· Perform relevant EE421 objectives

· Derive the Laplace transform from the Fourier transform

· Derive and apply properties of the Laplace transform

· Understand and evaluate the region of convergence of Laplace transforms

· Derive and apply the Final Value Theorem

· Understand the relationship between the Laplace transform and Fourier transform

· Find the Laplace transform of functions which are periodic for t > 0

· Find the Laplace transform of functions which are non-zero for all time

· Find frequency domain models of circuits and systems including models for initial conditions

· Find transfer functions directly from circuits including multi-input, multi-output circuits

· Obtain Bode plots from transfer functions

· Obtain Bode plots from steady-state sinusoidal plots

· Obtain Bode plots for underdamped second-order terms

· Obtain transfer functions from Bode plots

· Find sinusoidal steady-state responses from Bode plots

· Obtain block diagrams of systems

· Obtain transfer functions from block diagrams

· Utilize negative feedback to force systems to track reference inputs

· Employ feedback to stabilize systems

· Use MATLAB to:

· Determine the stability of systems from impulse responses and transfer functions

· Obtain the State-Variable model for linear circuits and systems

· Use the Laplace Transform to solve the State-Variable model)

· Find transfer functions from state variable models

· Find block diagrams and state variable models from transfer functions using:

· Understand and determine eigenvalues and eigenvectors of a square matrix A

· Understand the concepts of asymptotic stability, marginal stability and instability and determine the stability of state variable models

· Form the matrices P and S and find Ak=PSkP-1

· Find the state transition matrix, eAt=PeStP-1 , and solve the zero-input state variable model

· Use Laplace transforms to find the state transition matrix, eAt

· For an nxn matrix A, prove and apply the Cayley Hamilton theorem to find:

· Use two methods to find, eAt , when A has repeated eigenvalues

· Use the convolution theorem of the Laplace transform to derive and apply the time domain solution to the state variable model:

· Use the similarity transformation, x=Pz, to decouple a given state variable model

· From a decoupled state variable model, determine

· Find a state feedback control, w=-Kpvx, to set the eigenvalues of a single input system in phase variable form

· Find the controllability matrix, M=[B AB ... An-1B] and find a similarity transformation, Tpv=MMpv-1 which will put a controllable state model into phase variable form

· Design a state feedback control, w=-KpvTpv-1x, to force real systems to meet settling time specifications

· Implement your state feedback control using op-Amps

· Model the (ideal) sampling process and derive a frequency domain for Xs(f)

· From the frequency domain effects of sampling, understand the Nyquist Sampling theorem

· Model the quantizing process and derive an expression for the MSE due to sampling

· Derive the Z-transform from the Laplace transform of xs(t) and prove simple Z-transform theorems including the region of convergence

· Use two methods to discretize a continuous state variable model

· Use the Z-transform to solve the discrete state variable model and identify the zero-input and zero-state portion of the solution

· Find the Z-transform of discrete signals which are periodic after k=0

· Prove and apply the Z-transform Convolution Property

· Compute the discrete state transition matrix, Ak, using:

· Derive and apply the solution to the discrete state variable model

· Understand and apply the concepts of linearity and shift invariance to discrete systems

· Use the similarity transformation, x(kT)=Pz(kT), to decouple a discrete state variable model

· From a decoupled discrete state variable model, determine:

· Find a discrete state feedback control, w(kT)=-Kpvx(kT), to set the eigenvalues of a single input discrete system in phase variable form

· Find a similarity transformation, Tpv, which will put a controllable state model into phase variable form

· Design a discrete state feedback control, w(kT)=-KpvTpv-1x(kT), to force discrete systems to meet settling time specifications

· Find block diagrams and state variable models from discrete transfer functions using:

· Perform graphical convolution to find the output of digital filters with a given impulse response

· Implement IIR digital filters using microprocessors and A/D-D/A converters

· Derive and apply the bilinear transform starting with a trapezoidal integrator

· Understand and apply digital IIR filter (correct) invariant design techniques

· Obtain frequency responses of digital filters using z=ejw T

· Obtain approximate Bode plots of digital filters using the w-plane transformation

· Understand and apply the FIR design equation to obtain non-causal FIR filters

· Use shift and truncation techniques to obtain realizable FIR filters

· Use direct form II methods to realize FIR filters

· Implement FIR digital filters using microprocessors and A/D-D/A converters

· Use windows to reduce the effects of truncation

· Find the DFT or FFT of discrete signals

· Prove simple properties of the DFT

· Use the DFT in the following practical applications