Bode plot
In the frequency domain, the cause can be interpreted as due to the sharp (brick-wall) cut-off, and ringing reduced by using a filter with smoother roll-off. This is the case for the Gaussian filter, whose magnitude Bode plot is a downward opening parabola (quadratic roll-off), as its Fourier transform is again a Gaussian, hence (up to scale) formula_4 – taking logarithms yields formula_5