The Elegant Connection Between Polynomial Multiplication, Convolution, and Signal Processing
This post explores the connection between polynomial multiplication, convolution, and signal processing. It begins by visually explaining polynomial multiplication using tables and diagrams, revealing its fundamental nature as a convolution operation. The post then introduces discrete signals and systems, focusing on linear time-invariant (LTI) systems. It explains that any signal can be decomposed into a sequence of scaled and shifted impulse signals, and the response of an LTI system can be calculated using convolution. Finally, it briefly touches upon the properties of convolution and its relationship to the Fourier transform, highlighting that the Fourier transform of a convolution equals the product of the Fourier transforms of its operands, enabling efficient convolution computation.