Ringing artifacts

In signal processing, ringing artifacts are artifacts that appear as spurious signals near sharp transitions in a signal. In digital image processing, they appear as bands or "ghosts" near edges; audibly, they appear as "echos" near transients, particularly sounds from percussion instruments; most noticeable are the pre-echoes. The term "ringing" is because the output signal oscillates at a fading rate around a sharp transition in the input, similar to a bell after being struck. As with other artifacts, their minimization is a criterion in filter design.

In signal processing, ringing artifacts are artifacts that appear as spurious signals near sharp transitions in a signal. In digital image processing, they appear as bands or "ghosts" near edges; audibly, they appear as "echos" near transients, particularly sounds from percussion instruments; most noticeable are the pre-echoes. The term "ringing" is because the output signal oscillates at a fading rate around a sharp transition in the input, similar to a bell after being struck. As with other artifacts, their minimization is a criterion in filter design. In fact, if one takes a brick wall low-pass filter (sinc in time domain, rectangular in frequency domain) and truncates it (multiplies with a rectangular function in the time domain), this convolves the frequency domain with sinc (Fourier transform of the rectangular function) and causes ringing in the frequency domain, which is referred to as ripple. In symbols, \mathcal{F}(\mathrm{sinc}\cdot \mathrm{rect}) = \mathrm{rect} * \mathrm{sinc}. The frequency ringing in the stopband is also referred to as side lobes. Flat response in the passband is desirable, so one windows with functions whose Fourier transform has fewer oscillations, so the frequency domain behavior is better. Multiplication in the time domain corresponds to convolution in the frequency domain, so multiplying a filter by a window function corresponds to convolving the Fourier transform of the original filter by the Fourier transform of the window, which has a smoothing effect – thus windowing in the time domain corresponds to smoothing in the frequency domain, and reduces or eliminates overshoot and ringing. 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) e^{-x^2} – taking logarithms yields -x^2. In electronic filters, the trade-off between frequency domain response and time domain ringing artifacts is well-illustrated by the Butterworth filter: the frequency response of a Butterworth filter slopes down linearly on the log scale, with a first-order filter having slope of −6 dB per octave, a second-order filter –12 dB per octave, and an nth order filter having slope of -6n dB per octave – in the limit, this approaches a brick-wall filter. Thus, among these the, first-order filter rolls off slowest, and hence exhibits the fewest time domain artifacts, but leaks the most in the stopband, while as order increases, the leakage decreases, but artifacts increase.

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