What is mathematically wrong with this integration?

[EngWiPy]

Hello all,

I have the following integration:

\int_{-\infty}^{\infty}e^{-j2\pi f_ct[a_p-a_q]}g(t[1+a_p]-kT_s-\tau_p)g(t[1+a_q]-mT_s-\tau_q)\,dt

where g(t) is 1 in the interval [0,Ts]. This means that the integration has value when both function g(t[1+a_p]-kT_s-\tau_p) and g(t[1+a_q]-mT_s-\tau_q) are 1. Both are 1 when:

0\leq t[1+a_p]-kT_s-\tau_p\leq T_s

and

0\leq t[1+a_q]-mT_s-\tau_q\leq T_s

Which implies that both are 1 when:

t=\frac{(\tau_p-\tau_q)+(k-m)T_s}{a_p-a_q}

But the integration over a point is zero, which can be the answer of the physical problem I have in hand. Where did I go wrong in the process?

Thanks.

 

[the_wolfman]

Which implies that both are 1 when:

t=\frac{(\tau_p-\tau_q)+(k-m)T_s}{a_p-a_q}

This is where you went wrong. You can't use the two inequalities to evaluate t directly. You need to calculate the range of t where the two functions are both nonzero.

For example le\tau_p=\tau_q=a_p=a_q=0,
k=-1,
and
m=-2

Now
g(t[1+ap]−kTs−τp) =1 for t\in(0,1)
and
g(t[1+aq]−mTs−τq) =1 for t\in(0,.5)

As you can see there is still a range in t where both g are nonzero.

 

[EngWiPy]

This is where you went wrong. You can't use the two inequalities to evaluate t directly. You need to calculate the range of t where the two functions are both nonzero.

For example le\tau_p=\tau_q=a_p=a_q=0,
k=-1,
and
m=-2

Now
g(t[1+ap]−kTs−τp) =1 for t\in(0,1)
and
g(t[1+aq]−mTs−τq) =1 for t\in(0,.5)

As you can see there is still a range in t where both g are nonzero.

When I subtract the ranges of both functions I got something like:

0\leq x\leq 0

which implies that x=0. Right?

 

[EngWiPy]

But if I add the ranges I'll get a range of t! Which one is more correct? and why?