RF Sampling Mixer

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A simple passive sampling mixer is shown in Figure 1.
Figure 1. RF sampling mixer
Sample RF signal at much lower frequency (LO frequency need not be near to RF)
RF sampling mixer can be modeled as sampler followed by a ZOH circuit as shown in Figure 2.
A continuous time signal( $v_{RF}(t)$ ) sampled at uniform sampling frequency( $f_s$ ) is given by
(1) $\begin{equation*} v_{s,RF}(t) = v_{RF}(t)p(t) = \sum\limits_{n=-\infty}^{\infty} v_{RF}(nT_s) \delta(t-nT_s) \end{equation*}$

(2) $\begin{equation*} V_{RF}(\omega) = {1\over T_s}\sum\limits_{k=-\infty}^{\infty} V(\omega-k\omega_s) \end{equation*}$
Hold circuit
(3) $\begin{equation*} h(t) = \begin{cases} 1, & -T_h/2 < t < T_h/2 \\ 0, & \mbox{elsewhere} \end{cases} \end{equation*}$

Fourier transform is

(4) $\begin{equation*} H(\omega) = T_h {\sin(\omega T_h/2) \over \omega T_h/2} \end{equation*}$
The output after hold circuit is,
(5) $\begin{eqnarray*} v_{o}(t) &=& v_{s,RF}(t)*h(t) \\ &=& (T_h f_s) {\sin(\omega T_h/2) \over \omega T_h/2} \underbrace{\sum\limits_{k=-\infty}^{\infty} V(\omega-k\omega_s)}_{\text{spectral images}} \end{eqnarray*}$

The spectral images of RF signal are suppressed by the sinc function
At $f_{IF}$ , Conversion gain = $T_h f_s$