Patent Publication Number: US-6343304-B1

Title: Apparatus with selective fixed-coefficient filter for performing recursive discrete cosine transforms

Description:
FIELD OF THE INVENTION 
     The present invention relates to an apparatus with a selective fixed-coefficient filter, and more particularly to an apparatus with a selective fixed-coefficient filter for recursively computing N points discrete cosine transforms (DCTs). 
     BACKGROUND OF THE INVENTION 
     The discrete cosine transforms (DCTs) are widely applied to processing and compression of video and audio signals and other fields related to spectral analyses and parameter estimations. Therefore, the development of a simple implementation of discrete cosine transform becomes very important. Recently, the recursive implementation of orthogonal transforms gains a lot of attentions since they are characterized in a simple structure and local communication. Goertzel first used the periodicity of finite trigonometric sequences to realize the discrete Fourier transform in a simple recursive structure. Goertzel&#39;s recursive structure not only reduces the number of computation, but also simplifies the complexity of implementation. After that, there are several recursive algorithms innovated to achieve regular implementation of DCTs, for example, the recursive algorithms for the DCT and inverse DCT with arbitrary length, the structures of recursive algorithms of normal and inverse DCT, the DCT with the power of two length in the recursive structures of several groups, a general discrete sinusoidal transform, and a two dimensional DCT/IDCT VLSI design characterized in formulation, regularization, and rank modulation. 
     U.S. Pat. No. 4,023,028 disclosed the operation and the recursive structure of fully digital recursive computation DFT. U.S. Pat. No. 4,058,715 indicated that the algebraic manipulation units for computing FFT are serially combined and each of them is a recursive computation structure. U.S. Pat. No. 4,080,661 disclosed the algebraic manipulation unit in a type of recursive without needing a multiplier. A common type of second order recursive digital filter is disclosed in U.S. Pat. No. 4,569,030. In U.S. Pat. No. 4,612,626, two FFT computations of real number sequence inputs are implemented with two recursive FFT structures. These recursive computation algorithms provide some advantages for VLSI implementation. 
     A fixed-coefficient recursive structure for computing the prime length DCT has been proposed. However, the most common length of DCTs is power of two. A fixed-coefficient recursive structure for realizing the time domain aliasing cancellation (TDAC) filtering processes has also been discussed. 
     As the discrete cosine transform with the recursive algorithms are obtained by the above-mentioned method, the filter coefficients vary along with the corresponding frequency-index components in the recursive structures. Each set of filter coefficients can only compute a specific transformed output. Accordingly, a variable-coefficient multiplier in a recursive IIR filter is required to obtain all desirable transformed results. Therefore, a variable-coefficient multiplier needs a generally typical multiplier with N deposited multiplicands for a transformation length of N. Besides, some filter coefficients can shift an IIR filter to a nearly unstable state, resulting in several inaccurately recursive results, particularly in finite-word-length machines. This problem can be solved by increasing the word length in a processor. However, to own a high-speed multiplier with a long word length is pretty expensive. Practically, the shortcoming of a recursive structure is that it requires a very high speed and accurate processing unit, such as IIR filter multiplier. 
     Therefore, it is tried by the applicant to deal with the situation encountered with the prior art. 
     SUMMARY OF THE INVENTION 
     An object of the present invention is to provide a new category of fixed-coefficient recursive structures for computing discrete cosine transforms in power-of two length. By using number theorems, the fixed-coefficient recursive structures are developed from exploration of periodicity embedded in transform bases which form a complete residue system or a complete odd residue system. Based on permutation, sign change, and zero insertion of original inputs, the proposed filtering structures required fixed multipliers are better than the previous recursive methods which need general multipliers. In finite length machines, properly selected filter coefficients can avoid previous unstable state upon accessing and achieve low round-off errors in their transformations. 
     According to one aspect of the present invention, the apparatus with a selective fixed-coefficient filter for recursively computing N points discrete cosine transform (DCT) is provided. The apparatus includes a data manipulation unit for obtaining 2N new inputs through permutation, sign change, and zero insertion of original inputs for computing coefficient values of the N points discrete cosine transformn, and a second order filter electrically connected to the data manipulation unit for selecting a fixed-coefficient so as to perform the N points discrete cosine transform through 2N recursive operations based on the permuted 2N new inputs. 
     In a preferred embodiment, the N is a power of two. 
     In another preferred embodiment, the N points discrete cosine transform is a second type of discrete cosine transform (DCT-II). 
     Preferably, the second order filter has a fixed loop coefficient and a fixed output coefficient in accordance with the computation of the second type of discrete cosine transform. 
     Preferably, the fixed loop coefficient is 2 cos(qπ/2N) and the fixed output coefficient is −cos(qπ/4N). 
     Preferably, the q and the 2N are relatively prime and the q is a selected relatively larger positive integer but not greater than N−1. 
     Preferably, the data manipulation unit reads y[n] in sequence according to m, in which n and m satisfy q(2m+1)=(2n+1)(2k+1)mod 4N and a dummy zero data is appended to the y[n] which has a sign of (−1) r4  if n is greater than N−1, where r 4 =┌(2n+1)(2k+1)/4N┐. 
     Preferably, the second type of discrete cosine transform is commonly called discrete cosine transform (DCT). 
     In a preferred embodiment, the N points discrete cosine transform is a third type of discrete cosine transform (DCT-III). 
     Preferably, the second order filter has a fixed loop coefficient and a fixed output coefficient in accordance with the computation of the third type of discrete cosine transform. 
     Preferably, the fixed loop coefficient is 2 cos(qπ/2N) and the fixed output coefficient is cos(qπ/2N). 
     Preferably, the q and the 2N are relatively prime and the q is a selected relatively larger positive integer but not greater than N−1. 
     Preferably, the data manipulation unit reads y[n] in sequence according to m in accordance with the computation of the third type of discrete cosine transform, in which n and m satisfy n(2k+1)mod 2N=qm mod 2N, and a dummy zero data is appended to the y[n] which has a sign of (−1) r3  if n is greater than N−1, where r 3 =┌n(2k+1)/2N┐. 
     Preferably, the third type of discrete cosine transform is commonly called inverse discrete cosine transform (IDCT). 
     Preferably, the data manipulation unit includes an address generator based on a read only memory (ROM). 
     Preferably, the ROM contains computed data of n(2k+1)mod 2N=qm′ mod 2N and the mapping sequence and the sign change from n to m′ for completing an index mapping from m′ to n. 
     Preferably, the ROM contains computed data of (2n+1) (2k+1) mod 2N=q(2 m′+1)mod 2N and the mapping sequence and sign change from n to m′ for completing an index mapping from m′ to n. 
     Preferably, the data manipulation unit includes an address generator having a read only memory (ROM), an adder, a counter, a shifter, a comparator, and a first-in-first-out (FIFO) buffer. 
     In a preferred embodiment, the N points discrete cosine transform is a fourth type of discrete cosine transform (DCT-IV). 
     In another preferred embodiment, the second order filter has a fixed loop coefficient and a fixed output coefficient in accordance with the computation of the fourth type of discrete cosine transform. 
     Preferably, the fixed loop coefficient is 2 cos(qπ/2N) and the fixed output coefficient is −cos(qπ/4N). 
     Preferably, the q and the 2N are relatively prime and the q is a selected relatively larger positive integer but not greater than N−1. 
     Preferably, the data manipulation unit reads y[n] in sequence according to m, in which n and m satisfy q(2m+1)=(2n+1)(2k+1)mod 4N and a dummy zero data is appended to the y[n] which has a sign of (−1) r4  if n is greater than N−, where r 4 =┌(2n+1)(2k+1)/4N┐. 
     Preferably, the data manipulation unit includes an address generator based on a read only memory (ROM). 
     Preferably, the ROM contains computed data of n(2k+1)mod 2N=qm′ mod 2N and the mapping sequence and the sign change from n to m′ for completing an index mapping from m′, to n. 
     Preferably, the ROM contains computed data of (2n+1) (2k+1) mod 2N=q(2 m′+1)mod 2N and the mapping sequence and the sign change from n to m′ for completing an index mapping from m′ to n. 
     Preferably, the data manipulation unit includes an address generator having a read only memory (ROM), an adder, a counter, a shifter, a comparator, and a first-in-first-out (FIFO) buffer. 
     The present invention may best be understood through the following description with reference to the accompanying drawings, in which: 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 shows the fixed-coefficient recursive structure for realizing the DCT-III according to the present invention; 
     FIG. 2 shows the fixed-coefficient recursive structure for realizing the DCT-IV according to the present invention; 
     FIG. 3 shows the fixed-coefficient recursive structure for realizing the DCT-II according to the present invention; 
     FIG. 4 shows the implementation of fixed-coefficient multiplier 2 cos(47π/128) according to the present invention; 
     FIG. 5 shows the first realizing manner for the data manipulation unit according to the present invention; 
     FIG. 6 shows the second realizing manner for the data manipulation unit according to the present invention; 
     FIG. 7 shows the positions of the poles in a nearly unstable state of the recursive filter according to the present invention; and 
     FIG. 8 shows the positions of the poles in a stable state of the recursive filter according to the present invention. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The present invention will now be described more specifically with reference to the following embodiments. It is to be noted that the following descriptions of preferred embodiments of this invention are presented herein for purpose of illustration and description only; it is not intended to be exhaustive or to be limited to the precise form disclosed. 
     1. An Introduction for Discrete Cosine Transform 
     The discrete cosine transform (DCT) can be categorized into four types. The most well known DCT and inverse DCT are respectively DCT-II and DCT-III. Because of their similarity in algorithm expressions, an inverse transform of one type of DCT can be a normal transform of another type of DCT if the modified scale factor and the changed indices are ignored. For example, the inverse transform of DCT-II is the normal transform of DCT-III, while the inverse transform of DCT-III is the normal transform of DCT-II. For convenience sake, the scale factors of discrete cosine transform is ignored in the following derivations. 
     Therefore, the well-known DCT-II and DCT-III are given by                    DCT-II:                       Y   2          (   k   )         =       ∑     n   =   0       N   -   1                         y        (   n   )            cos        (         π        (       2      n     +   1     )          k       2      N       )             ,     k   =   0     ,   1   ,   …              ,     N   -   1     ,     
        and           (   1   )                     DCR-III:                       Y   3          (   k   )         =       ∑     n   =   0       N   -   1                         y        (   n   )            cos        (         π        (       2      k     +   1     )          n       2      N       )             ,     k   =   0     ,   1   ,   …              ,     N   -   1     ,           (   2   )                         
     To develop the DCT-II recursive algorithm, the recursive DCT-IV structure must be designed at first, where the DCT-IV is expressed by                    Y   4          (   k   )       =       ∑     n   =   0       N   -   1                         y        (   n   )            cos        (         π        (       2      n     +   1     )            (       2      k     +   1     )         4      N       )             ,     k   =   0     ,   1   ,   …              ,     N   -   1.             (   3   )                         
     As the discrete cosine transforms is considered by the prior recursive methods, the filter coefficients in the conventional recursive structure are related to the corresponding frequency components. In the following sections, a recursive structure with fixed-coefficient is developed to implement these discrete cosine transforms. 
     2. Recursive Cosine Structures with Fixed Coefficients 
     Because there are three different types of discrete cosine transforms which use different derivative procedures, their fixed-coefficient recursive structures will be described in the following sections, respectively. To fit the most common usage, fixed-coefficient recursive structures are derived by assuming the length of DCTs to be a power of two, i.e., N=2 M . 
     Since the indices of DCT-III bases can directly perform a complete residual system in modulo 2N, we first derive the fixed-coefficient recursive structure for the DCT-III. However, the indices of the DCT-II bases can not set up any complete residual or complete odd residual system in modulo 2N. In order to obtain a fixed-coefficient recursive structure for the DCT-II, the DCT-II is converted to the DCT-IV form, whose transform bases form a complete odd residual system. Before the derivation of the DCT-II recursive structure, the fixed-coefficient recursive structure for the DCT-IV is discussed first. 
     2-1. Fixed Recursive Structure for DCT-III 
     The N points DCT-III depicted in Eq. (2) includes a discrete cosine transform base (TB) cos(π(2k+1)n/2N) which is periodic with respect to indices n and k. The integer form, (2k+1)n/2N, is the most important index related to the recursive cosine transform base. Eq. (2) can be rewritten as                  Y   3          (   k   )       =       ∑     n   =   0       N   -   1                         y        (   n   )            cos        (       π                   r   3       +     π                     m   3     /   2        N       )                   (   4   )                         
     where r 3 =┌n(2k+1)/2N┐ and m 3 =[n(2k+1)mod 2N]. The rounding operator, ┌x┐, denotes the integer part of x and the modulo operator, [p mod 2N], expresses a positive integer less than 2N after the argument p is added or subtracted by a multiple of 2N. Because the transform base can be represented as cos(π(2k+1)n/2N)=(−1) r     3    cos(πm 3 /2N), the DCT-III in Eq. (4) becomes                  Y   3          (   k   )       =       ∑     n   =   0       N   -   1                         [         (     -   1     )       r   3            y        (   n   )         ]          cos        (     π                     m   3     /   2        N     )                   (   5   )                         
     For exploring the periodicity in TB functions, we shall first preview several properties and theorems in number theory. 
     Definition 1: A set of P integers a 1 , a 2  . . . , a p  is a complete residue (CR) system in modulo P if and only if a i ≠a j  (mod P) for i≠j. 
     Theorem 1: Let (a,P)=1 and r 1 , r 2  . . . , r p  is a complete residue system in modulo P. Then, ar 1 +b, ar 2 +b, . . . , ar p +b is also a CR system in modulo P for an arbitrary integer b. 
     In Definition 1, the complete residue sequence {a 1 , a 2  . . . , a p } describes all the possible values in modulo P. In Theorem 1, (p, q)=1 means that the greatest common factor between p and q is 1, i.e., p and q are relatively prime. 
     From Definition 1, it is obvious that {n mod 2N|n=0, 1, 2, . . . , 2N−1} represents a CR system in modulo 2N. In Theorem 1 (a=2k+1, P=2N, and b=0), we know that (2k+1, 2N)=1 since N is a power of two. Thus, {m 3 |for n=0, 1, 2, . . . , 2N−1} is also a CR system in modulo 2N. 
     To utilize the CR concept, the summation index in Eqs. (2), (4), and (5) is extended to 2N−1, 
     
       
           y ( n )=0 for  n=N, N+ 1, . . . , (2 N− 1)  (6) 
       
     
     By using these N dummy inputs and the above-mentioned properties, the summation index of the DCT-III can be changed from n to m as 
     
       
           m=m   3   =n (2 k+ 1)mod 2 N,   (7) 
       
     
     the DCT-III in Eq. (5) becomes                        Y   3          (   k   )       =       ∑     n   =   0         2      N     -   1                         [         (     -   1     )       r   3            y        (   n   )         ]          cos        (     π                     m   3     /   2        N     )                       =       ∑     m   =   0         2      N     -   1                           y   k          (   m   )            cos        (     π                   m   /   2        N     )                         (   8   )                         
     where 
     
       
           y   k ( m )=(−1) r     3     y ( n )  (9) 
       
     
     for k=0, 1, . . . , N−1. Since {m=n(2k+1)mod 2N|for n=0, 1, 2, . . . , 2N−1} exhibits a CR system in modulo 2N, m and n become a frequency-dependent one-to-one mapping defined by Eq. (7). Thus, the conversion of the summation index from n to m will not change final results. The DCT-III transform stated in Eqs. (8) and (9) becomes a fixed transformation once the frequency-dependent permuted, sign-changed, and zero-inserted data sequence of y k (m) is obtained. 
     Now, a fixed-coefficient DCT-III recursive structure is developed by replacing m with 2N−1-m in (8) to obtain                  Y   3          (   k   )       =       (     -   1     )            ∑     m   =   0         2      N     -   1                           y   k          (       2      N     -   1   -   m     )            cos        (         π        (     m   +   1     )       /   2        N     )                     (   10   )                         
     By letting θ=π/2N and T(θ m )=cos(mθ), we define                  Y   3          (     k   ,   j     )       =       ∑     m   =   0     j                         y   k          (     j   -   m     )            T        (     θ     m   +   1       )                   (   11   )                         
     By using the similar procedures described by Wang et al., we obtain the recursive formula as 
     
       
           Y   3 ( k,j )=cos(θ) y   k ( j )− y   k ( j −1)+2 cos(θ) Y   3 ( k,j− 1)− Y   3 ( k,j −2)  (12) 
       
     
     Based on the recursive index j, we obtain the z-transform of Eq. (12) as                    Y   3          (     k   ,   z     )           y   k          (   z   )         =         cos        (   θ   )       -     z     -   1           1   -     2        cos        (   θ   )            z     -   1         +     z     -   2                   (   12   )                         
     After 2N−1 recursive cycles depicted in Eq. (13), the transformed result can be obtained as Y 3 (k)=(−1)Y 3 (k, 2N−1). FIG. 1 shows the fixed-coefficient recursive structure for realizing the DCT-III transform according to the present invention. The fixed loop filter coefficient, 2 cos(π/2N), is executed in each of recursive cycles; however, the fixed output coefficient cos(π/2N) and the output sign coefficient −1 need to be performed only once for each frequency component. 
     2-2. Fixed Recursive Structure for DCT-IV 
     The N points DCT-IV depicted in Eq. (3) includes a discrete cosine transform base cos(π(2k+1)(2n+1)/2N) in which the indices n and k are periodic. Similarly, we only pay attention to the integer form, (2k+1)(2n+1)/2N. Then, the DCT-IV can be expressed as                  Y   4          (   k   )       =       ∑     n   =   0       N   -   1                         [       y        (   n   )              (     -   1     )       r   4         ]          cos        (     π                     m   4     /   4        N     )                   (   14   )                         
     where 
     
       
           r   4 =┌(2 n +1)(2 k+ 1)/4N┌ 
       
     
     and 
     
       
           m   4 =[(2 n +1)(2 k+ 1)mod 4N]. 
       
     
     For exploring its periodicity, we shall define another residue system. 
     Definition 2. A set of P odd integers a 1 , a 2 , . . . , a p  is the complete odd residue (COR) system in modulo 2P if and only if a i ≠a j  (mod 2P) for i≠j. 
     It is obvious that {(2n+1)mod 4N|n=0, 1, 2, . . . , 2N−1} sets up a COR system in modulo 4N. This complete odd residue system {1, 3, 5, . . . , 4N−1} is a reduced residue system. Because (2n+1)(2k+1) is an odd number for any integers n and k, (2n+1)(2k+1)mod 4N is also an odd number for any integers n and k. By Theorem 1 and letting a=2k+1, P=2N, and b=0, {(2n+1)(2k+1)mod 4N|n=0, 1, 2, . . . , 2N−1} which consists of 2N different odd integers, also sets up a COR system in modulo 4N. 
     However, {(2n+1)mod 4N|for n=0,1, . . . , N−1} which possesses N different odd integers r 0 , r 1 , . . . , r N−1 , supports only one half of the COR system in modulo 4N. From Theorem 1 and (2k+1, 4N)=1, it implies that {(2n+1)(2k+1)mod 4N for n=0,1, . . . , N−1} also sets up one half of the COR system in modulo 4N. It is similar to the discussion in the Eq. (6) so that the summation index in Eqs. (3) and (14) can be increased up to 2N−1 by appending dummy zero data. Thus, a new input sequence {tilde over (y)} k (2m+1) is defined as 
     
       
           {tilde over (y)}   k (2 m +1)=(−1) [(2n+1)(2k+1)/4N]   y ( n )  (15) 
       
     
     where 
     
       
         (2 m +1)=(2 n +1)(2 k +1)mod 4 N   (16) 
       
     
     for n=0, 1, . . . , 2N−1. For simplicity, we let y k (m)={tilde over (y)} k (2m+1). From Eqs. (15) and (16), after zero insertion, permutation, and sign change of inputs, the DCT-IV transform is expressed as 2N−1                  Y   4          (   k   )       =       ∑     m   =   0         2      N     -   1                           y   k          (   m   )            cos        (         π        (       2                 m     +   1     )       /   4        N     )                   (   17   )                         
     where k=0, 1, . . . , 2N−1. 
     To compute Eq. (17), we start to develop its recursive structure in the follows. By replacing m with 2N−1-m in Eq. (17), then                  Y   4          (   k   )       =       (     -   1     )            ∑     m   =   0         2      N     -   1                           y   k          (       2      N     -   1   -   m     )            cos        (         π        (     m   +     1   /   2       )       /   4        N     )                     (   18   )                         
     Let          θ   =     π     2      N         ,       θ   m     =       π        (     m   +     1   /   2       )         2      N         ,                   
     and T(θ m )=cos(θ m ), we define                  Y   4          (     k   ,   j     )       =       ∑     m   =   0     j                         y   k          (     j   -   m     )            T        (     θ   m     )                   (   19   )                         
     By using a similar method, the recursive formula is 
     
       
           Y   4 ( k,j )=cos(θ/2)[ y   k ( j )− y   k ( j −1)]+2 cos(θ) Y   4 ( k,j −1)− Y   4 ( k,j −2)  (20) 
       
     
     Based on the time recursive index j, the z-transform of the recursive formula is                    Y   4          (     k   ,   z     )           y   k          (   z   )         =         cos        (     θ   /   2     )            (     1   -     z     -   1         )         1   -     2        cos        (   θ   )            z     -   1         +     z     -   2                   (   21   )                         
     After 2N−1 recursive cycles depicted in (21), the transformed result can be obtained as Y 4  (k)=(−1) Y 4  (k, 2N−1). FIG. 2 shows the DCT-IV fixed-coefficient recursive structure according to the present invention. The fixed loop filter coefficient, 2 cos(π/2N), is executed in each of recursive cycles; however, the fixed output coefficient, −cos(π/4N), needs to be performed only once for each frequency component. ps 2-3. Fixed Recursive Structure for DCT-II 
     The N-point DCT-II stated in Eq. (1) possesses the cosine basis function as cos(π(2k+1)n/2N). Similarly, the integer factor (2k+1)n/2N will be investigated. Since (k, 2N)≠1 for even number k, it is not sure that {(2n+1)k mod 2N|n=0, 1, . . . , N−1} forms a CR or COR system in modulo 2N by Theorem 1. In order to obtain a fixed-coefficient recursive structure for the DCT-II, the DCT-II is converted to the DCT-IV first. 
     From the trigonometric property of cosine function, we can hold the equality as                cos                   (         (       2      n     +   1     )          (     k   +   1     )                   π       2                 N       )       =       2                 cos                   (         (       2      n     +   1     )        π       4      N       )        cos                   (         (       2      n     +   1     )          (       2      k     +   1     )        π       4      N       )       -     cos                   (         (       2      n     +   1     )        k                 π       2      N       )                 (   22   )                         
     By multiplying y(n) to each side of (22) and summing up every terms for both sides, another formula is obtained as follows,                  ∑     n   =   0       N   -   1              y        (   n   )          cos                   (         (       2      n     +   1     )          (     k   +   1     )                   π       2                 N       )         =         ∑     n   =   0       N   -   1            2        y        (   n   )                     cos                   (         (       2      n     +   1     )        π       4      N       )        cos                   (         (       2      n     +   1     )          (       2      k     +   1     )        π       4      N       )         -       ∑     n   =   0       N   -   1              y        (   n   )          cos                   (         (       2      n     +   1     )        k                 π       2      N       )                   (   23   )                         
     By using Eqs. (1) and (3) and letting 
     
       
           {tilde over (y)} ( n )=2 cos(θ n ) y ( n )  (24) 
       
     
     the relationship between DCT-II and DCT-IV is                  Y   2          (     k   +   1     )       =         ∑     n   =   0       N   -   1                y   ~          (   n   )                     cos                   (         (       2      n     +   1     )          (     k   +   1     )                   π       2                 N       )         -       Y   2          (   k   )                 (   25   )                         
     where 
     
       
         θ n =(2 n +1)π/4 N.   
       
     
     From Eq. (1), we know            Y   2          (   0   )       =       ∑     n   =   0       N   -   1              y        (   n   )       .                       
     If the fixed-coefficient DCT-IV recursive structure is employed to compute the DCT-IV of {tilde over (y)}(n), the DCT-II coefficients can be obtained recursively from Eq. (25). FIG. 3 shows the complete realization of fixed-coefficient recursive structure for computing the DCT-II. To obtain {tilde over (y)}(n), N multiplications are needed for a frequency-independent weighted procedure. However, this procedure can be merged into a normal window function so as to omit the computation of {tilde over (y)}(n). 
     3. Selected and Fixed Recursive Structures 
     In the previous section, the IIR filters depicted in Eqs. (13) and (21) are actually marginal stable since they have a pair of conjugate poles on the unit circle. The filter coefficient in the recursive loop for DCT-II, DCT-III, and DCT-IV is 2 cos(π/2N), which is closed to 2. In such a situation, the IIR filters have adjacent double poles on the unit circle, resulting in a nearly unstable condition. It is noted that this nearly unstable condition also happens in all prior recursive structures when they compute small frequency-index coefficients. This nearly unstable condition causes some extraordinarily magnified recursive errors, which produces inaccuracy of transformed results specially in finite-word-length machines. On the contrary, if we can select a much smaller loop coefficient, we can greatly improve the round-off error for the recursive DCT structures. In this section, the recursive structures are modified to compute the DCTs precisely. 
     3-1. Selected and Fixed Recursive Structure for DCT-III 
     By using number theory, we can further expand the structure developed in the section 2-1 for selecting a recursive filter so as to obtain more accurately transformed data in finite-word-length for the DCT-III fixed recursive structure. 
     From Theorem 1, we can further choose a=q, P=2N and b=0, in which q is relatively prime to 2N, i.e., (q, 2N)=1. Thus, {qm′ mod 2N|for m′=0, 1, . . . , 2N−1} also forms a CR system in modulo 2N. By extending Eqs. (8) and (9), a modified permuted input sequencey y′ k (m′) is obtained from Y k (m) as follows 
     
       
           y′   k ( m ′)=(−1) ┌qm′/2N┐   y   k ( m )  (26) 
       
     
     where 
     
       
           m=qm′  mod 2N  (27) 
       
     
     Thus, Eq. (8) is rewritten as                  Y   3          (   k   )       =       ∑       m   ′     =   0         2      N     -   1                y   k   ′          (     m   ′     )          cos                   (     π                     qm   ′     /   2        N     )                 (   28   )                         
     for k=1, 2, . . . , N−1. In Eq. (28), by substituting 2N−1−m′ for m′, the DCT-III becomes                  Y   3          (   k   )       =         (     -   1     )     q            ∑       m   ′     =   0         2      N     -   1                y   k   ′          (       2      N     -   1   -     m   ′       )          cos                   (     π                     q        (       m   ′     +   1     )       /   2        N     )                   (   29   )                         
     Let          θ   ⋒     =       q                 π       2      N                       
     and T(θ m′ )=cos(m′{circumflex over (θ)}), we define                  Y   3          (     k   ,   j     )       =       ∑       m   ′     =   0     j              y   k   ′          (     j   -     m   ′       )            T        (     θ       m   ′     +   1       )                   (   30   )                         
     By using similar procedure in Eq. (6), the recursive formula is obtained as follows, 
     
       
           Y   3 ( k,j )=cos({circumflex over (θ)}) y′   k ( j )− y′   k ( j −1)+2 cos({circumflex over (θ)}) Y   3 ( k,j −1)− Y   3 ( k,j −2)  (31) 
       
     
     Based on recursive index j, the z-transform of the recursive formula is obtained as                    Y   3          (     k   ,   z     )           y   k   ′          (   z   )         =         cos                   (     θ   ⋒     )       -     z     -   1           1   -     2                 cos                   (     θ   ⋒     )          z     -   1         +     z     -   1                   (   32   )                         
     After 2N−1 recursive cycles depicted in Eq. (32), we obtain Y 3 (k)=(−1) q Y 3 (k, 2N−1). The modified fixed-coefficient implementation of the DCT-III depicted in Eq. (23) is shown in FIG. 1, in which the fixed loop coefficient, the fixed output coefficient, and the output sign coefficient are replaced by 2 cos(qπ/2N), cos(qπ/2N), and(−1) q , respectively. 
     3-2. Selected and Fixed Recursive Structures for DCT-IV and DCT-II 
     Through the same procedure in the last section and Theorem 1, we now choose a=q, P=4N, and b=0, in which q is relative prime to 4N, i.e., (q, 4N)=1. Thus, we know that {q(2m′+1)mod 4N|m′=0, 1, . . . , 2N−1} also supports the COR system in modulo 4N. By extending Eqs. (15) and (16), the modified permuted input sequence {tilde over (y)}′ k (2m′+1) is obtained from {tilde over (Y)} k (2m+1) as 
     
       
           {tilde over (y)}′   k (2 m′ +1)=(−1) ┌q(2m′+1)/4N┐   {tilde over (y)}   k (2 m+ 1)  (33) 
       
     
     where the index mapping is depicted by 
     
       
         (2 m +1)= q (2 m′ +1)mod 4N  (34) 
       
     
     For simplicity, let y′ k (m′)={tilde over (y)}′ k (2m′+1), where m′=0, 1, . . . , 2N−1, and the modified DCT-IV is expressed by                  Y   4          (   k   )       =       ∑       m   ′     =   0         2      N     -   1                y   k   ′          (     m   ′     )          cos                   (     π                     q        (       2        m   ′       +   1     )       /   4        N     )                 (   35   )                         
     where k=1, 2, . . . , N−1. In Eq. (35), by substituting 2N−1−m′ for m′, the DCT-IV becomes                  Y   4          (   k   )       =         (     -   1     )     q            ∑       m   ′     =   0         2      N     -   1                y   k   ′          (       2      N     -   1   -     m   ′       )          cos                   (     π                     q   (       m   ′     +     1   /   2                  )     /   2        N     )                   (   36   )                         
     Letting            θ   ⋒     =       q                 π       2      N         ,       θ     m   ′       =         q        (       m   ′     +     1   /   2       )                     π       2                 N         ,                   
     and T(θ m′ )=cos(θ m′ ), we find 
     
       
           T (θ m′+1 )=2 cos({circumflex over (θ)}) T (θ m′ )− T (θ m′−1 )  (37) 
       
     
     Let                  Y   4          (     k   ,   j     )       =       ∑       m   ′     =   0     j              y   k     ′                       (     j   -     m   ′       )            T        (     θ     m   ′       )                   (   38   )                         
     By the same procedure as Eq. (6), a new recursive formula is obtained as follows 
     
       
           Y   4 ( k,j )=cos({circumflex over (θ)}/2)[ y′   k ( j )− y′   k ( j −1)]+2 cos({circumflex over (θ)}) Y   4 ( k,j −1)− Y   4 ( k,j −2)  (39) 
       
     
     Based on the recursive indexj, the z-transform of the recursive formula is expressed by              Y   4          (     k   ,   z     )           y   k   ′          (   z   )         =       cos                   (       θ   ⋒     /   2     )          (     1   -     z     -   1         )         1   -     2      cos                   (     θ   ⋒     )          z     -   1         +     z     -   2                           
     After 2N−1 recursive cycles depicted in Eq. (40), we obtain Y 4 (k)=(−1) q Y 4 (k, 2N−1), for k=0,1, . . . , N−1. The modified fixed-coefficient recursive DCT-IV depicted in Eq. (40) is shown in FIG. 2, in which the fixed loop and output coefficients are changed to 2 cos(qπ/2N) and (−1)q cos(qπ/4N), respectively. Hence, the modified fixed-coefficient DCT-II recursive structure as shown in FIG. 3 can be easily obtained by replacing the fixed loop and output coefficients with 2 cos(qπ/2N) and (−1)q cos(qπ/4N), respectively. 
     4. Comparisons of Accuracy and Complexity 
     In order to compare the precision and complexity, various DCT implementation algorithms are realized in finite-word-length machines by simulations. As the outputs of the direct DCTs calculated in 64-bit computers are treated as the noise-free results, Tables 1, 2, and 3 respectively show the averaged SNRs for computing 64-point, 16-point, and 8-point DCT-IIIs in different word lengths. All simulation results show that the proposed fixed-coefficient recursive algorithms with q=47(N=64), q=15(N=16), and q=7(N=8) can obtain the best precision in finite-word-length machines. When q=1, the proposed fixed-coefficient DCT recursive algorithms produce the most serious round-off errors specially for a long transformation length (N=64). 
     If the coefficient for recursive structure is selected improperly, the poles of recursive filter will be very close to each other, as shown in FIG. 7, and induce a nearly unstable state. After properly selecting coefficient for recursive structure and changing the coefficient to 2 cos(qπ/2N), the unstable condition can be avoided. In FIG. 8, the poles of a recursive filter are separated. on proper positions. The unstable situation will not be happened for a finite length of N recursive cycles. The direct implementation as depicted in Eqs. (1), (2), and (3) is a multiplication and addition structure, similar to an FIR filter structure, which is considered as one kind of recursive algorithms. There is a wide variety of cosine values of the multiplicand in the direct implementation causing larger round-off errors. After properly selecting fixed cosine values of multiplication, the recursive algorithm according to the present invention has a smaller and lower variety of cosine values, resulting in lower round-off errors. Therefore, the SNR values of the recursive algorithm according to the present invention are better than those of the direct implementation. 
     The traditional variable-coefficient recursive algorithms in average achieve considerably large round-off errors in the transformed results since they also have larger round-off errors in the transformed results in the smaller frequency-index components. In the variable-coefficient recursive algorithms, the quality of its first frequency component is similar to that of the fixed-coefficient recursive algorithms with q=1. It is noted that lower frequency components are usually more important than the larger ones in many analysis or compression applications. If the SNR of transformed results require about 80 dB, for example, an 18-bit machine for the proposed fixed-coefficient DCT-III recursive algorithm is chosen for the case of q=47 and N=64. From Table 1, it is found that the SNR of conventional recursive algorithms only reaches about 45.5 dB for the first frequency component. Thus, it needs 24-bit machines to maintain the SNR to obtain 80 dB. 
     With the above simulation results, we can evaluate the computational complexity and the throughput rate fairly. To be compared with conventional recursive algorithms which require 2N additions and N+1 multiplications for computing one frequency component, the proposed recursive computation needs 4N+1 additions and 2N+1 multiplications. The traditional recursive structure needs two general multipliers, two adders, and two delay buffers for DCT-III computation as described by Wang et al. The recursive scheme according to the present invention needs two fixed multipliers, three adders, two delay buffers, and the data manipulation unit for DCT-III computation. 
     To implement a 24-bit general multiplier, it requires 24 adders in 24-bit length for the conventional recursive structure to obtain 80 dB SNR. To realize a fixed coefficient multiplier, for example, with multiplicand 2 cos(47p/128)=0.810482628 10 =0.1100111101110111 2 , only 6 adders in 18-bit length as shown in FIG. 4 are taken to attain the same accurate performance. The fixed recursive structure requires a considerably low hardware complexity. 
     As to the data manipulation units for DCT-III and DCT-IV, please refer to FIG.  5 . The present invention provides the first kind of address generators completely based on an ROM. The input data are written into RAM in sequence. The ROM contains computed data of n(2k+1) mod 2N=qm′ mod 2N or (2n+1)(2k+1)mod 2N=q(2m′+1)mod 2N and the mapping sequence and the sign change from n to m′ for achieving an index mapping from m′ to n. A second kind of address generator according to the present invention is shown in FIG.  6 . The second kind of address generator includes an adder, a counter, a shifter, a comparator, an ROM, and a first-in-first-out (FIFO) buffer. Taking N=64 as an example, in a DCT-III situation, the output periodic sequence of the counter is {0, 1, 2, . . . , 127}, and the initial content of the FIFO buffer is {0, 1, 2, . . . , 127}. For a DCT-IV situation, the output periodic sequence of the counter is {1, 3, 5, . . . , 127} and the initial content of the FIFO buffer is {1, 3, 5, . . . , 127}. Actually, the modulo operation, which directly uses 7 least significant bits of the adders does not take any computation. Six LSBs are used to address the RAM and the carry bit is used to control the sign change. It is noted that the data manipulation unit can produce one pre-ordering input data per clock cycle before the recursive computation. 
     From the above discussions, it is obvious that the bottleneck of all DCT recursive structures is the multiplier associated with filter loop coefficient. The conventional 64-point DCT-III recursive structures normally require 64 iterations in use of a 24-bit general multiplier. The throughput rate of varied-coefficient recursive structures is limited by about 1536 24-bit additions for each transformed component. By performing 128 iterations, the proposed fixed-coefficient 64-point DCT-III recursive structures only need a fixed-coefficient multiplier (6 additions). The throughput rate of fixed-coefficient recursive structure is only limited by about 768 18-bit additions for each transformed component. Thus, the throughput rate of the fixed-coefficient recursive structures is overall two times faster than that of the varied-coefficient ones. Since the latency delay only happens in the recursive computation, the proposed recursive structures with twice throughput rate also only take one half of the latency delay as the varied-coefficient ones required. 
     5. Characteristics and Effectiveness 
     The present invention provides a new kind of fixed-coefficient recursive structures for computing discrete cosine transforms with a length of a power of two. By using number theorems, the fixed-coefficient recursive structures are developed from exploration of periodicity embedded in transform bases and form a complete residue system or a complete odd residue system. Based on permutation, sign change, and zero insertion of original inputs, the proposed filtering structures required fixed multipliers having low hardware complexity are better than the previous recursive methods which need general multipliers. In finite length machines, the present invention can properly select filter coefficients to avoid the nearly unstable state and to achieve low round-off errors in their transformations. 
     The discrete cosine transforms (DCTs) are widely applied to processing and compression of video and audio signals and other fields related to spectral analyses and parameter estimations. Besides, the recursive implementations are characterized in simple structure and local communication and are able to simplify the complexity in realization. A recursive DCT VLSI design is also characterized in formulation, regularization, and rank modulation. 
     While the invention has been described in terms of what are presently considered to be the most practical and preferred embodiments, it is to be understood that the invention needs not be limited to the disclosed embodiment. On the contrary, it is intended to cover various modifications and similar arrangements included within the spirit and scope of the appended claims which are to be accorded with the broadest interpretation so as to encompass all such modifications and similar structures. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Averaged SNRs (in dB) of the DCT-III Implementation 
               
               
                 Algorithms Realized in Finite-Word-Length Machines 
               
               
                 (N=64). 
               
            
           
           
               
               
               
            
               
                 Realization 
                 Word Length 
                   
               
            
           
           
               
               
               
               
               
            
               
                 Algorithms 
                 16 bits 
                 18 bits 
                 24 bits 
                 32 bits 
               
               
                   
               
               
                 Direct 
                 62.4488 
                 74.6779 
                 110.6337 
                 159.0172 
               
               
                 Implementation 
               
               
                 Variable 
                 48.7234 
                 58.8661 
                  94.2439 
                 142.3898 
               
               
                 Coefficient 
               
               
                 Fixed 
                 67.0644 
                 79.3099 
                 115.1087 
                 163.4046 
               
               
                 Coefficient(q=47) 
               
               
                 Fixed 
                 31.4115 
                 45.5038 
                  80.7936 
                 128.8874 
               
               
                 Coefficient(q=1) 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Averaged SNRs (in dB) of the DCT-III Implementation 
               
               
                 Algorithms Realized in Finite-Word-Length Machines 
               
               
                 (N=16). 
               
            
           
           
               
               
               
            
               
                 Realization 
                 Word Length 
                   
               
            
           
           
               
               
               
               
               
            
               
                 Algorithms 
                 16 bits 
                 18 bits 
                 24 bits 
                 32 bits 
               
               
                   
               
               
                 Direct 
                 74.4092 
                 86.3863 
                 122.5101 
                 171.6760 
               
               
                 Implementation 
               
               
                 Variable 
                 67.0540 
                 79.0705 
                 114.9538 
                 162.6671 
               
               
                 Coefficient 
               
               
                 Fixed 
                 75.4920 
                 87.4662 
                 124.2455 
                 172.4326 
               
               
                 Coefficient(q=15) 
               
               
                 Fixed 
                 60.5579 
                 73.1820 
                 108.5038 
                 156.6122 
               
               
                 Coefficient(q=1) 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 Averaged SNRs (in dB) of the DCT-III Implementation 
               
               
                 Algorithms Realized in Finite-Word-Length Machines (N=8). 
               
            
           
           
               
               
               
            
               
                 Realization 
                 Word Length 
                   
               
            
           
           
               
               
               
               
               
            
               
                 Algorithms 
                 16 bits 
                 18 bits 
                 24 bits 
                 32 bits 
               
               
                   
               
               
                 Direct 
                 86.0035 
                 101.2388 
                 134.3697 
                 184.4879 
               
               
                 Implementation 
               
               
                 Variable 
                 75.1634 
                  87.6756 
                 124.1262 
                 171.5373 
               
               
                 Coefficient 
               
               
                 Fixed 
                 88.9340 
                 101.7845 
                 138.7615 
                 183.7762 
               
               
                 Coefficient(q=7) 
               
               
                 Fixed 
                 59.3137 
                  70.4249 
                 106.9185 
                 155.0966 
               
               
                 Coefficient(q=1)