i) These are electronic circuits used for discrete time signal processing. It works by moving charges into and out of capacitors when switches are open and closed.

ii) The example of switched capacitor circuit is shown below, where 3 switches control the operation. $S_1$ and $S_3$ connect the left plate of $C_1$ to $V_{in}$ and ground respectively and $S_2$ provides unity gain feedback.

Switched Capacitor Integrator.

• Consider a continuous time integrator,

$V_{out}=\frac{-1}{RC_F}\int V_{in} dt$

• In continuous time, a resistor connected between 2 nodes, carries a current equal to $(V_A-V_B)/R$. See figure (2).

• The role of R is to take certain amount of charge from node (A) every second and move it to node (B).

• Suppose in the circuit(fig:(3)), capacitor $C_S$ is alternately connected to nodes A & B at clock rate $f_{clk}$, the average current flowing from A to B is then equal to the charge moved in 1 clock period.

• Let us now replace resistor R in fig(1) by its discrete time equivalent, arriving at the integrator of fig:4

• In every clock cycle, $C_S$ absorbs a change equal to $C_S\,\,V_{in}$ when $S_1$ -> ON and deposits charge on $C_F$ when $S_2$ -> ON.

• If $V_{in}$ is constant, output by $V_{in}$ $C_S/C_F$ every clock cycle (fig:4) and we obtain a staircase w/f, we note that the circuit behaves like integrator.

• The final value of $V_{out}$ in fig:4, after every clock cycle ca be written as

  $V_{out}(KT_{clk})=V_{out}[(K-1)T_{ck}]-V_{in}[(K-1)T_{ck}]\frac{C_S}{C_F}$