How to reconstruct RF signal using IQ data

Question

I have a complex IQ signal: $s = X + Yi$. I know my original $f_0$ was $5\textrm{ MHz}$.

I can calculate the magnitude/amplitude of $s$: $A = \sqrt{X^2 + Y^2}$ and phase, $\phi = \arctan\left(\frac YX\right) $

But how can I reconstruct the signal using this frequency, amplitude and phase data?

I need the output as a time varying signal which incorporates the amplitude and phase information from the IQ data. I have found this:

The reconstruction of RF-data from IQ-data is straightforward. It is a reversal of the complex demodulation in the previous section. The decimation is reversed by interpolation. The low- pass filter cannot be reversed, but should be chosen without loss of information in the first place. The down mixing is reversed by up mixing. At last, the RF-signal is found by taking the real-value of the complex up-mixed signal.

But I could really do with a worked example.

EDIT:

This MATLAB code appears to do what I want:

IQ = resample(IQ,downsampl,1); 

[Z,X] = size(IQ); 
time = [0:Z-1]'*1/F * ones(1,X); 

signal = IQ .* exp(+2*pi*i*time*carrier); 

signal = real(signal); 

where resample resamples IQ at downsampl/1 times.

It is not clear to me what I should use for downsampl and what F is?

Answer 1

The RF signal $r(t)$ is obtained from the complex baseband (IQ) signal $s(t)=x(t)+jy(t)$ in the following way:

$$r(t)=\text{Re}\{s(t)e^{j\omega_0t}\}\tag{1}$$

where $\omega_0$ is the carrier frequency (in rad/s). There are two other equivalent representations of $(1)$, where I use $s(t)=x(t)+jy(t)=a(t)e^{j\phi(t)}$:

$$\begin{align}r(t)&=x(t)\cos(\omega_0t)-y(t)\sin(\omega_0t)\tag{2}\\ r(t)&=a(t)\cos(\omega_0t+\phi(t))\tag{3}\end{align}$$

Eq. $(2)$ shows that the I- and Q-components are both modulated by orthogonal carriers, whereas $(3)$ shows that $r(t)$ generally exhibits amplitude modulation as well as phase modulation.

The quote in your question suggests that in the respective text the signals are all represented as discrete-time signals, hence the decimation and interpolation stages.

Answer 2

Just use the received I/Q signal $X(t) + jY(t)$ directly, and feed it back directly to your RF IQ mixer/upconverter.

Signal to transmit = $X(t)cos(2\pi f_0t) - Y(t)sin(2\pi f_0t)$