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
$ 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} $
Chapter 10
1. The z-Transform The z-Transform is the more general case of the discrete-time Fourier transform. For the DT Fourier transform $ z = e^{j\omega } $ with $ \omega $ real $ \Rightarrow |z| = 1 $. When z is not restricted to 1, it has the form $ re^{j\omega} $. This can be developed into the more general case of transform called z-Transform. The development of the z-Transform is outlined in Chapter 10.1 of the Oppenheim and Wilsky text.
- $ X(z) = \sum_{n = -\infty}^{+\infty}x[n]z^{-n} $
2. Region of Convergence for the z-Transform
3. The Inverse z-Transform
The derivation of the inverse z-Transform equation is outlined in chapter 10.3 of the text (pg 757-8).
- $ x[n] = \frac{1}{2\pi j} \oint X(z)z^{n-1}\,dz $
This is a closed loop integral around a CCW rotation centered at the origin with radius r. r can be any value for which X(z) converges.
| z-Transform Properties | |||
|---|---|---|---|
| Property | Signal | z-Transform | ROC |
| Linearity | $ \,\! ax_1[n] + bx_2[n] $ | $ \,\! aX_1(z)+bX_2(z) $ | At least $ R_1 \cap R_2 $ |
| Time Shifting | $ \,\! x[n-n_0] $ | $ z^{-n_0}X(z) $ | R, except for the possible addition or deletion of the origin |
| Scaling in the z-Domain | $ e^{j\omega_0 n}x[n] $ | $ X(e^{-j\omega_0}z) $ | R |
| $ z_0^nx[n] $ | $ X\Bigg( \frac{z}{z_0} \Bigg) $ | $ z_0 R $ | |
| $ \,\! a^nx[n] $ | $ \,\! X(a^{-1}z) $ | a|R= the set of point {|a|z} for z in R | |
| Time Reversal | $ \,\! x[-n] $ | $ \,\! X(z^{-1}) $ | Inverted R (i.e., R^-1= the set of point z^-1, where z is in R) |
| Time Expansion | $ x_{(k)}[n] = \begin{cases} x[r], & \mbox{if}n=rk\mbox{for}r\in \mathbb{Z}\\ 0, &\mbox{if }n\neq rk \mbox{ for} r\in \mathbb{Z}\end{cases} $ | $ \,\! X(z^k) $ | $ R^{1/k} $ (i.e., the set of points $ z^{1/k} $, where z is in R) |
| Conjugation | $ \,\! x^{*}[n] $ | $ \,\! X^{*}(z^{*}) $ | R |
| Convolution | $ \,\! x_1[n] * x_2[n] $ | $ \,\! X_1(z)X_2(z) $ | At least the intersection of R_1 and R_2 |
| First Difference | $ \,\! x[n] - x[n-1] $ | $ \,\! (1-z^{-1})X(z) $ | At least the intersection of R and $ |z| > 0 $ |
| Accumulation | $ \sum_{k = -\infty}^{n}x[k] $ | $ \frac{1}{1-z^{-1}}X(z) $ | At least the intersection of R and $ |z| > 1 $ |
| Differentiation in the z-Domain | $ \,\! nx[n] $ | $ -z\frac{dX(z)}{dz} $ | R |
Initial-Value Theorem
If x[n] = 0 for n < 0, then $ x[0] = \lim_{z\rightarrow \infty} X(z) $ | |||
| z-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 $ |
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.
