VLSI Computational Architectures for the Arithmetic Cosine Transform

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1 VLSI Computational Architectures for the Arithmetic Cosine Transform T.Anitha 1,Sk.Masthan 1 Jayamukhi Institute of Technological Sciences, Department of ECEWarangal , India Assistant ProfessorJayamukhi Institute of Technological Sciences, Department of ECE Abstract The discrete cosine transform (DCT) is a widely-used and important signal processing tool employed in a plethora of applications. Typical fast algorithms for nearly-exact computation of DCT require floating point arithmetic, are multiplier intensive, and accumulate round-off errors. Recently proposed fast algorithm arithmetic cosine transform (ACT) calculates the DCT exactly using only additions and integer constant multiplications, with very low area complexity, for null mean input sequences. The ACT can also be computed non-exactly for any input sequence, with low area complexity and low power consumption, utilizing the novel architecture described. However, as a trade-off, the ACT algorithm requires 10 non-uniformly sampled data points to calculate the eight-point DCT.This requirement can easily be satisfied for applications dealing with spatial signals such as image sensors and biomedical sensor arrays, by placing sensor elements in a non-uniform grid. In this work, a architecture for the computation of the null meanact is proposed, followed by a novel architectures that extend the ACT for non-null mean signals. All circuits are synthesized using the Xilinx ISE. Index Terms Discrete cosine transform, arithmetic cosine transform, fast algorithms, VLSI I. ITRODUCTIO THE Signal processing is a method to analyze the characteristics of a signal like storage, amplification, compression, reconstruction etc. The Discrete Cosine Transform (DCT) is a signal processing technique which converts a signal from its spatial domain in to frequency components. The existing DCT calculations include floating point operations which lead to computational errors caused by rounding off the values. The Arithmetic Cosine Transform (ACT) algorithm consists of only addition and constant multiplication which in turn reduces the computation error. The ACT can be used to calculate the exact and approximate value of the DCT for null mean and non null mean input sequences respectively. The algorithm is with less area complexity and consumes low power. The main application of DCT is in data compression. Its property states that the DCT coefficient contains most of the relevant information about the image so that it can be used in image compression applications. The DCT is mainly used in JPEG applications which are the algorithms for glossy image compression. Some applications include automatic surveillance, geospatial remote sensing, traffic cameras, satellite based imaging, automotives.this paper unfolds as follows. In Section II basic of discrete cosine transform. In Section III ACT Architecture. In SectionIV, the fundamental mathematical operations of the ACT are briefly described. Section V computing arithmetic mean.in section VI discuss Mertens Correction Factor and on-null mean input signals. In Section VII brings the simulation results and Conclusion furnished in Section VIII. II. DISCRETE COSIE TRASFORM The Discrete Cosine Transform is a conventional signal processing technique used in number of applications. It s property states that the DCT coefficients contains most of the relevant information about the image so that it can be used in image compression applications.the Arithmetic Cosine Transform algorithms (ACT) are used for quick computation of DCT. The ACT consists of only additions and multiplying with constant value. This overcomes the errors associated with rounding off the values when floating point operations are come in to picture. The exact evaluation of ACT is possible if the input data are non-uniformly sampled and has zero mean. This paper unfolds two main issues (i) calculation of mean value of input signal in case of non-uniformly sampled data and (ii) proposition of IJIRT ITERATIOAL JOURAL OF IOVATIVE RESEARCH I TECHOLOGY 51

2 efficient architectures of ACT for calculating the 8 point DCT when the input data are considered as only non-uniform samples. III. ARITHMETIC COSIE TRASFORM The Arithmetic Cosine Transforms (ACT) is a speedy algorithm for evaluating the DCT of non-uniformly sampled input data. The incoming signal to the DCT are generally treated as continuous signal u(t), that are uniformly sampled. This produces the column vector with dimension. Its DCT is represented by the vector. The corresponding non-uniform samples of the input sequence u (t) are required to compute the vector U. The sampling instants are given by s = r k 1 (1) Where k = 1, -1 and r = 0,1, k 1 Substituting for r = 0, k = 1 gives s = 1 / For r = 1, k = implies s = 15 / Substituting in the same way, all other values of set S is obtained. A set S with sampling points as the elements is defined as S = {All values of S} () For an 8-point DCT, s s = { 1, 5, 13, 7, 7, 57, 9, 59, 89, 15 } (3) Here the ACT algorithm is represented in two ways to compute the DCT for zero mean sequence and non-zero mean sequence. When considering zeromean sequence, the ACT averages as A K = 1 k 1 v k r=0 r (4) k 1 The above ACT averages can be used in the evaluation of DCT of non-uniform input samples by using the expression v k = [ K ] μ j. s j=1 kl (5) Where k = 1,,-1 and μ(.) is called the Mobius function. The ACT is derived by using the Mobius inversion formula. In case of non-null mean input signal, a correction term is subtracted from the equation of v k to calculate the DCT coefficients and it follows as v k [ K ] μ j. s j=1 kl u. M( 1 k ) (6) v = 1 8 w. v r (7) Withw as the interpolation weight n M (n) = μ(r) Where M (n) is the Mertens function IV. r=1 (8) ACT ARCHITECTURES This paper introduces architectures for the ACT which accepts only non-uniform samples as inputs and calculates the DCT with reduced area complexity and low power consumption. All the above explained methodologies are used for the design of these architectures. There are registers are introduced at different nodes for the temporary storage which gives a fully pipelined structure to the design. This pipelined structure reduces the critical path delay with a slight increase in the latency. Architecture I corresponds to the ACT architecture for computing the DCT of null mean input signals. This architecture can be realized using (4) and (5). This architecture is done with only additions and constant multiplication with integers which reduces the truncation error and complexity. The Architecture I shown in Fig.1 corresponds to =8 which takes 10 non-uniform samples as inputs according to the values of the set S given by (3). The applications dealing with zero mean input signal uses this architecture with the advantages of less complex computation and area. The second architecture is used for the calculation of DCT which has non-null mean in-put signals. It is desired to calculate the mean value of the incoming non-uniform samples. The Mertens correction function is included as per (6). Architecture II consists of Architecture I, mean calculation block and Mertens correction block as shown in Fig.4. Whereu is considered as the arithmetic average of the input uniform samples given by IJIRT ITERATIOAL JOURAL OF IOVATIVE RESEARCH I TECHOLOGY 5

3 VI. MERTES CORRECTIO FACTOR For non null mean input sequence, it is required to subtract a modification term in order to get the DCT coefficients. This is called the Mertens correction term, M(n). This term is the sum of the Mobius function. Fig. 1 Architecture I for ull mean input signalsv. V. COMPUTIG ARITHMETIC MEA The calculation of mean value is required if the incoming sig-nals are of non null mean type. The architecture in Fig.(3) is realized using (7). Here the input signals are scaled by the sampling instants and are given as input to the mean value calculation block. In the next step, each inputs are multiplied by the corresponding interpolation weight. The final step of mean value computation is to add all these values which will be the mean value of the incoming non null mean sequence. Fig. 3 Mertens correction block Fig. Mean value calculation Fig. 4 Architecture II for on-null mean input signals IJIRT ITERATIOAL JOURAL OF IOVATIVE RESEARCH I TECHOLOGY 53

4 VII. SIMULATIO RESULTS Fig. 7 Mean Calculation Block Fig. 5 on-null Mean input signals ACT Architecture Fig. 8 Mertens Correction block Fig. 6 ull Mean ACT Block VIII. COCLUSIO The ACT algorithm is suitable for calculating the eight point DCT coefficients exactly using only adders and integer constant multiplications, also with low computational complexity. ACT architectures for null mean inputs as well as for non-null mean inputs are proposed, synthesized and simulated on Xilinx ISE design suite. The average percentage error and PSR were adopted as figures of merit to assess the measured results. Results show that even for lower fixed point word-lengths, the implementations lead to acceptable margins of error. The resource utilization for various fixed point implementations indicate a trade-off between accuracy and device resources (chip area, speed,and power). It is the first step towards new research on low power and low IJIRT ITERATIOAL JOURAL OF IOVATIVE RESEARCH I TECHOLOGY 54

5 complexity computation of the DCT by means of the recently proposed ACT. REFERECES [1] ilanka Rajapaksha, Arjuna Madanayake, Renato J. Cintra, And Jithra Adikari VLSI Computational Architectures For The Arithmetic Cosine Transfor IEEE TRASACTIOS O COMPUTERS, VOL. 64, O. 9, SEPTEMBER 015 [] C. Chakrabarti and J. J_aJ_a, Systolic architectures for the computation of the discrete Hartley and the discrete cosine transforms based on prime factor decomposition, IEEE Trans. Comput., vol. 39, no. 11, pp , ov [3] F. A. Kamangar and K. R. Rao, Fast algorithms for the -D discrete cosine transform, IEEE Trans. Comput., vol. 31, no. 9,pp , Sep [4] H. Kitajima, A symmetric cosine transform, IEEE Trans. Comput.,vol. 1, no. 4, pp , Apr [5] S. Yu and E. Swartziander Jr,, DCT implementation with distributed arithmetic, IEEE Trans. Comput., vol. 50, no. 9, pp ,Sep [6] V. Britanak, P. Yip, and K. R. Rao, Discrete Cosine and Sine Transforms.Amsterdam, The etherlands: Academic Press, 007. [7]. Romaand L. Sousa, Efficient hybrid DCTdomain algorithm for video spatial downscaling, EURASIP J. Adv. Signal Process.,vol. 007, no., pp , 007. [8] H. Lin and W. Chang, High dynamic range imaging for stereoscopic scene representation, in Proc. 16th IEEE Int. Conf. Image Process., ov. 009, pp [9] E. Magli and D. Taubman, Image compression practices and standards for geospatial information systems, in Proc. IEEE Int.Geosci. Remote Sens. Symp., Jul. 003, vol. 1, pp [10] M. Bramberger, J. Brunner, B. Rinner, and H. Schwabach, Real time video analysis on an embedded smart camera for traffic surveillance, in Proc. 10th IEEE Real-Time Embedded Technol. Appl. Symp., May. 004, pp [11]. Ahmed, T. atarajan, and K. R. Rao, Discrete cosine transform, IEEE Trans. Comput., vol. 3, no. 1, pp , Jan IJIRT ITERATIOAL JOURAL OF IOVATIVE RESEARCH I TECHOLOGY 55