https://fweb.wallawalla.edu/class-wiki/api.php?action=feedcontributions&feedformat=atom&user=SmitryClass Wiki - User contributions [en]2026-04-05T23:17:24ZUser contributionsMediaWiki 1.43.0https://fweb.wallawalla.edu/class-wiki/index.php?title=User_talk:172.16.16.91&diff=4108User talk:172.16.16.912006-10-30T00:15:16Z<p>Smitry: </p>
<hr />
<div>*[[user:smitry]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=2006-2007_Assignments&diff=26382006-2007 Assignments2006-10-20T04:03:14Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
<br />
=Fall 2006 Homework Assignments=<br />
<br />
Assignments for this quarter will be listed here so that there is an easy place to look up the assignments. <br />
<br />
===HW #1===<br />
<br />
Initially Due: 10/2/06<br />
<br />
Follow Instructions Given in Handout<br />
<br />
===HW #2===<br />
<br />
Initially Due: 10/4/06<br />
<br />
Look at the Wiki & add your personal page. Add one thing to improve the Wiki.<br />
<br />
===HW #3===<br />
<br />
Finally Due: 10/11/06<br />
<br />
Spend at least 2 hours upgrading the Wiki. Must be done by Class on 10/11/06 to get credit. Our goal is to try and put the material we are covering into the Wiki. Please hand in a list of what you added and how long you spent working on the Wiki.<br />
<br />
===HW #4===<br />
<br />
Initially Due: ???<br />
<br />
Follow instructions on Handout Homework #3<br />
<br />
===HW #5===<br />
<br />
Finally Due: 10/13/06<br />
<br />
Spend at least 2 hours upgrading the Wiki. Must be done by Class on 10/13/06 to get credit. Our goal is to try and put the material we are covering into the Wiki. Please hand in a list of what you added and how long you spent working on the Wiki.<br />
<br />
===HW #6===<br />
<br />
Finally Due: At the Review session<br />
<br />
Make up one question for the test. Frohne will use at least one on the exam.<br />
<br />
===HW #7===<br />
<br />
Finally Due: 10/25/06 midnight<br />
<br />
Put in at least 2 hours on the wiki by midnight on the 25th.<br />
<br />
==TEST #1==<br />
<br />
Test will be on the 25th at the regular class time.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=2006-2007_Assignments&diff=26122006-2007 Assignments2006-10-20T04:03:03Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
<br />
Fall 2006 Homework Assignments<br />
<br />
Assignments for this quarter will be listed here so that there is an easy place to look up the assignments. <br />
<br />
===HW #1===<br />
<br />
Initially Due: 10/2/06<br />
<br />
Follow Instructions Given in Handout<br />
<br />
===HW #2===<br />
<br />
Initially Due: 10/4/06<br />
<br />
Look at the Wiki & add your personal page. Add one thing to improve the Wiki.<br />
<br />
===HW #3===<br />
<br />
Finally Due: 10/11/06<br />
<br />
Spend at least 2 hours upgrading the Wiki. Must be done by Class on 10/11/06 to get credit. Our goal is to try and put the material we are covering into the Wiki. Please hand in a list of what you added and how long you spent working on the Wiki.<br />
<br />
===HW #4===<br />
<br />
Initially Due: ???<br />
<br />
Follow instructions on Handout Homework #3<br />
<br />
===HW #5===<br />
<br />
Finally Due: 10/13/06<br />
<br />
Spend at least 2 hours upgrading the Wiki. Must be done by Class on 10/13/06 to get credit. Our goal is to try and put the material we are covering into the Wiki. Please hand in a list of what you added and how long you spent working on the Wiki.<br />
<br />
===HW #6===<br />
<br />
Finally Due: At the Review session<br />
<br />
Make up one question for the test. Frohne will use at least one on the exam.<br />
<br />
===HW #7===<br />
<br />
Finally Due: 10/25/06 midnight<br />
<br />
Put in at least 2 hours on the wiki by midnight on the 25th.<br />
<br />
==TEST #1==<br />
<br />
Test will be on the 25th at the regular class time.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=2006-2007_Assignments&diff=26112006-2007 Assignments2006-10-20T04:02:45Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
<br />
=Fall 2006 Homework Assignments=<br />
<br />
Assignments for this quarter will be listed here so that there is an easy place to look up the assignments. <br />
<br />
===HW #1===<br />
<br />
Initially Due: 10/2/06<br />
<br />
Follow Instructions Given in Handout<br />
<br />
===HW #2===<br />
<br />
Initially Due: 10/4/06<br />
<br />
Look at the Wiki & add your personal page. Add one thing to improve the Wiki.<br />
<br />
===HW #3===<br />
<br />
Finally Due: 10/11/06<br />
<br />
Spend at least 2 hours upgrading the Wiki. Must be done by Class on 10/11/06 to get credit. Our goal is to try and put the material we are covering into the Wiki. Please hand in a list of what you added and how long you spent working on the Wiki.<br />
<br />
===HW #4===<br />
<br />
Initially Due: ???<br />
<br />
Follow instructions on Handout Homework #3<br />
<br />
===HW #5===<br />
<br />
Finally Due: 10/13/06<br />
<br />
Spend at least 2 hours upgrading the Wiki. Must be done by Class on 10/13/06 to get credit. Our goal is to try and put the material we are covering into the Wiki. Please hand in a list of what you added and how long you spent working on the Wiki.<br />
<br />
===HW #6===<br />
<br />
Finally Due: At the Review session<br />
<br />
Make up one question for the test. Frohne will use at least one on the exam.<br />
<br />
===HW #7===<br />
<br />
Finally Due: 10/25/06 midnight<br />
<br />
Put in at least 2 hours on the wiki by midnight on the 25th.<br />
<br />
==TEST #1==<br />
<br />
Test will be on the 25th at the regular class time.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Ryan%27s_Resume&diff=4106Ryan's Resume2006-10-13T08:05:16Z<p>Smitry: /* '''EDUCATION''' */</p>
<hr />
<div>*[[User:Smitry|Ryan J Smith]]<br />
=RYAN J. SMITH=<br />
<br />
211 E Whitman Dr. #102<br />
<br />
College Place, WA 99324<br />
<br />
Phone: 509-525-9572<br />
<br />
Email: smitry@wwc.edu<br />
<br />
=='''WORK HISTORY'''==<br />
<br />
Grounds Dept. Supervisor<br />
<br />
Walla Walla College, College Place, WA<br />
<br />
August 2003 – Present<br />
<br />
Crew consists of 5 students; responsibilities include landscaping for college-owned apartments and part of main campus; various other types of special projects requiring the use of heavy equipment.<br />
<br />
<br />
Customer Service Rep. II<br />
<br />
Puget Sound Energy, Bellevue WA<br />
<br />
December 1999 – June 2003<br />
<br />
Worked in a Call Center answering phone—Helped customers with billing questions, starting and stopping of service, and power outage reporting.<br />
<br />
<br />
Office Support/Bidding Clerk<br />
<br />
Ohno Construction Co., Seattle WA<br />
<br />
January 1999 – December 1999<br />
<br />
Typed bidding documents—telephoned subcontractors to arrange sub-bids for projects; answered questions about jobs; typing; reception duties when assigned.<br />
<br />
=='''EDUCATION'''==<br />
<br />
Walla Walla College 2003-Present<br />
<br />
College Place, WA<br />
<br />
Electrical Engineering Major with Minor in Business & Math<br />
<br />
Anticipated Graduation Date: June 2007<br />
<br />
<br />
Whatcom Community College 2001-2003<br />
<br />
Bellingham, WA<br />
<br />
Generals<br />
<br />
<br />
Lane Community College 1997-1998<br />
<br />
Eugene, OR<br />
<br />
Business/Accounting Courses<br />
<br />
<br />
Milo Adventist Academy – 1997<br />
<br />
Days Creek, OR<br />
<br />
Math/Science Enriched Diploma (GPA 3.87)<br />
<br />
=='''REFERENCES'''== Available on Request</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Ryan%27s_Resume&diff=2608Ryan's Resume2006-10-13T08:04:39Z<p>Smitry: </p>
<hr />
<div>*[[User:Smitry|Ryan J Smith]]<br />
=RYAN J. SMITH=<br />
<br />
211 E Whitman Dr. #102<br />
<br />
College Place, WA 99324<br />
<br />
Phone: 509-525-9572<br />
<br />
Email: smitry@wwc.edu<br />
<br />
=='''WORK HISTORY'''==<br />
<br />
Grounds Dept. Supervisor<br />
<br />
Walla Walla College, College Place, WA<br />
<br />
August 2003 – Present<br />
<br />
Crew consists of 5 students; responsibilities include landscaping for college-owned apartments and part of main campus; various other types of special projects requiring the use of heavy equipment.<br />
<br />
<br />
Customer Service Rep. II<br />
<br />
Puget Sound Energy, Bellevue WA<br />
<br />
December 1999 – June 2003<br />
<br />
Worked in a Call Center answering phone—Helped customers with billing questions, starting and stopping of service, and power outage reporting.<br />
<br />
<br />
Office Support/Bidding Clerk<br />
<br />
Ohno Construction Co., Seattle WA<br />
<br />
January 1999 – December 1999<br />
<br />
Typed bidding documents—telephoned subcontractors to arrange sub-bids for projects; answered questions about jobs; typing; reception duties when assigned.<br />
<br />
=='''EDUCATION'''==<br />
<br />
Walla Walla College 2003-Present<br />
<br />
College Place, WA<br />
<br />
Electrical Engineering Major with Minor in Business<br />
<br />
Anticipated Graduation Date: June 2007<br />
<br />
<br />
Whatcom Community College 2001-2003<br />
<br />
Bellingham, WA<br />
<br />
Generals<br />
<br />
<br />
Lane Community College 1997-1998<br />
<br />
Eugene, OR<br />
<br />
Business/Accounting Courses<br />
<br />
<br />
Milo Adventist Academy – 1997<br />
<br />
Days Creek, OR<br />
<br />
Math/Science Enriched Diploma (GPA 3.87)<br />
<br />
=='''REFERENCES'''== Available on Request</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Ryan%27s_Resume&diff=2607Ryan's Resume2006-10-13T08:03:20Z<p>Smitry: </p>
<hr />
<div>=RYAN J. SMITH=<br />
<br />
211 E Whitman Dr. #102<br />
<br />
College Place, WA 99324<br />
<br />
Phone: 509-525-9572<br />
<br />
Email: smitry@wwc.edu<br />
<br />
=='''WORK HISTORY'''==<br />
<br />
Grounds Dept. Supervisor<br />
<br />
Walla Walla College, College Place, WA<br />
<br />
August 2003 – Present<br />
<br />
Crew consists of 5 students; responsibilities include landscaping for college-owned apartments and part of main campus; various other types of special projects requiring the use of heavy equipment.<br />
<br />
<br />
Customer Service Rep. II<br />
<br />
Puget Sound Energy, Bellevue WA<br />
<br />
December 1999 – June 2003<br />
<br />
Worked in a Call Center answering phone—Helped customers with billing questions, starting and stopping of service, and power outage reporting.<br />
<br />
<br />
Office Support/Bidding Clerk<br />
<br />
Ohno Construction Co., Seattle WA<br />
<br />
January 1999 – December 1999<br />
<br />
Typed bidding documents—telephoned subcontractors to arrange sub-bids for projects; answered questions about jobs; typing; reception duties when assigned.<br />
<br />
=='''EDUCATION'''==<br />
<br />
Walla Walla College 2003-Present<br />
<br />
College Place, WA<br />
<br />
Electrical Engineering Major with Minor in Business<br />
<br />
Anticipated Graduation Date: June 2007<br />
<br />
<br />
Whatcom Community College 2001-2003<br />
<br />
Bellingham, WA<br />
<br />
Generals<br />
<br />
<br />
Lane Community College 1997-1998<br />
<br />
Eugene, OR<br />
<br />
Business/Accounting Courses<br />
<br />
<br />
Milo Adventist Academy – 1997<br />
<br />
Days Creek, OR<br />
<br />
Math/Science Enriched Diploma (GPA 3.87)<br />
<br />
=='''REFERENCES'''== Available on Request</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Ryan%27s_Resume&diff=2606Ryan's Resume2006-10-13T08:02:25Z<p>Smitry: /* '''WORK HISTORY''' */</p>
<hr />
<div>=RYAN J. SMITH=<br />
<br />
211 E Whitman Dr. #102 <br />
College Place, WA 99324 <br />
Phone: 509-525-9572<br />
Email: smitry@wwc.edu<br />
<br />
=='''WORK HISTORY'''==<br />
<br />
Grounds Dept. Supervisor<br />
<br />
Walla Walla College, College Place, WA<br />
<br />
August 2003 – Present<br />
<br />
Crew consists of 5 students; responsibilities include landscaping for college-owned apartments and part of main campus; various other types of special projects requiring the use of heavy equipment.<br />
<br />
<br />
Customer Service Rep. II<br />
<br />
Puget Sound Energy, Bellevue WA<br />
<br />
December 1999 – June 2003<br />
<br />
Worked in a Call Center answering phone—Helped customers with billing questions, starting and stopping of service, and power outage reporting.<br />
<br />
<br />
Office Support/Bidding Clerk<br />
<br />
Ohno Construction Co., Seattle WA<br />
<br />
January 1999 – December 1999<br />
<br />
Typed bidding documents—telephoned subcontractors to arrange sub-bids for projects; answered questions about jobs; typing; reception duties when assigned.<br />
<br />
=='''EDUCATION'''==<br />
<br />
Walla Walla College 2003-Present<br />
College Place, WA<br />
Electrical Engineering Major with Minor in Business<br />
Anticipated Graduation Date: June 2007<br />
<br />
Whatcom Community College 2001-2003<br />
Bellingham, WA<br />
Generals<br />
<br />
Lane Community College 1997-1998<br />
Eugene, OR<br />
Business/Accounting Courses<br />
<br />
Milo Adventist Academy – 1997<br />
Days Creek, OR<br />
Math/Science Enriched Diploma (GPA 3.87)<br />
<br />
=='''REFERENCES'''== Available on Request</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Ryan%27s_Resume&diff=2605Ryan's Resume2006-10-13T08:01:40Z<p>Smitry: /* '''WORK HISTORY''' */</p>
<hr />
<div>=RYAN J. SMITH=<br />
<br />
211 E Whitman Dr. #102 <br />
College Place, WA 99324 <br />
Phone: 509-525-9572<br />
Email: smitry@wwc.edu<br />
<br />
=='''WORK HISTORY'''==<br />
<br />
Grounds Dept. Supervisor<br />
<br />
Walla Walla College, College Place, WA<br />
<br />
August 2003 – Present<br />
<br />
Crew consists of 5 students; responsibilities include landscaping for college-owned apartments and part of main campus; various other types of special projects requiring the use of heavy equipment.<br />
<br />
Customer Service Rep. II<br />
<br />
Puget Sound Energy, Bellevue WA<br />
<br />
December 1999 – June 2003<br />
<br />
Worked in a Call Center answering phone—Helped customers with billing questions, starting and stopping of service, and power outage reporting.<br />
<br />
Office Support/Bidding Clerk<br />
<br />
Ohno Construction Co., Seattle WA<br />
<br />
January 1999 – December 1999<br />
<br />
Typed bidding documents—telephoned subcontractors to arrange sub-bids for projects; answered questions about jobs; typing; reception duties when assigned.<br />
<br />
=='''EDUCATION'''==<br />
<br />
Walla Walla College 2003-Present<br />
College Place, WA<br />
Electrical Engineering Major with Minor in Business<br />
Anticipated Graduation Date: June 2007<br />
<br />
Whatcom Community College 2001-2003<br />
Bellingham, WA<br />
Generals<br />
<br />
Lane Community College 1997-1998<br />
Eugene, OR<br />
Business/Accounting Courses<br />
<br />
Milo Adventist Academy – 1997<br />
Days Creek, OR<br />
Math/Science Enriched Diploma (GPA 3.87)<br />
<br />
=='''REFERENCES'''== Available on Request</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Ryan%27s_Resume&diff=2604Ryan's Resume2006-10-13T08:01:17Z<p>Smitry: /* '''WORK HISTORY''' */</p>
<hr />
<div>=RYAN J. SMITH=<br />
<br />
211 E Whitman Dr. #102 <br />
College Place, WA 99324 <br />
Phone: 509-525-9572<br />
Email: smitry@wwc.edu<br />
<br />
=='''WORK HISTORY'''==<br />
<br />
Grounds Dept. Supervisor<br />
Walla Walla College, College Place, WA<br />
August 2003 – Present<br />
<br />
Crew consists of 5 students; responsibilities include landscaping for college-owned apartments and part of main campus; various other types of special projects requiring the use of heavy equipment.<br />
<br />
Customer Service Rep. II<br />
Puget Sound Energy, Bellevue WA<br />
December 1999 – June 2003<br />
<br />
Worked in a Call Center answering phone—Helped customers with billing questions, starting and stopping of service, and power outage reporting.<br />
<br />
Office Support/Bidding Clerk<br />
Ohno Construction Co., Seattle WA<br />
January 1999 – December 1999<br />
<br />
Typed bidding documents—telephoned subcontractors to arrange sub-bids for projects; answered questions about jobs; typing; reception duties when assigned.<br />
<br />
=='''EDUCATION'''==<br />
<br />
Walla Walla College 2003-Present<br />
College Place, WA<br />
Electrical Engineering Major with Minor in Business<br />
Anticipated Graduation Date: June 2007<br />
<br />
Whatcom Community College 2001-2003<br />
Bellingham, WA<br />
Generals<br />
<br />
Lane Community College 1997-1998<br />
Eugene, OR<br />
Business/Accounting Courses<br />
<br />
Milo Adventist Academy – 1997<br />
Days Creek, OR<br />
Math/Science Enriched Diploma (GPA 3.87)<br />
<br />
=='''REFERENCES'''== Available on Request</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Ryan%27s_Resume&diff=2603Ryan's Resume2006-10-13T07:59:26Z<p>Smitry: /* '''WORK HISTORY''' */</p>
<hr />
<div>=RYAN J. SMITH=<br />
<br />
211 E Whitman Dr. #102 <br />
College Place, WA 99324 <br />
Phone: 509-525-9572<br />
Email: smitry@wwc.edu<br />
<br />
=='''WORK HISTORY'''==<br />
<br />
<br />
----<br />
Grounds Dept. Supervisor<br />
Walla Walla College, College Place, WA <br />
August 2003 – Present<br />
<br />
Crew consists of 5 students; responsibilities include landscaping for college-owned apartments and part of main campus; various other types of special projects requiring the use of heavy equipment.<br />
<br />
----<br />
<br />
Customer Service Rep. II<br />
Puget Sound Energy, Bellevue WA <br />
December 1999 – June 2003<br />
<br />
Worked in a Call Center answering phone—Helped customers with billing questions, starting and stopping of service, and power outage reporting. <br />
<br />
----<br />
<br />
Office Support/Bidding Clerk<br />
Ohno Construction Co., Seattle WA <br />
January 1999 – December 1999<br />
<br />
Typed bidding documents—telephoned subcontractors to arrange sub-bids for projects; answered questions about jobs; typing; reception duties when assigned.<br />
<br />
=='''EDUCATION'''==<br />
<br />
Walla Walla College 2003-Present<br />
College Place, WA<br />
Electrical Engineering Major with Minor in Business<br />
Anticipated Graduation Date: June 2007<br />
<br />
Whatcom Community College 2001-2003<br />
Bellingham, WA<br />
Generals<br />
<br />
Lane Community College 1997-1998<br />
Eugene, OR<br />
Business/Accounting Courses<br />
<br />
Milo Adventist Academy – 1997<br />
Days Creek, OR<br />
Math/Science Enriched Diploma (GPA 3.87)<br />
<br />
=='''REFERENCES'''== Available on Request</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Ryan%27s_Resume&diff=2602Ryan's Resume2006-10-13T07:58:26Z<p>Smitry: /* '''WORK HISTORY''' */</p>
<hr />
<div>=RYAN J. SMITH=<br />
<br />
211 E Whitman Dr. #102 <br />
College Place, WA 99324 <br />
Phone: 509-525-9572<br />
Email: smitry@wwc.edu<br />
<br />
=='''WORK HISTORY'''==<br />
<nowiki><br />
Grounds Dept. Supervisor<br />
Walla Walla College, College Place, WA <br />
August 2003 – Present<br />
<br />
Crew consists of 5 students; responsibilities include landscaping for college-owned apartments and part of main campus; various other types of special projects requiring the use of heavy equipment.<br />
<br />
Customer Service Rep. II<br />
Puget Sound Energy, Bellevue WA <br />
December 1999 – June 2003<br />
<br />
Worked in a Call Center answering phone—Helped customers with billing questions, starting and stopping of service, and power outage reporting. <br />
<br />
Office Support/Bidding Clerk<br />
Ohno Construction Co., Seattle WA <br />
January 1999 – December 1999<br />
<br />
Typed bidding documents—telephoned subcontractors to arrange sub-bids for projects; answered questions about jobs; typing; reception duties when assigned.</nowiki><br />
<br />
=='''EDUCATION'''==<br />
<br />
Walla Walla College 2003-Present<br />
College Place, WA<br />
Electrical Engineering Major with Minor in Business<br />
Anticipated Graduation Date: June 2007<br />
<br />
Whatcom Community College 2001-2003<br />
Bellingham, WA<br />
Generals<br />
<br />
Lane Community College 1997-1998<br />
Eugene, OR<br />
Business/Accounting Courses<br />
<br />
Milo Adventist Academy – 1997<br />
Days Creek, OR<br />
Math/Science Enriched Diploma (GPA 3.87)<br />
<br />
=='''REFERENCES'''== Available on Request</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Ryan%27s_Resume&diff=2601Ryan's Resume2006-10-13T07:57:17Z<p>Smitry: </p>
<hr />
<div>=RYAN J. SMITH=<br />
<br />
211 E Whitman Dr. #102 <br />
College Place, WA 99324 <br />
Phone: 509-525-9572<br />
Email: smitry@wwc.edu<br />
<br />
=='''WORK HISTORY'''==<br />
<br />
Grounds Dept. Supervisor<br />
Walla Walla College, College Place, WA <br />
August 2003 – Present<br />
<br />
Crew consists of 5 students; responsibilities include landscaping for college-owned apartments and part of main campus; various other types of special projects requiring the use of heavy equipment.<br />
<br />
Customer Service Rep. II<br />
Puget Sound Energy, Bellevue WA <br />
December 1999 – June 2003<br />
<br />
Worked in a Call Center answering phone—Helped customers with billing questions, starting and stopping of service, and power outage reporting. <br />
<br />
Office Support/Bidding Clerk<br />
Ohno Construction Co., Seattle WA <br />
January 1999 – December 1999<br />
<br />
Typed bidding documents—telephoned subcontractors to arrange sub-bids for projects; answered questions about jobs; typing; reception duties when assigned.<br />
<br />
=='''EDUCATION'''==<br />
<br />
Walla Walla College 2003-Present<br />
College Place, WA<br />
Electrical Engineering Major with Minor in Business<br />
Anticipated Graduation Date: June 2007<br />
<br />
Whatcom Community College 2001-2003<br />
Bellingham, WA<br />
Generals<br />
<br />
Lane Community College 1997-1998<br />
Eugene, OR<br />
Business/Accounting Courses<br />
<br />
Milo Adventist Academy – 1997<br />
Days Creek, OR<br />
Math/Science Enriched Diploma (GPA 3.87)<br />
<br />
=='''REFERENCES'''== Available on Request</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=4085User:Smitry2006-10-13T07:49:56Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems Main page]]<br />
<br />
=Ryan Smith's Wiki Webpage=<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask. I try to keep that up to date.<br />
[[http://mask.wwc.edu/profile/show/755 The Mask]]<br />
<br />
===Family - Personal tid Bits ===<br />
<br />
I should be graduating in the summer of 2007. I got married the summer of 2006. My 7 year old son is still adjusting to the whole family structure thing again. My son Izak also just started the the 1st grade so he is really excited about that. It's been quite the summer and new school year.<br />
<br />
===Hobbies===<br />
<br />
Geocaching is a family favorite and I would recommend it to anyone the loves the good outdoors.<br />
[[http://www.geocaching.com Geocaching]]<br />
<br />
==Career Goals==<br />
<br />
When I graduate from college I would like to work in the Alternative Energy field. My primary experience is in the power industry but I would love to work in a variety of fields during my career. I have a interest in low impact Power generation and other ways to meet the needs of society without creating an environment that my kids will have to clean up.<br />
<br />
===Resume'===<br />
[[Ryan's Resume]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2600User:Smitry2006-10-13T07:31:35Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems Main page]]<br />
<br />
=Ryan Smith's Wiki Webpage=<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask. I try to keep that up to date.<br />
[[http://mask.wwc.edu/profile/show/755 The Mask]]<br />
<br />
===Family - Personal tid Bits ===<br />
<br />
I should be graduating in the summer of 2007. I got married the summer of 2006. My 7 year old son is still adjusting to the whole family structure thing again. My son Izak also just started the the 1st grade so he is really excited about that. It's been quite the summer and new school year.<br />
<br />
===Hobbies===<br />
<br />
Geocaching is a family favorite and I would recommend it to anyone the loves the good outdoors.<br />
[[http://www.geocaching.com Geocaching]]<br />
<br />
==Career Goals==<br />
<br />
When I graduate from college I would like to work in the Alternative Energy field. My primary experience is in the power industry but I would love to work in a variety of fields during my career. I have a interest in low impact Power generation and other ways to meet the needs of society without creating an environment that my kids will have to clean up.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2599User:Smitry2006-10-13T07:31:25Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems Main page]]<br />
<br />
Ryan Smith's Wiki Webpage<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask. I try to keep that up to date.<br />
[[http://mask.wwc.edu/profile/show/755 The Mask]]<br />
<br />
===Family - Personal tid Bits ===<br />
<br />
I should be graduating in the summer of 2007. I got married the summer of 2006. My 7 year old son is still adjusting to the whole family structure thing again. My son Izak also just started the the 1st grade so he is really excited about that. It's been quite the summer and new school year.<br />
<br />
===Hobbies===<br />
<br />
Geocaching is a family favorite and I would recommend it to anyone the loves the good outdoors.<br />
[[http://www.geocaching.com Geocaching]]<br />
<br />
==Career Goals==<br />
<br />
When I graduate from college I would like to work in the Alternative Energy field. My primary experience is in the power industry but I would love to work in a variety of fields during my career. I have a interest in low impact Power generation and other ways to meet the needs of society without creating an environment that my kids will have to clean up.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2598User:Smitry2006-10-13T07:30:46Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems Main page]]<br />
<br />
=Ryan Smith's Wiki Webpage=<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask. I try to keep that up to date.<br />
[[http://mask.wwc.edu/profile/show/755 The Mask]]<br />
<br />
===Family - Personal tid Bits ===<br />
<br />
I should be graduating in the summer of 2007. I got married the summer of 2006. My 7 year old son is still adjusting to the whole family structure thing again. My son Izak also just started the the 1st grade so he is really excited about that. It's been quite the summer and new school year.<br />
<br />
===Hobbies===<br />
<br />
Geocaching is a family favorite and I would recommend it to anyone the loves the good outdoors.<br />
[[http://www.geocaching.com Geocaching]]<br />
<br />
==Career Goals==<br />
<br />
When I graduate from college I would like to work in the Alternative Energy field. My primary experience is in the power industry but I would love to work in a variety of fields during my career. I have a interest in low impact Power generation and other ways to meet the needs of society without creating an environment that my kids will have to clean up.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Current_events&diff=2613Current events2006-10-13T07:16:32Z<p>Smitry: /* Class Announcements */</p>
<hr />
<div>==Class Announcements==<br />
<br />
We have no Current events so this page is BLANK!!!!</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2597User:Smitry2006-10-13T06:53:59Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems Main page]]<br />
<br />
=Ryan Smith's Wiki Webpage=<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask. I try to keep that up to date.<br />
[[http://mask.wwc.edu/profile/show/755 The Mask]]<br />
<br />
===Family - Personal tid Bits ===<br />
<br />
I should be graduating in the summer of 2007. I got married the summer of 2006. My 7 year old son is still adjusting to the whole family structure thing again. My son Izak also just started the the 1st grade so he is really excited about that. <br />
<br />
===Hobbies===<br />
<br />
Geocaching is a family favorite and I would recommend it to anyone the loves the good outdoors.<br />
[[http://www.geocaching.com Geocaching]]<br />
<br />
==Career Goals==<br />
<br />
When I graduate from college I would like to work in the Alternative Energy field. My primary experience is in the power industry but I would love to work in a variety of fields during my career. I have a interest in low impact Power generation and other ways to meet the needs of society without creating an environment that my kids will have to clean up.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2595User:Smitry2006-10-13T06:53:44Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems Main page]]<br />
<br />
=Ryan Smith's Wiki Webpage=<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask. I try to keep that up to date.<br />
[[http://mask.wwc.edu/profile/show/755 The Mask]]<br />
<br />
===Family - Personal tid Bits ====<br />
<br />
I should be graduating in the summer of 2007. I got married the summer of 2006. My 7 year old son is still adjusting to the whole family structure thing again. My son Izak also just started the the 1st grade so he is really excited about that. <br />
<br />
===Hobbies====<br />
<br />
Geocaching is a family favorite and I would recommend it to anyone the loves the good outdoors.<br />
[[http://www.geocaching.com Geocaching]]<br />
<br />
==Career Goals==<br />
<br />
When I graduate from college I would like to work in the Alternative Energy field. My primary experience is in the power industry but I would love to work in a variety of fields during my career. I have a interest in low impact Power generation and other ways to meet the needs of society without creating an environment that my kids will have to clean up.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2594User:Smitry2006-10-13T06:50:59Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems Main page]]<br />
<br />
=Ryan Smith's Wiki Webpage=<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask. I try to keep that up to date.<br />
[[http://mask.wwc.edu/profile/show/755 The Mask]]<br />
<br />
====Family - Personal tid Bits ====<br />
<br />
I should be graduating in the summer of 2007. I got married the summer of 2006. My 7 year old son is still adjusting to the whole family structure thing again. My son Izak also just started the the 1st grade so he is really excited about that. <br />
<br />
====Hobbies====<br />
<br />
Geocaching is a family favorite and I would recommend it to anyone the loves the good outdoors.<br />
[[http://www.geocaching.com Geocaching]]<br />
<br />
==Career Goals==<br />
<br />
When I graduate from college I would like to work in the Alternative Energy field. My primary experience is in the power industry but I would love to work in a variety of fields during my career. I have a interest in low impact Power generation and other ways to meet the needs of society without creating an environment that my kids will have to clean up.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2593User:Smitry2006-10-13T06:50:31Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems Main page]]<br />
<br />
==Ryan Smith's Wiki Webpage==<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask. I try to keep that up to date.<br />
[[http://mask.wwc.edu/profile/show/755 The Mask]]<br />
<br />
====Family - Personal tid Bits ====<br />
<br />
I should be graduating in the summer of 2007. I got married the summer of 2006. My 7 year old son is still adjusting to the whole family structure thing again. My son Izak also just started the the 1st grade so he is really excited about that. <br />
<br />
====Hobbies====<br />
<br />
Geocaching is a family favorite and I would recommend it to anyone the loves the good outdoors.<br />
[[http://www.geocaching.com Geocaching]]<br />
<br />
==Career Goals==<br />
<br />
When I graduate from college I would like to work in the Alternative Energy field. My primary experience is in the power industry but I would love to work in a variety of fields during my career. I have a interest in low impact Power generation and other ways to meet the needs of society without creating an environment that my kids will have to clean up.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2592User:Smitry2006-10-13T06:46:02Z<p>Smitry: /* About me */</p>
<hr />
<div>*[[Signals and systems|Signals and Systems Main page]]<br />
<br />
==Ryan Smith's Wiki Webpage==<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask. I try to keep that up to date.<br />
[[http://mask.wwc.edu/profile/show/755 The Mask]]<br />
<br />
====Family - Personal tid Bits ====<br />
<br />
I should be graduating in the summer of 2007. I got married the summer of 2006. My 7 year old son is still adjusting to the whole family structure thing again. My son Izak also just started the the 1st grade so he is really excited about that. <br />
<br />
====Hobbies====<br />
<br />
Geocaching is a family favorite and I would recommend it to anyone the loves the good outdoors.<br />
[[http://www.geocaching.com Geocaching]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Rayleigh%27s_Theorem&diff=4102Rayleigh's Theorem2006-10-13T06:36:58Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
Rayleigh's Theorem is derived from the equation for Energy<br />
*<math> W = \int_{-\infty}^{\infty}p(t)\,dt </math> <br />
If we assume that the circuit is a Voltage applied over a load then <math> p(t)=\frac{x^2(t)}{R_L}</math><br />
for matters of simplicity we can assume <math>R_L = 1\, \Omega</math><br />
<br><br />
This leaves us with<br />
*<math> W = \int_{-\infty}^{\infty}|x|^2(t)\,dt</math> <br />
This is the same as the dot product so to satisfy the condition for complex numbers it becomes<br />
*<math> W = \int_{-\infty}^{\infty}x(t)\,x^*(t)\,dt</math><br />
If we substitute <math> x(t) = \int_{-\infty}^{\infty}X(f)\,e^{j2\pi ft}\,df </math> and <math>x^*(t)= \int_{-\infty}^{\infty}X(f')\,e^{-j2\pi f't}\,df'</math><br />
<br><br />
<br>Substituting this back into the original equation makes it<br />
*<math>W = \int_{-\infty}^{\infty}\left(\int_{-\infty}^{\infty}X(f)\,e^{j2\pi ft}\,df\right) \,\left(\int_{-\infty}^{\infty}X^*(f')\,e^{-j2\pi f't}\,df'\right)\,dt</math><br />
*<math>W = \int_{-\infty}^{\infty}X(f)\,\int_{-\infty}^{\infty}X^*(f')\left(\int_{-\infty}^{\infty}e^{j2\pi (f-f')t}\,dt\right)\,df'\,df</math><br />
The time integral becomes <math> \delta (f-f') \,which \ is\ 0\ except\ for\ when\ f' = f</math><br />
This simplifies the above equation such that<br />
*<math>W = \int_{-\infty}^{\infty}X(f)\,\int_{-\infty}^{\infty}X^*(f')\left(\delta (f-f') \right)\,df'\,df</math><br />
*<math>W = \int_{-\infty}^{\infty}X(f)\,X^*(f)\,df</math><br />
Proving that the energy in the time domain is the same as that in the frequency domain<br />
*<math> W = \int_{-\infty}^{\infty}X(f)\,X^*(f)\,df = \int_{-\infty}^{\infty}x(t)\,x^*(t)\,dt</math></div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Signals_and_Systems&diff=2575Signals and Systems2006-10-12T04:47:33Z<p>Smitry: </p>
<hr />
<div>[http://www.wwc.edu/~frohro/ClassNotes/engr455index.htm Class notes for Signals & Systems]<br />
<br />
== Topics ==<br />
*[[Linear Time Invarient System]]<br />
*[[Orthogonal functions]]<br />
*[[Energy in a signal]]<br />
*[[Fourier series]]<br />
*[[Fourier transform]]<br />
*[[Rayleigh's Theorem]]<br />
*[[Sampling]]<br />
*[[The Game]]<br />
*[[Discrete Fourier transform]]<br />
*[[Fourier series - by Ray Betz|Signals and Systems - by Ray Betz]]<br />
*[[FIR Filter Example]]<br />
*[[2005-2006 Assignments]]<br />
*[[2006-2007 Assignments]]<br />
<br />
<br />
<br />
I couldn't figure out how to get to others Users pages easily so I decided to start posting them here, please add yours:<br />
<br />
[[User:Frohro|Rob Frohne]]<br />
<br />
==2004-2005 contributors==<br />
<br />
[[User:Barnsa|Sam Barnes]]<br />
<br />
[[User:Santsh|Shawn Santana]]<br />
<br />
[[User:Goeari|Aric Goe]]<br />
<br />
[[User:Caswto|Todd Caswell]]<br />
<br />
[[User:Andeda|David Anderson]]<br />
<br />
[[User:Guenan|Anthony Guenterberg]]<br />
<br />
==2005-2006 contributors==<br />
<br />
[[User:GabrielaV|Gabriela Valdivia]]<br />
<br />
[[User:SDiver|Raymond Betz]]<br />
<br />
[[User:chrijen|Jenni Christensen]]<br />
<br />
[[User:wonoje|Jeffrey Wonoprabowo]]<br />
<br />
[[User:wilspa|Paul Wilson]]<br />
<br />
==2006-2007 contributors==<br />
<br />
[[User:Smitry|Ryan J Smith]]<br />
<br />
[[User:Nathan|Nathan Ferch]]<br />
<br />
[[User:Andrew|Andrew Lopez]]<br />
<br />
[[User:Sherna|Nathan Sherman]]<br />
<br />
[[User:Adkich|Chris Adkins]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=2006-2007_Assignments&diff=25442006-2007 Assignments2006-10-08T21:52:45Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
<br />
==Fall 2006 Homework Assignments==<br />
<br />
Assignments for this quarter will be listed here so that there is an easy place to look up the assignments. <br />
<br />
===HW #1===<br />
<br />
Initially Due: 10/2/06<br />
<br />
Follow Instructions Given in Handout<br />
<br />
===HW #2===<br />
<br />
Initially Due: 10/4/06<br />
<br />
Look at the Wiki & add your personal page. Add one thing to improve the Wiki.<br />
<br />
===HW #3===<br />
<br />
Finally Due: 10/11/06<br />
<br />
Spend at least 2 hours upgrading the Wiki. Must be done by Class on 10/11/06 to get credit. Our goal is to try and put the material we are covering into the Wiki. Please hand in a list of what you added and how long you spent working on the Wiki.<br />
<br />
===HW #4===<br />
<br />
Initially Due: ???<br />
<br />
Follow instructions on Handout Homework #3</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=2005-2006_Assignments&diff=40792005-2006 Assignments2006-10-08T21:52:09Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
==Fall 2005 Homework Assignments==<br />
Assignments for this quarter will be listed here so that there is an easy place to look up the assignments. Below each assignment is the date that it was assigned.<br />
<br />
===HW #1===<br />
Look at the Wiki & add your personal page. Spend two hours.<br />
<br />
- 9/26/05<br />
<br />
===HW #2===<br />
Find the first three orthogonormal polynomials on <math>[-1,1]</math> (lowest order polynomials).<br />
<br />
- 9/30/05<br />
<br />
===HW #3===<br />
1) Work on the Wiki for two hours this week.<br />
<br />
2) Find the output of a periodic function, <br />
<br />
<math>x(t) = \sum_{k = -\infty}^\infty \alpha_k e^{j \pi k \frac{t}{T}}</math><br />
<br />
to an RC filter with RC = T.<br />
<br />
- 10/3/05<br />
<br />
===HW #4===<br />
Show how the real and imaginary parts of <math>\alpha_k</math> in the complex Fourier Series are related to the coefficients in the sine/cosine Fourier Series.<br />
<br />
10/5/05<br />
<br />
===HW #5===<br />
Individual Wiki pages on Fourier Transform.<br />
<br />
10/13/05<br />
<br />
===HW #6===<br />
1) Find <math>\mathcal{F} [x(t) sin(2 \pi f_o t + \theta)</math>.<br />
<br />
2) Convolve <math>u(t) - u(t-3)</math> with <math>cos(2 \pi t)[u(t-1)-u(t-2)]</math><br />
<br />
10/14/05<br />
<br />
===HW #7===<br />
1) Find <math>\mathcal{F} [u(t) cos(2 \pi f_o t)]</math><br />
<br />
2) Show <math>x(t)*\delta (t-t_o) = x(t-t_o)</math>.<br />
<br />
10/19/05<br />
<br />
===HW #8===<br />
There were two assignments that were labeled as HW #8.<br />
<br />
=====HW #8A=====<br />
Show that <br />
<br />
<math>\Phi_n (t) \equiv \frac{sin \left ( \frac{\pi (t - nT)}{T} \right)}{ \frac{\pi (t-nT)}{T} }</math><br />
<br />
form an orthogonal basis set. Tell me what functions these span.<br />
<br />
10/21/05<br />
<br />
=====HW #8B=====<br />
Do a Wiki page on how a 2x oversampling CD Player works.<br />
<br />
10/24/05<br />
<br />
===HW #9===<br />
Handout on FIR filter.<br />
<br />
11/2/05<br />
<br />
===HW #10===<br />
Put something on FIR filters on the Wiki.<br />
<br />
11/7/05<br />
<br />
===HW #11===<br />
Work on the Wiki. Read someone else's contribution & fix or extend it a little. Continue with FIR and do a DFT section if you have time.<br />
<br />
11/14/05<br />
<br />
===HW #12===<br />
Come see me to discuss your Wiki contributions. I will give you suggestions for edits, etc.<br />
<br />
11/30/05<br />
<br />
===HW #13===<br />
Write a Wiki page on adaptive FIR filters. Spend at least 2 hours by sundown Friday.<br />
<br />
11/30/05<br />
<br />
<br />
<br />
<br />
<small>Principle author: Jeffrey Wonoprabowo</small></div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=FIR_Filter_Example&diff=3841FIR Filter Example2006-10-08T21:51:51Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
The filter coefficients are given by:<br />
<math><br />
h_m = { T } \int_{-{1\over 4T}}^{{1\over 4T}} H(f)e^{j2\pi m f T}\,df<br />
</math><br />
where H(f) is the desired frequency response. See the notes at for October 31, 2005. As an example of this filter we will make a Matlab script that will computer the frequency response for a low pass filter having a cutoff frequency of 1/(4T), and using 2M+1 coefficients. Note that it is periodic with period 1/T. This is the case with all digital filters.<br />
<br />
Here is the plot the Matlab code produces:<br />
<br />
[[Image:Frequency_Response_Pict.png]]<br />
<br />
The Matlab code to see the frequency response is given below:<br />
<br />
% This shows how to find the frequency response for an FIR filter with cutoff 1/4/T and 2M+1 coefficients.<br />
<br />
clf;<br />
<br />
sum=0;<br />
<br />
T=1;<br />
<br />
fs=1/1000/T;<br />
<br />
f=-2/T:fs:2/T<br />
;<br />
M=100;<br />
<br />
for m=-M:M;<br />
<br />
if m==0<br />
<br />
h=1/2;<br />
<br />
else<br />
<br />
h=sin(pi*m/2)/(pi*m);<br />
<br />
end<br />
<br />
sum=sum+h*exp(-i*2*pi*f*m*T);<br />
<br />
end<br />
<br />
plot(f,20*log10(abs(sum)))<br />
<br />
title('Frequency Response of Our FIR Filter')<br />
<br />
xlabel('Frequency (1/T)')<br />
<br />
ylabel('Response (db)')<br />
<br />
text(-1.5,-5,'M = 100')<br />
<br />
% End of Matlab code.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Fourier_series_-_by_Ray_Betz&diff=3832Fourier series - by Ray Betz2006-10-08T21:51:27Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
==Fourier Series==<br />
If <br />
# <math> x(t) = x(t + T)</math><br />
*[[Dirichlet Conditions]] are satisfied<br />
then we can write<br />
<center><br />
<math> \bold x(t) = \sum_{k=-\infty}^\infty \alpha_k e^ \frac {j 2 \pi k t}{T}</math><br />
</center><br />
The above equation is called the complex Fourier Series. Given <math>x(t)</math>, we may determine <math> \alpha_k </math> by taking the [[inner product]] of <math>\alpha_k</math> with <math>x(t)</math>.<br />
Let us assume a solution for <math>\alpha_k</math> of the form <math>e^ \frac {j 2 \pi n t}{T}</math>. Now we take the inner product of <math>e^ \frac {j 2 \pi n t}{T}</math> with <math>x(t)</math> over the interval of one period, <math> T </math>.<br />
<math> <e^ \frac {j 2 \pi n t}{T}|x(t)> = <e^ \frac {j 2 \pi n t}{T}|\sum_{k=-\infty}^\infty \alpha_k e^ \frac {j 2 \pi k t}{T}> </math><br />
<math>= \int_{-\frac{T}{2}}^\frac{T}{2} x(t)e^ \frac {-j 2 \pi n t}{T} dt </math><br />
<math>= \int_{-\frac{T}{2}}^\frac{T}{2} \sum_{k=-\infty}^\infty \alpha_k e^ \frac {j 2 \pi k t}{T}e^ \frac {-j 2 \pi n t}{T} dt </math><br />
<math>= \sum_{k=-\infty}^\infty \alpha_k \int_{-\frac{T}{2}}^\frac{T}{2} e^ \frac {j 2 \pi (k-n) t}{T} dt </math><br />
<br />
If <math>k=n</math> then,<br />
<br />
<math> \int_{-\frac{T}{2}}^\frac{T}{2} e^ \frac {j 2 \pi (k-n) t}{T} dt = \int_{-\frac{T}{2}}^\frac{T}{2} 1 dt = T</math><br />
<br />
If <math>k \ne n </math> then,<br />
<br />
<math> \int_{-\frac{T}{2}}^\frac{T}{2} e^ \frac {j 2 \pi (k-n) t}{T} dt = 0 </math><br />
<br />
We can simplify the above two conclusions into one equation. (What is the [[delta function]] below?)<br />
<br />
<math> \sum_{k=-\infty}^\infty \alpha_k \int_{-\frac{T}{2}}^\frac{T}{2} e^ \frac {j 2 \pi (k-n) t}{T} dt = \sum_{k=-\infty}^\infty T \delta_{k,n} \alpha_k = T \alpha_n </math><br />
<br />
So, we conclude<br />
<math>\alpha_n = \frac{1}{T}\int_{-\frac{T}{2}}^\frac{T}{2} x(t) e^ \frac {-j 2 \pi n t}{T} dt </math><br />
<br />
==Orthogonal Functions==<br />
<br />
The function <math> y_n(t) </math> and <math> y_m(t) </math> are orthogonal on <math> (a,b) </math> if and only if <math> <y_n(t)|y_m(t)> = \int_{a}^{b} y_n^*(t)y_m(t) dt = 0 </math>. <br />
<br />
The set of functions are orthonormal if and only if <math> <y_n(t)|y_m(t)> = \int_{a}^{b} y_n^*(t)y_m(t) dt = \delta_{m,n} </math>.<br />
<br />
==Linear Systems==<br />
<br />
Let us say we have a linear time invarient system, where <math> x(t) </math> is the input and <math> y(t) </math> is the output. What outputs do we get as we put different inputs into this system? <br />
[[Image:Linear_System.JPG]]<br />
<br />
If we put in an impulse response, <math> \delta(t)</math>, then we get out <math>h(t)</math>. What would happen if we put a time delayed impulse signal, <math> \delta(t-u)</math>, into the system? The output response would be a time delayed <math>h(t)</math>, or <math>h(t-u)</math>, because the system is time invarient. So, no matter when we put in our signal the response would come out the same (just time delayed). <br />
<br />
What if we now multiplied our impulse by a coefficient? Since our system is linear, the proportionality property applies. If we put <math> x(u)\delta(t-u)</math> into our system then we should get out <math>x(u)h(t-u)</math>. <br />
<br />
By the superposition property(because we have a linear system) we may put into the system the integral of <math> x(u)\delta(t-u)</math> with respect to u and we would get out <math> \int_{-\infty}^\infty x(u)h(t-u) du</math>. This is because What would we get if we put <math> e^{j 2 \pi f t} </math> into our system? We could find out by plugging <math> e^{j 2 \pi f t} </math> in for <math> x(u) </math> in the integral that we just found the output for above. If we do a change of variables (<math> v = t-u </math>, and <math> dv = -du </math>) we get <math> \int_{-\infty}^\infty x(u)h(t-u) du = \int_{-\infty}^\infty e^{j 2 \pi f t} h(t-u) du = -\int_{\infty}^{-\infty} e^{j 2 \pi f (t-v)} h(v) dv = e^{j 2 \pi f t} \int_{-\infty}^\infty h(v)e^{-j 2 \pi f v} dv</math>. By pulling <math> e^{j 2 \pi f t} </math> out of the integral and calling the remaining integral <math> H_f </math> we get <math> e^{j 2 \pi f t} H_f</math>.<br />
<br />
<br />
<br />
<br />
{| style="width:600px; height:100px" border="1"<br />
|- <br />
| '''INPUT'''<br />
| '''OUTPUT'''<br />
| '''REASON'''<br />
|- <br />
| <math> \delta(t)</math><br />
| <math>h(t)</math> <br />
| Given<br />
|-<br />
| <math> \delta(t-u)</math><br />
| <math>h(t-u)</math> <br />
| Time Invarient<br />
|-<br />
| <math> x(u)\delta(t-u)</math><br />
| <math>x(u)h(t-u)</math> <br />
| Proportionality<br />
|-<br />
|<math> \int_{-\infty}^\infty x(u)\delta(t-u) du</math><br />
|<math> \int_{-\infty}^\infty x(u)h(t-u) du</math><br />
|Superposition<br />
|-<br />
|<math> \int_{-\infty}^\infty e^{j 2 \pi f t} h(t-u) du</math><br />
|<math> e^{j 2 \pi f t} \int_{-\infty}^\infty e^{j 2 \pi v t} h(v) dv</math><br />
|Superposition<br />
|-<br />
|<math> e^{j 2 \pi f t} </math><br />
|<math> e^{j 2 \pi f t} H_f</math><br />
|Superposition (from above)<br />
|}<br />
<br />
==Fourier Series (indepth)==<br />
<br />
I would like to take a closer look at <math> \alpha_k </math> in the Fourier Series. Hopefully this will provide a better understanding of <math> \alpha_k </math>.<br />
<br />
We will seperate x(t) into three parts; where <math> \alpha_k </math> is negative, zero, and positive. <br />
<math> \bold x(t) = \sum_{k=-\infty}^\infty \alpha_k e^ \frac {j 2 \pi k t}{T} = \sum_{k=-\infty}^{-1} \alpha_k e^ \frac {j 2 \pi k t}{T} + \alpha_0 + \sum_{k=1}^\infty \alpha_k e^ \frac {j 2 \pi k t}{T}</math><br />
<br />
Now, by substituting <math> n = -k </math> into the summation where <math> k </math> is negative and substituting <math> n = k </math> into the summation where <math> k </math> is positive we get:<br />
<math> \sum_{n=1}^{\infty} \alpha_{-n} e^ \frac {-j 2 \pi n t}{T} + \alpha_0 + \sum_{n=1}^\infty \alpha_n e^ \frac {j 2 \pi n t}{T} </math><br />
<br />
Recall that <math>\alpha_n = \frac{1}{T}\int_{-\frac{T}{2}}^\frac{T}{2} x(u) e^ \frac {-j 2 \pi n t}{T} dt </math><br />
<br />
If <math> x(t) </math> is real, then <math> \alpha_n^* = \alpha_{-n} </math>. Let us assume that <math> x(t) </math> is real.<br />
<br />
<math> x(t) = \alpha_0 +\sum_{n=1}^\infty (\alpha_n e^ \frac {j 2 \pi n t}{T} + \alpha_n^* e^ \frac {-j 2 \pi n t}{T}) </math><br />
<br />
Recall that <math> y + y^* = 2Re(y) </math> [[Here is further clarification on this property]]<br />
<br />
So, we may write:<br />
<br />
<math> x(t) = \alpha_0 +\sum_{n=1}^\infty 2Re(\alpha_n e^ \frac {j 2 \pi n t}{T}) </math><br />
<br />
In terms of cosine <math> x(t) = \alpha_0 +\sum_{n=1}^\infty 2 |\alpha_n| cos(\frac{2 \pi n t}{T} + \omega_n) </math> where <math> \omega_n </math> is an angle.<br />
<br />
==Fourier Transform==<br />
<br />
Fourier transforms emerge because we want to be able to make Fourier expransions of non-periodic functions. We can accomplish this by taking the limit of x(t).<br />
<br />
Remember that:<br />
<math>x(t)=x(t+T)= \sum_{k=-\infty}^\infty \alpha_k e^ \frac {j 2 \pi k t}{T} = \sum_{k=-\infty}^\infty 1/T \int_{-\frac{T}{2}}^\frac{T}{2} x(u)e^ \frac {-j 2 \pi k u }{T} du e^ \frac {j 2 \pi k t}{T} </math><br />
<br />
Let's substitute in x(t) for k/T substitute f, for 1/T substitute df, and for the summation substitute the integral. <br />
<br />
So, <br />
<math> \lim_{T \to \infty}x(t)= \int_{-\infty}^\infty (\int_{-\infty}^\infty x(u) e^{-j 2 \pi f u} du) e^{j 2 \pi f t} df</math><br />
<br />
From the above limit we define <math> x(t)</math> and <math> X(f) </math>.<br />
<br />
<math> x(t) = \mathcal{F}^{-1}[X(f)] = \int_{-\infty}^\infty X(f) e^ {j 2 \pi f t} df</math><br />
<br />
<math> X(f) = \mathcal{F}[x(t)] = \int_{-\infty}^\infty x(t) e^ {-j 2 \pi f t} dt</math><br />
<br />
By using the above transforms we can now change a function from the frequency domain to the time domain or vise versa. We are not limited to just one domain but can use both of them. <br />
<br />
We can take the derivitive of <math> x(t) </math> and then put it in terms of the reverse fourier transform. <br />
<br />
<math> \frac{dx}{dt} = \int_{-\infty}^\infty j 2 \pi f X(f) e^ {j 2 \pi f t} df = \mathcal{F}^{-1}[j 2 \pi f X(f)]<br />
<br />
</math><br />
<br />
What happens if we just shift the time of <math> x(t) </math>? <br />
<br />
<math> x(t-t_0) = \int_{-\infty}^\infty X(f) e^{j 2 \pi f(t-t_0)} df = \int_{-\infty}^\infty e^{-j 2 \pi f t_0} X(f) e^{j 2 \pi f t} df = \mathcal{F}^{-1}[e^{-j 2 \pi f t_0} X(f)] </math><br />
<br />
In the same way, if we shift the frequency we get:<br />
<br />
<math> X(f-f_0) = \int_{-\infty}^\infty x(t) e^{j 2 \pi (f-f_0)t} dt = \int_{-\infty}^\infty e^{-j 2 \pi t f_0} x(t) e^{j 2 \pi f t} df = \mathcal{F} [e^{-j 2 \pi t f_0} x(t)] </math><br />
<br />
What would be the Fourier transform of <math> cos(2 /pi f_0 t) x(t) </math>?<br />
<br />
<math> \mathcal{F} [cos(2 \pi f_0 t) x(t)] = \int_{-\infty}^\infty x(t) cos(2 \pi f_0 t) e^{-j 2 \pi f t} dt = \int_{-\infty}^\infty \frac{e^{j 2 \pi f_0 t} + e^{-j 2 \pi f_0 t}}{2} x(t) e^{-j 2 \pi f t} dt </math><br />
<br />
<math> = \frac{1}{2} \int_{-\infty}^\infty x(t) e^{-j 2 \pi (f-f_0) t} dt + \frac{1}{2} \int_{-\infty}^\infty x(t) e^{j 2 \pi (f+f_0) t} dt = \frac{1}{2} X(f-f_0) + \frac{1}{2} X(f+f_0)</math><br />
<br />
What would happen if we multiplied our time (time scaling) by a constant in <math> x(t) </math>? We will substitute <math> u=at </math> and <math> du = adt </math>. If <math> a \ne 0 </math>:<br />
<br />
<math> \mathcal{F} [x(a t)] = \int_{-\infty}^\infty x(at) e^{-j 2 \pi f t} dt = \int_{-\infty}^\infty x(u) e^\frac{-j 2 \pi f u}{a} \frac{du}{|a|} = \frac{1}{|a|} X(\frac{f}{a})</math><br />
<br />
Ok, lets take the fourier transform of the fourier series.<br />
<br />
<math> \mathcal{F} [\sum_{n=-\infty}^{\infty} \alpha_n e^\frac{j 2 \pi n t}{T}] = \int_{-\infty}^\infty \sum_{n=-\infty}^{\infty} \alpha_n e^\frac{j 2 \pi n t}{T} e^{-j 2 \pi f t} dt = \sum_{n=-\infty}^{\infty} \alpha_n \int_{-\infty}^\infty e^{-j 2 \pi (f-\frac{n}{T}) t} dt = \sum_{n=-\infty}^{\infty} \alpha_n\delta(f-\frac{n}{T}) </math><br />
<br />
Remember: <math> \delta (f) = \int_{-\infty}^\infty e^{-j 2 \pi f t} dt </math><br />
<br />
==CD Player==<br />
<br />
Below is a diagram of how the information on a CD player is read and processed. As you can see the information on the CD is processed by the D/A converter and then sent through a low pass filter and then to the speaker. If you were recording sound, the sound would be captured by a microphone. Then, it should be sent through a low pass filter. The reason you want a low-pass filter is to keep high frequencies (that you don't intend to record) from being recorded. If a high frequency was recorded at say 30 KHz and the maximum frequency you intended to record was 20KHz, then when you played back the recording you would here a tone at 10KHz. From the filter the signal goes onto the A/D converter and then it is ready to be put on the CD. Recording signals (as just described) is essentially the reverse of the operation pictured below.<br />
<br />
[[Image:CDsystem.jpg]]<br />
<br />
'''In Time Domain:'''<br />
<br />
Let's start with a signal <math> h(t) </math>, as shown in the below picture. In this signal there is an infinite amount of information. Obviously, we can't hold it all in a computer, but we could take samples every <math> T </math> seconds. Lets do that by multiplying <math> h(t) </math> by <math> \sum_{n=-\infty}^\infty \delta (t-nT) </math>. Since the magnitude of our delta function is one, we get a series of delta functions that record the value of <math> h(t) </math> at intervals of <math> T </math>. This gives us a result that looks like: <math> h(t)\sum_{n=-\infty}^\infty \delta (t-nT) = \sum_{n=-\infty}^\infty h(t) \delta (t-nT)</math><br />
<br />
'''In Frequency Domain:'''<br />
<br />
In the frequency domain we start with <math> H(f) </math>. Now we are in frequency, so we must convolve instead of multiply like we did in the time domain. We would have to convolve <math> H(f) </math> with <math> \mathcal{F}[ \sum_{n=-\infty}^\infty \delta (t-nT) ]</math>. <br />
<br />
Aside:<math> \mathcal{F}[ \sum_{n=-\infty}^\infty \delta (t-nT) ] = \int_{-\infty}^\infty \sum_{n=-\infty}^\infty \delta (t-nT) e^{j 2 \pi f t} dt = \sum_{n=-\infty}^\infty \int_{-\infty}^\infty \delta (t-nT) e^{j 2 \pi f t} dt = \sum_{n=-\infty}^\infty e^{j 2 \pi f n T}</math><br />
<br />
This result looks it could be a fourier series. We would like to get our result in terms of delta functions. As shown below, the periodic delta functions could be represented as a fourier series with coefficients <math> \alpha_m </math>.<br />
<br />
<math> \sum_{n=-\infty}^\infty \delta (t-nT) = \sum_{m=-\infty}^\infty \alpha_m e^ {j 2 \pi m t} </math><br />
<br />
Now we can solve for <math> \alpha_m </math>. <br />
<br />
<math> \alpha_m = \frac {1}{T} \int_{\frac{-T}{2}}^{\frac{T}{2}} \sum_{n=-\infty}^\infty \delta (t-nT) e^\frac {j 2 \pi m t}{T} dt = \frac {1}{T} \int_{\frac{-T}{2}}^{\frac{T}{2}} \delta (t) e^\frac {j 2 \pi m t}{T} dt = \frac {1}{T} </math><br />
<br />
Since the only delta function within the integration limits is the delta function at <math> t=0 </math>, we can take out the summation and just leave one delta function. Then, evaluating the integral at <math> t=0 </math> we get <math> \frac{1}{T} </math><br />
<br />
<math> \sum_{n=-\infty}^\infty \delta (t-nT) = \sum_{n=-\infty}^\infty \frac {1}{T} e^ \frac {j 2 \pi k t}{T} </math><br />
<math> \mathcal{F} [\sum_{n=-\infty}^\infty \delta (t-nT)] = \mathcal{F} [\sum_{n=-\infty}^\infty \frac {1}{T} e^ \frac {j 2 \pi k t}{T}] = \sum_{n=-\infty}^\infty \frac {1}{T} \int_{-\infty}^\infty e^ \frac {j 2 \pi k t}{T} e^ {-j 2 \pi f t} dt= \frac {1}{T} \sum_{n=-\infty}^\infty \int_{-\infty}^\infty e^ {-j 2 \pi (f-\frac{m}{T} t} dt = \frac {1}{T} \sum_{n=-\infty}^\infty \delta (f-\frac{n}{T})</math><br />
<br />
Now wer are ready to take the convolution. <br />
<br />
<math> H(f)* \frac {1}{T} \sum_{n=-\infty}^\infty \delta (f-\frac{n}{T}) = \frac{1}{T} \sum_{n=-\infty}^\infty H(f-\frac{n}{T})</math><br />
<br />
[[Image:barnsasample.jpg|Picture uploaded by Sam Barnes]]<br />
<br />
'''Time Domain'''<br />
<br />
In order to output as sound any of the signals that we have we must run them through a D/A converter. This is like convolving the below signal by a step function <math> p(t) = U(t+\frac{T}{2})- U(t-\frac{T}{2}) </math>.<br />
<br />
This gives us <math> \sum (nt)p(t-nT)</math>. This is what the signal looks like as it is output through the D/A converter.<br />
<br />
'''Frequency Domain'''<br />
<br />
To find out what we would multiply by in the frequency domain we just take the inverse fourier transform of <math> p(t) </math> and we get <math>P(f) = \frac{sin (\frac{\pi t}{T})}{\frac{\pi t}{T}} </math>.<br />
<br />
By multiplying <math> \frac {1}{T} \sum_{n=-\infty}^\infty X(f-\frac{n}{T})P(f) = X(f) </math>. This is hopefully close to what we started with for a signal. <br />
<br />
[[Image:barnsaDA.jpg|Picture uploaded by Sam Barnes]]<br />
<br />
For 2 times oversampling:<br />
<br />
In time, multiply: <math> \sum_{n=-\infty}^\infty x(nT)\delta(t-nT)</math> by <math> \sum_{n=-M}^M h(m \frac{T}{2}) \delta (t-\frac{mT}{2})</math>. This profides points that are interpolated and makes our output sound better because it looks closer to the original wave. <br />
<br />
In frequency, convolve: <math> \frac {1}{T} \sum_{n=-\infty}^\infty X(f- \frac{n}{T} ) </math> with <math> \sum_{m=-M}^M h(\frac{mT}{2}) e ^\frac{-j2 \pi m f}{\frac{2}{T}} </math>. The X(f) that you get is great because there is little distortion near the original frequency plot. This means that you can use a cheaper low-pass filter then you would otherwise have been able to.<br />
<br />
==Nyquist Frequency==<br />
<br />
If you are sampling at a frequency of 40 KHz, then the highest frequency that you can reproduce is 20 KHz. The nyquist frequency, would be 20 KHz, the highest frequency that can be reproduced for a given sampling rate.<br />
<br />
==FIR Filters==<br />
<br />
A finite impulse response filter (FIR filter) is a digital filter that is applied to data before sending it out a D/A converter. This type of filter allows for compensation of the signal before is it destorted so that it will look as it was originally recorded. Using an FIR filter also allows us to put a cheap low-pass filter on after the D/A converter because the signal has been compensated so it doesn't take an expensive low-pass filter, as it would without the FIR filter.<br />
<br />
The coefficients that are sent out to the D/A converter are:<br />
<br />
<math><br />
h_m = { T } \int_{T} H(f)e^{j2 \pi m f T}\,df<br />
</math><br />
<br />
where <math> H(f)=\sum_{m=-M}^{M}h(mT)e^{-j 2 \pi f m T} </math><br />
<br />
Example: Design a FIR low-pass filter to pass between <math> -\frac{1}{4T} < f < \frac{1}{4T} </math> and reject the rest. <br />
<br />
Our desired response is: <math> H_{hat} = 1 </math>, if |f| is less then or equal to <math> \frac{1}{4T} </math> or <math> H_{hat} = 0 </math> otherwise. <br />
<br />
So, <math> h(mT) = T \int_{} . . . </math><br />
<br />
Note: From the Circular Convolution we get: <math> y(n) = \sum_{m=0}^{N-1}h(m)x(n-m)</math><br />
<br />
==Discrete Fourier Transforms (DFTs)==<br />
<br />
The DFT allows us to take a sample of some signal that is not periodic with time and take the Fourier series of it. There is the DFT and the Inverse DFT listed below.<br />
<br />
'''DFT'''<br />
<br />
<math> x(m) = \sum_{n=0}^{N-1} x(n) e^{\frac{-j 2 \pi m n}{N}}</math><br />
<br />
'''IDFT'''<br />
<br />
<math> x(k) = \frac{1}{N}\sum_{n=0}^{N-1} x(n) e^{\frac{j 2 \pi k n}{N}}</math><br />
<br />
With the DFT all the negative frequency components are just the complex conjugate of the positive frequency components. <br />
<br />
One problem with the DFT is that if the sample taken does not begin and end at zero, (or the same point) then we get what is called leakage. Because the DFT is discrete, if the end of the sample is not at the same place it began then it will make a jump back to the point that it began (leakage). This is because the DFT repeats the recorded section of signal over and over. It is this periodic manner of the DFT that allows us to reproduce a discrete signal that is not periodic. The DFT and IDFT are periodic with period N. This can be easily proved by simplifying <math> x(n+N) </math>.<br />
<br />
==Adaptive FIR Filters==<br />
<br />
[[Image:Adaptive.JPG]]<br />
<br />
It should be noted that in the above diagram, <math> e(n)=y(n)-r(n) = [\sum_{k=0}^{N-1} h_n(k) x(n-k)] - r(n) </math>. The goal of an adaptive FIR filter is to drive the error, e(n), to zero. If we consider that this is a two coefficient filter and we have a contour plot of <math> e^2(n) </math> then we want to travel in the direction of the negative gradient to minimize the error. Let us say that <math> \mu </math> is the stepping size. So...<br />
<math> \triangle h_n(m) = - \frac{\partial (e^2(n))}{\partial h_n(m)} \mu = - \mu 2 e(n)\frac{\partial (e(n))}{\partial h_n(m)} = - 2 \mu e(n) x(n-m) </math><br />
<br />
What would <math> h_{n+1}(m) </math> look like? <br />
<br />
<math> h_{n+1}(m)= h_n(m) + \triangle h_n(m) = h_n(m) - 2 \mu (y(n)-r(n)) x(n-m) = h_n(m) - 2 \mu ([\sum_{k=0}^{N-1} h_n(k) x(n-k)] - r(n)) x(n-m)<br />
<br />
</math><br />
<br />
How might one find an unknown transfer function? Lets use the example of the tuner upper. The idea here is that we want to remove a sine wave from the signal and leave the original signal(voice) in place. <br />
<br />
[[Image:AdaptiveFilter.JPG]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Discrete_Fourier_transform&diff=2684Discrete Fourier transform2006-10-08T21:51:05Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
==Discrete Fourier Transform==</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Sampling&diff=3817Sampling2006-10-08T21:50:51Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
==Introduction==<br />
In this chapter we will look at sampling and how it affects signal processing. Sampling is the process of taking a continuous stream of data with infinite points of resolution and getting that data into a form that can be stored in a finite data set.<br />
<br />
==Sampling Overview==<br />
Imagine that we have a song that is being played in a studio. We would like to record the song and store it on a computer in form that will allow the song to be played back to sound as much like the original as possible. However, the song being played has an infinite set of points when it is played. We must throw out much of this data and yet keep enough to reconstruct the song. This is where sampling comes in. By taking evenly spaced out samples of the data stream, we can later reconstruct the stream of data. The sampler converts the continuous time signal x(t) into a discrete-time sequence x(n) by taking the values of x(t) at integer multiples of the sampling perioud, T. Typically, discrete signals are formed by periodically sampling a continuous time signal using the form:<br />
<br />
<math> \bold x(n) = x_a(n*T_s) </math><br />
<br />
[[Image:Signals_Sampling.JPG]]<br />
<br />
There are, however, a few things that we need to keep in mind when trying to sample signals with many frequencies in them. When we sample data, we want to be able to sample enough data to get a correct reconstruction (this is discussed further below), but we don’t want to sample too much data that our sets become unmanageable.<br />
<br />
==Sampling Theory==<br />
According to Sampling Theory, you must sample fast enough that you eliminate all possible aliases that can occur from sampling. Aliasing is what happens when you sample a signal that can later be reconstructed with two possible frequencies. This can be seen by the picture shown below. Another example of this can be seen in older western movies when watching the spoked wheels of a stagecoach rotate backward at a slow rate. This happens when each frame of film is taken at a slightly faster rate than that of the rotating wheel. In order to sample fast enough to eliminate all possible aliases, you must sample at a rate greater than twice the maximum frequency found in the signal to be sampled. <br />
<br />
[[Image:Sampling_pic1.JPG]]<br />
<br />
<br />
Going back to our example of audio, we know that humans can hear from about 20 Hz up to about 20,000 Hz. Therefore we know that we want to make sure that we sample fast enough that we will get this entire frequency spectrum. In order to do this, we apply the principle where F is our highest frequency and T is our sampling rate to get a value of T = 40,000 Hz. This is also known as the Nyquist Theorem. This explains why audio that is recorded onto a CD is sampled at a rate of 44,100 Hz (they sample slightly higher than necessary for a 20 KHz max frequency). <br />
Something to note here is the fact that even though we want to sample our signal for frequencies between 20 and 20,000 Hz, most signals have many more frequencies in them due to white noise. In order to make sure that only the desired frequency is sampled you can run your signal through a pre-filter (sometimes called an anti-aliasing filter). This guarantees that the sampled data system receives analog signals having a frequency spectrum no greater than those frequencies allowed by the filter.<br />
<br />
To see how the sampled data is reconstructed, refer to my section on how a [[User:santsh|CD Player]] works.<br />
<br />
==Sampling Process==<br />
<br />
In order to see how the sampling process works, imagine a periodic sequence of impulses,<br />
<br />
<math> s_a(t) = \sum_{n=-\infty}^\infty \delta(t-nT_s) </math><br />
<br />
Now multiply your signal by the impulses to form the sampled signal<br />
<br />
<math> x_s(t) = x_a(t) * s_a(t) = \sum_{n=-\infty}^\infty x_a(nT_s)*\delta(t-nT_s) </math><br />
<br />
After which the sampled signal is converted into a discrete-time signal by mapping the impulses that are spaced in time by <math> T_s </math> into a sequence x(n) where the sample values are indexed by the integer variable n:<br />
<br />
<br />
<math> \bold x(n) = x_a(nT_s) </math><br />
<br />
<br />
Principle author of this page: [[User:santsh|Shawn Santana]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Fourier_transform&diff=2637Fourier transform2006-10-08T21:50:37Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
==From the Fourier Transform to the Inverse Fourier Transform==<br />
An initially identity that is useful:<br />
<math><br />
<br />
X(f)=\int_{-\infty}^{\infty} x(t) e^{-j2\pi ft}\, dt<br />
<br />
</math><br />
<br><br />
<br><br />
Suppose that we have some function, say <math> \beta (t) </math>, that is nonperiodic and finite in duration.<br><br />
This means that <math> \beta(t)=0 </math> for some <math> T_\alpha < \left | t \right | </math><br />
<br><br><br />
Now let's make a periodic function<br />
<math><br />
\gamma(t)<br />
</math><br />
by repeating<br />
<math><br />
\beta(t)<br />
</math><br />
with a fundamental period<br />
<math><br />
T_\zeta<br />
</math>.<br />
Note that <br />
<math><br />
\lim_{T_\zeta \to \infty}\gamma(t)=\beta(t)<br />
</math><br />
<br><br />
The Fourier Series representation of <math> \gamma(t) </math> is<br />
<br><br />
<math><br />
\gamma(t)=\sum_{k=-\infty}^\infty \alpha_k e^{j2\pi fkt}<br />
</math><br />
where<br />
<math> <br />
f={1\over T_\zeta}<br />
</math><br />
<br>and<br />
<math><br />
\alpha_k={1\over T_\zeta}\int_{-{T_\zeta\over 2}}^{{T_\zeta\over 2}} \gamma(t) e^{-j2\pi kt}\,dt<br />
</math><br />
<br><br />
<math> \alpha_k </math> can now be rewritten as<br />
<math><br />
\alpha_k={1\over T_\zeta}\int_{-\infty}^{\infty} \beta(t) e^{-j2\pi kt}\,dt<br />
</math><br />
<br>From our initial identity then, we can write <math> \alpha_k </math> as<br />
<math><br />
\alpha_k={1\over T_\zeta}\Beta(kf)<br />
</math><br />
<br> and <br />
<math><br />
\gamma(t)<br />
</math><br />
becomes<br />
<math><br />
\gamma(t)=\sum_{k=-\infty}^\infty {1\over T_\zeta}\Beta(kf) e^{j2\pi fkt}<br />
</math><br />
<br><br />
Now remember that<br />
<math><br />
\beta(t)=\lim_{T_\zeta \to \infty}\gamma(t)<br />
</math><br />
and<br />
<math><br />
{1\over {T_\zeta}} = f.<br />
</math><br />
<br><br />
Which means that<br />
<math><br />
\beta(t)=\lim_{f \to 0}\gamma(t)=\lim_{f \to 0}\sum_{k=-\infty}^\infty f \Beta(kf) e^{j2\pi fkt}<br />
</math><br />
<br><br />
Which is just to say that<br />
<math><br />
\beta(t)=\int_{-\infty}^\infty \Beta(f) e^{j2\pi fkt}\,df<br />
</math><br />
<br><br />
<br><br />
So we have that<br />
<math><br />
\mathcal{F}[\beta(t)]=\Beta(f)=\int_{-\infty}^{\infty} \beta(t) e^{-j2\pi ft}\, dt<br />
</math><br />
<br><br />
Further<br />
<math><br />
\mathcal{F}^{-1}[\Beta(f)]=\beta(t)=\int_{-\infty}^\infty \Beta(f) e^{j2\pi fkt}\,df<br />
</math><br />
==Some Useful Fourier Transform Pairs==<br />
<math><br />
\mathcal{F}[\alpha(t)]=\frac{1}{\mid \alpha \mid}f(\frac{\omega}{\alpha})<br />
</math><br />
<br><br />
{|<br />
|-<br />
|<math>\mathcal{F}[c_1\alpha(t)+c_2\beta(t)]</math><br />
|<math>=\int_{-\infty}^{\infty} (c_1\alpha(t)+c_2\beta(t)) e^{-j2\pi ft}\, dt</math><br />
|-<br />
|<br />
|<math>=\int_{-\infty}^{\infty}c_1\alpha(t)e^{-j2\pi ft}\, dt+\int_{-\infty}^{\infty}c_2\beta(t)e^{-j2\pi ft}\, dt</math><br />
|-<br />
|<br />
|<math>=c_1\int_{-\infty}^{\infty}\alpha(t)e^{-j2\pi ft}\, dt+c_2\int_{-\infty}^{\infty}\beta(t)e^{-j2\pi ft}\, dt=c_1\Alpha(f)+c_2\Beta(f)</math><br />
|-<br />
|}<br />
<br><br />
<math><br />
\mathcal{F}[\alpha(t-\gamma)]=e^{-j2\pi f\gamma}\Alpha(f)<br />
</math><br />
<br><br />
<math><br />
\mathcal{F}[\alpha(t)*\beta(t)]=\Alpha(f)\Beta(f)<br />
</math><br />
<br><br />
<math><br />
\mathcal{F}[\alpha(t)\beta(t)]=\Alpha(f)*\Beta(f)<br />
</math><br />
<br><br />
<br />
==A Second Approach to Fourier Transforms==</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Fourier_series&diff=2636Fourier series2006-10-08T21:50:18Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
==Introduction==<br />
Born in Auxerre, France in 1768, Jean Baptiste Joseph Fourier was orphaned at the age of eight. Later he was instructed by Benedictine monks who taught at and ran a military college. <br />
<br />
Years later (1822) Fourier's genius became evident when he discovered that he could represent a periodic function as a sum of sinusoids. It may be interesting to note that what has come to be known as the Fourier series was invented while Fourier was studying heat flow.<br />
<br />
Even though Fourier had discovered a powerful tool, his peers were slow in accepting it. This may be because he provided to rigorous proof to show that his series was an accurate representation of a periodic function. Later, P.G.L. Dirichlet was able to present an acceptable proof.<br />
<br />
<br />
<small>Information used in the introduction has been adapted from <u>Linear Circuit Analysis</u> by DeCarlo & Lin and <u>Fundamentals of Electric Circuits</u> by Alexander and Sadiku.</small><br />
<br />
==Periodic Functions==<br />
A continuous time signal <math>x(t)</math> is said to be periodic if there is a positive nonzero value of T such that <br />
<center><br />
<math> s(t + T) = s(t)</math> for all <math>t</math><br />
</center><br />
==Dirichlet Conditions==<br />
The conditions for a periodic function <math>f</math> with period 2L to have a convergent Fourier series.<br />
<br />
===''Theorem:''===<br />
<br />
Let <math>f</math> be a piecewise regular real-valued function defined on some interval [-L,L], such that <math>f</math> has<br />
''only a finite number of discontinuities and extrema'' in [-L,L]. Then the Fourier series of this function converges to <math>f</math> when <math>f</math> is continuous and to the arithmetic mean of the left-handed and right-handed limit of <math>f</math> at a point where it is discontinuous.<br />
<br />
==The Fourier Series==<br />
A Fourier series is an expansion of a periodic function <math>f</math> in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions.<br />
<center><br />
<math> f(t) = \sum_{k= -\infty}^ \infty \alpha_k e^ \frac{j 2 \pi k t}{T} </math>. <br />
</center><br />
<br />
<br />
==See Also==<br />
*[[Orthogonal Functions]]<br />
*[[Fourier Transforms]]<br />
<br />
<small>If anyone knows how to make the Fourier transform link into an internal, please do. I wasn't able to do so.</small><br />
<br />
==Contributors==<br />
Principle author of this page: [[User:Goeari|Aric Goe]]<br />
<br />
Introduction added on 10/06/05 by [[User:wonoje|Jeff W]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Orthogonal_functions&diff=2588Orthogonal functions2006-10-08T21:50:05Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
==Introduction==<br />
In this article we will examine another viewpoint for functions than that traditionally taken. Normally we think of a function <math> f(t) </math> as a complicated entity <math> f() </math> in a simple environment (one dimension, or along the t axis). Now we want to think of a function as a vector or point (a simple thing) in a very complicated environment (possibly an infinite dimensional space).<br />
<br />
==Vectors==<br />
Recall that vectors consist of an ordered set of numbers. Often the numbers are Real numbers, but we shall allow them to be Complex for our purposes. The numbers represent the amount of the vector in the direction denoted by the position of the number in the list. Each position in the list is associated with a direction. For example, the vector<br />
<math> \vec \bold v = <1, 4, 3> </math> means that the vector <math> \vec \bold v </math> is one unit in the first direction (often the x direction), four units in the second direction (often the y direction), and three units in the last direction (often the z direction). We say the component of <math> \vec \bold v </math> in the second direction is 4. This is often written as <math> v_y = 4 </math>.<br />
====Vector notation====<br />
We don't have to use x, y, and z as the direction names; we can use numbers, like 1, 2, and 3 instead. The advantage of this is that it leads to more compact notation, and extends to more than three dimensions much better. For example we could say <math> v_2 = 4 </math> instead of <math> v_y = 4 </math>. Instead of writing <math> \vec \bold v = <1, 4, 3> </math> we can write <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math> where the <math>\hat \bold a_k </math> denotes a basis vector in the kth direction, <math>v_1 = 1,</math> <math> v_2 = 4, </math> and <math> v_3 = 3</math>. The idea of basis vectors was implicit in the notation <math> \vec \bold v = <1, 4, 3> </math>.<br />
<br />
===Inner products for vectors===<br />
When vectors are real, inner products (sometimes called dot products) give the component of one vector in another vector's direction, scaled by the magnitude (length) of the second vector. Inner products are useful to find components of vectors. We commonly use a dot as the symbol for inner product. For example, the inner product of <math> \vec \bold v </math> and <math> \vec \bold a_n </math> is written:<br />
<br />
<math> \vec \bold v \bullet \vec \bold a_n </math><br />
<br />
====Orthogonality for vectors====<br />
It is quite handy to pick the directions used so that they are perpendicular (or orthogonal). With this arrangement the basis vectors have no components in each other's directions, which means that <br />
<br />
<math>\vec \bold a_k \bullet \vec \bold a_n = w_k \delta_{k,n} </math><br />
<br />
where the <math> w_k </math> is the square of the length of <math> \vec \bold a_k </math> and the symbol <math> \delta_{k,n} </math>, known as the [http://en.wikipedia.org/wiki/Kronecker_delta Kronecker delta], is one when k = n and zero otherwise.<br />
=====Normalization=====<br />
When the <math> w_k = 1</math> we have an orthonormal basis set, meaning that it is both orthogonal and that the basis vectors are normalized to unity (or have length one). Orthonormal vector systems are very popular. In fact they are the most common vector systems you will find. The reason they are so handy is each direction is uncoupled from the others.<br />
<br />
For example, to find <math> v_n </math>, we take the inner product of the vector <math> \vec \bold v </math> with a unit vector in the nth direction, <math> \vec \bold a_n </math>. We write this operation like this: <br />
<br />
<math> \vec \bold v \bullet \vec \bold a_n = \sum_{k=1}^3 v_k \vec \bold a_k \bullet \vec \bold a_n = \sum_{k=1}^3 v_k \delta_{k,n} = v_n </math><br />
<br />
Suppose we have two vectors from an orthonormal system, <math> \vec \bold u </math> and <math> \vec \bold v </math>. Taking the inner product of these vectors, we get<br />
<br />
<math> \vec \bold u \bullet \vec \bold v = \sum_{k=1}^3 u_k \vec \bold a_k \bullet \sum_{m=1}^3 v_m \vec \bold a_m = \sum_{k=1}^3 u_k \sum_{m=1}^3 v_m \vec \bold a_k \bullet \vec \bold a_m = \sum_{k=1}^3 u_k \sum_{m=1}^3 v_m \delta_{k,m} = \sum_{k=1}^3 v_k u_k </math><br />
<br />
This shows that when we have an orthonormal vector space, inner products boil down to summing the products of like components. Also note that if we take the inner product of <math> \vec \bold v </math> with itself, we get<br />
<br />
<math> \vec \bold v \bullet \vec \bold v = \sum_{k=1}^3 v_k \vec \bold a_k \bullet \sum_{m=1}^3 v_m \vec \bold a_m = \sum_{k=1}^3 v_k \sum_{m=1}^3 v_m \vec \bold a_k \bullet \vec \bold a_m = \sum_{k=1}^3 v_k \sum_{m=1}^3 v_m \delta_{k,m} = \sum_{k=1}^3 v_k^2</math><br />
<br />
which is the magnitude of the vector <math> \vec \bold v </math> squared (<math> | \vec \bold v |^2 </math>) from the Pythagorean Theorem.<br />
<br />
====Changing vector basis sets====<br />
Sometimes in our studies we find it useful to change basis sets. For example, when solving a physics problem with cylindrical symmetry, it is often easier to use cylindrical coordinates, and the basis vectors that go with that system, rather than the more usual Cartesian coordinates and basis vectors. <br />
=====So, how do I change the basis set?=====<br />
If the new basis set is orthonormal, it is really pretty simple. You need to project the vector you want changed onto each of the new basis vectors. This means that the new components are just the inner product of the vector and the appropriate basis function. If the new basis set is not orthonormal, and if there are n dimensions in each basis set, you will have n linear coupled equations in n unknowns to solve.<br />
<br />
===More vector questions===<br />
<br />
[[Complex vector inner products|What if the vectors have complex components?]]<br />
<br />
[[Vector weighting functions|What if not all components of the vectors have the same units?]]<br />
<br />
[[Multiple dimensional vectors|What if there are more than three dimensions?]]<br />
<br />
==Functions and vectors, an analogy==<br />
We may think of the number of the direction, <math> k </math>, as the independent variable of a vector and the component in that direction, <math> v_k </math> as the dependent variable of the vector <math> \vec \bold v </math> in a similar way to the way we think of t as the independent variable of a function f(), where f(t) is the dependent variable of f. Probably the biggest difference here is that t often takes on real values from <math> - \infty </math> to <math> \infty </math>, and <math> k \in {1, 2, 3} </math>. Using this analogy, we may think of a function as a vector having an uncountably infinite number of dimensions. <br />
<br />
====Can we write functions in an analogous way compared to the way we write vectors?====<br />
<br />
Remember we wrote <math> \vec \bold v = \sum_{k=1}^3 v_k \hat \bold a_k </math>. Can we write something similar for a function, f(t) defined for a t element of the reals? Well maybe.... If the sum over the dummy index k becomes an integral over the dummy variable, x, and the unit vectors <math> \vec \bold a_k </math> are replaced with something like <math> \delta(x-t) </math>, the [http://en.wikipedia.org/wiki/Delta_function Dirac delta function]. The result would look something like this:<br />
<br />
<math> f(t) = \int_{- \infty}^\infty f(x) \delta (x-t) dx </math>.<br />
<br />
This works! The Dirac delta functions, playing the roll of the basis vectors, are called basis functions. The function f(x) plays the roll of the vector coefficients <math>v_k</math>. This gives us another way to think of the function f().<br />
<br />
===Inner products for functions===<br />
[[Orthogonal functions#Inner products for vectors|Above]] we found that a vector inner product between <math>\vec \bold u </math> and <math>\vec \bold v </math> could be written as <math> \vec \bold u \bullet \vec \bold v = \sum_{k=1}^3 u_k v_k </math>. If we follow our above analogy, we should be able to replace the sum over k with an integral over x. There is one little notational problem, and that is we don't want to confuse the functional inner product with a simple muliply, so we need some new notation to denote this new inner product. In [http://en.wikipedia.org/wiki/Quantum_mechanics quantum mechanics], physicists use the [http://en.wikipedia.org/wiki/Bra-ket_notation bra-ket] notation. Let's borrow that.<br />
<br />
<math> <u|v> = \int_{-\infty}^\infty u^*(x) v(x) dx </math><br />
<br />
Note the complex conjugate on the function u(x). That is in case u(x) is a complex valued function. For the analogous case with vectors see [[Complex vector inner products]].<br />
====Orthogonality for functions====<br />
Two functions, <math>u(t)</math> and <math>v(t)</math> are said to be orthogonal on the interval <math> (a,b) </math> with respect to the weighting function <math> w(t) </math> if and only if <br />
<math>\int_a^b w(x) u^*(x) v(x) dx = 0 </math>.<br />
The weighting function is often unity, but it is included so that different values of t can be weighted appropriately in analogy to the way the <math>w_k</math> weight was used when the vector basis set was orthogonal, but not orthonormal (that is, different basis vectors had different numerical lengths), as we discussed [[Vector weighting functions|here]]. Unless otherwise noted we will use <math> w(t) = 1 </math>, so that the defining relation for orthogonality of functions <math> u </math> and <math> v </math> becomes<br />
<math>\int_a^b u^*(x) v(x) dx = 0 </math>.<br />
<br />
====Changing basis sets with functions====<br />
====Examples====<br />
*[[Fourier series]]<br />
*[[Reconstructing bandlimited signals from sample points]]<br />
<br />
==Other resources on orthogonality==<br />
[http://en.wikipedia.org/wiki/Inner_product Wikipedia inner product]<br />
<br />
[http://en.wikipedia.org/wiki/Orthogonal Wikipedia Orthogonality]<br />
<br />
Principle author of this page: [[User:Frohro|Rob Frohne]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Linear_Time_Invarient_System&diff=2540Linear Time Invarient System2006-10-08T21:49:49Z<p>Smitry: </p>
<hr />
<div>*[[Signals and systems|Signals and Systems]]<br />
==LTI systems==<br />
LTI System theory is a powerful and widely used concept in electrical engineering. It has applictions in circuit anlysis, control theory , and our main topic of interest signal processing.<br />
<br />
===LTI system properties===<br />
A system is considered to be a Linear Time Invariant when it satisfies the two basic criteria implied in its name, one it must be linear and two it must be time invariant. A Linear system is characterized by two properties superposition (additivity) and scaling (homogeneity). The superposition principal says that for any linear system a linear combination of solutions to the system is also a solution to the same linear system. The principal of scaling implies that if you adjust your scale an input by N amount, your output will also be adjusted by N amount. An example of a linear system then would be,<br />
<br />
::<math>x_1(t)</math><br />
::<math>x_2(t)</math><br />
::<math>y_1(t) = H(x_1(t))</math><br />
::<math>y_2(t) = H(x_2(t))</math><br />
::<math>Ay_1(t) + By_2(t) = H(Ax_2(t) + Bx_1(t))</math><br />
for any scalar values of A and B.<br />
<br />
Time invariance of a system means that for any input <math>x(t)</math> by some amount of time T the out put will also be adjusted by that amount of time. This implies that for,<br />
<br />
::<math>x(t - T)</math><br />
::<math>y(t - T) = H(x(t - T))</math></div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Linear_Time_Invarient_System&diff=2514Linear Time Invarient System2006-10-08T21:48:35Z<p>Smitry: /* LTI system properties */</p>
<hr />
<div>==LTI systems==<br />
LTI System theory is a powerful and widely used concept in electrical engineering. It has applictions in circuit anlysis, control theory , and our main topic of interest signal processing.<br />
<br />
===LTI system properties===<br />
A system is considered to be a Linear Time Invariant when it satisfies the two basic criteria implied in its name, one it must be linear and two it must be time invariant. A Linear system is characterized by two properties superposition (additivity) and scaling (homogeneity). The superposition principal says that for any linear system a linear combination of solutions to the system is also a solution to the same linear system. The principal of scaling implies that if you adjust your scale an input by N amount, your output will also be adjusted by N amount. An example of a linear system then would be,<br />
<br />
::<math>x_1(t)</math><br />
::<math>x_2(t)</math><br />
::<math>y_1(t) = H(x_1(t))</math><br />
::<math>y_2(t) = H(x_2(t))</math><br />
::<math>Ay_1(t) + By_2(t) = H(Ax_2(t) + Bx_1(t))</math><br />
for any scalar values of A and B.<br />
<br />
Time invariance of a system means that for any input <math>x(t)</math> by some amount of time T the out put will also be adjusted by that amount of time. This implies that for,<br />
<br />
::<math>x(t - T)</math><br />
::<math>y(t - T) = H(x(t - T))</math></div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=2006-2007_Assignments&diff=25232006-2007 Assignments2006-10-08T21:32:40Z<p>Smitry: /* Fall 2006 Homework Assignments */</p>
<hr />
<div>*[[Signals and systems|Main Signals & Systems page]]<br />
<br />
==Fall 2006 Homework Assignments==<br />
<br />
Assignments for this quarter will be listed here so that there is an easy place to look up the assignments. <br />
<br />
===HW #1===<br />
<br />
Initially Due: 10/2/06<br />
<br />
Follow Instructions Given in Handout<br />
<br />
===HW #2===<br />
<br />
Initially Due: 10/4/06<br />
<br />
Look at the Wiki & add your personal page. Add one thing to improve the Wiki.<br />
<br />
===HW #3===<br />
<br />
Finally Due: 10/11/06<br />
<br />
Spend at least 2 hours upgrading the Wiki. Must be done by Class on 10/11/06 to get credit. Our goal is to try and put the material we are covering into the Wiki. Please hand in a list of what you added and how long you spent working on the Wiki.<br />
<br />
===HW #4===<br />
<br />
Initially Due: ???<br />
<br />
Follow instructions on Handout Homework #3</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=2006-2007_Assignments&diff=24762006-2007 Assignments2006-10-03T05:59:06Z<p>Smitry: /* Fall 2006 Homework Assignments */</p>
<hr />
<div>==Fall 2006 Homework Assignments==<br />
<br />
Assignments for this quarter will be listed here so that there is an easy place to look up the assignments. <br />
<br />
'''HW #1'''<br />
<br />
9/21/06-10/02/06<br />
<br />
Follow Instructions Given in Handout<br />
<br />
'''HW #2'''<br />
<br />
9/29/06<br />
<br />
Look at the Wiki & add your personal page. Add one thing to improve the Wiki.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=2006-2007_Assignments&diff=24692006-2007 Assignments2006-10-03T05:58:42Z<p>Smitry: /* Fall 2006 Homework Assignments */</p>
<hr />
<div>==Fall 2006 Homework Assignments==<br />
<br />
Assignments for this quarter will be listed here so that there is an easy place to look up the assignments. <br />
<br />
'''HW #1'''<br />
<br />
9/21/06-10/02/06<br />
<br />
Follow Instructions Given in Handout<br />
<br />
'''HW #2'''<br />
<br />
Look at the Wiki & add your personal page. Add one thing to improve the Wiki.</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=2006-2007_Assignments&diff=24682006-2007 Assignments2006-10-03T05:55:20Z<p>Smitry: /* Fall 2006 Homework Assignments */</p>
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<div>==Fall 2006 Homework Assignments==<br />
<br />
Assignments for this quarter will be listed here so that there is an easy place to look up the assignments. <br />
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'''HW #1'''<br />
<br />
9/21/06-10/02/06<br />
<br />
Follow Instructions Given in Handout</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2505User:Smitry2006-10-01T18:50:05Z<p>Smitry: /* About me */</p>
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<div>==Ryan Smith's Wiki Webpage==<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask.<br />
[[http://mask.wwc.edu/profile/show/755 The Mask]]<br />
<br />
====Hobbies====<br />
<br />
Geocaching is a family favorite and I would recommend it to anyone the loves the good outdoors.<br />
[[http://www.geocaching.com Geocaching]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2438User:Smitry2006-10-01T18:49:23Z<p>Smitry: </p>
<hr />
<div>==Ryan Smith's Wiki Webpage==<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask.<br />
[[http://mask.wwc.edu/profile/show/755]]<br />
<br />
====Hobbies====<br />
<br />
Geocaching is a family favorite and I would recommend it to anyone the loves the good outdoors.<br />
[[http://www.geocaching.com]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2437User:Smitry2006-10-01T18:47:17Z<p>Smitry: /* About me */</p>
<hr />
<div>==Ryan Smith's Wiki Webpage==<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask.<br />
[[http://mask.wwc.edu/profile/show/755]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2436User:Smitry2006-10-01T18:39:31Z<p>Smitry: /* Ryan Smith's Wiki Webpage */</p>
<hr />
<div>==Ryan Smith's Wiki Webpage==<br />
[[Image:Family.JPG]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask.<br />
[[http://mask.wwc.edu]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=User:Smitry&diff=2435User:Smitry2006-10-01T18:38:53Z<p>Smitry: </p>
<hr />
<div>==Ryan Smith's Wiki Webpage==<br />
[[Image:Family.jpg]]<br />
<br />
==About me==<br />
<br />
Want to contact me just look me up in the Mask.<br />
[[http://mask.wwc.edu]]</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=File:Family.JPG&diff=4084File:Family.JPG2006-10-01T18:37:34Z<p>Smitry: Family Picture Ryan Smith</p>
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<div>Family Picture Ryan Smith</div>Smitryhttps://fweb.wallawalla.edu/class-wiki/index.php?title=Signals_and_Systems&diff=2449Signals and Systems2006-10-01T18:11:29Z<p>Smitry: </p>
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<div>[http://www.wwc.edu/~frohro/ClassNotes/engr455index.htm Class notes for Signals & Systems]<br />
<br />
== Topics ==<br />
*[[Orthogonal functions]]<br />
*[[Fourier series]]<br />
*[[Fourier transform]]<br />
*[[Sampling]]<br />
*[[Discrete Fourier transform]]<br />
*[[Fourier series - by Ray Betz|Signals and Systems - by Ray Betz]]<br />
*[[FIR Filter Example]]<br />
*[[2005-2006 Assignments]]<br />
<br />
I couldn't figure out how to get to others Users pages easily so I decided to start posting them here, please add yours:<br />
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[[User:Frohro|Rob Frohne]]<br />
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==2004-2005 contributors==<br />
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[[User:Barnsa|Sam Barnes]]<br />
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[[User:Santsh|Shawn Santana]]<br />
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[[User:Goeari|Aric Goe]]<br />
<br />
[[User:Caswto|Todd Caswell]]<br />
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[[User:Andeda|David Anderson]]<br />
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[[User:Guenan|Anthony Guenterberg]]<br />
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==2005-2006 contributors==<br />
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[[User:GabrielaV|Gabriela Valdivia]]<br />
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[[User:SDiver|Raymond Betz]]<br />
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[[User:chrijen|Jenni Christensen]]<br />
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[[User:wonoje|Jeffrey Wonoprabowo]]<br />
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[[User:wilspa|Paul Wilson]]<br />
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==2006-2007 contributors==<br />
<br />
[[User:Smitry|Ryan J Smith]]</div>Smitry