(Chapter 9)
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| align="right" style="padding-right: 1em;"|Integration in the Time Domain || <math>\int_{-\infty}^{t}x(\tau)\,d\tau</math> || <math>\frac{1}{s}X(s)</math> || At least <math>R \cap \lbrace \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 \rbrace</math>
 
| align="right" style="padding-right: 1em;"|Integration in the Time Domain || <math>\int_{-\infty}^{t}x(\tau)\,d\tau</math> || <math>\frac{1}{s}X(s)</math> || At least <math>R \cap \lbrace \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 \rbrace</math>
 
|-  
 
|-  
| colspan="4"| Initial- and Final-Value Theorem
+
| colspan="4" style="border:2px solid gray;"| Initial- and Final-Value Theorem
 
  If <math>x(t) = 0 </math> for t < 0 and <math>x(t)</math> contains  
 
  If <math>x(t) = 0 </math> for t < 0 and <math>x(t)</math> contains  
 
no impulses or higher-order singularities at t = 0, then
 
no impulses or higher-order singularities at t = 0, then
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| align="right" style="padding-right: 1em;"|15 || <math>u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}}</math> || <math>s^{n}</math> || <math>All\,\, s</math>
 
| align="right" style="padding-right: 1em;"|15 || <math>u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}}</math> || <math>s^{n}</math> || <math>All\,\, s</math>
 
|-  
 
|-  
| align="right" style="padding-right: 1em;"|16 || <math>u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n times}</math> || <math>\frac{1}{s^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace  > 0 </math>
+
| align="right" style="padding-right: 1em;"|16 || <math>u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times}</math> || <math>\frac{1}{s^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace  > 0 </math>
 
|}
 
|}
  

Revision as of 06:10, 5 December 2008

Exam 3 Material Summary

Chapter 7

  1. Sampling
    1. Impulse Train Sampling
    2. The Sampling Theorem 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)

Recommended Exercises: 7.1, 7.2, 7.3, 7.4, 7.5, 7.7, 7.10, 7.22, 7.29, 7.31, 7.33

Chapter 8

  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)

Recommended Exercises: 8.1, 8.2, 8.3, 8.5, 8.8, 8.10, 8.11, 8.12, 8.21, 8.23

Chapter 9

1. The Laplace Transform "Here I come to save the day!"

$ X(s) = \int_{-\infty}^{+\infty}x(t)e^{-st}\, dt $

s is a complex number of the form $ \sigma + j\omega $ and if $ \sigma = 0 $ then this equation reduces to the Fourier Transform of $ x(t) $. Indeed, the LT can be viewed as the FT of the signal $ x(t)e^{-\sigma t} $ as follows:

$ \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 (To Infinity or Converge!)

3. The Inverse Laplace Transform

$ x(t) = \frac{1}{2\pi}\int_{\sigma - j\infty}^{\sigma + j\infty} X(s)e^{st}\,ds $

for values of $ s = \sigma + j\omega $ in the ROC. The formal evaluation of the integral requires contour integration in the complex plane which is beyond the scope of this course.

3.1 The Laplace Transforms we will consider will fall into several categories that can be inverted using tables.
$ X(s) = \sum_{i=1}^{m} \frac{A_i}{s+a_i} $
Laplace Transform Properties
Property Signal Laplace Transform ROC
Linearity $ ax_1(t) + bx_2(t) $ $ aX_1(s)+bX_2(s) $ At least $ R_1 \cap R_2 $
Time Shifting $ x(t-t_0) $ $ e^{-st_0}X(s) $ R
Shifting in the s-Domain $ e^{s_0 t}x(t) $ $ X(s-s_0) $ Shifted version of R (i.e., s is in the ROC if $ s - s_0 $ is in R)
Time scaling $ x(at) $ $ \frac{1}{|a|}X\Bigg( \frac{s}{a} \Bigg) $ Scaled ROC (i.e., s is in the ROC if s/a is in R)
Conjugation $ x^{*}(t) $ $ X^{*}(s^{*}) $ R
Convolution $ x_1(t)*x_2(t) $ $ X_1(s)X_2(s) $ At least $ R_1 \cap R_2 $
Differentiation in the Time Domain $ \frac{d}{dt}x(t) $ $ sX(s) $ At least R
Differentiation in the s-Domain $ -tx(t) $ $ \frac{d}{ds}X(s) $ R
Integration in the Time Domain $ \int_{-\infty}^{t}x(\tau)\,d\tau $ $ \frac{1}{s}X(s) $ At least $ R \cap \lbrace \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 \rbrace $
Initial- and Final-Value Theorem
If $ x(t) = 0  $ for t < 0 and $ x(t) $ contains 

no impulses or higher-order singularities at t = 0, then

$ x(0^{+}) = \lim_{x\rightarrow \infty} sX(s) $

$ \lim_{t\rightarrow \infty} x(t) = \lim_{s\rightarrow 0}sX(s) $


Laplace Transform Pairs
Transform Pair Signal Transform ROC
1 $ \delta(t) $ $ 1 $ $ All\,\, s $
2 $ u(t) $ $ \frac{1}{s} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $
3 $ -u(-t) $ $ \frac{1}{s} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $
4 $ \frac{t^{n-1}}{(n-1)!}u(t) $ $ \frac{1}{s^{n}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $
5 $ -\frac{t^{n-1}}{(n-1)!}u(-t) $ $ \frac{1}{s^{n}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 $
6 $ e^{-\alpha t}u(t) $ $ \frac{1}{s+\alpha} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $
7 $ -e^{-\alpha t}u(-t) $ $ \frac{1}{s+\alpha} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $
8 $ \frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t) $ $ \frac{1}{(s+\alpha )^{n}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $
9 $ -\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t) $ $ \frac{1}{(s+\alpha )^{n}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha $
10 $ \delta (t - T) $ $ e^{-sT} $ $ All\,\, s $
11 $ [cos( \omega_0 t)]u(t) $ $ \frac{s}{s^2+\omega_0^{2}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $
12 $ [sin( \omega_0 t)]u(t) $ $ \frac{\omega_0}{s^2+\omega_0^{2}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $
13 $ [e^{-\alpha t}cos( \omega_0 t)]u(t) $ $ \frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $
14 $ [e^{-\alpha t}sin( \omega_0 t)]u(t) $ $ \frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha $
15 $ u_n(t) = \frac{d^{n}\delta (t)}{dt^{n}} $ $ s^{n} $ $ All\,\, s $
16 $ u_{-n}(t) = \underbrace{u(t) *\dots * u(t)}_{n\,\,times} $ $ \frac{1}{s^{n}} $ $ \mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 $

Recommended Exercises: 9.2, 9.3, 9.4, 9.6, 9.8, 9.9, 9.21, 9.22

Chapter 10

Recommended Exercises: 10.1, 10.2, 10.3, 10.4, 10.6, 10.7, 10.8, 10.9, 10.10, 10.11, 10.13, 10.15, 10.21, 10.22, 10.23, 10.24, 10.25, 10.26, 10.27, 10.30, 10.31, 10.32, 10.33, 10.43, 10.44.

Note: If a problem states that you should use “long division”, feel free to use the geometric series formula instead.

Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett