At the completion of EE422, you should be able to perform the following objectives:

· Perform relevant EE421 objectives (Lecture #1 and review HW#1)

· Derive the Laplace transform from the Fourier transform (Lecture #2)

· Derive and apply properties of the Laplace transform (Lecture #2, HW#2 and HW#3)

· Understand and evaluate the region of convergence of Laplace transforms (Lecture #3 and HW#3)

· Derive and apply the Final Value Theorem (HW#2)

· Understand the relationship between the Laplace transform and Fourier transform ( Lecture #3 and HW#3)

· Find the Laplace transform of functions which are periodic for t > 0 (Lecture #3 and HW#3)

· Find the Laplace transform of functions which are non-zero for all time (Lecture #3 and HW#3)

· Find frequency domain models of circuits and systems including models for initial conditions (Lecture #4 and HW#4)

· Find transfer functions directly from circuits including multi-input, multi-output circuits (Lecture #4 and HW#4)

· Obtain Bode plots from transfer functions (Lecture #5 and HW#5)

· Obtain Bode plots from steady-state sinusoidal plots (Lecture #5 and HW#5)

· Obtain Bode plots for underdamped second-order terms (Lecture #5 and HW#5)

· Obtain transfer functions from Bode plots (Lecture #5 and HW#5)

· Find sinusoidal steady-state responses from Bode plots (HW#5 and HW#7)

· Obtain block diagrams of systems (Lecture #6 and HW#6)

· Obtain transfer functions from block diagrams (Lecture #6 and HW#6)

· Utilize negative feedback to force systems to track reference inputs (HW#6)

· Employ feedback to stabilize systems (HW#7)

· Use MATLAB to (Lecture #7 and HW#7):

- Find Bode Plots and determine bandwidth

- Determine the step response of systems and determine steady-state error and settling time (2%)

- Plot root loci and thereby determine gain values to meet control specifications

· Determine the stability of systems from impulse responses and transfer functions (Lecture #8 and HW#8)

· Obtain the State-Variable model for linear circuits and systems (Lecture #8 and HW#8)

· Use the Laplace Transform to solve the State-Variable model (Lecture #9)

· Find transfer functions from state variable models (Lecture #9 and HW#9)

· Find block diagrams and state variable models from transfer functions using (Lecture #9 and HW#9):

- Parallel decomposition

- Cascade decomposition

- Phase variables

· Understand and determine eigenvalues and eigenvectors of a square matrix A.( Lecture #10 and HW#10)

· Understand the concepts of asymptotic stability, marginal stability and instability and determine the stability of state variable models ( Lecture #10 and HW#10)

· Form the matrices P and S and find A^k=PS^kP^-1 ( Lecture #10 and HW#10)

· Find the state transition matrix, e^At =Pe^StP^-1, and solve the zero-input state variable model ( Lecture #10 and HW#10)

· Use Laplace transforms to find the state transition matrix, e^At (Lecture #11 and HW#11)

· For an nxn matrix A, prove and apply the Cayley Hamilton theorem to find (Lecture #11 and HW#11):

- A^k =b₀A⁰+b₁A¹ +...+b_n-1A^n-1

- e^At =c₀A⁰+c₁A¹ +...+c_n-1A^n-1

· Use two methods to find, e^At , when A has repeated eigenvalues (Lecture #11 and HW#11)

· Use the convolution theorem of the Laplace transform to derive and apply the time domain solution to the state variable model (Lecture #12 and HW#12):

· Use the similarity transformation, x=Pz, to decouple a given state variable model (Lecture #12 and HW#12)

· From a decoupled state variable model, determine(Lecture 12 and HW#12)

- the solution, x(t)=Pz(t)

- controllable and uncontrollable eigenvalues

- observable and unobservable eigenvalues

· Find a state feedback control, w=-K_pvx, to set the eigenvalues of a single input system in phase variable form( Lecture #13 and HW#13-14)

· Find the controllability matrix, M=[B AB ... A^n-1 B] and find a similarity transformation, T_pv =MM_pv^-1 which will put a controllable state model into phase variable form (Lecture #13 and HW#13-14)

· Design a state feedback control, w=-K_pvT_pv^-1x, to force real systems to meet settling time specifications (Lecture #13 and HW#13-14)

· Implement your state feedback control using op-Amps (HW#13-14)

· Model the (ideal) sampling process and derive a frequency domain for Xs(f) (Lecture #14 and HW#15)

· From the frequency domain effects of sampling, understand the Nyquist Sampling theorem (Lecture #14 and HW#15)

· Model the quantizing process and derive an expression for the MSE due to sampling (Lecture #15 and HW#16)

· Derive the Z-transform from the Laplace transform of xs(t) and prove simple Z-transform theorems including the region of convergence (Lecture #16 and HW#17-18)

· Use two methods to discretize a continuous state variable model (Lecture #17 and HW#19)

· Use the Z-transform to solve the discrete state variable model and identify the zero-input and zero-state portion of the solution (Lecture #17 and HW#19)

· Find the Z-transform of discrete signals which are periodic after k=0 (Lecture #17 and HW#19)

· Prove and apply the Z-transform Convolution Property (Lecture#18 and HW#20)

· Compute the discrete state transition matrix, A^k, using (Lecture#18 and HW#20):

- Eigenvectors and eigenvalues

- Cayley-Hamilton

- Z-transforms

· Derive and apply the solution to the discrete state variable model ( Lecture #18 and HW#20)

· Understand and apply the concepts of linearity and shift invariance to discrete systems ( Lecture #18 and HW#20)

· Use the similarity transformation, x(kT)=Pz(kT), to decouple a discrete state variable model (Lecture #18 and HW#20)

· From a decoupled discrete state variable model, determine: (Lecture 18 and HW#20)

- the solution, x(kT)=Pz(kT)

- controllable and uncontrollable eigenvalues

- observable and unobservable eigenvalues

· Find a discrete state feedback control, w(kT)=-K_pvx(kT), to set the eigenvalues of a single input discrete system in phase variable form (Lecture #18)

· Find a similarity transformation, T_pv, which will put a controllable state model into phase variable form (Lecture #18 and HW#20)

· Design a discrete state feedback control, w(kT)=-K_pvT_pv^-1x (kT), to force discrete systems to meet settling time specifications (Lecture #18 and HW#20)

· Find block diagrams and state variable models from discrete transfer functions using (Lecture #19 and HW#21):

- Parallel decomposition

- Cascade decomposition

- Phase variables

- Direct Form I

- Direct Form II

· Perform graphical convolution to find the output of digital filters with a given impulse response ( Lecture #19 and HW#21)

· Implement IIR digital filters using microprocessors and A/D-D/A converters ( Lecture #20 and Lab3)

· Derive and apply the bilinear transform starting with a trapezoidal integrator (Lecture #20 and Lab3 and HW#24)

· Understand and apply digital IIR filter (correct) invariant design techniques (Lecture #20 and Lab3 and HW#24)

· Obtain frequency responses of digital filters using z=e^{jw T} (Lecture #20 and HW#24)

· Obtain approximate Bode plots of digital filters using the w-plane transformation (Lecture #20 and HW#24)

· Understand and apply the FIR design equation to obtain non-causal FIR filters (Lecture #21 and HW#25)

· Use shift and truncation techniques to obtain realizable FIR filters (Lecture #21 and Lab3 and HW#25)

· Use direct form II methods to realize FIR filters (Lecture #21 and Lab3 and HW#25)

· Implement FIR digital filters using microprocessors and A/D-D/A converters ( Lecture #21 and HW#25)

· Use windows to reduce the effects of truncation (Lecture #22)

· Find the DFT or FFT of discrete signals (Lecture #23 and Lab4)

· Prove simple properties of the DFT (Lecture #23 and Lab4)

· Use the DFT in the following practical applications ( Lecture #23 and Lab4)

- Estimate the energy contained in a sampled signal

- Illustrate the Nyquist Sampling Theorem

- Perform system identification

- Perform signal detection