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

Result: Linear phase e simply a rigid shift in time, no distortion Nonlinear phase e distortion as well as shift DT

1. DTFT Properties and Examples 2. Duality in fs& ft 3. Magnitude/Phase of Transforms and Frequency Responses

Convergence Issues Synthesis equation: None. since integrating over a finite interval Analysis Equation: Need conditions analogous to CTFT, e. g

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

Convolution Property 0(t)=h(t)*(t)←→Y(j)=H(ju)X( where h(t)←→H(ju) A consequence of the eigenfunction property

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

Another(important! ) example: Periodic Impulse train n(t)=∑(t Sampling function important for sampling .2T

Representation of ct signals Approximate any input x(t) as a sum of shifted, scaled

Portrait of Jean Baptiste Joseph Fourier Image removed due to copyright considerations Signals& Systems, 2nd ed. Upper Saddle River, N.J. Prentice Hall, 1997, p 179