Sinc function is a sinusoidal activation function in neural networks. In contrast to other common activation functions, it has rises and falls. However, the function saturated and its output converges to zero for large positive and negative inputs.

The definition of the function is sine x over x. Dividing something to zero makes the equation undefined. So, the function would be undefined for x is equal to zero. That’s why, there is an exception point for function definition where x is equal to zero. sinc(0) is defined as value of 1 as an exception.

f(x) = sin(x)/x for x≠0

f(x) = 1 for x=0

So, the function is illustrated below. As seen, its output closes to zero when x increased positively or negatively.

Actually, the function is manipulated version of cosine function. Its output decreases in ratio of the distance from the origin.

Funnily, name of the function comes from cardinal sine. The widest area of the graph resembles the cardinal with a hat.

## Derivative

An activation function can be useful if it is differentiable. Its derivative would be involved in backpropagation while neural networks learn.

As mentioned before, the function has an exception point for x = 0. Derivative of constant value is equal to zero. We need to find the derivative for x ≠ 0.

y = sin(x)/x

Quotient rule says that two differantiable function can be expressed as the following form.

(f/g)’ = (f’g – fg’) / g^{2}

Apply the rule to sinc function.

f'(x) = (sin(x)/x)’ = (sin'(x).x – sin(x).x’)/x^{2} = (cos(x).x – sin(x).1)/x^{2}

Derivative can be expressed in simpler form

f'(x) = cos(x)/x – sin(x)/x^{2}

y = sin(x)/x for x ≠ 0

y = 1 for x = 0

dy/dx = cos(x)/x – sin(x)/x

^{2}, if x ≠ 0dy/dx = 0, if x = 0

So, periodic functions such as sine or cosine are not popular in neural networks as transfer function. Even though sinc is periodic, it would be saturated when input increases positively or negatively just like other common activation functions such as sigmoid or tanh. That’s why, cardinal sine is powerful alternative for activation unit in neural networks.