- #1
epsilonjon
- 58
- 0
Question:
Derive the relationship
[tex]\int^t_{- \infty} f(\tau) d \tau \Leftrightarrow \frac{F(\omega)}{j \omega} + \pi F(0) \delta (\omega)[/tex]
(where [itex]\Leftrightarrow[/itex] means "Fourier transforms into").
Attempt:
I have already proved the relationship
[tex]\frac{dg(t)}{dt} \Leftrightarrow j \omega G( \omega)[/tex]
so define [itex]h(t) = \frac{dg(t)}{dt}[/itex]. Then we have
[tex]\int^t_{- \infty} h(\tau) d \tau = \int^t_{- \infty} \frac{dg}{d \tau} d \tau = [g(\tau)]^t_{- \infty} = g(t) - g(-\infty)[/tex]
so [itex]g(t) = \int ^t _{- \infty} h(\tau) d \tau + g(- \infty)[/itex]
Using the Fourier differentiation relationship above we get
[tex]\mathcal{F}[h(t)] = j \omega \mathcal{F}[g(t)] = j \omega \mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau + g(- \infty) \right ] = j \omega \mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau \right ] + j \omega \mathcal{F}[g(- \infty)] [/tex]
[tex]= j \omega \mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau \right ] + j \omega 2 \pi g(-\infty) \delta(\omega)[/tex]
so
[tex]\mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau \right ] = \frac{\mathcal{F}[h(t)]}{j \omega} - 2 \pi g(-\infty) \delta(\omega)[/tex]
I'm not sure if I've done something wrong here or if somehow this is equivalent to the correct relationship? Could someone help please?
Thanks!
Jon.
Derive the relationship
[tex]\int^t_{- \infty} f(\tau) d \tau \Leftrightarrow \frac{F(\omega)}{j \omega} + \pi F(0) \delta (\omega)[/tex]
(where [itex]\Leftrightarrow[/itex] means "Fourier transforms into").
Attempt:
I have already proved the relationship
[tex]\frac{dg(t)}{dt} \Leftrightarrow j \omega G( \omega)[/tex]
so define [itex]h(t) = \frac{dg(t)}{dt}[/itex]. Then we have
[tex]\int^t_{- \infty} h(\tau) d \tau = \int^t_{- \infty} \frac{dg}{d \tau} d \tau = [g(\tau)]^t_{- \infty} = g(t) - g(-\infty)[/tex]
so [itex]g(t) = \int ^t _{- \infty} h(\tau) d \tau + g(- \infty)[/itex]
Using the Fourier differentiation relationship above we get
[tex]\mathcal{F}[h(t)] = j \omega \mathcal{F}[g(t)] = j \omega \mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau + g(- \infty) \right ] = j \omega \mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau \right ] + j \omega \mathcal{F}[g(- \infty)] [/tex]
[tex]= j \omega \mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau \right ] + j \omega 2 \pi g(-\infty) \delta(\omega)[/tex]
so
[tex]\mathcal{F} \left [ \int ^t _{- \infty} h(\tau) d \tau \right ] = \frac{\mathcal{F}[h(t)]}{j \omega} - 2 \pi g(-\infty) \delta(\omega)[/tex]
I'm not sure if I've done something wrong here or if somehow this is equivalent to the correct relationship? Could someone help please?
Thanks!
Jon.