Revision as of 07:00, 5 December 2008

Exam 3 Material Summary

Chapter 7

Sampling
1. Impulse Train Sampling
2. The Sampling Theorem and the Nyquist
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)
Undersampling: Aliasing
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
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.
1. Complex exponential carrier signal: $c(t) = e^{\omega_c t + \theta_c}$
2. Sinusoidal carrier signal: $c(t) = cos(\omega_c t + \theta_c )$
Recovering the Information Signal $ x(t) $ Through Demodulation
1. Synchronous
2. 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
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

$X(e^{j\omega}) = \sum_{n=-\infty}^{+\infty} x[n]e^{-j\omega n}$

$x[n] = \frac{1}{2\pi}\int_{2\pi} X(e^{j\omega})e^{j\omega n}$

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	$$ ax_1(t) + bx_2(t) $$	$$ aX_1(s)+bX_2(s) $$	At least $R_1 \cap R_2$
Time Shifting	$$ x(t-t_0) $$	$e^{-st_0}X(s)$	R
Scaling in the z-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 Reversal	$$ x(at) $$	$\frac{1}{\|a\|}X\Bigg( \frac{s}{a} \Bigg)$	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 $x(0^{+}) = \lim_{x\rightarrow \infty} sX(s)$ $\lim_{t\rightarrow \infty} x(t) = \lim_{s\rightarrow 0}sX(s)$

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.

@@ Line 129: / Line 129: @@
 Recommended Exercises: 9.2, 9.3, 9.4, 9.6, 9.8, 9.9, 9.21, 9.22
+<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>
 ==Chapter 10==
+. '''The z-Transform'''
+The z-Transform is the more general case of the discrete-time Fourier transform. For the DT Fourier transform <math>z = e^{j\omega }</math> with <math>\omega</math> real <math>\Rightarrow |z| = 1</math>. When z is not restricted to 1, it has the form <math>re^{j\omega}</math>. 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.
+:<math>X(z) = \sum_{n = -\infty}^{+\infty}x[n]z^{-n}</math>
+. '''Region of Convergence for the z-Transform'''
+. '''The Inverse z-Transform'''
+The derivation of the inverse z-Transform equation is outlined in chapter 10.3 of the text (pg 757-8).
+:<math>x[n] = \frac{1}{2\pi j} \oint X(z)z^{n-1}\,dz</math>
+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.
+{|style="width:75%; background: none; text-align: center; border:1px solid gray;" align="center"
+|-
+! colspan="4" align="left" style="background: #b79256; font-size: 120%;" | z-Transform Properties
+|- style="background: #e4bc7e; font-size: 110%;" align="center"
+! Property !! Signal
+! width="170px"|z-Transform
+! width="170px"|ROC
+|-
+| align="right" style="padding-right: 1em;"| 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="right" style="padding-right: 1em;"|Time Shifting || <math>x(t-t_0)</math> || <math>e^{-st_0}X(s)</math> || R
+|-
+| align="right" style="padding-right: 1em;"|Scaling in the z-Domain || <math>e^{s_0 t}x(t)</math> || <math>X(s-s_0)</math> || Shifted version of R (i.e., s is in the ROC if <math> s - s_0</math> is in R)
+|-
+| align="right" style="padding-right: 1em;"|Time Reversal || <math>x(at)</math> || <math>\frac{1}{|a|}X\Bigg( \frac{s}{a} \Bigg)</math>  || Scaled ROC (i.e., s is in the ROC if s/a is in R)
+|-
+| align="right" style="padding-right: 1em;"|Conjugation || <math>x^{*}(t)</math> || <math>X^{*}(s^{*})</math>  || R
+|-
+| align="right" style="padding-right: 1em;"|Convolution || <math>x_1(t)*x_2(t)</math> || <math>X_1(s)X_2(s)</math>  || At least <math>R_1 \cap R_2</math>
+|-
+| align="right" style="padding-right: 1em;"|Differentiation in the Time Domain || <math>\frac{d}{dt}x(t)</math> || <math>sX(s)</math> || At least R
+|-
+| align="right" style="padding-right: 1em;"|Differentiation in the s-Domain || <math>-tx(t)</math> || <math>\frac{d}{ds}X(s)</math> || R
+|-
+| 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" 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
+no impulses or higher-order singularities at t = 0, then
+<math>x(0^{+}) = \lim_{x\rightarrow \infty} sX(s)</math>
+<math>\lim_{t\rightarrow \infty} x(t) = \lim_{s\rightarrow 0}sX(s)</math>
+|}
+{|style="width:75%; background: none; text-align: center; border:1px solid gray;" align="center"
+|-
+! colspan="4" align="left" style="background: #b79256; font-size: 120%;" | z-Transform Pairs
+|- style="background: #e4bc7e; font-size: 110%;" align="center"
+! width="75px"|Transform Pair !! Signal
+! width="170px"|Transform
+! width="170px"|ROC
+|-
+| align="right" style="padding-right: 1em;"|1 || <math>\delta(t)</math> || <math>1</math> ||  <math>All\,\, s</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="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="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="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="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="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="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="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="right" style="padding-right: 1em;"|10 || <math>\delta (t - T)</math> || <math>e^{-sT}</math> || <math>All\,\, s</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="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>
+|}
 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.

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