1. 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

2. 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?)
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3. 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.
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4. 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.
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5. 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.
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6. 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.
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7. 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