Corner-turn complexity properties of polynomial transform 2D convolution methods

This paper examines the matrix data re-order requirements of a variety of polynomial transform 2D convolution methods which can be employed to efficiently accommodate large field problems. The results indicate that several power-of-2 length polynomial transform methods developed by Nussbaumer allow one to totally avoid the row-column data corner-turn commonly encountered in Fourier Transform 2D convolution methods, while also providing significantly reduced computational complexity. Execution time comparison with an FFT reference base is made assuming the use of general register and array processor units and use of recently developed matrix-transpose methods by Eklundh and Ari to support 2D Fourier Transform corner-turn requirements. These results demonstrate a 2-4 times throughput performance improvement for use of the polynomial transform method in place of the 2D Fourier Transform approach to circularly convolve large 2D fields in the range 1024×1024 to 8192×8192.