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Evaluate this integral - HW1 - Revision history
2026-04-06T16:18:51Z
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Max.Woesner at 23:31, 28 October 2009
2009-10-28T23:31:24Z
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Homework #1 - Evaluate this integral ===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>=== Homework #1 - Evaluate this integral ===</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Evaluate the integral <math>\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt\!</math><br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Evaluate the integral <math>\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt\!</math><br></div></td></tr>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><b>Solution</b><br></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For <math> n=m, \int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \int_{-\frac{T}{2}}^{\frac{T}{2}}1 dt = T \Bigg|_{\frac{-T}{2}}^{\frac{T}{2}} = \frac{T}{2} - \frac{-T}{2} = T \!</math><br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For <math> n=m, \int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \int_{-\frac{T}{2}}^{\frac{T}{2}}1 dt = T \Bigg|_{\frac{-T}{2}}^{\frac{T}{2}} = \frac{T}{2} - \frac{-T}{2} = T \!</math><br></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For <math> n\neq m,\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \frac{e^{j2 \pi (n-m)t/T}}{\frac{j2 \pi (n-m)}{T}} \Bigg|_{\frac{-T}{2}}^{\frac{T}{2}} = \frac{e^{j \pi (n-m)} - e^{-j \pi (n-m)}}{\frac{j2 \pi (n-m)}{T}} = 0 \!</math><br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For <math> n\neq m,\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \frac{e^{j2 \pi (n-m)t/T}}{\frac{j2 \pi (n-m)}{T}} \Bigg|_{\frac{-T}{2}}^{\frac{T}{2}} = \frac{e^{j \pi (n-m)} - e^{-j \pi (n-m)}}{\frac{j2 \pi (n-m)}{T}} = 0 \!</math><br></div></td></tr>
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Max.Woesner
https://fweb.wallawalla.edu/class-wiki/index.php?title=Evaluate_this_integral_-_HW1&diff=5089&oldid=prev
Max.Woesner at 05:27, 19 October 2009
2009-10-19T05:27:00Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 22:27, 18 October 2009</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For <math> n=m, \int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \int_{-\frac{T}{2}}^{\frac{T}{2}}1 dt = T \Bigg|_{\frac{-T}{2}}^{\frac{T}{2}} = \frac{T}{2} - \frac{-T}{2} = T \!</math><br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For <math> n=m, \int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \int_{-\frac{T}{2}}^{\frac{T}{2}}1 dt = T \Bigg|_{\frac{-T}{2}}^{\frac{T}{2}} = \frac{T}{2} - \frac{-T}{2} = T \!</math><br></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For <math> n\neq m,\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \frac{e^{j2 \pi (n-m)t/T}}{\frac{j2 \pi (n-m)}{T}} \Bigg|_{\frac{-T}{2}}^{\frac{T}{2}} = \frac{e^{j \pi (n-m)} - e^{-j \pi (n-m)}}{\frac{j2 \pi (n-m)}{T}} = 0 \!</math><br></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For <math> n\neq m,\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \frac{e^{j2 \pi (n-m)t/T}}{\frac{j2 \pi (n-m)}{T}} \Bigg|_{\frac{-T}{2}}^{\frac{T}{2}} = \frac{e^{j \pi (n-m)} - e^{-j \pi (n-m)}}{\frac{j2 \pi (n-m)}{T}} = 0 \!</math><br></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">So, <math>\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \begin{cases} T, & \mbox{for }n=m \\ 0, & \mbox{for }n\neq m \end{cases}\!</math><br><br></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Alternate method: <br></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math>\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \int_{-\frac{T}{2}}^{\frac{T}{2}}[cos(2 \pi (n-m)t/T)+jsin(2 \pi (n-m)t/T)]dt\!</math><br></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">For <math> n=m, \int_{-\frac{T}{2}}^{\frac{T}{2}}[cos(2 \pi (n-m)t/T)+jsin(2 \pi (n-m)t/T)]dt = \int_{-\frac{T}{2}}^{\frac{T}{2}}1 dt = T \Bigg|_{\frac{-T}{2}}^{\frac{T}{2}} = \frac{T}{2} - \frac{-T}{2} = T \!</math><br></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">For <math> n\neq m,\int_{-\frac{T}{2}}^{\frac{T}{2}}[cos(2 \pi (n-m)t/T)+jsin(2 \pi (n-m)t/T)]dt = 0\!</math><br></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>So, <math>\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \begin{cases} T, & \mbox{for }n=m \\ 0, & \mbox{for }n\neq m \end{cases}\!</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>So, <math>\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \begin{cases} T, & \mbox{for }n=m \\ 0, & \mbox{for }n\neq m \end{cases}\!</math></div></td></tr>
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Max.Woesner
https://fweb.wallawalla.edu/class-wiki/index.php?title=Evaluate_this_integral_-_HW1&diff=5087&oldid=prev
Max.Woesner: New page: == Max Woesner == Back to my Home Page === Homework #1 - Evaluate this integral === Evaluate the integral <math>\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt\!</...
2009-10-19T05:16:48Z
<p>New page: == Max Woesner == <a href="/class-wiki/index.php/Max_Woesner" title="Max Woesner">Back to my Home Page</a> === Homework #1 - Evaluate this integral === Evaluate the integral <math>\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt\!</...</p>
<p><b>New page</b></p><div>== Max Woesner ==<br />
[[Max Woesner|Back to my Home Page]]<br />
<br />
=== Homework #1 - Evaluate this integral ===<br />
<br />
Evaluate the integral <math>\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt\!</math><br><br />
For <math> n=m, \int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \int_{-\frac{T}{2}}^{\frac{T}{2}}1 dt = T \Bigg|_{\frac{-T}{2}}^{\frac{T}{2}} = \frac{T}{2} - \frac{-T}{2} = T \!</math><br><br />
For <math> n\neq m,\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \frac{e^{j2 \pi (n-m)t/T}}{\frac{j2 \pi (n-m)}{T}} \Bigg|_{\frac{-T}{2}}^{\frac{T}{2}} = \frac{e^{j \pi (n-m)} - e^{-j \pi (n-m)}}{\frac{j2 \pi (n-m)}{T}} = 0 \!</math><br><br />
So, <math>\int_{-\frac{T}{2}}^{\frac{T}{2}}e^{j2 \pi (n-m)t/T}dt = \begin{cases} T, & \mbox{for }n=m \\ 0, & \mbox{for }n\neq m \end{cases}\!</math></div>
Max.Woesner