H(z) is a transfer function of a real system. When a signal

H(z) is a transfer function of a real system. When a signal x[n] = (t + j)ⁿ is the input to such a system, the output is zero. Further, the Region Of Convergence (ROC) of
1 - 1 z^-1
2

H(z) is the entire Z-plane (except z = 0). It can then be inferred that H(z) can have a minimum of

1. one pole and one zero
2. one pole and two zero
3. two poles and one zero
4. two poles and two zeros

It will have minimum 1 pole and 2 zeros.

ROC =		1 -	1	z^-1		H(z) =	(2z - 1)	H(z)
			2				2z

∴ H(z) will be in the form of	zⁿ	,
	(2z - 1)

Hence for minimum realization, n = 2

H(z) =	z²	,
	(2z - 1)