301: A Log on DSP

The mathematical definition of ‘not blowing up’ is that

there is an Element of M (an element of R, positive valued) Such that for all time, the absolute value of y(t) is less than M.

An LTI system is BIBO Stable iff:

There exists some M, an element of R, such that the integral of the absolute value of the impulse response over all time is less than M. Essentially, the impulse response must be an element of L1, meaning that it is absolutely integrable. Now a proof:
BIBO if h has an absolute integral.
given h is ai, we have a BIBO System
We most prove that for an arbitrary x(t) (Bounded), y(t) Bounded
The Integral over all time of x(t)*h(t-tao)dtao=abs(y(t))<=abs(x(t))*abs(h(t-tao))dtao=int(B*H)

Principles of proofs:
Iff: If and only if.
Given constraints, we want to choose the optimal choice.