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Signal demodulation

In practice the signal deduced in the previous section is not visualised as it is, but instead the NMR or MRI instrument eliminates the frequency offset $\omega_0$ by demodulating the signal. It is done by multiplying the signal with harmonics of frequency $\Omega$ that is close to the Larmor frequency $\omega_0$. That is, $\Omega = \omega_0 + \delta \omega$. In this way a complex signal is created: the real part will be when the sign is multiplied with $\mathrm{sin} (\Omega t)$, and the imaginary part arises as the multiplication of the signal and $\mathrm{cos} (\Omega t)$.

The real part of the demodulated signal (often called as the "real channel") will be:

\mathrm{sin} \big ( \omega_0 t + \theta_B - \phi_0 \big ) \mathrm{sin} \Big ( ( \omega_0 + \delta \omega ) t \Big ) = \frac{1}{2} \bigg [ \mathrm{cos} \big ( \delta \omega t - \theta_B + \phi_0 \big ) - \mathrm{cos} \Big ( ( 2 \omega_0 + \delta \omega ) t + \theta_b - \phi_0 \Big ) \bigg ](1)

After the multiplication the part with frequency $ 2 \omega_0 + \delta \omega $ is removed with a lowpass filter, and the remaining low-frequency term forms the real channel of the signal.

S_{\mathrm{Re}} \propto \frac{1}{2} \mathrm{cos} \big ( \delta \omega t - \theta_b + \phi_0 \big ) = \frac{1}{2} \Re \Big ( \mathrm{e}^{ - \mathrm{i} \delta \omega t - \mathrm{i} \theta_B + \mathrm{i} \phi_0 } \Big )(2)

The imaginary channel is quite similar. After the multiplication with $\mathrm{cos} \big (\Omega t \big )$ and the lowpass filtering:

S_{\mathrm{Im}} \propto \frac{1}{2} \mathrm{sin} \big ( \delta \omega t - \theta_b + \phi_0 \big ) = \frac{1}{2} \Im \Big ( \mathrm{e}^{ - \mathrm{i} \delta \omega t - \mathrm{i} \theta_B + \mathrm{i} \phi_0 } \Big )(3)


So the detected signal managed as complex is:

S \propto \omega_0 \int  \mathrm{d^3r} \hspace{3pt} \mathrm{e}^{ - \tfrac{t}{T_2 (\mathbf{r})} } M_{\bot} (\mathbf{r} ) \mathcal{B}^{rec}_{\bot} (\mathbf{r}, 0 ) \mathrm{e}^{ - \mathrm{i} \big ( \delta \omega t + \theta_B (\mathbf{r} ) - \phi_0 (\mathbf{r} ) \big ) }(4)

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