Question: Are there sets of basic\ signals so that a) We can represent rich classes of signals as linear combinations of these building block signals b) The response of ltI Systems to these basic signals are both simp and insightful Fact: For LtI Systems(CT or dt) there are two natural choices for these building blocks Focus for now:

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

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

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

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

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

More efficient to transmit e&m signals at higher frequencies Transmitting multiple signals through the same medium using different carriers Transmitting through\channels, with limited passbands

1.aM with an Arbitrary Periodic Carrier 2. Pulse Train Carrier and Time-Division Multiplexing 3. Sinusoidal Frequency modulation 4. DT Sinusoidal am 5. DT Sampling, Decimation,d Interpolation

Properties of ct Rational system functions a) However, if H(s)is rational, then The system is causal The roc of H(s) is to the right of the rightmost pole

Why use feedback? Reducing effects of nonidealities Reducing Sensitivity to Uncertainties and variability Stabilizing Unstable Systems Reducing Effects of Disturbances Tracking Shaping system response Characteristics(bandwidth/speed