Recap: Chapters 1-7: Signals and Systems Chapter 2: Continuous-Time Signals Types of CT signals: sine, cosine, exponential, polynomials, impulse, unit-step, ramp, rectangle, triangle, and ... Using the impulse to sample (why?) Building new signals (combinations) Why? to model real world signals Add, Multiply, Amp Shift&Scaling, Ratios Time shifting and scaling Integration and derivatives Characteristics of Signals even and odd components periodic, aperiodic, combinations power and energy signals real, imaginary, magnitude, phase and converting between and how these relate to sine, cosine, exponential. why is this important? chapters 6 and 7. Chapter 3: Discrete-Time Signals (WHY) Types of DT signals: sine, cosine, exponential, polynomials, unit- impulse, unit-step, ramp, rectangle, triangle, and ... (periodic signals are special in DT) Using the unit-impulse to sample. Building new signals (combinations) Add, Multiply, Amp Shift&Scale, Ratios Time shifting and scaling (losing info) what is different about DT vs CT Accumulation and Difference Characteristics even and odd components periodic, aperiodic, combinations power and energy real, imaginary, magnitude and phase REVIEW 11/7/2017

Chapter 4: Modeling Systems System Equations Block Diagrams CT systems differential equations Inputs and outputs DT systems difference equations Components and symbols Properties of Systems Homogeneity, Additivity, and Linearity (Superposition) Time Invariance Other Properties Stability, Causality, Memory, Invertibility (system inverse) Dynamics of Systems LTI CT systems respond well to est and LTI DT systems respond well to zn What does “respond well” mean if x(t) = est then y(t) is a scaled version y(t) = Hest; and if x[n] = zn then y[n] is a scaled version y[n] = Hzn The homogeneous solution takes on these forms where the specific values for s and z are the eigenvalues. Can be used to determine if the system is stable: real(s) needs to be negative and mag(z) needs to be <1. LTI Systems (Why?) REVIEW 11/7/2017

Chapter 5: Time Domain Analysis of LTI Systems Concepts: For LTI systems – The input x can be represented by a sum (additivity) of scaled (homogeneity) time shifted (time-invariance) impulses as shown by the superposition integral or summation. We can find the response of the system due to an impulse in both CT and DT The output y can be directly calculated as the sum (additivity) of the same scaling factors (homogeneity) and same time-shifted (time-invariance) impulse responses of the system. BECAUSE THE SYSTEM IS LINEAR and TIME INVARIANT This calculation uses the CONVOLUTION operator any linear operator can be used in a similar way Not just restricted to impulses - If we knew the rect(t/2) response – then we could find the response to scaled and time shifted combinations of rect(t/2), Akrect((t-tk)/2) What do we need to know – besides the concepts How to find the impulse response of a system in both CT and DT\ what form the impulse response will take based on the differential/difference Equations How to perform a convolution in both CT and DT; properties of convolutions. REVIEW 11/7/2017

Chapter 6 and 7 Fourier Methods In the previous two slides, two bullets are of particular interest: LTI CT systems respond well to est and LTI DT systems respond well to zn if x(t) = est then y(t) = Hest; and if x[n] = zn then y[n] = Hzn ; H(s) and H[z] are easy to find ... no time shifting involved!!! Not just restricted to impulses: If we knew the rect(t/2) response; then we could find the response to scaled and time shifted combinations of rect(t/2), Akrect((t-tk)/2) It makes sense to find the est or zn response, and then find the output as scaled combinations of est or zn ... no time shifting involved. REVIEW 11/7/2017

Chapter 6 and 7 Fourier Methods Copying (rephrasing) again from the chapter 5 concepts (2 slides ago) The input x can be represented by a sum (additivity) of scaled (homogeneity) time shifted (time-invariance) impulses est or zn as shown by the superposition integral or summation the Fourier Transforms. We can easily find the response of the system due to an impulse est or zn in both CT and DT The output y can be directly calculated as the sum (additivity) of the same scaling factors (homogeneity) and same time-shifted (time-invariance) impulse est or zn responses of the system. What do we need to know – besides the concepts Fourier Transforms: CTFS, CTFT, DTFS, DTFT properties and pairs and how to use them. Finding H(s) or H(z) Getting things in the same units. REVIEW 11/7/2017

Preview – Chapters 8 and 9 The CT and DT Fourier methods are limited: est and zn are restricted to complex exponentials. imaginary axis for s; and unit circle for z; Certain signals cannot be represented with this restriction and we must evaluate the transform integrals over different paths to ensure the integral converges The region-of-convergence (ROC) defines where these paths can exist. More insight into the systems can be gained by expanding our view over the entire s or z plane of complex values. Especially useful in Stability analysis and Filter design. Generalized transforms are described in terms of: s for the Laplace transform z for the z-transform. REVIEW 11/7/2017

Chapter 10: Sampling Converting between analog and digital signals (CT and DT) The Process Hardware Block Diagrams Mathematical Descriptions Analysis Requirements. What happens to the signals. CT to DT DT to CT In both time and frequency REVIEW 11/7/2017