General Reversible Integer Transform Conversion

In this paper, we introduce an algorithm, which is named the triangular matrix scheme, to convert every reversible discrete linear transform into a reversible integer transform. The integer transform is a special case of the discrete linear transforms whose entries can be expressed as a summation of 2 k. It is much more efficient than the non-integer transform since it can be implemented by a fix-point processor and no floating-point processor is required. Moreover, for a binary valued sequence, it is unnecessary to convert it into a floating-point one before doing the transform. The 2 k-point integer discrete Fourier, sine, cosine, Hartley, and wavelet transforms have been derived in pervious works by the prototype matrix method and the lifting scheme. In this paper, with the proposed triangular matrix scheme, we can convert any reversible discrete linear transform into a reversible integer transform. The triangular matrix scheme is very flexible. Except for reversibility, there is no other constraint for the original non-integer transform to be converted. We also show some examples that use the triangular matrix scheme to derive the 7-point and 210-point integer DFT and the integer Karhunen Loeve Transform.