Parseval's theorem

Parseval’s theorem

This is a pretty useful theorem in signal processing field as well as in courses related with random processes. Sometimes a signal or a random process in time domain is hard to calculate, thus we want to transform it into transformation domain(e.g. frequency domain, z domain, etc.) to facilitate math. In Oppenheim’s Signals, Systems & Inference and Therrien’s random process we talked about some random processes in time domain which were sent pass through a linear system, and we need to calculate some features of output signal regarding the input signal, or vice versa.

For the energy or power of a certain signal, we might take advantage of Parseval’s theorem, since it yields some insights of how we understand the power of a signal in different domains. However, sometimes it need a little deduction to figure it out. Hereby I paste my primary dedction of it.