A system with transfer function H(z) has impulse response h(n) defined as h(2) = 1, h(3) = -1 and h(k) = 0 otherwise. Consider the following statements. S1 : H(z) is a low-pass filter. S2 : H(z) is an FIR filter. Which of the following is correct?

The correct answer is: C. Both S1 and S2 are true, and S2 is a reason for S1.

A low-pass filter is a filter that passes low frequencies and attenuates high frequencies. An FIR filter is a filter whose impulse response is finite in length. The impulse response of the system in the question is $h(2) = 1$, $h(3) = -1$, and $h(k) = 0$ otherwise. This impulse response is finite in length, so the system is an FIR filter. The system also passes low frequencies and attenuates high frequencies, so it is a low-pass filter. Therefore, both S1 and S2 are true, and S2 is a reason for S1.

Here is a brief explanation of each option:

• Option A: Only S2 is true. This is incorrect because S1 is also true.
• Option B: Both S1 and S2 are false. This is incorrect because S1 is true.
• Option C: Both S1 and S2 are true, and S2 is a reason for S1. This is the correct answer.
• Option D: Both S1 and S2 are true, but S2 is not a reason for S1. This is incorrect because S2 is a reason for S1.