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Absolute Value Integration

  • Thread starter Xkaliber
  • Start date
  • #1
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0

Homework Statement



[tex]
\int_{-3}^{3}|t|e^{-jwt}dt
[/tex]


The Attempt at a Solution



I am not sure if I need to break this into two regions due to the abs value...
 

Answers and Replies

  • #2
CompuChip
Science Advisor
Homework Helper
4,302
47
Yes, that is one approach:

[tex]\int_{-3}^3 |t| e^{-j \omega t} \, dt = \int_{0}^3 t e^{-j \omega t} \, dt - \int_{-3}^0 t e^{-j \omega t} \, dt[/tex]
and then you can solve both integrals with a trick (write the integrand as a derivative w.r.t. omega, for example).

Alternatively, you can use Euler's identity to write the integral as

[tex]\int_{-3}^3 |t| \cos(\omega t) \, dt + j \int_{-3}^3 |t| \sin(\omega t) \, dt[/tex]
and use (anti)-symmetries to reduce the problem before taking care of the absolute value.
 

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