I/Q of Signals and Hilbert Transform

[ashah99]

Homework Statement

Please see below for problem statement.

Relevant Equations

In-phase component: [S(f) + S*(-f)]/2
Quadrature phase component: [S(f) - S*(-f)] / (2j)
Hilbert Transform: H(f) = -j*sign(f)

Hello, would anyone be willing to provide help to the following problem? I can find the Fourier Transform of the complex envelope of s(t) and the I/Q can be found by taking the Real and imaginary parts of that complex envelope, but how can I approach the actual question of finding the carrier f0? Do I multiply by -j*sign(f) of the Q part in the frequency domain and set it equal to I and solve for f0? I appreciate any help.

 

[Master1022]

Hi, just thought I would give this a go because it is unanswered.

I can find the Fourier Transform of the complex envelope of s(t) and the I/Q can be found by taking the Real and imaginary parts of that complex envelope, but how can I approach the actual question of finding the carrier f0?

Apologies, would you mind explaining how you find the in-phase and quadrature-phase components of the signal. If I had to make an educated guess (let me know if I am wrong), I thought we would try to resolve onto the in-phase and quadrature signals. That is, if we defined cos as in-phase and sin as quadrature-phase something along the lines of:

$$\text{In-phase component } s_{I}(t) = \int s(t) cos(2\pi f_0 t) dt$$
and
$$\text{Quadrature-phase component } s_{Q}(t) = \int s(t) sin(2\pi f_0 t) dt$$

(or we could define them the other way around)

Before I type out the rest of the what I did does that seem reasonable (otherwise, I don't want to lead you astray)?

IF that seems fair, then those integrals look like Fourier transform integrals evaluated at $f = 0$... and we know that the Fourier transform of a product in the time domain is the product of the Fourier transforms in the frequency domain...
- By duality we can find Fourier transform of s(t)
- We know the Fourier transforms of the $sin(2 \pi f_0 t)$ and $cos(2\pi f_0 t)$ for a general $f_0$
- We could calculate the convolution for both in-phase and quadrature-components

If you agree with the above (feel free to correct and/or disagree with me), then we can apply the Hilbert transform visually to look where the two