# Integral Approximation

1. Mar 21, 2013

### S_David

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

2. Mar 23, 2013

### 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 $T/(1+\alpha)$.

3. Mar 23, 2013

### S_David

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

4. Mar 23, 2013

### jashua

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