# Báo cáo hóa học: " Research Article Novel VLSI Algorithm and Architecture with Good Quantization Properties for a "

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## Báo cáo hóa học: " Research Article Novel VLSI Algorithm and Architecture with Good Quantization Properties for a "

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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Novel VLSI Algorithm and Architecture with Good Quantization Properties for a

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## Nội dung Text: Báo cáo hóa học: " Research Article Novel VLSI Algorithm and Architecture with Good Quantization Properties for a "

3. EURASIP Journal on Advances in Signal Processing 3 for k = 1, . . . , N where π ⎧ with α = . (2) (N − 1) ⎪ 2N ⎨ϕ (k ) if ϕ (k) ≤ , ψ (k) = ⎪ 2 √ (7) ⎩ To simplify the presentation, the constant coeﬃcient 2/N ϕ (N − 1 + k ) otherwise, will be dropped from (1) that represents the deﬁnition of DCT; a multiplier will be added at the end of the VLSI array to scale the output sequence with this constant. with We will reformulate relation (1) as a parallel decomposi- ⎧ tion based on a cycle and a pseudo-cycle convolution forms ⎨ϕ(k ) if k > 0, using a single new input restructuring sequence as opposed ϕ (k ) = ⎩ to [16] where two such auxiliary input sequences were used. ϕ(N − 1 + k) otherwise , (8) Further, we will use the proprieties of DCT kernel and those of the Galois Field of indexes to appropriately permute the ϕ(k) = g k N , auxiliary input and output sequences. Thus, we will introduce an auxiliary input sequence {xa (i) : i = 0, 1, . . . , N − 1}. It can be recursively deﬁned where x N denotes the result of x modulo N and g is the as follows: primitive root of indexes. We have also used the properties of the Galois Field of xa (N − 1) = x(N − 1), (3) indexes to convert the computation of DCT as a convolution form. xa (i) = (−1)i x(i) + xa (i + 1), (4) for i = N − 2, . . . , 0. 3. Examples Using this restructuring input sequence, we can reformu- late (1) as follows: To illustrate our approach, we will consider two examples for 1D DCT, one with the length N = 7 and the primitive root N −1 g = 3 and the other with the length N = 11 and the primitive (−1)ϕ(i) X (0) = xa (0) + 2 root g = 2. i=0 (N − 1) 3.1. DCT Algorithm with Length N = 7. We recursively × xa ϕ(i) − xa ϕ i + , 2 compute the following input auxiliary sequence {xa (i) : i = 0, . . . , N − 1} as follows: X (k) = [xa (0) + T (k)] · cos(kα), for k = 1, . . . , N − 1. (5) xa (6) = x(6), The new auxiliary output sequence {T (k) : k = 1, 2, (9) xa (i) = (−1)i x(i) + xa (i + 1), . . . , N − 1} can be computed in parallel as a cycle and pseudo- for i = 5, . . . , 0. cycle convolutions if the transform length N is a prime number as follows: Using the auxiliary input sequence {xa (i) : i = 0, . . . , N − 1}, (N −1)/ 2 we can write (6) in a matrix-vector product form as (−1)ξ (k,i) T (δ (k)) = i=1 ⎡ ⎤ T (4) ⎢ ⎥ (N − 1) ⎢ ⎥ · xa ϕ(i − k ) − xa ϕ i − k + ⎢T (2)⎥ ⎣ ⎦ 2 T (6) × 2 · cos ψ (i) × 2α , ⎡ ⎤ −[xa (3) − xa (4)] −[xa (2) − xa (5)] −[xa (6) − xa (1)] ⎢ ⎥ (N −1)/ 2 ⎢ xa (3) − xa (4) −[xa (2) − xa (5)]⎥ (−1)ζ (i) = ⎢ [xa (6) − xa (1)] ⎥ T γ(k) = ⎣ ⎦ i=1 xa (2) − xa (5) −[xa (6) − xa (1)] xa (3) − xa (4) (N − 1) ⎡ ⎤ · xa ϕ(i − k ) + xa ϕ i − k + c(2) 2 ⎢ ⎥ ⎢ ⎥ · ⎢c(1)⎥, ⎣ ⎦ (N − 1) × 2 · cos ψ (i) × 2α for k = 0, 1, . . . , , c(3) 2 (6) (10)
4. 4 EURASIP Journal on Advances in Signal Processing ⎡ ⎤ T (3) Finally, the output sequence {X (k) : k = 1, 2, . . . , N − 1} can ⎢ ⎥ ⎢ ⎥ ⎢T (5)⎥ be computed as ⎣ ⎦ T (1) X (1) = [xa (0) + T (1)] · cos[α], ⎡ ⎤ X (2) = [xa (0) + T (2)] · cos[2α], xa (3) + xa (4) xa (2) + xa (5) xa (6) + xa (1) ⎢ ⎥ ⎢ ⎥ = ⎢xa (6) + xa (1) xa (3) + xa (4) xa (2) + xa (5)⎥ X (3) = [xa (0) + T (3)] · cos[3α], (11) ⎣ ⎦ xa (2) + xa (5) xa (6) + xa (1) xa (3) + xa (4) X (4) = [xa (0) + T (4)] · cos[4α], ⎡ ⎤ X (5) = [xa (0) + T (5)] · cos[5α], c(2) (13) ⎢ ⎥ ⎢ ⎥ X (6) = [xa (0) + T (6)] · cos[6α], · ⎢−c(1)⎥, ⎣ ⎦ −c(3) 3 (−1)ϕ(i) X (0) = xa (0) + 2 where we have noted c(k) for 2 · cos(2kα). i=1 The index mappings δ (i) and γ(i) realize a partition into (N − 1) two groups of the permutation of indexes {1, 2, 3, 4, 5, 6}. × xa ϕ(i) − xa ϕ i + . 2 They are deﬁned as follows: 3.2. DCT Algorithm with Length N = 11. We recursively {δ (i) : 1 − 4, 2 − 2, 3 − 6}, → → → compute the following input auxiliary sequence {xa (i) : i = (12) γ(i) : 1 −→ 3, 2 −→ 5, 3 −→ 1 . 0, . . . , N − 1} as follows: The functions ξ (k, i) and ζ (i) deﬁne the sign of terms in xa (10) = x(10), (10), respectively, as follows: (14) xa (i) = (−1)i x(i) + xa (i + 1) for i = 9, . . . , 0. 111 ξ (k, i) is deﬁned by the matrix and 001 Using the auxiliary input sequence {xa (i) : i = 0, . . . , N − 1} 010 ζ (i) is deﬁned by the vector [0 1 1]. we can write (6) in a matrix-vector product form as ⎡ ⎤ ⎡ ⎤ T (2) xa (2) − xa (9) −(xa (4) − xa (7)) xa (8) − xa (3) xa (5) − xa (6) xa (10) − xa (1) ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ T (4) ⎥ ⎢ xa (10) − xa (1) −(xa (2) − xa (9)) −(xa (4) − xa (7)) −(xa (8) − xa (3)) −(xa (5) − xa (6))⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ T (8) ⎥ = ⎢−xa (5) − xa (6) −(xa (10) − xa (1)) −(xa (2) − xa (9)) xa (8) − xa (3) ⎥ −(xa (4) − xa (7)) ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ T (6) ⎥ ⎢ xa (8) − xa (3) xa (4) − xa (7) ⎥ xa (5) − xa (6) −(xa (10) − xa (1)) −(xa (2) − xa (9)) ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ T (10) xa (4) − xa (7) −(xa (8) − xa (3)) xa (5) − xa (6) −(xa (10) − xa (1)) xa (2) − xa (9) ⎡ ⎤ c(4) ⎢ ⎥ ⎢ ⎥ ⎢c(3)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ · ⎢c(5)⎥, (15) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢c(1)⎥ ⎢ ⎥ ⎣ ⎦ c(2) ⎡ ⎤ ⎡ ⎤⎡ ⎤ T (9) xa (2) + xa (9) xa (4) + xa (7) xa (8) + xa (3) xa (5) + xa (6) xa (10) + xa (1) c(4) ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢T (7)⎥ ⎢xa (10) + xa (1) xa (2) + xa (9) xa (4) + xa (7) xa (8) + xa (3) (5) + xa (6) ⎥ ⎢−c(3)⎥ xa ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢T (3)⎥ = ⎢ xa (5) + xa (6) xa (10) + xa (1) xa (2) + xa (9) xa (4) + xa (7) xa (8) + xa (3) ⎥ · ⎢−c(5)⎥, ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢T (5)⎥ ⎢ xa (8) + xa (3) xa (5) + xa (6) xa (10) + xa (1) xa (2) + xa (9) xa (4) + xa (7) ⎥ ⎢−c(1)⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦ T (1) xa (4) + xa (7) xa (8) + xa (3) xa (5) + xa (6) xa (10) + xa (1) xa (2) + xa (9) c(2)