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Convolutions, Polynomials and Flipped Kernels
This is a post about multiplying polynomials, convolution sums and the connection between them. Multiplying polynomials Suppose we want to multiply one polynomial by another: This is basic middle-school math - we start by cross-multiplying: And then collect all the terms together by adding up the coefficients: Let's look at a slightly different way to achieve the same result.
The theory of signals and systems is a large topic (typically taught for one or two semesters in undergraduate engineering degrees), but here I want to focus on just one aspect of it which I find really elegant. We'll combine these and use linearity again (note that in the following are just constants); the response to a signal decomposed into a sum of shifted and scaled impulses: Just like with polynomials , the reason why one of the inputs is flipped is clear from the definition of the convolution sum, where one of the the indices increases ( k), while the other decreases ( n-k).
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