Integral Approximation: Tau <<T

[EngWiPy]

Hello,

If tau<<T which of the following relations are true:

$$\int_{\tau/(1+a)}^{(T+\tau)/(1+a)}v(t)e^{-j2\pi f_0 at}e^{-j2\pi\frac{k}{T}t[1+a]}\,dt=\int_{0}^{T/(1+a)}v(t)e^{-j2\pi f_0 at}e^{-j2\pi\frac{k}{T}t[1+a]}\,dt$$

or

$$\int_{\tau/(1+a)}^{(T+\tau)/(1+a)}v(t)e^{-j2\pi f_0 at}e^{-j2\pi\frac{k}{T}t[1+a]}\,dt\simeq\int_{0}^{T/(1+a)}v(t)e^{-j2\pi f_0 at}e^{-j2\pi\frac{k}{T}t[1+a]}\,dt$$

Thanks

 

[jashua]

Your assumtion is not enough to show one of the relations is true. It seems you need to show that the integrand is periodic with $T/(1+\alpha)$.

 

[EngWiPy]

Your assumtion is not enough to show one of the relations is true. It seems you need to show that the integrand is periodic with $T/(1+\alpha)$.

OK, the integral after substituting for v(t) will look like:

$$\int_{\tau/(1+a)}^{(T+\tau)/(1+a)}e^{j2\pi\frac{m-k}{T}t[1+a]}\,dt$$

Is that enough to tell now?

Thanks

 

[jashua]

If that is the final version of the integral then it is enough to say that the equality holds.