This course reviews the notation for roots of polynomial expressions describing Linear-Time Invariant (LTI) systems in the frequency domain, and relates the operator notation to the time-domain response using complex exponential notation. A single pole circuit is introduced and responses analyzed in the frequency domain and time domain. An ideal delay is introduced for comparison and sets a reference for step response behaviors.

Polynomial root locations are described in the complex s-plane and complex conjugate pairs plotted and described using (w0, z) notation as well as (t0, Q) notation. Phasor notation is introduced for evaluation of steady-state sinusoidal excitation of transfer functions. Second-order, complex conjugate pole pairs are introduced and the asymptotic behaviors developed and contrasted to the single-pole behaviors in magnitude, phase, and group delay attributes. Straight-line approximations are produced and the errors of approximation discussed.

Classical Butterworth, Chebyshev, and Bessel filters are introduced and the construction formulae developed. The Cauer filter is also illustrated, but mathematical development using elliptic functions is not included. Frequency domain and time domain responses are developed using a 4th design form as representative of even-order forms and a 5th order design as representative of odd-order forms. Only the 5th order Bessel filter example is synthesized from the equations. A 5th order equalizer is examined for the 5th order Chebyshev and Cauer filter and shown to provide equivalent results for both Chebyshev and Cauer filters. An additional pole pair is added to the 5th order equalizer and the justification and improvements noted. Transformations are discussed to convert low-pass prototype designs to high-pass and band-pass filters.

The course is designed for a practicing engineer seeking a capability for designing and specifying filters and equalizers for frequency domain and time domain applications.