What symmetry does system function H(z) have if h[n] is real?

1. What is the real part of conjugate symmetric signal?
2. What is the Z transform H Z of the impulse response of this filter?

What is the real part of conjugate symmetric signal?

Every real signal, therefore, consists of an equal contribution of positive and negative frequency components.

What is the Z transform H Z of the impulse response of this filter?

The impulse response of the discrete-time filter is expressed by the z transform function as: H ( z ) = 1 − z − 1 1 + 0.5 z − 1.