5.2 Discrete Cosine Transform

dct(X[, type=2])

Replaces the columns of a dense real matrix with their discrete cosine transforms. The second argument, an integer between 1 and 4, denotes the type of transform (DCT-I, DCT-II, DCT-III, DCT-IV). The DCT-I transform requires that the row dimension of X is at least 2. These transforms are defined as follows (for a matrix with n rows).

                                 k            n∑- 2
 DCT -I:   X [k,:] :=  X [0,:]+ (- 1)X [n - 1,:]+ 2    X[j,:]cos(πjk∕(n - 1)),   k = 0,...,n - 1.
                                               j=1
                       n∑-1
DCT -II:   X [k,:] :=  2   X [j,:]cos(π(j + 1∕2)k∕n),  k = 0,...,n - 1.
                       j=0
                               n∑-1
DCT -III:   X [k,:] :=  X [0,:]+ 2    X [j,:]cos(πj(k + 1∕2)∕n),    k = 0,...,n - 1.
                               j=1
                       n∑-1
DCT -IV:    X [k,:] :=  2   X [j,:]cos(π(j + 1∕2)(k + 1∕2)∕n),   k = 0,...,n- 1.
                       j=0

idct(X[, type=2])

Replaces the columns of a dense real matrix with the inverses of the discrete cosine transforms defined above.