The correct answer is: A. h[n] is real for all n.
The pole-zero diagram of a causal and stable discrete-time system is shown in the figure. The zero at the origin has multiplicity 4. The impulse response of the system is h[n]. If h[0] = 1, we can conclude that h[n] is real for all n.
The impulse response of a causal and stable discrete-time system is the inverse Laplace transform of the system’s transfer function. The transfer function of a system with a zero at the origin has a pole at the origin with multiplicity 4. The inverse Laplace transform of a function with a pole at the origin with multiplicity 4 is a polynomial of degree 4 with real coefficients. Therefore, the impulse response of a causal and stable discrete-time system with a zero at the origin has multiplicity 4 is a real polynomial of degree 4.
Since h[0] = 1, the constant term of the impulse response is 1. Therefore, the impulse response of a causal and stable discrete-time system with a zero at the origin has multiplicity 4 is a real polynomial of degree 4 with a constant term of 1. This means that h[n] is real for all n.
Option B is incorrect because the impulse response of a causal and stable discrete-time system with a zero at the origin has multiplicity 4 is a real polynomial of degree 4, not a purely imaginary polynomial.
Option C is incorrect because the impulse response of a causal and stable discrete-time system with a zero at the origin has multiplicity 4 is a real polynomial of degree 4, not a real polynomial for only even n.
Option D is incorrect because the impulse response of a causal and stable discrete-time system with a zero at the origin has multiplicity 4 is a real polynomial of degree 4, not a purely imaginary polynomial for only odd n.