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:<math>X(s) = \sum_{i=1}^{m} \frac{A_i}{s+a_i} </math> | :<math>X(s) = \sum_{i=1}^{m} \frac{A_i}{s+a_i} </math> | ||
| − | {|style="background: none; text-align: center; border:1px solid gray;" | + | {|style="width:75%; background: none; text-align: center; border:1px solid gray;" align="center" |
|- | |- | ||
! colspan="4" align="left" style="background: #b79256; font-size: 120%;" | Laplace Transform Properties | ! colspan="4" align="left" style="background: #b79256; font-size: 120%;" | Laplace Transform Properties | ||
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|} | |} | ||
| − | + | ||
| − | {| style="width: | + | {|style="width:75%; background: none; text-align: center; border:1px solid gray;" align="center" |
| − | |- align="center" | + | |- |
| − | + | ! colspan="4" align="left" style="background: #b79256; font-size: 120%;" | Laplace Transform Pairs | |
| − | |- align=" | + | |- style="background: #e4bc7e; font-size: 110%;" align="center" |
| − | | 1 || <math>\delta(t)</math> || <math>1</math> || <math>All\,\, s</math> | + | ! width="75px"|Transform Pair !! Signal |
| − | |- align=" | + | ! width="170px"|Transform |
| − | | 2 || <math>u(t)</math> || <math>\frac{1}{s}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> | + | ! width="170px"|ROC |
| − | |- align=" | + | |- |
| − | | 3 || <math>-u(-t)</math> || <math>\frac{1}{s}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 </math> | + | | align="right" style="padding-right: 1em;"|1 || <math>\delta(t)</math> || <math>1</math> || <math>All\,\, s</math> |
| − | |- align=" | + | |- |
| − | | 4 || <math>\frac{t^{n-1}}{(n-1)!}u(t)</math> || <math>\frac{1}{s^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> | + | | align="right" style="padding-right: 1em;"|2 || <math>u(t)</math> || <math>\frac{1}{s}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> |
| − | |- align=" | + | |- |
| − | | 5 || <math>-\frac{t^{n-1}}{(n-1)!}u(-t)</math> || <math>\frac{1}{s^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 </math> | + | |align="right" style="padding-right: 1em;"| 3 || <math>-u(-t)</math> || <math>\frac{1}{s}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 </math> |
| − | |- align=" | + | |- |
| − | | 6 || <math>e^{-\alpha t}u(t)</math> || <math>\frac{1}{s+\alpha}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math> | + | | align="right" style="padding-right: 1em;"|4 || <math>\frac{t^{n-1}}{(n-1)!}u(t)</math> || <math>\frac{1}{s^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> |
| − | |- align=" | + | |- |
| − | | 7 || <math>-e^{-\alpha t}u(-t)</math> || <math>\frac{1}{s+\alpha}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha </math> | + | | align="right" style="padding-right: 1em;"|5 || <math>-\frac{t^{n-1}}{(n-1)!}u(-t)</math> || <math>\frac{1}{s^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < 0 </math> |
| − | |- align=" | + | |- |
| − | | 8 || <math>\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t)</math> || <math>\frac{1}{(s+\alpha )^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math> | + | | align="right" style="padding-right: 1em;"|6 || <math>e^{-\alpha t}u(t)</math> || <math>\frac{1}{s+\alpha}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math> |
| − | |- align=" | + | |- |
| − | | 9 || <math>-\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t)</math> || <math>\frac{1}{(s+\alpha )^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha </math> | + | | align="right" style="padding-right: 1em;"|7 || <math>-e^{-\alpha t}u(-t)</math> || <math>\frac{1}{s+\alpha}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha </math> |
| − | |- align=" | + | |- |
| − | | 10 || <math>\delta (t - T)</math> || <math>e^{-sT}</math> || <math>All\,\, s</math> | + | | align="right" style="padding-right: 1em;"|8 || <math>\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(t)</math> || <math>\frac{1}{(s+\alpha )^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math> |
| − | |- align=" | + | |- |
| − | | 11 || <math>[cos( \omega_0 t)]u(t)</math> || <math>\frac{s}{s^2+\omega_0^{2}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> | + | | align="right" style="padding-right: 1em;"|9 || <math>-\frac{t^{n-1}}{(n-1)!}e^{-\alpha t}u(-t)</math> || <math>\frac{1}{(s+\alpha )^{n}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace < -\alpha </math> |
| − | |- align=" | + | |- |
| − | | 12 || <math>[sin( \omega_0 t)]u(t)</math> || <math>\frac{\omega_0}{s^2+\omega_0^{2}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> | + | | align="right" style="padding-right: 1em;"|10 || <math>\delta (t - T)</math> || <math>e^{-sT}</math> || <math>All\,\, s</math> |
| − | |- align=" | + | |- |
| − | | 13 || <math>[e^{-\alpha t}cos( \omega_0 t)]u(t)</math> || <math>\frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math> | + | |align="right" style="padding-right: 1em;"| 11 || <math>[cos( \omega_0 t)]u(t)</math> || <math>\frac{s}{s^2+\omega_0^{2}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> |
| − | |- align=" | + | |- |
| − | | 14 || <math>[e^{-\alpha t}sin( \omega_0 t)]u(t)</math> || <math>\frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math> | + | | align="right" style="padding-right: 1em;"|12 || <math>[sin( \omega_0 t)]u(t)</math> || <math>\frac{\omega_0}{s^2+\omega_0^{2}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > 0 </math> |
| − | |- align=" | + | |- |
| − | | 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;"| 13 || <math>[e^{-\alpha t}cos( \omega_0 t)]u(t)</math> || <math>\frac{s+\alpha}{(s+\alpha)^{2}+\omega_0^{2}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </math> |
| − | |- align=" | + | |- |
| − | | 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;"| 14 || <math>[e^{-\alpha t}sin( \omega_0 t)]u(t)</math> || <math>\frac{\omega_0}{(s+\alpha)^{2}+\omega_0^{2}}</math> || <math>\mathcal{R} \mathfrak{e} \lbrace s \rbrace > -\alpha </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> | ||
|} | |} | ||
Revision as of 06:08, 5 December 2008
Exam 3 Material Summary
Chapter 7
- Sampling
- Impulse Train Sampling
- The Sampling Theorem and the Nyquist
- Signal Reconstruction Using Interpolation: the fitting of a continuous signal to a set of sample values
- Sampling with a Zero-Order Hold (Horizontal Plateaus)
- Linear Interpolation (Connect the Samples)
- Undersampling: Aliasing
- Processing CT Signals Using DT Systems (Vinyl to CD)
- Analog vs. Digital: The Show-down (A to D conversion -> Discrete-Time Processing System -> D to A conversion
- 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
- 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.
- Complex exponential carrier signal: $ c(t) = e^{\omega_c t + \theta_c} $
- Sinusoidal carrier signal: $ c(t) = cos(\omega_c t + \theta_c ) $
- Recovering the Information Signal $ x(t) $ Through Demodulation
- Synchronous
- Asynchronous
- Frequency-Division Multiplexing (Use the Entire Width of that Frequency Band!)
- Single-Sideband Sinusoidal Amplitude Modulation (Save the Bandwidth, Save the World!)
- AM with a Pulse-Train Carrier Digital Airwaves
- $ c(t) = \sum_{k=-\infty}^{+\infty}\frac{sin(k\omega_c \Delta /2)}{\pi k}e^{jk\omega_c t} $
- 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.
