Class Wiki - User contributions [en]

Max Woesner

2011-12-23T04:17:30Z

Max.Woesner:

[[Image:Mask08.jpg |thumb|This is me]]
[[Image:Diving_Turtle.jpg |thumb|Diving in Guam]]

=== The Purpose of This Page ===
This page was created to document my work in ENGR455 - Signals and Systems. I hope that both current and future students will be able to benefit from my work here. Since I am a student learning as I go along, I cannot guarantee the complete accuracy of the content on this page. 

[[Signals and systems|Back to the class homepage]]

=== About Me ===
I am a senior electrical engineering major also minoring in business and math. For more info check out the [http://mask.wallawalla.edu/profile/show/7065 Mask]

=== Contact Info ===
*Email: maxwell.woesner@wallawalla.edu
*Messenger: maximiliaknow@hotmail.com
*[http://www.facebook.com/max.woesner Facebook]

=== Class Resources ===
[[User:Frohro|Dr. Frohne]] 
[[Signals and systems|Class Wiki]] 
[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR455/2009/Keystone/index.php Class Notes (2009)] 
[http://dlearn2.wallawalla.edu/moodle/ Moodle] 
[http://wikipedia.org/ Wikipedia]

=== Homework ===
[[Evaluate this integral - HW1|Homework #1 - Evaluate this integral]] 
[[Something interesting from class - HW2|Homework #2 - Something interesting from class]] 
[[Class lecture notes October 5 - HW3|Homework #3 - Class lecture notes October 5]] 
[[Fourier Transform Properties|Homework #4 - Fourier transform properties]] 
[[Fourier Transform Property review|Homework #5 - Review a Fourier transform property]] 
[[Fourier Transform Properties|Homework #6c - Another Fourier transform property]] 
[[Sampling - HW7|Homework #7 - Sampling]] 
[[How a CD player works - HW8|Homework #8 - How a CD player works]] 
[[Third harmonic sampling and QSD - HW9|Homework #9 - Third harmonic sampling and QSD]] 
[[Quadrature sampling waveform plot - HW10|Homework #10 - Quadrature sampling waveform plot]] 
[[DFT example using MATLAB - HW11|Homework #11 - DFT example using MATLAB]] 
[[Automatic Antenna Phasing Project|Class Project - Automatic antenna phasing]]

Power Electronics

2011-06-06T06:10:53Z

Max.Woesner: /* How to Use LTSpice with the supplied PSpice Models */

[[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/engr432index.htm Class Notes]]

==Feedback Analysis of Switched Mode Power Supplies==

====How to Use LTSpice with the supplied PSpice Models====
*[[http://denethor.wlu.ca/ltspice/#models A page showing how to import models]]
*[[http://www.electronicspoint.com/pspice-ltspice-switchercad-schematic-conversion-t28041.html Forum results that cued me in to the fact you can open PSpice .sch schematics in LTSpice.]] (You need to select *.* (All Files) in the navigator window.)

====Class Project 2011====
*[http://mtjungle.com/pe/ 120 VAC LED Light Project]

Power Electronics

2011-06-06T06:08:59Z

Max.Woesner: /* Homework */

Power Electronics HW5

2011-04-26T22:05:00Z

Max.Woesner: Created page with 'Back to the Power Electronics home page === Homework #5 === Problem Statement Either figure out how to make use of LTSpice with the PSpice…'

[[Power Electronics|Back to the Power Electronics home page]]

=== Homework #5 ===

 Problem Statement 

Either figure out how to make use of LTSpice with the PSpice libraries or develop LTSpice or QUCs models for the switching power pole in the Buck mode. 

Solution

Power Electronics

2011-04-26T22:04:22Z

Max.Woesner: /* Homework */

[[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/engr432index.htm Class Notes]]

==Feedback Analysis of Switched Mode Power Supplies==

====How to Use LTSpice with the supplied PSpice Models====
*[[http://denethor.wlu.ca/ltspice/#models A page showing how to import models]]
*[[http://www.electronicspoint.com/pspice-ltspice-switchercad-schematic-conversion-t28041.html Forum results that cued me in to the fact you can open PSpice .sch schematics in LTSpice.]] (You need to select *.* (All Files) in the navigator window.)

=== Homework ===
*[[Power Electronics HW5|Homework #5 - Max]]

Power Electronics

2011-04-26T21:57:17Z

Max.Woesner: /* How to Use LTSpice with the supplied PSpice Models */

[[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/engr432index.htm Class Notes]]

==Feedback Analysis of Switched Mode Power Supplies==

====How to Use LTSpice with the supplied PSpice Models====
*[[http://denethor.wlu.ca/ltspice/#models A page showing how to import models]]
*[[http://www.electronicspoint.com/pspice-ltspice-switchercad-schematic-conversion-t28041.html Forum results that cued me in to the fact you can open PSpice .sch schematics in LTSpice.]] (You need to select *.* (All Files) in the navigator window.)

=== Homework ===
*[[Power Electronics HW5|Homework #5]]

Max Woesner

2009-12-18T17:19:28Z

Max.Woesner:

[[Image:Mask08.jpg |thumb|This is me]]
[[Image:Diving_Turtle.jpg |thumb|Diving in Guam]]

=== The Purpose of This Page ===
This page was created to document my work in ENGR455 - Signals and Systems. I hope that both current and future students will be able to benefit from my work here. Since I am a student learning as I go along, I cannot guarantee the complete accuracy of the content on this page. 

[[Signals and systems|Back to the class homepage]]

=== About Me ===
I am a senior electrical engineering major also minoring in business and math. For more info check out the [http://mask.wallawalla.edu/profile/show/7065 Mask]

=== Contact Info ===
*Phone: (509) 524-6978
*Email: maxwell.woesner@wallawalla.edu
*Messenger: maximiliaknow@hotmail.com
*[http://www.facebook.com/max.woesner Facebook]

=== Class Resources ===
[[User:Frohro|Dr. Frohne]] 
[[Signals and systems|Class Wiki]] 
[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR455/2009/Keystone/index.php Class Notes (2009)] 
[http://dlearn2.wallawalla.edu/moodle/ Moodle] 
[http://wikipedia.org/ Wikipedia]

=== Homework ===
[[Evaluate this integral - HW1|Homework #1 - Evaluate this integral]] 
[[Something interesting from class - HW2|Homework #2 - Something interesting from class]] 
[[Class lecture notes October 5 - HW3|Homework #3 - Class lecture notes October 5]] 
[[Fourier Transform Properties|Homework #4 - Fourier transform properties]] 
[[Fourier Transform Property review|Homework #5 - Review a Fourier transform property]] 
[[Fourier Transform Properties|Homework #6c - Another Fourier transform property]] 
[[Sampling - HW7|Homework #7 - Sampling]] 
[[How a CD player works - HW8|Homework #8 - How a CD player works]] 
[[Third harmonic sampling and QSD - HW9|Homework #9 - Third harmonic sampling and QSD]] 
[[Quadrature sampling waveform plot - HW10|Homework #10 - Quadrature sampling waveform plot]] 
[[DFT example using MATLAB - HW11|Homework #11 - DFT example using MATLAB]] 
[[Automatic Antenna Phasing Project|Class Project - Automatic antenna phasing]]

Automatic Antenna Phasing Project

2009-12-16T23:38:33Z

Max.Woesner:

== Max Woesner ==
[[Max Woesner|Back to my Home Page]]

=== Class Project - Automatic Antenna Phasing ===

====Team Members:====
[[Joshua Sarris]] 
[[Kevin Starkey]] 
[[Max Woesner]] 
[[Nick Christman]]

 Problem Statement 
Design and implement an algorithm to determine the amount of delay necessary to automatically add two signals from different antennas in phase. Also determine the relative amplitudes of the two signals. 

====Tools:====
Software: 
* MATLAB/Octave
* GnuRadio
* Dttsp
* Sdr-Shell
Hardware: 
* Dual Softrock Receiver
* Two Antennas
Data: 
Files of real world data gathered using the above equipment will be provided. 

References: 
[http://www.kkn.net/dayton2008/beam_steering_on_160_meters.pdf Beam Steering on 160 Meters] 
[http://k1lt.com/ More K1LT Notes] 
[http://johnsogg.blogspot.com/2009/11/installing-gnu-radio-on-ubuntu-910.html Install gnuradio on Ubuntu 9.10 using the package manager] 
[http://sdrblog.wordpress.com/gnuradio-installation/ Install gnuradio on Ubuntu 9.10 from GIT repositories] 

Other Information: 
So Gnu Radio will work with Jack, you need to do the following: 
<pre>
Edit ~/.gnuradio/config.conf

Add these three lines to the file:

[audio]

verbose = True

audio_module = audio_jack
</pre> 
The IF of the signals is the sample_rate/4. The bandwidth of SSB signals is about 3 KHz, CW (Morse code) about 200 Hz. The format of the IF signals is I/Q (one on each channel). 

This project should be useful with dual antenna receivers to detect the polarization or angle of arrival of a signal, and in general to improve the signals received. 

 Approach to Solution 
We first downloaded the files with the real world data and used [http://audacity.sourceforge.net/ Audacity] to shorten the data to a workable length to analyze in MATLAB. 

After obtaining a reasonable length of data (~5 seconds worth) we used MATLAB's 'spectrogram' function to create a spectrograph of our signal in order to obtain a reasonable frequency. Here is the code used and the results (shown below):

<pre>
close all;
clear all;

sound1 = 'Alaska_mid-1.wav';
sound2 = 'Alaska_mid-2.wav';
sound3 = 'Alaska_mid-3.wav';
sound4 = 'Alaska_mid-4.wav';

% Opens wave files above and labels them appropriately
% while also cheking dimensions and scaling the matrices
% appropriately.
[I1 Q1 I2 Q2] = dimCheck(sound1,sound2,sound3,sound4);

% Grabs the sampling frequency of each sound file.
[FI1 FQ1 FI2 FQ2] = freqGet(sound1,sound2,sound3,sound4);

% Creates a spectrograph of each signal while also
% outputing the number of samples (S), the frequecnies
% (F), and the times at which the spectrogram is
% computed.

[S1,F1,T1] = spectrogram(I1);
[S2,F2,T2] = spectrogram(I2);
[S3,F3,T3] = spectrogram(Q1);
[S4,F4,T4] = spectrogram(Q2);

Fn = F1/(F1'*F1);

figure(1)
spectrogram(I1)
title('Spectrogram of I1')

figure(2)
spectrogram(Q1)
title('Spectrogram of Q1')

figure(3)
spectrogram(I2)
title('Spectrogram of I2')

figure(4)
spectrogram(Q2)
title('Spectrogram of Q2')

</pre>

[[Image:Sp_i1.jpg]]
[[Image:Sp_q1.jpg]]
[[Image:Sp_i2.jpg]]
[[Image:Sp_q2.jpg]]

As you can see, it appears that we need to create a bandpass (FIR) filter that will filter around .5 to .57 <math>\scriptstyle \pi</math>rad/s.

Unfortunately, due to time constraints, we were unable to proceed further. However, if time provided, we would have proceeded as follows: 

* Create a bandpass (FIR) filter as described above.
* Create a cross-correlation algorithm that would obtain the phase difference between I1, I2 and Q1, Q2.

Automatic Antenna Phasing Project

2009-12-16T20:36:27Z

Max.Woesner:

== Max Woesner ==
[[Max Woesner|Back to my Home Page]]

=== Class Project - Automatic Antenna Phasing ===

====Team Members:====
[[Kevin Starkey]] 
[[Max Woesner]] 
[[Nick Christman]] 
[[Joshua Sarris]]

 Problem Statement 
Design and implement an algorithm to determine the amount of delay necessary to automatically add two signals from different antennas in phase. Also determine the relative amplitudes of the two signals. 

====Tools:====
Software: 
* MATLAB/Octave
* GnuRadio
* Dttsp
* Sdr-Shell
Hardware: 
* Dual Softrock Receiver
* Two Antennas
Data: 
Files of real world data gathered using the above equipment will be provided. 

References: 
[http://www.kkn.net/dayton2008/beam_steering_on_160_meters.pdf Beam Steering on 160 Meters] 
[http://k1lt.com/ More K1LT Notes] 
[http://johnsogg.blogspot.com/2009/11/installing-gnu-radio-on-ubuntu-910.html Install gnuradio on Ubuntu 9.10 using the package manager] 
[http://sdrblog.wordpress.com/gnuradio-installation/ Install gnuradio on Ubuntu 9.10 from GIT repositories] 

Other Information: 
So Gnu Radio will work with Jack, you need to do the following: 
<pre>
Edit ~/.gnuradio/config.conf

Add these three lines to the file:

[audio]

verbose = True

audio_module = audio_jack
</pre> 
The IF of the signals is the sample_rate/4. The bandwidth of SSB signals is about 3 KHz, CW (Morse code) about 200 Hz. The format of the IF signals is I/Q (one on each channel). 

This project should be useful with dual antenna receivers to detect the polarization or angle of arrival of a signal, and in general to improve the signals received. 

 Approach to Solution 
We first downloaded the files with the real world data and used [http://audacity.sourceforge.net/ Audacity] to shorten the data to a workable length to analyze in MATLAB. 
Unfortunately, due to time constraints, we were unable to proceed further. However, if time provided, we would have proceeded as follows: 
*

Automatic Antenna Phasing Project

2009-12-09T02:57:09Z

Max.Woesner:

== Max Woesner ==
[[Max Woesner|Back to my Home Page]]

=== Class Project - Automatic Antenna Phasing ===

 Problem Statement 
Design and implement an algorithm to determine the amount of delay necessary to automatically add two signals from different antennas in phase. Also determine the relative amplitudes of the two signals. 

====Tools:====
Software: 
* MATLAB/Octave
* GnuRadio
* Dttsp
* Sdr-Shell
Hardware: 
* Dual Softrock Receiver
* Two Antennas
Data: 
Files of real world data gathered using the above equipment will be provided. 

References: 
[http://www.kkn.net/dayton2008/beam_steering_on_160_meters.pdf Beam Steering on 160 Meters] 
[http://k1lt.com/ More K1LT Notes] 
[http://johnsogg.blogspot.com/2009/11/installing-gnu-radio-on-ubuntu-910.html Install gnuradio on Ubuntu 9.10 using the package manager] 
[http://sdrblog.wordpress.com/gnuradio-installation/ Install gnuradio on Ubuntu 9.10 from GIT repositories] 

Other Information: 
So Gnu Radio will work with Jack, you need to do the following: 
<pre>
Edit ~/.gnuradio/config.conf

Add these three lines to the file:

[audio]

verbose = True

audio_module = audio_jack
</pre> 
The IF of the signals is the sample_rate/4. The bandwidth of SSB signals is about 3 KHz, CW (Morse code) about 200 Hz. The format of the IF signals is I/Q (one on each channel). 

This project should be useful with dual antenna receivers to detect the polarization or angle of arrival of a signal, and in general to improve the signals received. 

 Approach to Solution 
We first downloaded the files with the real world data and used [http://audacity.sourceforge.net/ Audacity] to shorten the data to a workable length to analyze in MATLAB.

Automatic Antenna Phasing Project

2009-12-09T01:46:47Z

Max.Woesner:

Automatic Antenna Phasing Project

2009-12-08T21:35:25Z

Max.Woesner:

Max Woesner

2009-12-04T19:21:18Z

Max.Woesner:

[[Image:Mask08.jpg |thumb|This is me]]
[[Image:Diving_Turtle.jpg |thumb|Up close and personal]]

=== The Purpose of This Page ===
This page was created to document my work in ENGR455 - Signals and Systems. I hope that both current and future students will be able to benefit from my work here. Since I am a student learning as I go along, I cannot guarantee the complete accuracy of the content on this page. 

[[Signals and systems|Back to the class homepage]]

=== About Me ===
I am a senior electrical engineering major also minoring in business and math. I just got back from a year in Guam working as a volunteer engineer with Adventist World Radio. I love scuba diving but probably won't be doing much of that around these parts. For more info check out the [http://mask.wallawalla.edu/profile/show/7065 Mask]

=== Contact Info ===
*Phone: (509) 524-6978
*Email: maxwell.woesner@wallawalla.edu
*Messenger: maximiliaknow@hotmail.com
*[http://www.facebook.com/max.woesner Facebook]

=== Class Resources ===
[[User:Frohro|Dr. Frohne]] 
[[Signals and systems|Class Wiki]] 
[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR455/2009/Keystone/index.php Class Notes (2009)] 
[http://dlearn2.wallawalla.edu/moodle/ Moodle] 
[http://wikipedia.org/ Wikipedia]

=== Homework ===
[[Evaluate this integral - HW1|Homework #1 - Evaluate this integral]] 
[[Something interesting from class - HW2|Homework #2 - Something interesting from class]] 
[[Class lecture notes October 5 - HW3|Homework #3 - Class lecture notes October 5]] 
[[Fourier Transform Properties|Homework #4 - Fourier transform properties]] 
[[Fourier Transform Property review|Homework #5 - Review a Fourier transform property]] 
[[Fourier Transform Properties|Homework #6c - Another Fourier transform property]] 
[[Sampling - HW7|Homework #7 - Sampling]] 
[[How a CD player works - HW8|Homework #8 - How a CD player works]] 
[[Third harmonic sampling and QSD - HW9|Homework #9 - Third harmonic sampling and QSD]] 
[[Quadrature sampling waveform plot - HW10|Homework #10 - Quadrature sampling waveform plot]] 
[[DFT example using MATLAB - HW11|Homework #11 - DFT example using MATLAB]] 
[[Automatic Antenna Phasing Project|Class Project - Automatic antenna phasing]]

User:Max.Woesner

2009-12-03T19:14:36Z

Max.Woesner:

[[Max Woesner|Max Woesner's Wiki]]

Max Woesner

2009-12-03T19:10:25Z

Max.Woesner:

[[Image:Mask08.jpg |thumb|This is me]]
[[Image:Diving_Turtle.jpg |thumb|Up close and personal]]

=== The Purpose of This Page ===
This page was created to document my work in ENGR455 - Signals and Systems. I hope that both current and future students will be able to benefit from my work here. Since I am a student learning as I go along, I cannot guarantee the complete accuracy of the content on this page. 

[[Signals and systems|Back to the class homepage]]

=== About Me ===
I am a senior electrical engineering major also minoring in business and math. I just got back from a year in Guam working as a volunteer engineer with Adventist World Radio. I love scuba diving but probably won't be doing much of that around these parts. For more info check out the [http://mask.wallawalla.edu/profile/show/7065 Mask]

=== Contact Info ===
*Phone: (509) 524-6978
*Email: maxwell.woesner@wallawalla.edu
*Messenger: maximiliaknow@hotmail.com
*[http://www.facebook.com/max.woesner Facebook]

=== Class Resources ===
[[User:Frohro|Dr. Frohne]] 
[[Signals and systems|Class Wiki]] 
[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR455/2009/Keystone/index.php Class Notes (2009)] 
[http://dlearn2.wallawalla.edu/moodle/ Moodle] 
[http://wikipedia.org/ Wikipedia]

=== Homework ===
[[Evaluate this integral - HW1|Homework #1 - Evaluate this integral]] 
[[Something interesting from class - HW2|Homework #2 - Something interesting from class]] 
[[Class lecture notes October 5 - HW3|Homework #3 - Class lecture notes October 5]] 
[[Fourier Transform Properties|Homework #4 - Fourier Transform properties]] 
[[Fourier Transform Property review|Homework #5 - Review a Fourier transform property]] 
[[Fourier Transform Properties|Homework #6c - Another Fourier transform Property]] 
[[Sampling - HW7|Homework #7 - Sampling]] 
[[How a CD player works - HW8|Homework #8 - How a CD player works]] 
[[Third harmonic sampling and QSD - HW9|Homework #9 - Third harmonic sampling and QSD]] 
[[Quadrature sampling waveform plot - HW10|Homework #10 - Quadrature sampling waveform plot]] 
[[DFT example using MATLAB - HW11|Homework #11 - DFT example using MATLAB]] 
[[Automatic Antenna Phasing Project|Class Project - Automatic Antenna Phasing]]

Max Woesner

2009-12-02T17:45:47Z

Max.Woesner:

[[Image:Mask08.jpg |thumb|This is me]]
[[Image:Diving_Turtle.jpg |thumb|Up close and personal]]

=== The Purpose of This Page ===
This page was created to document my work in ENGR455 - Signals and Systems. I hope that both current and future students will be able to benefit from my work here. Since I am a student learning as I go along, I cannot guarantee the complete accuracy of the contents on this page. 

[[Signals and systems|Back to the class homepage]]

=== About Me ===
I am a senior electrical engineering major also minoring in business and math. I just got back from a year in Guam working as a volunteer engineer with Adventist World Radio. I love scuba diving but probably won't be doing much of that around these parts. For more info check out the [http://mask.wallawalla.edu/profile/show/7065 Mask]

=== Contact Info ===
*Phone: (509) 524-6978
*Email: maxwell.woesner@wallawalla.edu
*Messenger: maximiliaknow@hotmail.com
*[http://www.facebook.com/max.woesner Facebook]

=== Class Resources ===
[[User:Frohro|Dr. Frohne]] 
[[Signals and systems|Class Wiki]] 
[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR455/2009/Keystone/index.php Class Notes (2009)] 
[http://dlearn2.wallawalla.edu/moodle/ Moodle] 
[http://wikipedia.org/ Wikipedia]

=== Homework ===
[[Evaluate this integral - HW1|Homework #1 - Evaluate this integral]] 
[[Something interesting from class - HW2|Homework #2 - Something interesting from class]] 
[[Class lecture notes October 5 - HW3|Homework #3 - Class lecture notes October 5]] 
[[Fourier Transform Properties|Homework #4 - Fourier Transform properties]] 
[[Fourier Transform Property review|Homework #5 - Review a Fourier transform property]] 
[[Fourier Transform Properties|Homework #6c - Another Fourier transform Property]] 
[[Sampling - HW7|Homework #7 - Sampling]] 
[[How a CD player works - HW8|Homework #8 - How a CD player works]] 
[[Third harmonic sampling and QSD - HW9|Homework #9 - Third harmonic sampling and QSD]] 
[[Quadrature sampling waveform plot - HW10|Homework #10 - Quadrature sampling waveform plot]] 
[[DFT example using MATLAB - HW11|Homework #11 - DFT example using MATLAB]] 
[[Automatic Antenna Phasing Project|Class Project - Automatic Antenna Phasing]]

Automatic Antenna Phasing Project

2009-12-02T17:22:42Z

Max.Woesner:

== Max Woesner ==
[[Max Woesner|Back to my Home Page]]

=== Class Project - Automatic Antenna Phasing ===

 Problem Statement 
Design and implement an algorithm to determine the amount of delay necessary to automatically add two signals from different antennas in phase. Also determine the relative amplitudes of the two signals. 

====Tools:====
Software: 
* MATLAB/Octave
* GnuRadio
* Dttsp
* Sdr-Shell
Hardware: 
* Dual Softrock Receiver
* Two Antennas
Data: 
Files of real world data gathered using the above equipment will be provided. 

References: 
[http://www.kkn.net/dayton2008/beam_steering_on_160_meters.pdf Beam Steering on 160 Meters] 
[http://k1lt.com/ More K1LT Notes] 
[http://johnsogg.blogspot.com/2009/11/installing-gnu-radio-on-ubuntu-910.html Install gnuradio on Ubuntu 9.10 using the package manager] 
[http://sdrblog.wordpress.com/gnuradio-installation/ Install gnuradio on Ubuntu 9.10 from GIT repositories] 

Other Information: 
The IF of the signals is the sample_rate/4. The bandwidth of SSB signals is about 3 KHz, CW (Morse code) about 200 Hz. The format of the IF signals is I/Q (one on each channel). 
This project should be useful with dual antenna receivers to detect the polarization or angle of arrival of a signal, and in general to improve the signals received.

Automatic Antenna Phasing Project

2009-12-02T17:07:35Z

Max.Woesner:

Automatic Antenna Phasing Project

2009-12-02T17:02:33Z

Max.Woesner: New page: == Max Woesner == Back to my Home Page === Class Project - Automatic Antenna Phasing === Problem Statement Design and implement an algorithm to determine t...

Max Woesner

2009-12-02T17:00:45Z

Max.Woesner:

[[Image:Mask08.jpg |thumb|This is me]]
[[Image:Diving_Turtle.jpg |thumb|Up close and personal]]

=== The Purpose of This Page ===
This page was created to document my work in ENGR455 - Signals and Systems. I hope that both current and future students will be able to benefit from my work here. Since I am a student learning as I go along, I cannot guarantee the complete accuracy of the contents on this page. 

[[Signals and systems|Back to the class homepage]]

=== About Me ===
I am a senior electrical engineering major also minoring in business and math. I just got back from a year in Guam working as a volunteer engineer with Adventist World Radio. I love scuba diving but probably won't be doing much of that around these parts. For more info check out the [http://mask.wallawalla.edu/profile/show/7065 Mask]

=== Contact Info ===
*Phone: (509) 524-6978
*Email: maxwell.woesner@wallawalla.edu
*Messenger: maximiliaknow@hotmail.com
*[http://www.facebook.com/max.woesner Facebook]

=== Class Resources ===
[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR455/2009/Keystone/index.php Class notes] 
[http://dlearn2.wallawalla.edu/moodle/ Moodle]

=== Homework ===
'''Sources:'''
Class Notes,
Class Wiki,
Wikipedia

[[Evaluate this integral - HW1|Homework #1 - Evaluate this integral]] 
[[Something interesting from class - HW2|Homework #2 - Something interesting from class]] 
[[Class lecture notes October 5 - HW3|Homework #3 - Class lecture notes October 5]] 
[[Fourier Transform Properties|Homework #4 - Fourier Transform properties]] 
[[Fourier Transform Property review|Homework #5 - Review a Fourier transform property]] 
[[Fourier Transform Properties|Homework #6c - Another Fourier transform Property]] 
[[Sampling - HW7|Homework #7 - Sampling]] 
[[How a CD player works - HW8|Homework #8 - How a CD player works]] 
[[Third harmonic sampling and QSD - HW9|Homework #9 - Third harmonic sampling and QSD]] 
[[Quadrature sampling waveform plot - HW10|Homework #10 - Quadrature sampling waveform plot]] 
[[DFT example using MATLAB - HW11|Homework #11 - DFT example using MATLAB]] 
[[Automatic Antenna Phasing Project|Class Project - Automatic Antenna Phasing]]

Max Woesner

2009-12-02T06:49:47Z

Max.Woesner:

[[Image:Mask08.jpg |thumb|This is me]]
[[Image:Diving_Turtle.jpg |thumb|Up close and personal]]

=== The Purpose of This Page ===
This page was created to document my work in ENGR455 - Signals and Systems. I hope that both current and future students will be able to benefit from my work here. Since I am a student learning as I go along, I cannot guarantee the complete accuracy of the contents on this page. 

[[Signals and systems|Back to the class homepage]]

=== About Me ===
I am a senior electrical engineering major also minoring in business and math. I just got back from a year in Guam working as a volunteer engineer with Adventist World Radio. I love scuba diving but probably won't be doing much of that around these parts. For more info check out the [http://mask.wallawalla.edu/profile/show/7065 Mask]

=== Contact Info ===
*Phone: (509) 524-6978
*Email: maxwell.woesner@wallawalla.edu
*Messenger: maximiliaknow@hotmail.com
*[http://www.facebook.com/max.woesner Facebook]

=== Class Resources ===
[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR455/2009/Keystone/index.php Class notes] 
[http://dlearn2.wallawalla.edu/moodle/ Moodle]

=== Homework ===
'''Sources:'''
Class Notes,
Class Wiki,
Wikipedia

[[Evaluate this integral - HW1|Homework #1 - Evaluate this integral]] 
[[Something interesting from class - HW2|Homework #2 - Something interesting from class]] 
[[Class lecture notes October 5 - HW3|Homework #3 - Class lecture notes October 5]] 
[[Fourier Transform Properties|Homework #4 - Fourier Transform properties]] 
[[Fourier Transform Property review|Homework #5 - Review a Fourier transform property]] 
[[Fourier Transform Properties|Homework #6c - Another Fourier transform Property]] 
[[Sampling - HW7|Homework #7 - Sampling]] 
[[How a CD player works - HW8|Homework #8 - How a CD player works]] 
[[Third harmonic sampling and QSD - HW9|Homework #9 - Third harmonic sampling and QSD]] 
[[Quadrature sampling waveform plot - HW10|Homework #10 - Quadrature sampling waveform plot]] 
[[DFT example using MATLAB - HW11|Homework #11 - DFT example using MATLAB]]

Quadrature sampling waveform plot - HW10

2009-12-02T06:31:14Z

Max.Woesner:

== Max Woesner ==
[[Max Woesner|Back to my Home Page]]

=== Homework #10 - Quadrature sampling waveform plot ===

 Problem Statement 
Plot <math> \ \frac{2}{T} \sum_{n=1}^\infty sin\bigg(\frac{2 \pi nt}{T}\bigg) \!</math> 
Solution 

While we can't sum to infinity in the computer, we can get a close approximation summing over a large enough range of <math> n \!</math> 

I found summing over <math> n = 1:1000 \!</math> was about the most the computer could handle reasonably. 

The following script was written in MATLAB to produce the desired plot. 
<pre>
clear all;
close all;
sum = 0;
T = 1;
t = -T:0.0001:T;
N = 1000;
for n = 1:N;
if n==0
h = 0;
else
h = 2/T;
end
sum = sum+h*sin(2*pi*n*t/T);
end
plot(t,sum)
title('Quadrature Sampling Waveform')
xlabel('time(T)')
ylabel('Sampling Waveform')
</pre> 
Running the MATLAB script above gives us the following plot. 

[[Image:Quadrature sampling.jpg]] 

Summing over a smaller range of <math> n \!</math> would look like the following. 
<pre>
clear all;
close all;
sum = 0;
T = 1;
t = -T:0.001:T;
N = 100;
for n = 1:N;
if n==0
h = 0;
else
h = 2/T;
end
sum = sum+h*sin(2*pi*n*t/T);
end
plot(t,sum)
title('Quadrature Sampling Waveform')
xlabel('time(T)')
ylabel('Sampling Waveform')
</pre> 

[[Image:Quadrature sampling2.jpg]]

File:Quadrature sampling2.jpg

2009-12-02T06:28:38Z

Max.Woesner:

Max Woesner

2009-12-02T05:07:29Z

Max.Woesner:

[[Image:Mask08.jpg |thumb|This is me]]
[[Image:Diving_Turtle.jpg |thumb|Up close and personal]]

=== The Purpose of This Page ===
This page was created to document my work in ENGR455 - Signals and Systems. I hope that both current and future students will be able to benefit from the work on this page. Since I am a student learning as I go along, I cannot guarantee the complete accuracy of the contents on this page. 

[[Signals and systems|Back to the class homepage]]

=== About Me ===
I am a senior electrical engineering major also minoring in business and math. I just got back from a year in Guam working as a volunteer engineer with Adventist World Radio. I love scuba diving but probably won't be doing much of that around these parts. For more info check out the [http://mask.wallawalla.edu/profile/show/7065 Mask]

=== Contact Info ===
*Phone: (509) 524-6978
*Email: maxwell.woesner@wallawalla.edu
*Messenger: maximiliaknow@hotmail.com
*[http://www.facebook.com/max.woesner Facebook]

=== Class Resources ===
[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR455/2009/Keystone/index.php Class notes] 
[http://dlearn2.wallawalla.edu/moodle/ Moodle]

=== Homework ===
'''Sources:'''
Class Notes,
Class Wiki,
Wikipedia

[[Evaluate this integral - HW1|Homework #1 - Evaluate this integral]] 
[[Something interesting from class - HW2|Homework #2 - Something interesting from class]] 
[[Class lecture notes October 5 - HW3|Homework #3 - Class lecture notes October 5]] 
[[Fourier Transform Properties|Homework #4 - Fourier Transform properties]] 
[[Fourier Transform Property review|Homework #5 - Review a Fourier transform property]] 
[[Fourier Transform Properties|Homework #6c - Another Fourier transform Property]] 
[[Sampling - HW7|Homework #7 - Sampling]] 
[[How a CD player works - HW8|Homework #8 - How a CD player works]] 
[[Third harmonic sampling and QSD - HW9|Homework #9 - Third harmonic sampling and QSD]] 
[[Quadrature sampling waveform plot - HW10|Homework #10 - Quadrature sampling waveform plot]] 
[[DFT example using MATLAB - HW11|Homework #11 - DFT example using MATLAB]]

Max Woesner

2009-12-02T05:00:59Z

Max.Woesner:

[[Image:Mask08.jpg |thumb|This is me]]
[[Image:Diving_Turtle.jpg |thumb|Up close and personal]]

=== The Purpose of This Page ===
This page was created to document my work in ENGR455 - Signals and Systems. I hope that both current and future students will be able to benefit from the work on this page. Since I am a student learning as I go along, I cannot guarantee the complete accuracy of the contents on this page. 

[[Signals and systems|Back to the class homepage]]

=== About Me ===
I am a senior electrical engineering major also minoring in business and math. I just got back from a year in Guam working as a volunteer engineer with Adventist World Radio. I love scuba diving but probably won't be doing much of that around these parts. For more info check out the [http://mask.wallawalla.edu/profile/show/7065 Mask]

=== Contact Info ===
*Phone: (509) 524-6978
*Email: maxwell.woesner@wallawalla.edu
*Messenger: maximiliaknow@hotmail.com
*[http://www.facebook.com/max.woesner Facebook]

=== Class Resources ===
[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR455/2009/Keystone/index.php Class notes] 
[http://dlearn2.wallawalla.edu/moodle/ Moodle]

=== Homework ===
'''Sources:'''
Class Notes,
Class Wiki,
Wikipedia

[[Evaluate this integral - HW1|Homework #1 - Evaluate this integral]] 
[[Something interesting from class - HW2|Homework #2 - Something interesting from class]] 
[[Class lecture notes October 5 - HW3|Homework #3 - Class lecture notes October 5]] 
[[Fourier Transform Properties|Homework #4 - Fourier Transform properties]] 
[[Fourier Transform Property review|Homework #5 - Review a Fourier transform property]] 
[[Fourier Transform Properties|Homework #6c - Another Fourier transform Property]] 
[[Sampling - HW7|Homework #7 - Sampling]] 
[[How a CD player works - HW8|Homework #8 - How a CD player works]] 
[[Third harmonic sampling and QSD - HW9|Homework #9 - Third harmonic sampling and QSD]] 
[[Quadrature sampling waveform plot - HW10|Homework #10 - Quadrature sampling waveform plot]] 
[[DFT example using MATLAB - HW11|Homework #11 - DFT example using MATLAB]] 
[[Automatic Antenna Phasing Project|Class Project - Automatic Antenna Phasing]]

Sampling - HW7

2009-12-02T04:42:01Z

Max.Woesner:

== Max Woesner ==
[[Max Woesner|Back to my Home Page]]

=== Homework #7 - Sampling ===

 Problem Statement 
Figure out what happens if your sampled signal, <math> x(t)\!</math>, has frequency components only for <math> \frac{f_s}{2}<f<f_s\!</math>. Can you recover the original signal from it? If so, find the expression for <math> x(t)\!</math> in terms of <math> x(nT)\!</math>. 

Solution 
For frequency components <math> \frac{f_s}{2}<f<f_s\!</math>, our sampled signal <math> x(t)\!</math> in the frequency domain, or <math> X(f)\!</math>, is going to look like this. 
[[Image:Sampling1a.jpg]] 
After sampling with frequency <math> f_s\!</math>, the signal is going to be shifted over by <math> \frac{1}{T}\!</math>, since <math> f_s = \frac{1}{T}\!</math>. It will look like this. 
[[Image:Sampling2.jpg]] 
To recover the original signal <math> x(t)\!</math>, we need to a use bandpass filter to filter out the parts we don't want, as indicated by the red line in the figure below. (Note: not to scale.) 
[[Image:Sampling3.jpg]] 
This leaves us with the original signal, as shown below. 
[[Image:Sampling1a.jpg]] 
The transfer function of the bandpass filter that will accomplish this for us is <math>H(f)=\begin{cases} T, & \mbox{ } \frac{1}{2T}<|f|< \frac{1}{T} \\ 0, & \mbox{ else } \end{cases}\!</math> 
To find the expression for <math> h(t)\!</math>, or the expression for the bandpass filter in the time domain, we can take the inverse Fourier transform of <math>H(f)\!</math>. 
<math> h(t) = \mathcal{F}^{-1}[H(f)] = \int_{-\frac{1}{T}}^{-\frac{1}{2T}}Te^{j2 \pi ft}df\ + \int_{\frac{1}{2T}}^{\frac{1}{T}}Te^{j2 \pi ft}df = \frac{Te^{j2 \pi ft}}{j2 \pi t} \Bigg|_{f=-\frac{1}{T}}^{-\frac{1}{2T}} + \ \frac{Te^{j2 \pi ft}}{j2 \pi t} \Bigg|_{f=\frac{1}{2T}}^{\frac{1}{T}}\!</math> 
<math> h(t) = T \frac{e^{-j \pi t/T} - e^{-j2 \pi t/T}}{j2( \pi t)} \ + \ T \frac{e^{j2 \pi t/T} - e^{j \pi t/T}}{j2( \pi t)} = \frac{T}{j2}\Bigg[\frac{e^{-j \pi t/T} - e^{-j2 \pi t/T}+e^{j2 \pi t/T} - e^{j \pi t/T}}{ \pi t}\Bigg]\!</math> 
<math> h(t) = \frac{T}{j2}\Bigg[\frac{- e^{j \pi t/T} + e^{-j \pi t/T} + e^{j2 \pi t/T} - e^{-j2 \pi t/T}}{ \pi t}\Bigg] = \frac{T}{j2}\Bigg[\frac{e^{j2 \pi t/T} - e^{-j2 \pi t/T}}{ \pi t}\Bigg] - \ \frac{T}{j2}\Bigg[\frac{e^{j \pi t/T} - e^{-j \pi t/T}}{ \pi t}\Bigg] \!</math> 
Recall <math> sin(\theta) = \frac{1}{j2}(e^{j\theta} - e^{-j\theta})\!</math>, so 
<math> h(t) = \frac{T}{j2}\Bigg[\frac{e^{j2 \pi t/T} - e^{-j2 \pi t/T}}{ \pi t}\Bigg] - \ \frac{T}{j2}\Bigg[\frac{e^{j \pi t/T} - e^{-j \pi t/T}}{ \pi t}\Bigg] =\frac{T}{ \pi t}sin \Bigg(\frac{2 \pi t}{T} \Bigg) \ - \ \frac{T}{ \pi t}sin \Bigg(\frac{ \pi t}{T} \Bigg) = \frac{2sin \Big(\frac{2 \pi t}{T} \Big)}{\frac{2 \pi t}{T}} \ - \ \frac{sin \Big(\frac{ \pi t}{T} \Big)}{\frac{ \pi t}{T}} \!</math> 
Also recall <math> sinc(\theta) = \frac{sin(\pi \theta)}{\pi \theta}\!</math>, so 
<math> h(t) = \frac{2sin \Big(\frac{2 \pi t}{T} \Big)}{\frac{2 \pi t}{T}} \ - \ \frac{sin \Big(\frac{ \pi t}{T} \Big)}{\frac{ \pi t}{T}} = 2sinc \Bigg(\frac{2t}{T} \Bigg) - sinc \Bigg(\frac{t}{T} \Bigg)\!</math> 
Now that we know <math> h(t) \!</math>, we can find <math> x(t) \!</math> by convolving the function for <math> x(t) \!</math> after sampling, or <math> \sum_{n=-\infty}^\infty x(nT)\delta(t-nT) \!</math>, with <math> h(t) \!</math>. 
<math> x(t) = \sum_{n=-\infty}^\infty x(nT)\delta(t-nT) * \Bigg[2sinc \Bigg(\frac{2t}{T} \Bigg) - sinc \Bigg(\frac{t}{T} \Bigg)\Bigg] = \sum_{n=-\infty}^\infty x(nT)\Bigg[2sinc \Bigg(\frac{2(t-nT)}{T} \Bigg) \ - \ sinc \Bigg(\frac{t-nT}{T} \Bigg)\Bigg]\!</math> 
Or, if you prefer, <math> x(t) = \sum_{n=-\infty}^\infty x(nT)\Bigg[\frac{2sin\Big(\frac{2\pi (t-nT)}{T}\Big)}{\frac{2\pi (t-nT)}{T}} \ - \ \frac{sin\Big(\frac{\pi (t-nT)}{T}\Big)}{\frac{\pi (t-nT)}{T}}\Bigg]\!</math> 

Alternatively, a different bandpass filter can be used that will simplify the math we have to do, such as the one indicated by the red line in the figure below. (Note: not to scale.) 
[[Image:Sampling4.jpg]] 
Leaving us with the signal shown below. 
[[Image:Sampling5.jpg]] 
The transfer function of this bandpass filter is <math>H(f)=\begin{cases} T, & \mbox{ } |f|< \frac{1}{2T} \\ 0, & \mbox{ else } \end{cases}\!</math> 
To find the expression for <math> h(t)\!</math>, or the expression for the bandpass filter in the time domain, we can take the inverse Fourier transform of <math>H(f)\!</math>. 
<math> h(t) = \mathcal{F}^{-1}[H(f)] = \int_{-\frac{1}{2T}}^{\frac{1}{2T}}Te^{j2 \pi ft}df\ = \frac{Te^{j2 \pi ft}}{j2 \pi t} \Bigg|_{f=-\frac{1}{2T}}^{\frac{1}{2T}} = \frac{T}{j2}\Bigg[\frac{e^{j \pi t/T} - e^{-j \pi t/T}}{ \pi t}\Bigg]\!</math> 
Recall again that <math> sin(\theta) = \frac{1}{j2}(e^{j\theta} - e^{-j\theta})\!</math>, so 
<math> h(t) = \frac{T}{j2}\Bigg[\frac{e^{j \pi t/T} - e^{-j \pi t/T}}{ \pi t}\Bigg] = \frac{T}{ \pi t}sin \Bigg(\frac{ \pi t}{T} \Bigg) \ = \frac{sin \Big(\frac{ \pi t}{T} \Big)}{\frac{ \pi t}{T}} \!</math> 
Also recall again that <math> sinc(\theta) = \frac{sin(\pi \theta)}{\pi \theta}\!</math>, so 
<math> h(t) = \frac{sin \Big(\frac{ \pi t}{T} \bigg)}{\frac{ \pi t}{T}} = sinc \bigg(\frac{t}{T} \bigg) \!</math> 
Now that we know <math> h(t) \!</math>, we can find <math> x(t) \!</math> by convolving the function for <math> x(t) \!</math> after sampling, or <math> \sum_{n=-\infty}^\infty x(nT)\delta(t-nT) \!</math>, with <math> h(t) \!</math>. 

<math> x(t) = \sum_{n=-\infty}^\infty x(nT)\delta(t-nT) * \Bigg[sinc \bigg(\frac{t}{T} \bigg)\Bigg] = \sum_{n=-\infty}^\infty x(nT)(sinc \Bigg(\frac{t-nT}{T} \Bigg)\!</math> 
Or, if you prefer, <math> x(t) = \sum_{n=-\infty}^\infty x(nT)\Bigg[\frac{sin\Big(\frac{\pi (t-nT)}{T}\Big)}{\frac{\pi (t-nT)}{T}}\Bigg]\!</math>

File:Sampling5.jpg

2009-12-02T03:58:16Z

Max.Woesner:

File:Sampling4.jpg

2009-12-02T03:58:05Z

Max.Woesner:

Fourier Transform Properties

2009-12-01T06:26:19Z

Max.Woesner:

[http://cnx.org/content/m0045/latest/ Some properties to choose from if you are having difficulty....]

==[[Max Woesner|Max Woesner]] ==

'''Look carefully at the signs in section 2 for <math>f^'</math> and <math>f^{''}</math> after the * operation. I think the signs are backwards but it ends up working out just fine because <math>\delta (f^{''}-f^')=\delta (f^'-f^{''})</math> -Brandon''' 
'''Corrected -Max'''

1. '''Find <math>\mathcal{F}[cos(w_0t)g(t)]\!</math> '''
Recall <math> w_0 = 2\pi f_0\!</math>, so <math>\mathcal{F}[cos(w_0t)g(t)] = \mathcal{F}[cos(2\pi f_0t)g(t)] = \int_{-\infty}^{\infty}cos(2\pi f_0t)g(t)e^{-j2\pi ft}dt\!</math> 
Also recall <math> cos(\theta) = \frac{1}{2}(e^{j\theta} + e^{-j\theta})\!</math>,so <math>\int_{-\infty}^{\infty}cos(2\pi f_0t)g(t)e^{-j2\pi ft}dt = \int_{-\infty}^{\infty} \frac{1}{2}[e^{j2\pi f_0t}+e^{-j2\pi f_0t}]g(t)e^{-j2\pi ft}dt\!</math> 
Now <math>\int_{-\infty}^{\infty} \frac{1}{2}[e^{j2\pi f_0t}+e^{-j2\pi f_0t}]g(t)e^{-j2\pi ft}dt = \frac{1}{2}\int_{-\infty}^{\infty}e^{-j2\pi (f-f_0)t}g(t)dt \ + \ \frac{1}{2}\int_{-\infty}^{\infty}e^{-j2\pi (f+f_0)t}g(t)dt = \frac{1}{2}G(f-f_0) \ + \ \frac{1}{2}G(f+f_0)\!</math> 
So <math>\mathcal{F}[cos(w_0t)g(t)] = \frac{1}{2}[G(f-f_0)+ G(f+f_0)]\!</math> 

reviewed by [[Joshua Sarris]]

2. '''Find <math>\mathcal{F}\bigg[\int_{-\infty}^{\infty}g(t) h^*(t) dt\bigg]\!</math> '''
Recall <math> g(t)= \mathcal{F}^{-1}[G(f)] = \int_{-\infty}^{\infty}G(f)e^{j2\pi ft}df\!</math> 
Similarly, <math> h(t)= \mathcal{F}^{-1}[H(f)] = \int_{-\infty}^{\infty}H(f)e^{j2\pi ft}df\!</math> 
So <math>\mathcal{F}\bigg[\int_{-\infty}^{\infty}g(t) h^*(t) dt\bigg] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}G(f^')e^{j2\pi f^'t}df^' \Bigg(\int_{-\infty}^{\infty}H(f^{''})e^{j2\pi f^{''}t}df^{''}\Bigg)^* dt \!</math> 
Now <math>\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}G(f^')e^{j2\pi f^'t}df^' \Bigg(\int_{-\infty}^{\infty}H(f^{''})e^{j2\pi f^{''}t}df^{''}\Bigg)^* dt = \int_{-\infty}^{\infty}G(f^')\int_{-\infty}^{\infty}H^*(f^{''})\int_{-\infty}^{\infty}e^{j2\pi (f^'-f^{''})t}dt df^{''} df^' \!</math> 
Note that <math>\int_{-\infty}^{\infty}e^{j2\pi (f^'-f^{''})t}dt = \delta (f^'-f^{''}) \!</math> 
Added step per Nick's suggestion 
Substituting gives us <math>\int_{-\infty}^{\infty}G(f^')\int_{-\infty}^{\infty}H^*(f^{''})\int_{-\infty}^{\infty}e^{j2\pi (f^'-f^{''})t}dt df^{''} df^' = \int_{-\infty}^{\infty}G(f^')\int_{-\infty}^{\infty}H^*(f^{''})\delta (f^'-f^{''}) df^{''} df^' \!</math> 
And <math> \int_{-\infty}^{\infty}G(f^')\int_{-\infty}^{\infty}H^*(f^{''})\delta (f^'-f^{''}) df^{''} df^' = \int_{-\infty}^{\infty}G(f^')H^*(f^')df^' \!</math> 
Since <math> f^' \!</math> is a simply a dummy variable, we can conclude that: 
<math>\mathcal{F}\bigg[\int_{-\infty}^{\infty}g(t) h^*(t) dt\bigg] = \int_{-\infty}^{\infty}G(f)H^*(f)df \!</math> 

"I was going to make a comment on the delta identity, but after looking at it closer I think it is fine. One comment I have is that you might consider adding one more step, showing the delta function in the integral and pulling the integrands together to make it look like a double integral -- it isn't necessary and I understood the transition, but it helps the proof/identity look a little more complete. Good job!"

Example: <math>\int_{a1}^{a2}\int_{b1}^{b2}X(s')Y(s'') \delta (s''-s') \,ds' \,ds'' = \int_{a1}^{a2}X(s')Y(s') \,ds' </math>

Reviewed by [[Nick Christman]]

----

==[[Nick Christman|Nick Christman]] ==
Note: After scratching my head for a couple of hours, I decided that I would try a different Fourier Property. In fact, I chose a property that would need to be defined in order to show my second property.

1. '''Find <math>\mathcal{F} \left[ g(t)e^{j2 \pi f_{0}t} \right] </math> '''

This is a fairly straightforward property and is known as ''complex modulation'' 

<math>
\mathcal{F} \left[ g(t)e^{j2 \pi f_{0}t} \right] = \int_{- \infty}^{\infty} \left[ g(t)e^{j2 \pi f_{0}t} \right] e^{-j2 \pi ft} \,dt
</math>

Combining terms, we get:

<math>
\int_{- \infty}^{\infty} \left[ g(t)e^{j2 \pi f_{0}t} \right] e^{-j2 \pi ft} \,dt = \int_{- \infty}^{\infty} g(t)e^{-j2 \pi (f-f_{0})t} \,dt
</math>
 

Now let's make the following substitution <math> \displaystyle \theta = f-f_{0}</math>

This now gives us a surprisingly familiar function:

<math>
\int_{- \infty}^{\infty} g(t)e^{-j2 \pi (f-f_{0})t} \,dt = \int_{- \infty}^{\infty} g(t)e^{-j2 \pi \theta t} \,dt
</math>
 

This looks just like <math> \displaystyle G(\theta )</math>!

We can now conclude that:
 

<math>
\mathcal{F} \left[ g(t)e^{j2 \pi f_{0}t} \right] = G(\theta ) = G(f-f_{0})
</math>
 
 

'''PLEASE ENTER PEER REVIEW HERE'''

 
 

2. '''Find <math>\mathcal{F} \left[ g(t-t_{0})e^{j2 \pi f_{0}t} \right]</math> '''

-- Using the above definition of ''complex modulation'' and the definition from class of a ''time delay'' (a.k.a "the slacker function"), I will attempt to show a hybrid of the two...
 

By definition we know that:

<math>
\mathcal{F} \left[ g(t-t_{0})e^{j2 \pi f_{0}t} \right] = \int_{- \infty}^{\infty} \left[ g(t-t_{0})e^{j2 \pi f_{0}t} \right] e^{-j2 \pi ft} \,dt
</math>

Rearranging terms we get:

<math>
\int_{- \infty}^{\infty} \left[ g(t-t_{0})e^{j2 \pi f_{0}t} \right] e^{-j2 \pi ft} \,dt = \int_{- \infty}^{\infty} g(t-t_{0})e^{-j2 \pi (f-f_{0})t} \,dt
</math>
 

Now lets make the substitution <math>\lambda = t-t_{0} \rightarrow t = \lambda + t_{0}</math>.
 
This leads us to:

<math>
\int_{- \infty}^{\infty} g(t-t_{0})e^{-j2 \pi (f-f_{0})t} \,dt = \int_{- \infty}^{\infty} g(\lambda )e^{-j2 \pi (f-f_{0})(\lambda + t_{0})} \,d \lambda
</math>

After some simplification and rearranging terms, we get:

<math>
\int_{- \infty}^{\infty} g(\lambda )e^{-j2 \pi (f-f_{0})(\lambda + t_{0})} \,d \lambda = \int_{- \infty}^{\infty} g(\lambda )e^{-j2 \pi (f-f_{0})\lambda } e^{-j2 \pi (f-f_{0})t_{0}} \,d \lambda
</math>

Rearranging the terms yet again, we get:

<math>
\int_{- \infty}^{\infty} g(\lambda )e^{-j2 \pi (f-f_{0})\lambda } e^{-j2 \pi (f-f_{0})t_{0}} \,d \lambda = e^{-j2 \pi (f-f_{0})t_{0}} \left[ \int_{- \infty}^{\infty} g(\lambda )e^{-j2 \pi (f-f_{0})\lambda } \,d \lambda \right]
</math>

We know that the exponential in terms of <math>\displaystyle t_{0}</math> is simply a constant and because of the Fourier Property of ''complex modualtion'', we finally get:

<math>
\mathcal{F} \left[ g(t)e^{j2 \pi f_{0}t} \right] = G(f-f_{0})e^{-j2 \pi (f-f_{0})t_{0}}
</math>

 

'''Reviewed by [[Kevin Starkey]] --> add <math>\, d \lambda</math> above... other than that it looks good.'''

 
 

----

==[[Joshua Sarris|Joshua Sarris]] ==
'''Find <math>\mathcal{F}[sin(w_0t)g(t)]\!</math> '''

Recall
<math> w_0 = 2\pi f_0\!</math>,

so expanding we have,

<math>\mathcal{F}[sin(w_0t)g(t)] = \mathcal{F}[sin(2\pi f_0t)g(t)] = \int_{-\infty}^{\infty}sin(2\pi f_0t)g(t)e^{-j2\pi ft}dt\!</math> 

Also recall
<math> sin(\theta) = \frac{1}{j2}(e^{j\theta} - e^{-j\theta})\!</math>,

so we can convert to exponentials.

<math>\int_{-\infty}^{\infty}sin(2\pi f_0t)g(t)e^{-j2\pi ft}dt = \int_{-\infty}^{\infty} \frac{1}{j2}[e^{j2\pi f_0t}-e^{-j2\pi f_0t}]g(t)e^{-j2\pi ft}dt\!</math> 

Now integrating gives us,

<math>\int_{-\infty}^{\infty} \frac{1}{j2}[e^{j2\pi f_0t}-e^{-j2\pi f_0t}]g(t)e^{-j2\pi ft}dt = \frac{1}{j2}\int_{-\infty}^{\infty}e^{-j2\pi (f-f_0)t}g(t)dt-\frac{1}{j2}\int_{-\infty}^{\infty}e^{-j2\pi (f+f_0)t}g(t)dt = \frac{1}{j2}G(f-f_0)- \frac{1}{j2}G(f+f_0)\!</math> 

So we now have the identity,

<math>\mathcal{F}[sin(w_0t)g(t)] = \frac{1}{j2}[G(f-f_0)- G(f+f_0)]\!</math>

or rather

<math>\mathcal{F}[sin(w_0t)g(t)] =\frac{1}{2}j[G(f+f_0)+ G(f-f_0)]\!</math>

[[Fourier Transform Property review|Reviewed by Max Woesner]]

Also reviewed by [[Nick Christman]]
-- Looks good.

'''Find <math>\mathcal{F}[\frac{d}{dt} x(t)] \!</math> '''

We begin by finding the Fourier of x(t).

<math>\mathcal{F}[\frac{d}{dt} x(t)] = \frac{d}{dt} [ \int_{-\infty}^{\infty} x(t)e^{-j2 \pi f t} df]</math>

We can then pull the derivitive into the integral and carry out its opperation.

<math> \int_{-\infty}^{\infty} x(t)\frac{d}{dt} e^{-j2 \pi ft} dt = -j2\pi f \int_{-\infty}^{\infty} x(t) e^{-j2\pi f t} dt</math>

Since we know <math>\int_{-\infty}^{\infty} x(t) e^{-j2\pi f t} dt</math> is X(f) we can simplify.

<math>\mathcal{F}[\frac{d}{dt} x(t)]= -j2\pi f X(f)</math>

----

==[[Kevin Starkey|Kevin Starkey]] ==
1. Find <math> \mathcal{F}\left[\int_{-\infty}^ \infty s(t)dt\right]</math> 
First we know that <math> \mathcal{F}\left[\int_{-\infty}^ \infty s(t)dt\right] = \int_{-\infty}^\infty\left(\int_{-\infty}^ \infty s(t)dt\right)e^{-j2\pi ft} dt </math> 
We also know that <math> \mathcal{F}\left[s(t)\right] = S(f) \mbox{ and } \int_{-\infty}^ \infty e^{-j2\pi ft} dt = \delta(f) </math> 
(+) Which gives us <math> \int_{-\infty}^\infty\left(\int_{-\infty}^ \infty s(t)dt\right)e^{-j2\pi ft} dt = \int_{-\infty}^ \infty S(f) \delta (f)df </math> 
Since <math> \int_{-\infty}^ \infty \delta (f) df </math> is only non-zero at f = 0 this yeilds 
<math> \int_{-\infty}^ \infty S(f) \delta (f)df = S(0) </math> 
So <math> \mathcal{F}\left[\int_{-\infty}^ \infty s(t)dt\right] = S(0)</math> 

Reviewed by [[Nick Christman]]
-- I fixed one typo (needed a minus sign in the exponential). I'm not sure about the step (+). I would like to believe it, but I'm just not sure that it works... if you are sure it works, maybe add a little comment to explain it a little better. Other than that, it looks good! 

2. Find <math>\mathcal{F}\left[e^{j2\pi f_0t}s(t)\right] </math> 
First <math>\mathcal{F}\left[e^{j2\pi f_0t}s(t)\right] = \int_{- \infty}^{\infty}e^{j2\pi f_0t}s(t)e^{-j2\pi ft} </math> 
or rearranging we get <math> \int_{- \infty}^{\infty}e^{j2\pi f_0t}s(t)e^{-j2\pi ft}dt = \int_{- \infty}^{\infty}s(t)e^{-j2\pi t(f -f_0)}dt</math> 
Which leads to <math> \int_{- \infty}^{\infty}s(t)e^{-j2\pi t(f -f_0)}dt = S(f-f_0)</math> 
So <math>\mathcal{F}\left[e^{j2\pi f_0t}s(t)\right] = S(f-f_0) </math> 

'''PLEASE ENTER PEER REVIEW HERE'''

How a CD player works - HW8

2009-12-01T06:10:23Z

Max.Woesner:

== Max Woesner ==
[[Max Woesner|Back to my Home Page]]

=== Homework #8 - How a CD player works ===

This page describes how a CD player works with no oversampling but with digital filtering, i.e. 1x oversampling. 

When music for an audio CD is produced, the music has infinite data points and can be expressed as as a continuous function of time, or <math> x(t)\!</math>, such as the one shown below. 
[[Image:x_continuous3.jpg]] 
In the frequency domain, the signal <math> X(f)\!</math> looks like this. 
[[Image:x_frequency1b.jpg]] 
Since a CD has a finite amount of storage space, it would be impossible to store <math> x(t)\!</math> on a CD. To solve this problem, the data is sampled at periodic intervals, creating a discrete function of time <math> x(nT)\!</math>, where <math> n\!</math> is and integer and <math> T\!</math> is the period between samples. 
The sample rate <math> f_s = \frac{1}{T} \!</math>. Aliasing can occur when you don't sample a signal fast enough to be able to reconstruct it accurately after sampling. To make sure this doesn't happen, we must follow Nyquist's theorem, which states that we must sample at a rate at least twice the fastest frequency of the signal we are sampling. Since human hearing can typically range from 20 Hz to 22 kHz, we want to sample at a rate greater than twice the highest frequency, or 44 kHz. 
The discrete function of time <math> x(nT)\!</math> can be expressed mathematically as <math>\sum_{n=-\infty}^\infty x(nT) \delta (t-nT) \!</math>. It might look like this. 
[[Image:x_discrete_a.jpg]] 
In the frequency domain, the function can be expressed either as <math>\frac{1}{T}\sum_{n=-\infty}^\infty X\bigg(f-\frac{n}{T}\bigg)</math> or <math> \sum_{n=-\infty}^\infty x(nT)e^{-j2\pi fnT} \!</math> and would look like this. 
[[Image:x_frequency2a.jpg]] 
Oversampling can be used to create a smoother discrete function by filling in the gaps with more data points. For example, 8x oversampling, which is fairly common, would decrease the sampling period by a factor of eight, giving us a more accurate function of the original signal. For this page, however, we will focus on 1x oversampling. While no additional data points are added with 1x oversampling, the same process can be used as with 2x, 4x, or 8x oversampling by running the signal through a digital filter. 
To do this, we want to convolve our discrete function by a standard finite impulse response (FIR) filter we will call <math> h(t)\!</math>. 
<math> h(t)\!</math> can be defined as <math> h(t) = \sum_{m=-M}^M h\bigg(\frac{mT}{n'}\bigg)\,\delta \bigg(\frac{t-mT}{n'}\bigg)\!</math>, where <math> n'\!</math> is the oversampling rate. 
In this case, <math> n'=1\!</math>, so 
<math> h(t) = \sum_{m=-M}^M h(mT)\,\delta (t-mT)\!</math> . Since we are using 1x oversampling, the function won't look any different than the original discrete function. 
[[Image:x_discrete_a.jpg]] 
The frequency response of this is <math>H(f)=\mathcal{F}\Bigg[\sum_{m=-M}^M h(mT)\,\delta (t-mT)\Bigg] = \sum_{m=-M}^M h(mT)e^{-j2\pi fmT}</math>, which looks like this. 
[[Image:x_frequency3a.jpg]] 
The convolution of our discrete function and <math> h(t) \!</math> can be expressed mathematically as 
<math> x(t)*h(t) \ = \sum_{n=-\infty}^\infty x(nT) \delta (t-nT) * \sum_{m=-M}^M h(mT)\,\delta (t-mT) \!</math> 
::::<math> = \sum_{n=-\infty}^\infty x(nT) \sum_{m=-M}^M h(mT)\,\delta (t-mT-nT)\!</math> 
Let <math> l = m + n \!</math>, so <math> n = l - m \!</math>, and 
<math> x(t)*h(t) \ = \sum_{n=-\infty}^\infty \sum_{m=-M}^M x[(l-m)T] h(mT)\,\delta (t-lT)\!</math> 
Let us define <math>\sum_{m=-M}^M x[(l-m)T] h(mT)\!</math> as the function <math> y(lT)\!</math>, so 
<math> x(t)*h(t) \ = \sum_{l=-\infty}^\infty y(lT) \delta (t-lT)\!</math>. It would look something like this. 
[[Image:x_discrete_a.jpg]] 
Note that <math> y(lT)\!</math> is both a discrete convolution sum and a matrix multiply. 
The frequency equivalent would be <math> \sum_{l=-\infty}^\infty y(lT)e^{-j2\pi flT}\!</math>. It can also be expressed as 
<math> \frac{1}{T}\sum_{n=-\infty}^\infty X\bigg(f-\frac{n}{T}\bigg)\cdot H(f)</math>, or <math> \ \frac{1}{T}\sum_{n=-\infty}^\infty X\bigg(f-\frac{n}{T}\bigg)\cdot \sum_{m=-M}^M h(mT)e^{-j2\pi fmT}\!</math> 
This is what it would look like. 
[[Image:x_frequency4a.jpg]] 
This pre-emphasis exaggerates the high frequencies so that when the signal goes through the D/A converter and the low pass filter, both of which de-emphasize the high frequencies, the signal coming out will be close to the original signal. 
Now we are ready to convolve our function with a pulse function <math> p(t) \!</math>, which is done by the D/A converter in a CD player. 
<math> p(t) = u\bigg(t+\frac{T}{2}\bigg) - u\bigg(t-\frac{T}{2}\bigg)\!</math> and would look like this 
[[Image:p_t.jpg]] 
The convolution looks like this: <math> \sum_{l=-\infty}^\infty y(lT) \delta (t-lT) * p(t) = \sum_{l=-\infty}^\infty y(lT) p(t-lT)\!</math> 
This convolution gives us a step function, such as the one below. 
[[Image:x_step.jpg]] 
The frequency equivalent of <math> p(t) \!</math>, or <math> P(f) \!</math>, can be expressed as <math> P(f) = T sinc(fT)\!</math> and will look something like this. 
[[Image:P_f.jpg]] 
Since the equivalent to convolution in time is multiplication in frequency, we will multiply <math> P(f)\!</math> by <math>\sum_{n=-\infty}^\infty x(nT)e^{-j2\pi fnT}\!</math>, or 
<math>T sinc(fT) \cdot \sum_{n=-\infty}^\infty x(nT)e^{-j2\pi fnT} \!</math>. It would look like this. 
[[Image:x_frequency5.jpg]] 
Now we can simply run the signal through a low pass filter and send it to the speakers for playback. 
In the time domain, an RC low pass filter, <math> g(t) \!</math>, might look like this. 
[[Image:lowpass_g_t.jpg]] 
In time, we want to convolve our function with the low pass filter, or <math> \sum_{l=-\infty}^\infty y(lT) p(t-lT) * g(t)\!</math> 
The frequency response of the lowpass filter, <math> G(f) \!</math>, would look something like this. 
[[Image:lowpass_G_f.jpg]] 
In frequency, we would want to multiply our function with the low pass filter, or <math> T sinc(fT) \cdot \sum_{n=-\infty}^\infty x(nT)e^{-j2\pi fnT} \cdot G(f)\!</math> 
And that is probably more than you ever wanted to know about how a CD player works.

DFT example using MATLAB - HW11

2009-11-17T07:52:47Z

Max.Woesner:

== Max Woesner ==
[[Max Woesner|Back to my Home Page]]

=== Homework #11 - DFT example using MATLAB ===

 Problem Statement 
Present an Octave (or MATLAB) example using the discrete Fourier transform (DFT). 
Solution 

I decided to demonstrate aliasing for my MATLAB example using the DFT.

Aliasing occurs when you don't sample a signal fast enough to be able to reconstruct it accurately after sampling. To avoid this problem, we must follow Nyquist's theorem, which states that we must sample at a rate at least twice the fastest frequency of the signal we are sampling.

First, let's look at aliasing. Consider the MATLAB script below.

<pre>
clear all;
close all;
f = 2; % sine wave frequency
fs = 3; % sample frequency
tmax = 2; % go to 2 seconds
T = 1/fs;
t = 0:0.001:tmax;
x = sin(2*pi*f*t);
ts = 0:T:tmax;
xs = sin(2*pi*f*ts);
xa = sin(2*pi*(f-fs)*t);
figure(1)
plot(t,x)
xlabel('Time (s)')
ylabel('Signal')
title('Original Signal x(t)')
figure(2)
plot(t, x, '-b', ts, xs, 'go', t, xa, 'r')
legend('Original signal x(t)', 'Sample points', 'Reconstructed signal');
xlabel('Time (s)')
ylabel('Signal')
title('Orignial Signal with Aliasing')
</pre> 
Here is the original signal to be sampled. 
[[Image:Signalorig.jpg]] 
Here is the sampled signal added. Notice that the frequency is 1.5 times the highest frequency of the original signal, thus aliasing occurs, and the reconstructed signal is significantly different than the original signal. 
[[Image:Signalaliased.jpg]] 
Now let's consider the discrete Fourier transform of the original signal.

<pre>
clear all;
close all;
f = 2; % sine wave frequency
tmax = 2; % go to 2 seconds
T = 0.01;
t = 0:T:tmax;
N = length(t);
x = sin(2*pi*f*t); % The original signal
X = fft(x); % The discrete Fourier transform
Xact = 0.*t; % The actual Fourier transform
Xact((N+1)/2-round(f*N*T)+1) = 0.5; % Simulates the delta function
Xact((N+1)/2+round(f*N*T)) = 0.5;
Xshft = fftshift(X); % To get the correct frequencies
f = -1/(2*T):1/(N*T):1/(2*T)-1/(N*T);
figure(1)
plot(t,x)
xlabel('Time (s)')
ylabel('Signal')
title('Original Signal x(t)')
figure(2)
plot(f, abs(X))
title('Unshifted Discrete Fourier Transform of x(t)')
xlabel('Frequency (s)')
ylabel('Magnitude of X(f)')
figure(3)
plot(f, abs(Xshft))
title('Shifted Discrete Fourier Transform of x(t)')
xlabel('Frequency (s)')
ylabel('Magnitude of X(f)')
figure(4)
plot(f, Xact)
title('Actual Fourier Transform of x(t)')
xlabel('Frequency (s)')
ylabel('Magnitude of X(f)')
</pre> 
Here is the original signal again. 
[[Image:1_signal.jpg]] 
Using the fast Fourier transform (FFT) to obtain the discrete Fourier transform gives us this plot. 
[[Image:2_dftunshifted.jpg]] 
Since the FFT only shows the positive frequencies, we need to shift the graph to get the correct frequencies. The plot looks like this. 
[[Image:3_dftshifted.jpg]] 
We can compare the DFT to the actual Fourier transform and see that they are very similar. The magnitude is different, and the DFT contains some extra frequencies from sampling. 
[[Image:4_actualft.jpg]]

File:4 actualft.jpg

2009-11-17T07:41:29Z

Max.Woesner:

File:3 dftshifted.jpg

2009-11-17T07:39:47Z

Max.Woesner:

File:2 dftunshifted.jpg

2009-11-17T07:37:01Z

Max.Woesner:

File:1 signal.jpg

2009-11-17T07:34:52Z

Max.Woesner:

File:Signalaliased.jpg

2009-11-17T07:32:15Z

Max.Woesner:

File:Signalorig.jpg

2009-11-17T07:31:38Z

Max.Woesner:

File:Sigwaliasing.jpg

2009-11-17T02:18:23Z

Max.Woesner:

File:Origsignal.jpg

2009-11-17T02:18:13Z

Max.Woesner:

DFT example using MATLAB - HW11

2009-11-17T01:18:34Z

Max.Woesner: New page: == Max Woesner == Back to my Home Page === Homework #11 - DFT example using MATLAB === Problem Statement Present an Octave (or MATLAB) example using th...

Max Woesner

2009-11-17T01:10:25Z

Max.Woesner:

[[Image:Mask08.jpg |thumb|This is me]]
[[Image:Diving_Turtle.jpg |thumb|Up close and personal]]

=== About Me ===
I am a senior electrical engineering major also minoring in business and math. I just got back from a year in Guam working as a volunteer engineer with Adventist World Radio. I love scuba diving but probably won't be doing much of that around these parts. For more info check out the [http://mask.wallawalla.edu/profile/show/7065 Mask]

=== Contact Info ===
*Phone: (509) 524-6978
*Email: maxwell.woesner@wallawalla.edu
*Messenger: maximiliaknow@hotmail.com
*[http://www.facebook.com/max.woesner Facebook]

=== Class Resources ===
[http://people.wallawalla.edu/~Rob.Frohne/ClassNotes/ENGR455/2009/Keystone/index.php Class notes] 
[http://dlearn2.wallawalla.edu/moodle/ Moodle]

=== Homework ===
'''Sources:'''
Class Notes,
Class Wiki,
Wikipedia

[[Evaluate this integral - HW1|Homework #1 - Evaluate this integral]] 
[[Something interesting from class - HW2|Homework #2 - Something interesting from class]] 
[[Class lecture notes October 5 - HW3|Homework #3 - Class lecture notes October 5]] 
[[Fourier Transform Properties|Homework #4 - Fourier Transform properties]] 
[[Fourier Transform Property review|Homework #5 - Review a Fourier transform property]] 
[[Fourier Transform Properties|Homework #6c - Another Fourier transform Property]] 
[[Sampling - HW7|Homework #7 - Sampling]] 
[[How a CD player works - HW8|Homework #8 - How a CD player works]] 
[[Third harmonic sampling and QSD - HW9|Homework #9 - Third harmonic sampling and QSD]] 
[[Quadrature sampling waveform plot - HW10|Homework #10 - Quadrature sampling waveform plot]] 
[[DFT example using MATLAB - HW11|Homework #11 - DFT example using MATLAB]]

Third harmonic sampling and QSD - HW9

2009-11-16T01:20:50Z

Max.Woesner:

== Max Woesner ==
[[Max Woesner|Back to my Home Page]]

=== Homework #9 - Third harmonic sampling and QSD ===

This page describes how third harmonic sampling and a QSD (quadrature sampling detector) work. 
References: 
[http://groups.yahoo.com/group/softrock40 SoftRock-40 SDR interest group] 
[http://www.flex-radio.com/News.aspx?topic=publications FlexRadio SDR-1000 QEX articles] 

====Third Harmonic Sampling====
Third harmonic sampling is used in some applications, such as the SoftRock-40 software defined radio, to divide the local oscillator frequency by three. 

Consider the simple representation of part of a software defined radio below. 
[[Image:simpleradio2.jpg]] 

The local oscillator can be defined as <math> sgn(v_{LO}(t)) \!</math>, giving us a square wave. Note that <math> sgn(v_{LO}(t)) = \begin{cases} 1, & \mbox{ } v_{LO}(t)>0 \\ 0, & \mbox{ } v_{LO}(t)=0 \\ -1, & \mbox{ } v_{LO}(t)<0 \end{cases} \!</math> 

The output voltage of the radio, or <math> v_O \!</math> is <math> v_O = v_{RF}(t) \cdot sgn(v_{LO}(t)) \!</math> 
Now <math> sgn(v_{LO}(t)) \!</math> can be written as the series <math> cos(\omega _{LO}t) + \frac{1}{2}cos(2 \omega _{LO}t) + \frac{1}{3}cos(3 \omega _{LO}t) + ... \!</math> 
We are interested in the third harmonic term, or <math> \frac{1}{3}cos(3 \omega _{LO}t) \!</math> 
This frequency is three times that of <math> cos(\omega _{LO}t) \!</math>. Note that the period decreases by a factor of three. 
[[Image:thirdharmoniccosine.jpg]] 
We can use the third harmonic to drive the mixer in the radio at a frequency three time that of the local oscillator. 

The advantage of using the third harmonic rather than using a higher speed oscillator is that it costs less to buy both a lower frequency oscillator as well as the parts that are driven by the oscillator. 

The disadvantage is that there is some loss in the signal when the third harmonic is used, specifically <math> 20 log_{10}(1/3) \ db = -9.54 \ db \!</math> 

We can now apply the same concept to a radio with parallel doubly balanced mixers. The radio could be represented such as in the diagram below. 
[[Image:Simpleradioouble2.jpg]] 
We could use a mixer like this one. 
[[Image:dbmixer.jpg]] 

====Quadrature Sampling Detector====
If you read the above section about third harmonic sampling, then you might be wondering how we input both <math> sine \!</math> and <math> cosine \!</math> into the parallel mixers. The answer is that we can use a quadrature sampling detector, or QSD. 

The local oscillator <math> (sine) \!</math> is directly fed to the lower-channel mixer and is delayed 90<math>^\circ (cosine) \!</math> to feed the upper-channel mixer. The upper channel provides the in-phase signal <math> I(t) \!</math>, and the lower channel provides the quadrature signal <math> Q(t) \!</math>. 
[[Image:Simpleradioouble3.jpg]] 
A common QSD is the Tayloe detector, named after designer Dan Tayloe. The Tayloe detector can be thought of as a four-position rotary switch. The switch revolves at the same rate as the carrier frequency. Each of the four switch positions is connected to a sampling capacitor, and the 50 ohm antenna impedance is connected to the rotor. Each capacitor will track the carrier's amplitude for exactly one-quarter of the cycle since the switch rotor is turning at exactly the RF carrier frequency, causing the switch to sample the signal at 0°, 90°, 180° and 270°. If the switch is off, each capacitor will hold its value for the remainder of the cycle. The 180° and 270° outputs carry redundant information with the 0° and 90° outputs respectively. Therefore the 0° and 180° outputs can be summed differentially to produce the in-phase signal (I). Similarly, the 90° and 270° can be summed to form the quadrature signal (Q). 
[[Image:Tayloe max.jpg]]

File:Tayloe max.jpg

2009-11-16T01:19:14Z

Max.Woesner:

Third harmonic sampling and QSD - HW9

2009-11-16T00:20:23Z

Max.Woesner:

Quadrature sampling waveform plot - HW10

2009-11-13T06:17:04Z

Max.Woesner:

File:Quadrature sampling.jpg

2009-11-13T06:09:45Z

Max.Woesner:

Quadrature sampling waveform plot - HW10

2009-11-13T01:49:24Z

Max.Woesner: New page: == Max Woesner == Back to my Home Page === Homework #10 - Quadrature sampling waveform plot === Problem Statement Plot <math> \ \frac{2}{T} \sum_{n=1}^...

Max Woesner

2009-11-13T01:12:29Z

Max.Woesner:

Third harmonic sampling and QSD - HW9

2009-11-13T01:06:37Z

Max.Woesner:

== Max Woesner ==
[[Max Woesner|Back to my Home Page]]

=== Homework #9 - Third harmonic sampling and QSD ===

This page describes how third harmonic sampling and a QSD (quadrature sampling detector) work. 
References: 
[http://groups.yahoo.com/group/softrock40 SoftRock-40 SDR interest group] 
[http://www.flex-radio.com/News.aspx?topic=publications FlexRadio SDR-1000 QEX articles] 

====Third Harmonic Sampling====
Third harmonic sampling is used in some applications, such as the SoftRock-40 software defined radio, to divide the local oscillator frequency by three. 

Consider the simple representation of part of a software defined radio below. 
[[Image:simpleradio2.jpg]] 

The local oscillator can be defined as <math> sgn(v_{LO}(t)) \!</math>, giving us a square wave. Note that <math> sgn(v_{LO}(t)) = \begin{cases} 1, & \mbox{ } v_{LO}(t)<0 \\ 0, & \mbox{ } v_{LO}(t)=0 \\ -1, & \mbox{ } v_{LO}(t)<0 \end{cases} \!</math> 

The output voltage of the radio, or <math> v_O \!</math> is <math> v_O = v_{RF}(t) \cdot sgn(v_{LO}(t)) \!</math> 
Now <math> sgn(v_{LO}(t)) \!</math> can be written as the series <math> cos(\omega _{LO}t) + \frac{1}{2}cos(2 \omega _{LO}t) + \frac{1}{3}cos(3 \omega _{LO}t) + ... \!</math> 
We are interested in the third harmonic term, or <math> \frac{1}{3}cos(3 \omega _{LO}t) \!</math> 
This frequency is three times that of <math> cos(\omega _{LO}t) \!</math>. Note that the period decreases by a factor of three. 
[[Image:thirdharmoniccosine.jpg]] 
We can use the third harmonic to drive the mixer in the radio at a frequency three time that of the local oscillator. 

The advantage of using the third harmonic rather than using a higher speed oscillator is that it costs less to buy both a lower frequency oscillator as well as the parts that are driven by the oscillator. 

The disadvantage is that there is some loss in the signal when the third harmonic is used, specifically <math> 20 log_{10}(1/3) \ db = -9.54 \ db \!</math> 

We can now apply the same concept to a radio with parallel doubly balanced mixers. The radio could be represented such as in the diagram below. 
[[Image:Simpleradioouble2.jpg]] 
We could use a mixer like this one. 
[[Image:dbmixer.jpg]] 

====Quadrature Sampling Detector====
If you read the above section about third harmonic sampling, then you might be wondering how we input both <math> sine \!</math> and <math> cosine \!</math> into the parallel mixers. The answer is that we can use a quadrature sampling detector, or QSD. 

The local oscillator <math> (sine) \!</math> is directly fed to the lower-channel mixer and is delayed 90<math>^\circ (cosine) \!</math> to feed the upper-channel mixer. The upper channel provides the in-phase signal <math> I(t) \!</math>, and the lower channel provides the quadrature signal <math> Q(t) \!</math>. 
[[Image:Simpleradioouble3.jpg]] 
A common QSD is the Tayloe detector, named after designer Dan Tayloe. The Tayloe detector can be thought of as a four-position rotary switch. The switch revolves at the same rate as the carrier frequency. Each of the four switch positions is connected to a sampling capacitor, and the 50 ohm antenna impedance is connected to the rotor. Each capacitor will track the carrier's amplitude for exactly one-quarter of the cycle since the switch rotor is turning at exactly the RF carrier frequency, causing the switch to sample the signal at 0°, 90°, 180° and 270°. If the switch is off, each capacitor will hold its value for the remainder of the cycle. The 180° and 270° outputs carry redundant information with the 0° and 90° outputs respectively. Therefore the 0° and 180° outputs can be summed differentially to produce the in-phase signal (I). Similarly, the 90° and 270° can be summed to form the quadrature signal (Q). 
[[Image:Tayloe detector.JPG]] 
 Image from [http://www.flex-radio.com/Data/Doc/qex1.pdf http://www.flex-radio.com/Data/Doc/qex1.pdf]

Third harmonic sampling and QSD - HW9

2009-11-12T08:34:04Z

Max.Woesner:

== Max Woesner ==
[[Max Woesner|Back to my Home Page]]

=== Homework #9 - Third harmonic sampling and QSD ===

This page describes how third harmonic sampling and a QSD (quadrature sampling detector) work. 
References: 
[http://groups.yahoo.com/group/softrock40 SoftRock-40 SDR interest group] 
[http://www.flex-radio.com/News.aspx?topic=publications FlexRadio SDR-1000 QEX articles] 

====Third Harmonic Sampling====
Third harmonic sampling is used in some applications, such as the SoftRock-40 software defined radio, to divide the local oscillator frequency by three. 

Consider the simple representation of part of a software defined radio below. 
[[Image:simpleradio2.jpg]] 

The local oscillator can be defined as <math> sgn(v_{LO}(t)) \!</math>, giving us a square wave. Note that <math> sgn(v_{LO}(t)) = \begin{cases} 1, & \mbox{ } v_{LO}(t)<0 \\ 0, & \mbox{ } v_{LO}(t)=0 \\ -1, & \mbox{ } v_{LO}(t)<0 \end{cases} \!</math> 

The output voltage of the radio, or <math> v_O \!</math> is <math> v_O = v_{RF}(t) \cdot sgn(v_{LO}(t)) \!</math> 
Now <math> sgn(v_{LO}(t)) \!</math> can be written as the series <math> cos(\omega _{LO}t) + \frac{1}{2}cos(2 \omega _{LO}t) + \frac{1}{3}cos(3 \omega _{LO}t) + ... \!</math> 
We are interested in the third harmonic term, or <math> \frac{1}{3}cos(3 \omega _{LO}t) \!</math> 
This frequency is three times that of <math> cos(\omega _{LO}t) \!</math>. Note that the amplitude and period decrease by a factor of three. 
[[Image:thirdharmoniccosine.jpg]] 
We can use the third harmonic to drive the mixer in the radio at a frequency three time that of the local oscillator. 

The advantage of using the third harmonic rather than using a higher speed oscillator is that it costs less to buy both a lower frequency oscillator as well as the parts that are driven by the oscillator. 

The disadvantage is that there is some loss in the signal when the third harmonic is used, specifically <math> 20 log_{10}(1/3) \ db = -9.54 \ db \!</math> 

We can now apply the same concept to a radio with parallel doubly balanced mixers. The radio could be represented such as in the diagram below. 
[[Image:Simpleradioouble2.jpg]] 
We could use a mixer like this one. 
[[Image:dbmixer.jpg]] 

====Quadrature Sampling Detector====
If you read the above section about third harmonic sampling, then you might be wondering how we input both <math> sine \!</math> and <math> cosine \!</math> into the parallel mixers. The answer is that we can use a quadrature sampling detector, or QSD. 

The local oscillator <math> (sine) \!</math> is directly fed to the lower-channel mixer and is delayed 90<math>^\circ (cosine) \!</math> to feed the upper-channel mixer. The upper channel provides the in-phase signal <math> I(t) \!</math>, and the lower channel provides the quadrature signal <math> ((t) \!</math>. 
[[Image:Simpleradioouble3.jpg]] 
A common QSD is the Tayloe detector, named after designer Dan Tayloe. The Tayloe detector can be thought of as a four-position rotary switch. The switch revolves at the same rate as the carrier frequency. Each of the four switch positions is connected to a sampling capacitor, and the 50 ohm antenna impedance is connected to the rotor. Each capacitor will track the carrier's amplitude for exactly one-quarter of the cycle since the switch rotor is turning at exactly the RF carrier frequency, causing the switch to sample the signal at 0°, 90°, 180° and 270°. If the switch is off, each capacitor will hold its value for the remainder of the cycle. The 180° and 270° outputs carry redundant information with the 0° and 90° outputs respectively. Therefore the 0° and 180° outputs can be summed differentially to produce a single Inphase (I) signal. Similarly, the 90° and 270° can be summed to form a Quadrature (Q) signal. 
[[Image:Tayloe detector.JPG]] 
 Image from [http://www.flex-radio.com/Data/Doc/qex1.pdf http://www.flex-radio.com/Data/Doc/qex1.pdf]