(→Table of Properties) |
(→Table of Properties) |
||
| Line 68: | Line 68: | ||
| Property || Signal || Laplace Transform || ROC | | Property || Signal || Laplace Transform || ROC | ||
|- align="center" | |- align="center" | ||
| − | | Linearity || <math> ax_1(t) + bx_2(t)</math> || <math>aX_1(s)+bX_2(s)</math> || <math> | + | | Linearity || <math> ax_1(t) + bx_2(t)</math> || <math>aX_1(s)+bX_2(s)</math> || At least <math>R_1 \cap R_2</math> |
|- align="center" | |- align="center" | ||
| Time Shifting || <math>x(t-t_0)</math> || <math>e^{-st_0}X(s)</math> || R | | Time Shifting || <math>x(t-t_0)</math> || <math>e^{-st_0}X(s)</math> || R | ||
| Line 80: | Line 80: | ||
| Convolution || <math>x_1(t)*x_2(t)</math> || <math>X_1(s)X_2(s)</math> || | | Convolution || <math>x_1(t)*x_2(t)</math> || <math>X_1(s)X_2(s)</math> || | ||
|- align="center" | |- align="center" | ||
| − | | Differentiation in the Time Domain || || || | + | | Differentiation in the Time Domain || <math>\frac{d}{dt}x(t)</math> || <math>sX(s)</math> || |
|- align="center" | |- align="center" | ||
| − | | Differentiation in the s-Domain || || || | + | | Differentiation in the s-Domain || <math>-tx(t)</math> || <math>\frac{d}{ds}X(s)</math> || |
|- align="center" | |- align="center" | ||
| − | | Integration in the Time Domain || || || | + | | Integration in the Time Domain || <math>\int_{-\infty}^{t}x(\tau)\,d\tau</math> || <math>\frac{1}{s}X(s)</math> || |
|- align="center" | |- align="center" | ||
| − | | Initial-Value Theorem | + | | colspan="4" | Initial- and Final-Value Theorem |
| − | + | 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 | |
| − | + | <math>x(0^{+}) = \lim_{x\rightarrow \infty} sX(s)</math> | |
|} | |} | ||
Revision as of 03:35, 5 December 2008
Suggested problems from Oppenheim and Willsky
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)
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} $
4. Properties of the Laplace Transform
- Linearity
- Time Shifting
- Shifting in the s-Domain
- Time Scaling
- Conjugation
- Convolution
- Differentiation in the Time Domain
- Differentiation in the s-Domain
- Integration in the Time Domain
- The Initial- and Final-Value Theorems
Table of 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) $ | |
| Time scaling | $ x(at) $ | $ \frac{1}{|a|}X\Bigg( \frac{s}{a} \Bigg) $ | |
| Conjugation | $ x^{*}(t) $ | $ X^{*}(s^{*}) $ | |
| Convolution | $ x_1(t)*x_2(t) $ | $ X_1(s)X_2(s) $ | |
| Differentiation in the Time Domain | $ \frac{d}{dt}x(t) $ | $ sX(s) $ | |
| Differentiation in the s-Domain | $ -tx(t) $ | $ \frac{d}{ds}X(s) $ | |
| Integration in the Time Domain | $ \int_{-\infty}^{t}x(\tau)\,d\tau $ | $ \frac{1}{s}X(s) $ | |
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) $ | |||
5. Some Laplace Transform Pairs
9.2, 9.3, 9.4, 9.6, 9.8, 9.9, 9.21, 9.22
Chapter 10
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.
