Computing running DCTs and DSTs based on their second-order shift properties

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University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering Information Sciences 000 Computing running DCTs DSTs based on their second-order

au Joe F. Chicharo University of Wollongong, chicharo@uow.edu.

1 University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering Information Sciences 000 Computing running DCTs DSTs based on their second-order shift properties Jiangtao Xi University of Wollongong, jiangtao@uow.edu.au Joe F. Chicharo University of Wollongong, chicharo@uow.edu.au Publication Details This article was originally published as: Xi, J & Chicharo, JF, Computing running DCTs DSTs based on their second-order shift properties, IEEE Transactions on Circuits Systems-I: Fundamental Theory Applications, May 000, 47(5), Copyright IEEE 000. Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: research-pubs@uow.edu.au

2 Computing running DCTs DSTs based on their second-order shift properties Abstract This paper presents a set of second-order recursive equations which are referred to as the second-order shift (SOS) properties of the discrete cosine transform (DCT) the discrete sine transform (DST). The proposed SOS properties enable independent updating of the respective DCT DST coefficients. This is in direct contrast with existing methodology for computing the running DCT DST there is an inherent interdependency between the DCT DST coefficients. The SOS properties provide more efficient algorithms in terms of computational burden memory requirements when implementing running DCTs DSTs. Keywords DCT, discrete cosine transform, discrete sine transform Disciplines Physical Sciences Mathematics Publication Details This article was originally published as: Xi, J & Chicharo, JF, Computing running DCTs DSTs based on their second-order shift properties, IEEE Transactions on Circuits Systems-I: Fundamental Theory Applications, May 000, 47(5), Copyright IEEE 000. This journal article is available at Research Online:

3 IEEE TRASACTIOS O CIRCUITS AD SYSTEMS I: FUDAMETAL THEORY AD APPLICATIOS, VOL. 47, O. 5, MAY REFERECES [] B. Razavi, Challenges in portable RF transceiver design, IEEE Circuits Devices, pp. 6, Sept [] F. H. Raab, Idealized operation of the class E tuned power amplifier, IEEE Trans. Circuits Syst., vol. CAS-4, pp , Dec [3], Effects of variations on the class E tuned power amplifier, IEEE J. Solid-State Circuits, vol. SC-3, pp , Apr [4] J. A. Blanchard J. S. Yuan, Effects of collector current exponential decay on power efficiency for class E tuned power amplifier, IEEE Trans. Circuits Syst.I, vol. 4, pp. 69 7, Jan [5] M. J. Chudobiak, The use of parasitic nonlinear capacitors in class E amplifiers, IEEE Trans. Circuits Syst. I, vol. 4, pp , Dec [6] T. Sowlati et al., Low voltage, high efficiency GaAs class E power amplifier for wireless transmitters, IEEE J. Solid-State Circuits, vol. 30, pp , Oct [7]. O. Sokal F. H. Raab, Harmonic output of class E RF power amplifier load coupling network design, IEEE J. Solid-State Circuits, vol. SC-, pp , Feb [8]. M. guyen R. G. Meyer, Si IC-compatible inductors LC passive filters, IEEE J. Solid-State Circuits, vol. 5, pp , Aug [9] C. Li Y. Yam, Maximum frequency optimum performance of class E power amplifiers, Proc. Inst. Elect. Eng., vol. 4, no. 3, June 994. [0] J. Chang, A. A. Abidi, M. Gaitan, Large suspended inductors on silicon their use in a -m CMOS RF amplifier, IEEE Electron. Device Lett., vol. 4, May 993. []. O. Sokal A. D. Sokal, Class E, a new class of high efficiency tuned single-ended switching power amplifiers, IEEE J. Solid-State Circuits, vol. SC-0, pp , June 975. Computing Running DCT s DST s Based on Their Second-Order Shift Properties Jiangtao Xi Joe F. Chicharo Abstract This paper presents a set of second-order recursive equations which are referred to as the second-order Shift (SOS) properties of the discrete cosine transform (DCT) the discrete sine transform (DST). The proposed SOS properties enable independent updating of the respective DCT DST coefficients. This is in direct contrast with existing methodology for computing the running DCT DST there is an inherent interdependency between the DCT DST coefficients. The SOS properties provide more efficient algorithms in terms of computational burden memory requirements when implementing running DCT s DST s. Index Terms DCT, discrete cosine transform, DST. I. ITRODUCTIO The discrete cosine transform (DCT) [] has been successfully applied to the fields of speech image processing. In order to compute the DCT efficiently, various fast efficient block- based algorithms have been proposed (for example, see [] [5] their references). However, in the case the DCT parameters need to be updated for every new signal sample, the running DCT implementation is required. As indicated by [7], the computational burden for the running DCT is rather intensive. One approach for computing the running DCT s discrete sine transforms (DST s) is to use their shift properties, as derived by Yip Rao [6]. The shift properties are a set of recursive equations that can be used for updating the DCT s DST s coefficients. However, this approach is not very efficient in terms of computation. A source of the excessive computational burden is the dependency between a DCT coefficient its corresponding DST coefficient. The process of updating a DCT (or DST) requires updating the corresponding DST (or DCT). To alleviate this problem, Murthy Swamy [7] proposed an approach for DCT-II, DST-II, DCT-IV, DST-IV. In [7] each transform member was represented as the real part of a complex function recursive equations were derived for updating these complex functions. In other words, the approach updates complex functions rather than transform coefficients. It is obvious that there is still some excessive unnecessary computation. This paper proposes a more efficient class of running algorithms for DCT s DST s by deriving a set of recursive equations which enable the independent updating of each DCT DST member respectively. The proposed recursive equations are called second-order shift (SOS) properties due to their second-order nature. By contrast, the shift properties derived by Yip Rao [6] are referred to as the first-order shift (FOS) properties, as they are in effect first-order recursive equations. This paper is organized as follows.: Section II reviews the FOS properties of DCT s DST s. The SOS properties for the DCT s DST s are derived presented in Section III. A performance analysis of running DCT s DST s based on SOS properties is undertaken in Section IV the results are compared with the most recent approaches reported in literature thus far. Finally, Section V concludes the paper. II. FIRST-ORDER SHIFT PROPERTIES OF DCT s AD DST s As defined in [8], the family of DCT s DST s for the signal block x(n 0 ), x(n 0 +); ;x(n) is given as follows [8] DCT-I: DCT-II: P m k P m x(n 0 + m)cos k ; for k =0; ; ; () P k 0 x(n 0 + m) cos m + k ; for k =0; ; ;0 () 0 P m x(n 0 + m) cos m k + ; for k =0; ; ;0 (3) Manuscript received September 6, 997; revised March 6, 999. This paper was recommended by Associate Editor J. Götze. The authors are with the School of Electrical, Computer, Telecommunications Engineering, The University of Wollongong, Wollongong, SW 5, Australia. Publisher Item Identifier S 057-7(00) P k 0 x(n0 +m) cos m + k + for k =0; ; ;0 (4) ; 057 7/00$ IEEE

4 780 IEEE TRASACTIOS O CIRCUITS AD SYSTEMS I: FUDAMETAL THEORY AD APPLICATIOS, VOL. 47, O. 5, MAY 000 DST-I: x(n 0 + m)sin m k ; for k =; ; ;0 (5) DST-II: s(n +;k)=a k s(n; k) 0 B k c(n; k) + P kd k x(n 0 ) 0 (0) k x(n) (0b) DST-II: DST-III: P k x(n 0 + m) sin m 0 k ; for k =; ; ; (6) P mx(n 0 + m) sin m k + ; for k =; ; ; (7) c(n +;k) = E k c(n; k) +F k s(n; k) + 0 p E k x(n 0 )+ p 0 x(n 0 +) +(0) k 0 p F k x(n) : (a) s(n +;k) = E k s(n; k) 0 F k c(n; k) + DST-IV: 0 x(n0 +m) sin m + k + ; for k =0; ; 0 (8) p F k x(n 0 )+(0) k 0 p + (0)k p x(n +) E k x(n) (b) P j = if j 6= 0nor p if j =0or is the block length of the transform. For the case when the input signal is a stream of samples, c(n; k) s(n; k) are referred to as the running DCT s DST s at time n. When a new sample x(n + ) becomes available, the signal block of interest shifts one sample in time to include x(n +) exclude the sample x(n 0 +).In this case, the running DCT s at the instant n +is determined by the following expressions [6]: DCT-I: c(n +;k) DST-I: = A k c(n; k) +P k B k s(n; k) + P k 0 p A k x(n 0 )+ p 0 x(n 0 +) +(0) k 0 p kx(n) +(0) k p x(n +) (9a) s(n +;k)=a k s(n; k) 0 B k c(n; k) + p x(n 0 )+ 0 p DCT-II: c(n +;k)=a k c(n; k) +B k s(n; k) B k (0) k x(n) (9b) + P kc k (0) k x(n) 0 x(n 0 ) (0a) c(n +;k)=e k c(n; k) +F k s(n; k) + 0G k x(n 0 )+(0) k H k x(n) : (a) s(n +;k)=e k s(n; k) 0 F k c(n; k) + H k x(n 0 )+(0) k G k x(n) (b) A k =cos k B k = sin k C k = cos k D k = sin k (k +) E k = cos (k +) F k = sin (k +) G k = cos 4 (k +) H k = sin : 4 Equations (9) () were derived by Yip Rao [6] were called the shift properties, which can be used to update the transform coefficients. Consider the DCT-II as a particular example. Clearly (0a) can be used to update the DCT-II coefficients. However, the updated DCT-II is related to the DST- II coefficients. In other words, updating

5 IEEE TRASACTIOS O CIRCUITS AD SYSTEMS I: FUDAMETAL THEORY AD APPLICATIOS, VOL. 47, O. 5, MAY the DCT-II also requires the updating of DST-II, this can be performed by (0b). Consequently, both DCT-II DST-II must be updated at the same time, even in the case when only one of them is required. This problem exists for other transforms as well is a source of excessive computational burden. III. SECOD-ORDER SHIFT PROPERTIES OF DCT s AD DST s As mentioned in Section I it is desirable to derive recursive equations which enable the independent updating of DCT s DST s, respectively. We present the derivation for DCT-II DST-II in detail. Similar derivations can be performed for the remaining transforms the final results will be listed without proof in the interest of brevity. Let us consider the DCT-II DST-II. The objective is to derive two independent second-order one-variable recursive equations which are equivalent to (0a) (b). We begin by taking the z transform of (0a) (b) as follows: zc(z; k) =A k C(z; k) +B k S(z; k) + P kc k (0) k 0 z 0 X(z) (3a) zs(z; k) =A k S(z; k) 0 B k C(z; k) + P kd k z 0 0 (0) k X(z) (3b) C(z; k), S(z; k), X(z) are the z transforms of c(n; k), s(n; k) x(n), respectively. From (3a) (3b) we can obtain C(z; k) = B kz 0 S(z; k) 0 A k z0 + P kc k (0) k 0 z 0 z 0 S(z; k) = 0B kz 0 C(z; k) 0 A k z0 0 A k z 0 X(z) (4a) P kd k z 0 0 (0) k z 0 + X(z): (4b) 0 A k z 0 Substituting (4a) into (4b) after some manipulation we have ( 0 A k z 0 + z 0 )C(z; k) = P kc k (0) k z 0 0 (0) k z 0 0 z 0(+) + z 0(+) X(z): (5a) In the same way, substituting (4b) into (4a) after some manipulation we have ( 0 A k z 0 + z 0 )S(z; k) = P kd k (0) k z 0 0 (0) k z 0 + z 0(+) + z 0(+) X(z): Taking the inverse z transform of (5a) (5b) yields c(n +;k)=a k c(n; k) 0 c(n 0 ; k)+t k ; (0) k x(n) 0 (0) k x(n 0 ) (5b) 0x(n 0 )+x(n 0 0 ) (6) s(n +;k)=a k s(n; k) 0 s(n 0 ; k)+u k ; 0(0) k x(n) 0 (0) k x(n 0 ) +x(n 0 )+x(n 0 0 ) (7) T k ; = P kc k U k ; = P kd k : Equations (6) (7) represent the second-order shift properties for DCT-II DST-II. Close examination of (6) (7) indicate that they are independent of each other, hence, can be used to recursively update the DCT-II DST-II, respectively. It is interesting to note that there are similarities between (6) (7) the frequency sampling structure for FIR filter implementation [9]. The second-order shift properties for other members of the DCT DST families can be obtained using a similar approach the results are listed as follows: DCT-I: c(n +;k) =A k c(n; k) 0 A k + P k B k c(n 0 ; k) + T k ; x(n +)+T k ; x(n) +T k ; 3x(n 0 ) 0 T k ; 4x(n 0 +)+T k ; 5x(n 0 ) + T k ; 6x(n 0 0 ) (8) T k ; =(0) k T k ; =(0) 0k 0 p A k P k T k ; 3 = 0(0) 0k 0 T k ; 4 = 0 P k T k ; 5 = 0 p A k P k P k T k ; 6 = P k : (9) ote that in (8) the coefficient, [A k + P kbk ], is always equal to one when k 6= 0 k 6=. This means that this coefficient introduces a multiplication only when k =0or k = DST-I: s(n +;k) =A k s(n; k) 0 A k + P k B k s(n 0 ; k) + U k ; x(n) +U k ; x(n 0 ) + U k ; 3x(n 0 ) + U k ; 4x(n 0 0 ) (0) U k ; =(0) 0k B k U k ; =(0) k 0 U k ; 3 = B k 0 P k p + 0 U k ; 4 = A k B k (P k 0 ) 0 ; A k ( 0 P k )B k ; P k ; : ()

COMPUTATIOAL BURDE TABLE II COMPARISO OF MEMORY LOCATIOS DST-III: s(n +;k)=e k s(n; k) 0 s(n 0 ; k)+u3; k x(n +) +U3; k x(n) +U3; k 3x(n 0 ) + U3; k 4x(n 0 ) (4) DST-IV: U3; k =(0) k U3; k =(0) k 0 U

6 78 IEEE TRASACTIOS O CIRCUITS AD SYSTEMS I: FUDAMETAL THEORY AD APPLICATIOS, VOL. 47, O. 5, MAY 000 c(n +;k) =E k c(n; k) 0 c(n 0 ; k)+t k 3; x(n) + T3; k x(n 0 +)+T3; k 3x(n 0 ) + T3; k 4x(n 0 0 ) () T k 3; =(0) k F k T k 3; = 0 T k 3; 3 = 0 p E k T k 3; 4 = : (3) TABLE I COMPARISO OF COMPUTATIOAL BURDE TABLE II COMPARISO OF MEMORY LOCATIOS DST-III: s(n +;k)=e k s(n; k) 0 s(n 0 ; k)+u3; k x(n +) +U3; k x(n) +U3; k 3x(n 0 ) + U3; k 4x(n 0 ) (4) DST-IV: U3; k =(0) k U3; k =(0) k 0 U k 3; 3 = 0 (0) k 0 F k 0 E k F k + E k F k U k 3; 4 = 0 F k: (5) s(n +;k) c(n +;k) =E k c(n; k) 0 c(n 0 ; k) + T k 4; [x(n) +x(n 0 )] + T k 4; [x(n 0 ) 0 x(n 0 0 )] (6) T k 4; =(0) k H k T k 4; =0G k : (7) =E k s(n; k) 0 s(n 0 ; k)+u4; k [x(n) 0 x(n 0 )] + U4; k [x(n 0 ) 0 x(n 0 0 )] (8) U4; k =(0) k G U 4; k = H k : (9) Equations (6) (9) provide a new approach for implementing a running DCT DST algorithm. IV. PERFORMACE AALYSIS The computation of running DCT s DST s based on SOS properties is expected to provide certain advantages over the first-order approach proposed by Yip Rao [6] as well as the algorithm proposed by Murthy Swamy [7]. Let us first compare the computational burden associated with the proposed approaches against those by Yip Rao [6] Murthy Swamy [7]. Taking DCT-II as the key example, from (9a) (b) it is clear that the first-order approach needs the following computation: 6 multiplications with coefficients; 4 addition of terms; additions for [(0) k x(n) 0 x(n 0 )] for even odd k. Similarly, from (6) the proposed second-order approach requires the following operations: multiplications with coefficients; additions of terms; four additions for (0) k [x(n) 0 x(n 0 )] 0 x(n 0 )+x(n 0 0 ) for odd even k. Hence, for DCT-II the second-order approach can reduce the number of multiplications by 67% when compared with the first-order approach. Table I lists the results for all the transform members. Clearly, in all cases the computational burden associated with the proposed approaches is significantly reduced when compared to both references [6] [7]. ow let us consider the memory requirements. ote that in the case of [7] there is no improvement in terms of the memory requirements when compared with the first-order approach [6]. This is because both the real imaginary part of the complex function need storage locations. Hence, we only compare the proposed algorithm with the firstorder approach [6]. For both algorithms, memory is required to store ) the coefficients of the recursive equations; ) transform results; 3) the input signal samples. Table II depicts the comparison between these two methods. As expected, the memory requirements for the proposed approach are either equal to or better than the first-order algorithm [6].

7 IEEE TRASACTIOS O CIRCUITS AD SYSTEMS I: FUDAMETAL THEORY AD APPLICATIOS, VOL. 47, O. 5, MAY V. COCLUSIO In this paper, we have derived the SOS properties for DCT s DST s. The result is a new approach for computing the running DCT s DST s. The computational burden memory requirements associated with the proposed approach have been investigated compared with the first-order approach proposed by Yip Rao [6], as well as the approach proposed by Murthy Swamy [7]. It has been shown that the proposed approach is characterized by reduced computational burden generally lower memory storage requirements. REFERECES []. Ahmed, T. atarajan, K. R. Rao, Discrete cosine transform, IEEE Trans. Comput., vol. C-3, pp , Jan [] W. H. Chen, C. H. Smith, S. C. Fralick, A fast computational algorithm for the discrete cosine transform, IEEE Trans. Commun., vol. COM-5, pp , Sept [3] B. G. Lee, A new algorithm to compute the discrete cosine transform, IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-3, pp , Dec [4] M. Malvar, Fast computation of discrete cosine transform through fast Hartley transform, Electron. Lett., vol., no. 7, pp , March 986. [5] Y.-H. Chan W.-C. Siu, Mixed-radix discrete cosine transform, IEEE Trans. Signal Processing, vol. 4, pp , ov [6] P. Yip K. R. Rao, On the shift properties of DCT s DST s, IEEE Trans. Acoust., Speech, Signal Processing, vol. 35, pp , Mar [7]. R. Murthy M.. S. Swamy, On the computation of running discrete Cosine Sine transforms, IEEE Trans. Signal Processing, vol. 40, pp , June 99. [8] Z. Wang, Fast algorithms for the discrete W transform for the discrete Fourier transform, IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-3, pp , Aug [9] A. V. Oppenheim R. W. Schafer, Digital Signal Processing. Englewood Cliffs, J: Prentice-Hall, 975.

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