Can be continuous Trajectory of a space shuttle Mass density in a cross-section of a brain · Can be discrete dNa base sequence Digital image pixels Can be 1-D,2-D,·N-D For this course: Focus on a single(1-D) independent variable which we call“time
Fouriers derivation of the ct fourier transform x(t)-an aperiodic signal view it as the limit of a periodic signal as t→∞ For a periodic signal the harmonic components are spaced Oo=2π/ T apart. AsT→∞,Obo→>0, and harmonic components are space
Fourier series: Periodic signals and lti Systems ()=∑H(k k= ak一→H(ko)ak “g Soak-→|H(jkco)lkl H(7k)=1H(k0e∠B(ko) or powers of signals get modified through filter/system ncludes both amplitude phase akeJhwon
SAMPLING We live in a continuous-time world most of the signals we encounter are CT signals, e.g. x(). How do we convert them into Dt signals x[n? Sampling, taking snap shots of x(t) every Seconds
Motivation for the Laplace transform CT Fourier transform enables us to do a lot of things, e. g Analyze frequency response of lTi systems Sampling Modulation Why do we need yet another transform? One view of Laplace Transform is as an extension of the Fourier
