An active area of research in supercomputing is concerned with mapping certain sums, such as discrete Fourier transforms (DFT) and discrete cosine transforms (DCT) to multi-processor arrays. This paper presents two kinds of systolic arrays for 2D-DFT using the flow diagram of row-column de composition algorithm. One is a linear array of N₁ processors (if the DFT is N₁×N₂), and it takes O(N₁N₂)time steps. The speed-up of this array over the sequential implementation of the row-column decomposition algorithm on a single processor is O(N)(IF N₁=N₂=N). This result is currently optimal not only in PE numbers, but a1so in time cost. The other is a rectagular array of N₁×N₂ processors and it takes O(N₁+N₂)steps. The specdup of it over the sequential implementation of the row-cotumn decomposition algorithm on a single processor is O(N²)(IF N= N₁=N₂). At last, the paper gives two systolic arrays of 2D-DCT,which are similar to that of 2D-DFT. Furthermore, these systolic arrays can be easily generalized to multi-dimensional cases.