Inverse Fourier Transform of cos(4ω + pi/3)

[jkface]

Homework Statement

Find the inverse Fourier transform of F(jω) = cos(4ω + pi/3)

Homework Equations

δ(t) <--> 1
δ(t - t_o) <--> exp(-j*ω_o*t)
cos(x) = 1/2 (exp(jx) + exp(-jx))

The Attempt at a Solution

So first I turned the given equation into its complex form using Euler's Formula.

F(jω) = 1/2 (exp(j*4*ω + j*pi/3) + exp(-j*4*ω - j*pi/3))

And using the relevant equation above, I get..

exp(j*4*ω) <--> δ(t + 4) and exp(j*4*ω) <--> δ(t - 4)

I'm not exactly sure how to do inverse Fourier transformation on exp(j*pi/3) and exp(-j*pi/3). My guess is you simply get simply exp(j*pi/3)*δ(t) and exp(-j*pi/3)*δ(t).

Assuming I'm correct in the above step, I now multiply the resulting expressions.

1/2 ( exp(j*pi/3)*δ(t)*δ(t + 4) + exp(-j*pi/3)*δ(t)*δ(t - 4) )

Is my solution correct? I feel like I would have to simply the final answer I got but I'm not really sure how.

 

[Mandelbroth]

Homework Statement

Find the inverse Fourier transform of F(jω) = cos(4ω + pi/3)

Homework Equations

δ(t) <--> 1
δ(t - t_o) <--> exp(-j*ω_o*t)
cos(x) = 1/2 (exp(jx) + exp(-jx))

The Attempt at a Solution

So first I turned the given equation into its complex form using Euler's Formula.

F(jω) = 1/2 (exp(j*4*ω + j*pi/3) + exp(-j*4*ω - j*pi/3))

And using the relevant equation above, I get..

exp(j*4*ω) <--> δ(t + 4) and exp(j*4*ω) <--> δ(t - 4)

I'm not exactly sure how to do inverse Fourier transformation on exp(j*pi/3) and exp(-j*pi/3). My guess is you simply get simply exp(j*pi/3)*δ(t) and exp(-j*pi/3)*δ(t).

Assuming I'm correct in the above step, I now multiply the resulting expressions.

1/2 ( exp(j*pi/3)*δ(t)*δ(t + 4) + exp(-j*pi/3)*δ(t)*δ(t - 4) )

Is my solution correct? I feel like I would have to simply the final answer I got but I'm not really sure how.

Well...I don't think so. Are you saying that $\cos(4\omega + \frac{\pi}{3})$ is the Fourier transform and you are attempting to find $\mathcal{F}^{-1}[\cos(4\omega + \frac{\pi}{3})](t)$?

 

[Mute]

Homework Statement

Find the inverse Fourier transform of F(jω) = cos(4ω + pi/3)

Homework Equations

δ(t) <--> 1
δ(t - t_o) <--> exp(-j*ω_o*t)
cos(x) = 1/2 (exp(jx) + exp(-jx))

The Attempt at a Solution

So first I turned the given equation into its complex form using Euler's Formula.

F(jω) = 1/2 (exp(j*4*ω + j*pi/3) + exp(-j*4*ω - j*pi/3))

And using the relevant equation above, I get..

exp(j*4*ω) <--> δ(t + 4) and exp(j*4*ω) <--> δ(t - 4)

I'm not exactly sure how to do inverse Fourier transformation on exp(j*pi/3) and exp(-j*pi/3). My guess is you simply get simply exp(j*pi/3)*δ(t) and exp(-j*pi/3)*δ(t).

Assuming I'm correct in the above step, I now multiply the resulting expressions.

1/2 ( exp(j*pi/3)*δ(t)*δ(t + 4) + exp(-j*pi/3)*δ(t)*δ(t - 4) )

Is my solution correct? I feel like I would have to simply the final answer I got but I'm not really sure how.

Your factors of $\exp(\pm i \pi/3)$ are constants multiplying the factors $\exp(\pm i 4\omega)$.

What is the inverse Fourier transform $\mathcal F^{-1}[c \hat{F}(\omega)]$, where $c$ is a constant (or any function, really)? It's not $\mathcal F^{-1}[c]\mathcal F^{-1}[ \hat{F}(\omega)]$, but that's what you implicitly assumed in your final answer.