The causal realization of a system transfer function H( )s having poles at (2, –1), (–2, 1) and zeroes at (2, 1), (–2, –1) will be

The correct option is Unstable, complex, all pass.

To determine the characteristics of the causal realization of the system transfer function $H\left(s\right)$, we analyze the given positions of poles and zeros in the complex s-plane.

The coordinates of poles and zeros are given as (Real, Imaginary):
Poles:
$p_1=2-j$
$p_2=-2+j$
Zeros:
$z_1=2+j$
$z_2=-2-j$

1. Stability Analysis:
A causal system is stable if and only if all of its poles lie in the left half of the s-plane, which means their real parts must be negative.
Here, the pole $p_1=2-j$ has a real part of +2, which lies in the right half of the s-plane. Because there is at least one pole in the right-half plane, the causal realization of the system is unstable.

2. Real vs. Complex Realization:
For a system to have a real impulse response (and thus a real realization), its poles and zeros must occur in complex conjugate pairs.
Let's check the poles:
The complex conjugate of $p_1=2-j$ is $2+j$, but the other pole is $p_2=-2+j$. Since the poles do not form a conjugate pair, the coefficients of the transfer function $H\left(s\right)$ will be complex numbers. Therefore, the system realization is complex.

3. Frequency Response Analysis (All Pass Property):
The transfer function of the system is:
$H\left(s\right)=K\frac{\left(s-z_1\right)\left(s-z_2\right)}{\left(s-p_1\right)\left(s-p_2\right)}=K\frac{\left(s-\left(2+j\right)\right)\left(s-\left(-2-j\right)\right)}{\left(s-\left(2-j\right)\right)\left(s-\left(-2+j\right)\right)}$

To find the frequency response, we substitute $s=j\omega$:
$H\left(j\omega \right)=K\frac{\left(j\omega -2-j\right)\left(j\omega +2+j\right)}{\left(j\omega -2+j\right)\left(j\omega +2-j\right)}=K\frac{\left(-2+j\left(\omega -1\right)\right)\left(2+j\left(\omega +1\right)\right)}{\left(-2+j\left(\omega +1\right)\right)\left(2+j\left(\omega -1\right)\right)}$

Now, we compute the magnitude of $H\left(j\omega \right)$:
$\left|H\left(j\omega \right)\right|=\left|K\right|\frac{\sqrt{{\left(-2\right)}^2+{\left(\omega -1\right)}^2}·\sqrt{2^2+{\left(\omega +1\right)}^2}}{\sqrt{{\left(-2\right)}^2+{\left(\omega +1\right)}^2}·\sqrt{2^2+{\left(\omega -1\right)}^2}}$

Simplifying the terms in the fraction:
$\left|H\left(j\omega \right)\right|=\left|K\right|\frac{\sqrt{4+{\left(\omega -1\right)}^2}·\sqrt{4+{\left(\omega +1\right)}^2}}{\sqrt{4+{\left(\omega +1\right)}^2}·\sqrt{4+{\left(\omega -1\right)}^2}}=\left|K\right|$
Since the magnitude of the transfer function is a constant value independent of frequency $\omega$, the system behaves as an all pass filter.

Combining all three findings, the causal realization of the system is unstable, complex, and all pass.