The integer transforms analogous to discrete trigonometric transforms

The integer transform (such as the Walsh transform) is the discrete transform that all the entries of the transform matrix are integer. It is much easier to implement because the real number multiplication operations can be avoided, but the performance is usually worse. On the other hand, the noninteger transform, such as the DFT and DCT, has a good performance, but real number multiplication is required. W derive the integer transforms analogous to some popular noninteger transforms. These integer transforms retain most of the performance quality of the original transform, but the implementation is much simpler. Especially, for the two-dimensional (2-D) block transform in image/video, the saving can be huge using integer operations. In 1989, Cham had derived the integer cosine transform. Here, we will derive the integer sine, Hartley, and Fourier transforms. We also introduce the general method to derive the integer transform from some noninteger transform. Besides, the integer transform derived by Cham still requires real number multiplication for the inverse transform. We modify the integer transform introduced by Cham and introduce the complete integer transform. It requires no real number multiplication operation, no matter what the forward or inverse transform. The integer transform we derive would be more efficient than the original transform. For example, for the 8-point DFT and IDFT, there are in total four real numbers and eight fixed-point multiplication operations required, but for the forward and inverse 8-point complete integer Fourier transforms, there are totally 20 fixed-point multiplication operations required. However, for the integer transform, the implementation is simpler, and many of the properties of the original transform are kept.