- #1
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Can anyone show me how to evaluate an integral like this by hand? I believe such integrals have an analytic solution, but I can't figure out how to find them. Mathematica seems unable to help (the Integrate command runs forever) but I believe this can be done by hand. It's a sort of integral commonly found in communications theory. I actually don't think it's supposed to be very hard...
[itex]
\int_{ - \infty }^\infty {\left[ {\frac{1}
{{\sqrt T }}\operatorname{sinc} \left( {\frac{t}
{T}} \right) - \frac{1}
{{2\sqrt T }}\operatorname{sinc} \left( {\frac{{t - T}}
{T}} \right)} \right]^2 dt}
[/itex]
where
[itex]
\operatorname{sinc} \left( t \right) \triangleq \frac{{\sin \left( {\pi t} \right)}}
{{\pi t}}
[/itex]
Obviously I can expand the binomial out, but I'm left with products of sinc's with different arguments, and I don't know how to continue.
- Warren
[itex]
\int_{ - \infty }^\infty {\left[ {\frac{1}
{{\sqrt T }}\operatorname{sinc} \left( {\frac{t}
{T}} \right) - \frac{1}
{{2\sqrt T }}\operatorname{sinc} \left( {\frac{{t - T}}
{T}} \right)} \right]^2 dt}
[/itex]
where
[itex]
\operatorname{sinc} \left( t \right) \triangleq \frac{{\sin \left( {\pi t} \right)}}
{{\pi t}}
[/itex]
Obviously I can expand the binomial out, but I'm left with products of sinc's with different arguments, and I don't know how to continue.
- Warren