Patent ID: 6868188

Claim:
An encoder having means for calculating the DCT of a sequence of length N/2, N being a positive, even integer, and having means for calculating a DCT of length N directly from two sequences of length N/2 representing the first and second half of an original sequence of length N, characterised in that the means for calculating DCTs of length N/2 are arranged to calculate the even coefficients of a DCT of length N as: X 2 ⁢ k = ⁢ 2 N ⁢ ɛ 2 ⁢ k ⁢ ∑ n = 0 N - 1 ⁢ x n ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ 2 ⁢ κπ 2 ⁢ N = ⁢ 2 N ⁢ ɛ k ⁢ { ∑ n = 0 N 2 - 1 ⁢ x n ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ κπ 2 ⁢ ( N / 2 ) + ∑ n = N / 2 N - 1 ⁢ x n ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ κπ 2 ⁢ ( N / 2 ) } = ⁢ 2 N ⁢ ɛ k ⁢ { ∑ n = 0 N 2 - 1 ⁢ y n ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ κπ 2 ⁢ ( N / 2 ) + ⁢ ∑ n = 0 N 2 - 1 ⁢ x N - 1 - n ⁢ cos ⁢ [ [ 2 ⁢ ( N - 1 - n ) + ⁢ 1 ] ⁢ κπ ⁢ 2 ⁢ ( N / 2 ) ] } = ⁢ 1 2 ⁢ { 2 N / 2 ⁢ ɛ k ⁢ ∑ n = 0 N 2 - 1 ⁢ y n ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ κπ 2 ⁢ ( N / 2 ) + ⁢ 2 N / 2 ⁢ ɛ k ⁢ ∑ n = 0 N 2 - 1 ⁢ z n ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ κπ 2 ⁢ ( N / 2 ) } = ⁢ 1 2 ⁡ [ Y k + ( - 1 ) k ⁢ Z k ] = 1 2 ⁡ [ Y k + Z k ′ ] ⁢ ⁢ k = 0 , 1 , … ⁢ , ( N / 2 ) - 1. ⁢ where ⁢ ⁢ Z k ′ ⁢ ⁢ is ⁢ ⁢ the ⁢ ⁢ DCT ⁢ ⁢ of ⁢ ⁢ z n ′ = x N - 1 - n , n = 0 , 1 , … ⁢ , ( N / 2 ) - 1. And the odd index coefficients as X 2 ⁢ k + 1 + X 2 ⁢ k - 1 == ⁢ 2 N ⁢ { ∑ n = 0 N - 1 ⁢ x n ⁢ ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ ⁢ ( 2 ⁢ k + 1 ) ⁢ π 2 ⁢ N + ⁢ ∑ n = 0 N - 1 ⁢ x n ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ ⁢ ( 2 ⁢ k - 1 ) ⁢ π 2 ⁢ N } = ⁢ 1 ɛ k ⁢ 1 2 ⁢ { 2 N / 2 ⁢ ɛ k ⁢ ∑ n = 0 N 2 - 1 ⁢ ( y n - z n ′ ) ⁢ 2 ⁢ ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ π 2 ⁢ N ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ k ⁢ ⁢ π 2 ⁢ ( N / 2 ) } ⁢ ⁢ or ⁢ X 2 ⁢ k + 1 = 1 ɛ k ⁢ 1 2 ⁢ { 2 N / 2 ⁢ ɛ k ⁢ ∑ n = 0 N 2 - 1 ⁢ r n ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ k ⁢ ⁢ π 2 ⁢ ( N / 2 ) } - X 2 ⁢ k - 1 , ⁢ where ⁢ ⁢ k = 0 , 1 , … ⁢ , ( N / 2 ) - 1 ⁢ ⁢ and r n = ( y n - z n ′ ) ⁢ 2 ⁢ ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ π 2 ⁢ N = { 2 N / 2 ⁢ ∑ l = 0 N 2 - 1 ⁢ ɛ l ⁡ ( Y l - Z l ′ ) ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ l ⁢ ⁢ π 2 ⁢ ( N / 2 ) } ⁢ 2 ⁢ cos ⁢ ⁢ ( 2 ⁢ n + 1 ) ⁢ π 2 ⁢ N .