dB,correlation coefficient has value close to one, and magnitude and phase responses a of the quantized filters are closely approximate the magnitude and phase responses of the unquantized filters. This wordlength is found to be 6 bits as indicated by the PSNR readings taken for the designed filter banks using a group of gray scale test images and minimum error value between two magnitude and phase responses. Figure 10 shows the magnitude responses for quantized low-pass filter. Thus BLWDF coefficients after quantization are = 0.34375 , = 0.03125, .

V. FPGA Implementations
The proposed design is described first with VHDL language. Then an FPGA device is used to implement the filter design. That device is a 500,000-gate Xilinx Spartan-3E XC3S500E in a 320-ball Fine-Pitch Ball Grid Array package (XC3S500EFG320).

A. Implementation of one-level analysis FBs
An efficient hardware implementation of the BLWDF bank structure (shown in Fig. 2) with a 9 th order discrete wavelet transform can be accomplished using number of Processing Elements (PEs). Each PE in Fig. 11 is a generic term that refers to a hardware element that executes a stream of instructions included in a 1 st order all-pass section. For example, PE-A and PE-B represent the two processing elements that construct the 1 st order all-pass section ( ) The functions of PE-A and PE-B can, respectively be represented the following two difference equations: where represents the even pixels of the input image which also represents the input to PE-A. [ ] represents the output of PE-A which also represents the input to PE-B.
[ ] represents the output of PE-B.
represents the coefficient of PE-A, i.e., the first part of all-pass filter ( ). represents the coefficient of PE-B, i.e., the second part of all-pass filter ( ). Fig. 11 Structure of one-level wavelet transform analysis side using 9 th order BLWDF.
The FPGA device utilization summary for analysis side is illustrated in Table1.

B. Implementation of three-level analysis FBs
After designing one-level analysis side structures depending on 9 th order BLWDF as wavelet filters. They can be used to construct three-level analysis filter banks structures for such DWT using pipeline implementation. The block diagram of the proposed analysis side scalable structure for three-level DWT is shown in Fig. 12. The main units in the structure are the Processing Elements (PEs). Each PE consists of one-level analysis filter bank of the same 9 th order BLWDF type shown in Fig. 11. Therefore, the PE structure depends on the chosen filter, the filter order and the selected structure of the filter banks. The FPGA device utilization summary for three-levelanalysis side is illustrated in Table 2.  Table 2: Device utilization summary forthree-level analysis side.

C. Implementation of one-level synthesis filter bank
The two parallel all-pass filters used in this section are ( ) with processing elements PEr-A and PEr-B, and ( ) ,with processing elements PEr-C and PEr-D, as shown in Fig. 13. These filters can be implemented as in ( ) and ( ) all-pass filters, respectively, by reversing the inputs and outputs of these filters using the last in first out (LIFO) register [18] as shown in Fig. 14. This leads to an overall wavelet reconstructed signal with exact linear phase. The FPGA device utilization summary for one-level synthesis side is illustrated in Table 3.  Device utilization summary:

D. Implementation of three-level synthesis filter bank
The wavelet filters constructed from 9 th order BLWDF for one-level synthesis side structure (shown in Fig. 13) can also be used for three-level synthesis side filter banks structure of DWTas shown in Fig. 15, where the main processing element that refers to onelevel synthesis side filter bankis PEr. The PEr structure depends on the chosen filter.The FPGA device utilization summary for three-level synthesis side is illustrated in Table 4.

VI. A Comparative Study
A comparison between the implementation of the proposed design of the order intermediate IIR wavelet filter bank and that of biorthogonal 9/7 lifting scheme wavelet structure in Ref. [23] is made using Spartan-3E FPGA device. This comparison is shown in Table 5.  Table 5 indicates that the utilization ratios of the proposed filter bank are of the same order of the Bio. 9/7 lifting scheme in Ref. [23], while the magnitude response of the proposal filter bank is more selective ( see Fig. 9), since it contains IIR filters.

VII. Conclusions
Linear phase 9 th order IIR wavelet filter bank utilizing bireciprocal lattice wave digital filters (BLWDFs) has been designed and realized by all-pass filters. Filter coefficients has been quantized then realized in a multiplierless manner taking into account the lowcoefficient sensitivity property of these wave structures.The resulting coefficients can be achieved by shift and add processes only highlighting a multiplierless realization. Multiplierless FPGA implementations of one-and three-level of bireciprocal lattice wave digital wavelet filter banks have been obtained with less complexity and high speed. Linear phase processing of such structures indicates their perfect reconstruction property.
It has been shown that, the resultimg less-complex high-speed implementation characterstics of the proposed 9 th order bireciprocal lattice wave discrete wavelet filter banks are almost of the same orders of those in lifting scheme Bio. 9/7 wavelet structures in Ref. [24]. Moreover, Half-band filters in a wavelet transform using 9 th order BLWDF possess superior band discriminations and perfect roll-off frequency characteristics compared to those in Bio. 9/7 wavelet filters.

APPENDIX
The final form of (13) can be written as Substituting the above coefficient in equations (11) and (12)