Nonuniqueness of the integration operator

Integration can be thought of as $1/(-i\omega)$.The implied division by zero at $\omega =0$ warns us that this filter is not quite normal. For example, $1/(-i\omega)$appears to be an imaginary, antisymmetric function of $\omega$.This implies that the time function is the real antisymmetric signum function, namely, $\sgn(t)= t / \vert t\vert$.The signum is not usually thought of as an integration operator, but by adding a constant we have a step function, and that is causal integration. By subtracting a constant we have anticausal integration. We can play games with the constant because it is at zero frequency that the definition contains zero division.

EXERCISES:

1. Show that the mean of the input of leaky() is the mean of the output, which demonstrates that the gain of the filter is unity at zero frequency.