Sampled signals as a vector space
The sampled signal is supposed to look like a vector, just as you learned in linear algebra. It just happens to be infinitely long. You can still treat it as you would regular vectors: add, subtract, pointwise multiply, scalar multiply, and take inner products. For instance, write
It is a nuisance to keep writing all these zeros in the infinite vectors, so sometimes we shorten things to condensed vectors, like and . In this form, we have the 0-th entry in as the first number that appears in the short vector. So , and the rest are zero. We do need to be careful when working with two or more vectors, that we continue to align the appropriate entries. So for instance must align with , must align with , and so on.