Input–Output Stability

In this recitation, we review the ∞-norm, 1-norm, and 2-norm on the vector spaces L n p (IR), where p = ∞, 1, or 2 respectively (or n for discrete time signals) discussed in lecture. The Fourier p Transform is defined and a proof for Parseval's relation is presented. A dynamic system could be thought of as an operator that acts on a set of signals, just as a m × n matrix is an operator that acts on the vector space IR n. We are interested in considering the effect of our dynamic system on particular sets of signals, specifically, vector spaces whose elements are certain signals and on which we can define a notion of the size of a signal (i.e. a norm). Once we have defined our vector space and norm, we can define a notion of size or " maximum amplification " of the dynamic system, just as we had done for matrices (recall that the induced norm was a measure of maximum amplification of a matrix). The size or induced norm of the system will depend on the norm we pick, so we quickly review signal norms below.