How to clip a portion of waveform and pull it to zero level?
If I understand you correctly, this is a circuit to solve your problem:







simulate this circuit &ndash; Schematic created using CircuitLab

and here is what the simulated output looks like:

Where blue is the input sawtooth signal, and orange is the output signal of the circuit.

If the input voltage is out of the range of 1.5 V to 2.5 V, the output is 0 V, and if the input voltage is between 1.5 V and 2.5 V, Output Voltage = Input Voltage.
Depending on your background in electronics, this circuit may seem pretty convoluted, so here is what is happening in the circuit:
1) R4 and R5 form a voltage divider with an output of 2.5 V
2) R2 and R3 form a voltage divider with an output of 1.5 V
3) OA8 compares the 2.5 V (from 1)) with the input signal, has a high output if the input signal is greater than 2.5 V
4) OA9 compares the 1.5 V (from 2)) with the input signal, has a high output if the input signal is less than 1.5 V
5) If either OA8 or OA9 have a high output signal, M1 and M2 pull the 'processed signal' node to ground.
6) R1 allows M1 and M2 to pull 'processed signal' to ground without damaging the signal source or M1 and M2  by drawing too much current
7) OA2 is an analog buffer, to lower the output impedance.

If anyone knows why the little bump just after 2s on the transient graph is there, please let me know :)