Gilbert cell:

i) Small signal gain of differential pair is a function of tail current.

ii) Fig(a) shows a differential pair whose gain can be varied by a control voltage.

iii) Control Voltage varies, tail current changes, therefore gain changes.

iv) In this circuit, $A_v=V_{out}/V_{in}$ varies from zero (if $I_{D_{3}}=0$) to a maximum value given by control and device dimensions.

v) Now, consider 2 differential pair, as per fig (b)

We have, $\frac{V_{out_{1}}}{V_{in}}=-g_m\,R_D$ and $\frac{V_{out_{2}}}{V_{in}}=+g_m\,R_D$

vi) Now, $V_{out_{1}}$ and $V_{out_{2}}$ can be combined, $I_1\,\,,\,\,I_2$ can be replaced by MOSFET's $M_5$ and $M_6$ controlled by control voltages.

vii) In the circuit $V_{const}$ varies $I_1\,,\,I_2$ in opposite direction such that the gain of the amplifier changes monotonically. This is known as "Gilbert cell".