1. 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: 11 X 2 &it; k = 2 N &it; &varepsilon; 2 &it; k &it; &Sum; n = 0 N - 1 &it; x n &it; cos &it; &it; ( 2 &it; n + 1 ) &it; 2 &it; κπ 2 &it; N = 2 N &it; &varepsilon; k &it; { &Sum; n = 0 N 2 - 1 &it; x n &it; cos &it; &it; ( 2 &it; n + 1 ) &it; κπ 2 &it; ( N / 2 ) + &Sum; n = N / 2 N - 1 &it; x n &it; cos &af; ( 2 &it; n + 1 ) &it; κπ 2 &it; ( N / 2 ) } = 2 N &it; &varepsilon; k &it; { &Sum; n = 0 N 2 - 1 &it; y n &it; cos &af; ( 2 &it; n + 1 ) &it; κπ 2 &it; ( N / 2 ) + &Sum; n = 0 N 2 - 1 &it; x N - 1 - n &it; cos &it; [ [ 2 &it; ( N - 1 - n ) + 1 ] &it; κπ 2 &it; ( N / 2 ) ] } = 1 2 &it; { 2 N / 2 &it; &varepsilon; k &it; &Sum; n = 0 N 2 - 1 &it; y n &it; cos &it; &it; ( 2 &it; n + 1 ) &it; κπ 2 &it; ( N / 2 ) + 2 N / 2 &it; &varepsilon; k &it; &Sum; n = 0 N 2 - 1 &it; z n &it; cos &it; &it; ( 2 &it; n + 1 ) &it; κπ 2 &it; ( N / 2 ) } = 1 2 &af; [ Y k + ( - 1 ) k &it; Z k ] = 1 2 &af; [ Y k + Z k ′ ] &it; &NewLine; &it; k = 0, 1, … &it;, ( N / 2 ) - 1 &it; &NewLine; &it; where &it; &it; Z k ′ &it; &it; is &it; &it; the &it; &it; DCT &it; &it; of &it; &it; z n ′ = x N - 1 - n, n = 0, 1, … &it;, ( N / 2 ) - 1. 12 X 2 &it; k + 1 + X 2 &it; k - 1 = &it; 2 N &it; { &Sum; n = 0 N - 1 &it; x n &it; &it; cos &it; &it; ( 2 &it; n + 1 ) &it; &it; ( 2 &it; k + 1 ) &it; π 2 &it; N + &it; &Sum; n = 0 N - 1 &it; x n &it; cos &it; &it; ( 2 &it; n + 1 ) &it; &it; ( 2 &it; k - 1 ) &it; π 2 &it; N } = &it; 1 &varepsilon; k &it; 1 2 &it; { 2 N / 2 &it; &varepsilon; k &it; &Sum; n = 0 N 2 - 1 &it; ( y n - z n ′ ) &it; 2 &it; &it; cos &it; &it; ( 2 &it; n + 1 ) &it; π 2 &it; N &it; cos &it; &it; ( 2 &it; n + 1 ) &it; k &it; &it; π 2 &it; ( N / 2 ) } &it; &it; or &it; X 2 &it; k + 1 = 1 &varepsilon; k &it; 1 2 &it; { 2 N / 2 &it; &varepsilon; k &it; &Sum; n = 0 N 2 - 1 &it; r n &it; cos &it; &it; ( 2 &it; n + 1 ) &it; k &it; &it; π 2 &it; ( N / 2 ) } - X 2 &it; k - 1, &NewLine; &it; where &it; &it; k = 0, 1, … &it;, ( N / 2 ) - 1 &it; &it; and r n = ( y n - z n ′ ) &it; 2 &it; &it; cos &it; &it; ( 2 &it; n + 1 ) &it; π 2 &it; N = { 2 N / 2 &it; &Sum; l = 0 N 2 - 1 &it; &varepsilon; l &af; ( Y l - Z l ′ ) &it; cos &it; &it; ( 2 &it; n + 1 ) &it; l &it; &it; π 2 &it; ( N / 2 ) } &it; 2 &it; cos &it; &it; ( 2 &it; n + 1 ) &it; π 2 &it; N