Q$1.(10$ Marks$)$

Consider the signal$-$flow graph of the perceptron shown in the above figure. The

activation function, $(v),$ where $v$ is the induced local field, can be designed by the

user. If the activation function is chosen as hard limiter $($i$.$e$.$ step function$),$ then it

becomes the classical perceptron, and the decision boundary is shown to be a

hyperplane. In this problem, let's explore other choices of the activation function, and

its effect on the decision boundary. Let's assume that the classification decision made

by the perceptron is simply a threshold rule defined as follows:

where $$ is a user$-$defined threshold; otherwise, $x$ belongs to class $C_2.$

Consider the following three choices of activation function:

The activation function is a quadratic function: $(v)=(v-a{)}^2+c$;

The activation function is the hyperbolic tangent function: $(v)=\frac{1-e^{-v}}{1+e^{-v}}$;

The activation function is the Bell$-$shaped Gaussian function: $(v)=e^{-\frac{(v-m{)}^2}{2}}.$

For each case, investigate whether the resulting decision boundary is a hyper$-$plane or

not.