- #1
muzialis
- 166
- 1
Hi there,
I am trying to get some practice with Fourier Transforms, there is a long way to go.
For example, let me consider the function $$ \gamma (t) = \int_{-\infty}^{t} C(t-\tau) \sigma(\tau) \mathrm{d}{\tau}$$
Defining the Fourier Transform as
$$ \gamma(\omega) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \gamma(t) e^{i\omega t} \mathrm{d}t$$
I am supposed to compute with ease that
$$ \gamma(\omega) = \sigma(\omega) \int_0 ^{\infty} C(t) e^{i\omega t} \mathrm{d}t$$,
but I am struggling, because I can not apply the convolution theorem (as the first equation is a convolution only to actual time).
I tried to use the definition writing
$$\gamma(\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \int_{-\infty}^{t} C(t-\tau) \sigma(\tau) \mathrm{d}{\tau} e^{i\omega t} \mathrm{d}t$$
hoping to invertt integration order and express the inner integral as a Fourier Transform, but again I am not getting anywhere. I tried a variable change $$t-\tau = u$$ and that helps in changing the integration limits to 0 and infinity, but still does not bring me to the desired result, any advice or hint?
Many thanks as usual
I am trying to get some practice with Fourier Transforms, there is a long way to go.
For example, let me consider the function $$ \gamma (t) = \int_{-\infty}^{t} C(t-\tau) \sigma(\tau) \mathrm{d}{\tau}$$
Defining the Fourier Transform as
$$ \gamma(\omega) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} \gamma(t) e^{i\omega t} \mathrm{d}t$$
I am supposed to compute with ease that
$$ \gamma(\omega) = \sigma(\omega) \int_0 ^{\infty} C(t) e^{i\omega t} \mathrm{d}t$$,
but I am struggling, because I can not apply the convolution theorem (as the first equation is a convolution only to actual time).
I tried to use the definition writing
$$\gamma(\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \int_{-\infty}^{t} C(t-\tau) \sigma(\tau) \mathrm{d}{\tau} e^{i\omega t} \mathrm{d}t$$
hoping to invertt integration order and express the inner integral as a Fourier Transform, but again I am not getting anywhere. I tried a variable change $$t-\tau = u$$ and that helps in changing the integration limits to 0 and infinity, but still does not bring me to the desired result, any advice or hint?
Many thanks as usual