(→‎Chapter 5: DT Fourier Transform)
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===[[Chapter 1_ECE301Fall2008mboutin]]: CT and DT Signals and Systems===
 
===[[Chapter 1_ECE301Fall2008mboutin]]: CT and DT Signals and Systems===
 +
#CT and DT Signals
 +
#Transformations of the Independent Variable
 +
#Exponential and Sinusoidal Signals
 +
#The Unit Impulse and Unit Step Functions
 +
#CT and DT Systems
 +
#Basic System Properties
 +
##Memory
 +
##Invertibility and Inverse Systems
 +
##Causality
 +
##Stability
 +
##Time Invariance
 +
##Linearity
 +
 
===[[Chapter 2_ECE301Fall2008mboutin]]: Linear Time-Invariant Systems===
 
===[[Chapter 2_ECE301Fall2008mboutin]]: Linear Time-Invariant Systems===
 +
#Convolution in DT
 +
#Convolution in CT
 +
#LTI System Properties
 +
##Commutative
 +
##Distributive
 +
##Associative
 +
##LTI Systems Memory
 +
##LTI Invertibility
 +
##LTI Causality
 +
##LTI Stability
 +
##The Unit Step Response of an LTI System
 +
#Causal LTI Systems Described by Differential and Difference Equations
 +
 
===[[Chapter 3_ECE301Fall2008mboutin]]: Fourier Series Representation of Period Signals===
 
===[[Chapter 3_ECE301Fall2008mboutin]]: Fourier Series Representation of Period Signals===
 +
DT Fourier Series Pair
 +
 +
:<math>x[n] = \sum_{k = <N>}a_ke^{jk\omega_0 n} = \sum_{k = <N>}a_ke^{jk(2\pi / N) n}</math>
 +
:<math>a_k = \frac{1}{N}\sum_{k = <N>}x[n]e^{-jk\omega_0 n} = \frac{1}{N}\sum_{k = <N>}x[n]e^{-jk(2\pi /N) n}</math>
 +
 +
CT Fourier Series Pair
 +
 +
:<math>x(t) = \sum_{k = -\infty}^{+\infty} a_k e^{jk\omega_0 t}= \sum_{k = -\infty}^{+\infty} a_k e^{jk(2\pi) /T t}</math>
 +
:<math>a_k = \frac{1}{T}\int_{T} x(t)e^{-jk\omega_0 t}\, dt = \frac{1}{T}\int_{T} x(t)e^{-jk(2\pi /T) t}\, dt</math>
 +
 +
#Response of LTI Systems to Complex Exponentials
 +
#FS Representations of CT Periodic Signals
 +
#FS Representations of DT Periodic Signals
 +
#Properties of FS
 +
#FS and LTI Systems
 +
#Filtering
 +
 
===[[Chapter 4_ECE301Fall2008mboutin]]: CT Fourier Transform===
 
===[[Chapter 4_ECE301Fall2008mboutin]]: CT Fourier Transform===
  
  
−
<math> \mathcal{X}(\omega) = \int_{-\infty}^{+\infty}x(t)e^{-j\omega t} \,dt </math>
+
:<math> \mathcal{X}(\omega) = \int_{-\infty}^{+\infty}x(t)e^{-j\omega t} \,dt </math>
  
−
<math> x(t) = \frac{1}{2\pi}\int_{-\infty}^{+\infty}\mathcal{X}(\omega)e^{j\omega t} \,dt </math>
+
:<math> x(t) = \frac{1}{2\pi}\int_{-\infty}^{+\infty}\mathcal{X}(\omega)e^{j\omega t} \,dt </math>
 +
 
 +
#Representing Aperiodic Signals: CT Fourier Transform
 +
#FT for Periodic Signals
 +
#Properties of the CT FT
 +
## Linearity
 +
## Time Shifting
 +
## Conjugation and Conjugate Symmetry
 +
## Differentiating and Integrating
 +
## Time and Frequency Scaling
 +
## Duality
 +
## Parseval's Relationship
 +
#The Convolution Property
 +
#The Multiplication Property
 +
#Systems Characterized by Linear Constant-Coefficient Differential Equations
  
 
===[[Chapter 5_ECE301Fall2008mboutin]]: DT Fourier Transform===
 
===[[Chapter 5_ECE301Fall2008mboutin]]: DT Fourier Transform===
  
−
<math> X(e^{j\omega}) = \sum_{n=-\infty}^{+\infty} x[n]e^{-j\omega n} </math>
+
:<math> X(e^{j\omega}) = \sum_{n=-\infty}^{+\infty} x[n]e^{-j\omega n} </math>
  
−
<math> x[n] = \frac{1}{2\pi}\int_{2\pi} X(e^{j\omega})e^{j\omega n} </math>
+
:<math> x[n] = \frac{1}{2\pi}\int_{2\pi} X(e^{j\omega})e^{j\omega n} </math>
 +
 
 +
#Representing Aperiodic Signals: DT Fourier Transform
 +
#FT for Periodic Signals
 +
#Properties of the CT FT
 +
## Periodicity of the Discrete-Time FT
 +
## Linearity
 +
## Time Shifting and Frequency Scaling
 +
## Conjugation and Conjugate Symmetry
 +
## Differencing and Accumulation
 +
## Time Reversal
 +
## Time Expansion
 +
## Differentiation in Frequency
 +
## Parseval's Relationship
 +
#The Convolution Property
 +
#The Multiplication Property
 +
#Duality
 +
#Systems Characterized by Linear Constant-Coefficient Difference Equations
  
 
===[[Chapter 7_ECE301Fall2008mboutin]]: Sampling===
 
===[[Chapter 7_ECE301Fall2008mboutin]]: Sampling===
 
#'''Sampling'''
 
#'''Sampling'''
 
##Impulse Train Sampling
 
##Impulse Train Sampling
−
##The Sampling Theorem and the Nyquist  
+
##The [[Sampling theorem_ECE301Fall2008mboutin]] and the Nyquist  
 
#'''Signal Reconstruction Using Interpolation:''' the fitting of a continuous signal to a set of sample values
 
#'''Signal Reconstruction Using Interpolation:''' the fitting of a continuous signal to a set of sample values
 
##Sampling with a Zero-Order Hold (Horizontal Plateaus)
 
##Sampling with a Zero-Order Hold (Horizontal Plateaus)
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===[[Chapter 8_ECE301Fall2008mboutin]]: Communication Systems===
 
===[[Chapter 8_ECE301Fall2008mboutin]]: Communication Systems===
−
==Chapter 8==
 
 
#'''Complex Exponential and Sinusoidal Amplitude Modulation''' (You Can Hear the Music on the Amplitude Modulation Radio -''Everclear'') Systems with the general form <math> y(t) = x(t)c(t) </math> where <math>c(t)</math> is the ''carrier signal'' and <math>x(t)</math> is the ''modulating signal''. The ''carrier signal'' has its amplitude multiplied (modulated) by the information-bearing ''modulating signal''.
 
#'''Complex Exponential and Sinusoidal Amplitude Modulation''' (You Can Hear the Music on the Amplitude Modulation Radio -''Everclear'') Systems with the general form <math> y(t) = x(t)c(t) </math> where <math>c(t)</math> is the ''carrier signal'' and <math>x(t)</math> is the ''modulating signal''. The ''carrier signal'' has its amplitude multiplied (modulated) by the information-bearing ''modulating signal''.
 
##Complex exponential ''carrier signal'': <math>c(t) = e^{\omega_c t + \theta_c}</math>  
 
##Complex exponential ''carrier signal'': <math>c(t) = e^{\omega_c t + \theta_c}</math>  
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:2. '''The Region of Convergence for Laplace Transforms'''
 
:2. '''The Region of Convergence for Laplace Transforms'''
−
 
  
 
:3. '''The Inverse Laplace Transform'''
 
:3. '''The Inverse Laplace Transform'''

Latest revision as of 09:05, 11 December 2008

ExamReviewNav

Summary of Information for the Final

ABET Outcomes

(a) an ability to classify signals (e.g. periodic, even) and systems (e.g. causal, linear) and an understanding of the difference between discrete and continuous time signals and systems. [1,2;a]
(b) an ability to determine the impulse response of a differential or difference equation. [1,2;a]
(c) an ability to determine the response of linear systems to any input signal convolution in the time domain. [1,2,4;a,e,k]
(d) an understanding of the deffinitions and basic properties (e.g. time-shifts,modulation, Parseval's Theorem) of Fourier series, Fourier transforms, bi-lateral Laplace transforms, Z transforms, and discrete time Fourier transforms and an ability to compute the transforms and inverse transforms of basic examples using methods such as partial fraction expansion_ECE301Fall2008mboutin. [1,2;a]
(e) an ability to determine the response of linear systems to any input signal by transformation to the frequency domain, multiplication, and inverse transformation to the time domain. [1,2,4;a,e,k]
(f) an ability to apply the Sampling theorem_ECE301Fall2008mboutin, reconstruction, aliasing, and Nyquist_ECE301Fall2008mboutin theorem to represent continuous-time signals in discrete time so that they can be processed by digital computers. [1,2,4;a,e,k]

Chapter 1_ECE301Fall2008mboutin: CT and DT Signals and Systems

  1. CT and DT Signals
  2. Transformations of the Independent Variable
  3. Exponential and Sinusoidal Signals
  4. The Unit Impulse and Unit Step Functions
  5. CT and DT Systems
  6. Basic System Properties
    1. Memory
    2. Invertibility and Inverse Systems
    3. Causality
    4. Stability
    5. Time Invariance
    6. Linearity

Chapter 2_ECE301Fall2008mboutin: Linear Time-Invariant Systems

  1. Convolution in DT
  2. Convolution in CT
  3. LTI System Properties
    1. Commutative
    2. Distributive
    3. Associative
    4. LTI Systems Memory
    5. LTI Invertibility
    6. LTI Causality
    7. LTI Stability
    8. The Unit Step Response of an LTI System
  4. Causal LTI Systems Described by Differential and Difference Equations

Chapter 3_ECE301Fall2008mboutin: Fourier Series Representation of Period Signals

DT Fourier Series Pair

$ x[n] = \sum_{k = <N>}a_ke^{jk\omega_0 n} = \sum_{k = <N>}a_ke^{jk(2\pi / N) n} $
$ a_k = \frac{1}{N}\sum_{k = <N>}x[n]e^{-jk\omega_0 n} = \frac{1}{N}\sum_{k = <N>}x[n]e^{-jk(2\pi /N) n} $

CT Fourier Series Pair

$ x(t) = \sum_{k = -\infty}^{+\infty} a_k e^{jk\omega_0 t}= \sum_{k = -\infty}^{+\infty} a_k e^{jk(2\pi) /T t} $
$ a_k = \frac{1}{T}\int_{T} x(t)e^{-jk\omega_0 t}\, dt = \frac{1}{T}\int_{T} x(t)e^{-jk(2\pi /T) t}\, dt $
  1. Response of LTI Systems to Complex Exponentials
  2. FS Representations of CT Periodic Signals
  3. FS Representations of DT Periodic Signals
  4. Properties of FS
  5. FS and LTI Systems
  6. Filtering

Chapter 4_ECE301Fall2008mboutin: CT Fourier Transform

$ \mathcal{X}(\omega) = \int_{-\infty}^{+\infty}x(t)e^{-j\omega t} \,dt $
$ x(t) = \frac{1}{2\pi}\int_{-\infty}^{+\infty}\mathcal{X}(\omega)e^{j\omega t} \,dt $
  1. Representing Aperiodic Signals: CT Fourier Transform
  2. FT for Periodic Signals
  3. Properties of the CT FT
    1. Linearity
    2. Time Shifting
    3. Conjugation and Conjugate Symmetry
    4. Differentiating and Integrating
    5. Time and Frequency Scaling
    6. Duality
    7. Parseval's Relationship
  4. The Convolution Property
  5. The Multiplication Property
  6. Systems Characterized by Linear Constant-Coefficient Differential Equations

Chapter 5_ECE301Fall2008mboutin: DT Fourier Transform

$ X(e^{j\omega}) = \sum_{n=-\infty}^{+\infty} x[n]e^{-j\omega n} $
$ x[n] = \frac{1}{2\pi}\int_{2\pi} X(e^{j\omega})e^{j\omega n} $
  1. Representing Aperiodic Signals: DT Fourier Transform
  2. FT for Periodic Signals
  3. Properties of the CT FT
    1. Periodicity of the Discrete-Time FT
    2. Linearity
    3. Time Shifting and Frequency Scaling
    4. Conjugation and Conjugate Symmetry
    5. Differencing and Accumulation
    6. Time Reversal
    7. Time Expansion
    8. Differentiation in Frequency
    9. Parseval's Relationship
  4. The Convolution Property
  5. The Multiplication Property
  6. Duality
  7. Systems Characterized by Linear Constant-Coefficient Difference Equations

Chapter 7_ECE301Fall2008mboutin: Sampling

  1. Sampling
    1. Impulse Train Sampling
    2. The Sampling theorem_ECE301Fall2008mboutin and the Nyquist
  2. Signal Reconstruction Using Interpolation: the fitting of a continuous signal to a set of sample values
    1. Sampling with a Zero-Order Hold (Horizontal Plateaus)
    2. Linear Interpolation (Connect the Samples)
  3. Undersampling: Aliasing
  4. Processing CT Signals Using DT Systems (Vinyl to CD)
    1. Analog vs. Digital: The Show-down (A to D conversion -> Discrete-Time Processing System -> D to A conversion
  5. Sampling DT Signals (CD to MP3 albeit a complicated sampling algorithm, MP3 is less dense signal)

Chapter 8_ECE301Fall2008mboutin: Communication Systems

  1. Complex Exponential and Sinusoidal Amplitude Modulation (You Can Hear the Music on the Amplitude Modulation Radio -Everclear) Systems with the general form $ y(t) = x(t)c(t) $ where $ c(t) $ is the carrier signal and $ x(t) $ is the modulating signal. The carrier signal has its amplitude multiplied (modulated) by the information-bearing modulating signal.
    1. Complex exponential carrier signal: $ c(t) = e^{\omega_c t + \theta_c} $
    2. Sinusoidal carrier signal: $ c(t) = cos(\omega_c t + \theta_c ) $
  2. Recovering the Information Signal $ x(t) $ Through Demodulation
    1. Synchronous
    2. Asynchronous
  3. Frequency-Division Multiplexing (Use the Entire Width of that Frequency Band!)
  4. Single-Sideband Sinusoidal Amplitude Modulation (Save the Bandwidth, Save the World!)
  5. AM with a Pulse-Train Carrier Digital Airwaves
    1. $ c(t) = \sum_{k=-\infty}^{+\infty}\frac{sin(k\omega_c \Delta /2)}{\pi k}e^{jk\omega_c t} $
    2. Time-Division Multiplexing "Dost thou love life? Then do not squander time; for that's the stuff life is made of." -Benjamin Franklin)

Chapter 9_ECE301Fall2008mboutin: Laplace Transformation

1. The Laplace Transform
$ X(s) = \int_{-\infty}^{+\infty}x(t)e^{-st}\, dt $
$ \mathcal{F}\lbrace x(t)e^{-\sigma t} \rbrace = \mathcal{X}(\omega) = \int_{-\infty}^{+\infty}x(t)e^{-\sigma t}e^{-j\omega t}\, dt $
2. The Region of Convergence for Laplace Transforms
3. The Inverse Laplace Transform
$ x(t) = \frac{1}{2\pi}\int_{\sigma - j\infty}^{\sigma + j\infty} X(s)e^{st}\,ds $

Chapter 10_ECE301Fall2008mboutin: z-Transformation

1. The z-Transform
$ X(z) = \sum_{n = -\infty}^{+\infty}x[n]z^{-n} $
2. Region of Convergence for the z-Transform
3. The Inverse z-Transform
$ x[n] = \frac{1}{2\pi j} \oint X(z)z^{n-1}\,dz $
4. z-Transform Properties
5. z-Transform Pairs
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