Can we use both input and output at the Gate, if we use common source transistor

I am designing low noise amplifier using common source transistor. My Question is, can Common source transistors Gate be used as both input and output. Is it possible for common source transistor.

I am not using single transistor GATE for connecting both input and output at the transistor. I am using different common source transistors for giving input at the Gate and taking output at the Gate.

I want to connect the common source transistors Gate to the input and output as shown in the screen shot circuit

Your input is at the source of M1, not the gate.
Your output is at the source of M3, not the gate.
M1 and M2 are a variation of a Cascode amplifier.

M1 and M2 are common-gate amplifiers.
M3 is a common drain (source-follower) amplifier.

I am so sorry. I should have mentioned that can we give input at the source and take output at the source for the common source transistor.

Example diagram as shown in the posting # 1. Can we do like that for the common source transistor. Usually Common transistor input is at the gate and output at the drain. My doubt is, can we give input at the source and take output at the source.

Then the output is also the input and the transistor is not needed.

You make no sense. Your circuit does not have a common source transistor! In a common source transistor the source pin is usually grounded.
Your schematic shows the input at the source pin of M1 and the output at the source pin of M3.