Evaluate this integral - HW1: Difference between revisions

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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\!</...
 
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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>
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>
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>
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>
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>
Alternate method: <br>
<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>
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>
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>
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>
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>

Revision as of 22:27, 18 October 2009

Max Woesner

Back to my Home Page

Homework #1 - Evaluate this integral

Evaluate the integral ∫−T2T2ej2π(n−m)t/Tdt
For n=m,∫−T2T2ej2π(n−m)t/Tdt=∫−T2T21dt=T|−T2T2=T2−−T2=T
For n≠m,∫−T2T2ej2π(n−m)t/Tdt=ej2π(n−m)t/Tj2π(n−m)T|−T2T2=ejπ(n−m)−e−jπ(n−m)j2π(n−m)T=0
So, ∫−T2T2ej2π(n−m)t/Tdt={T,for n=m0,for n≠m

Alternate method:
∫−T2T2ej2π(n−m)t/Tdt=∫−T2T2[cos(2π(n−m)t/T)+jsin(2π(n−m)t/T)]dt
For n=m,∫−T2T2[cos(2π(n−m)t/T)+jsin(2π(n−m)t/T)]dt=∫−T2T21dt=T|−T2T2=T2−−T2=T
For n≠m,∫−T2T2[cos(2π(n−m)t/T)+jsin(2π(n−m)t/T)]dt=0
So, ∫−T2T2ej2π(n−m)t/Tdt={T,for n=m0,for n≠m

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