# What is mathematically wrong with this integration?

## Main Question or Discussion Point

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.

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.

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?

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