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.
Introduction
The main cause of ringing artifacts is due to a signal being bandlimited (specifically, not having high frequencies) or passed through a low-pass filter; this is the frequency domain description. In terms of the time domain, the cause of this type of ringing is the ripples in the sinc function, which is the impulse response (time domain representation) of a perfect low-pass filter. Mathematically, this is called the Gibbs phenomenon. One may distinguish overshoot (and undershoot), which occurs when transitions are accentuated – the output is higher than the input – from ringing, where after an overshoot, the signal overcorrects and is now below the target value; these phenomena often occur together, and are thus often conflated and jointly referred to as "ringing". The term "ringing" is most often used for ripples in the time domain, though it is also sometimes used for frequency domain effects: windowing a filter in the time domain by a rectangular function causes ripples in the frequency domain for the same reason as a brick-wall low pass filter (rectangular function in the frequency domain) causes ripples in the time domain, in each case the Fourier transform of the rectangular function being the sinc function. There are related artifacts caused by other frequency domain effects, and similar artifacts due to unrelated causes.
Causes
Description
By definition, ringing occurs when a non-oscillating input yields an oscillating output: formally, when an input signal which is monotonic on an interval has output response which is not monotonic. This occurs most severely when the impulse response or step response of a filter has oscillations – less formally, if for a spike input, respectively a step input (a sharp transition), the output has bumps. Ringing most commonly refers to step ringing, and that will be the focus. Ringing is closely related to overshoot and undershoot, which is when the output takes on values higher than the maximum (respectively, lower than the minimum) input value: one can have one without the other, but in important cases, such as a low-pass filter, one first has overshoot, then the response bounces back below the steady-state level, causing the first ring, and then oscillates back and forth above and below the steady-state level. Thus overshoot is the first step of the phenomenon, while ringing is the second and subsequent steps. Due to this close connection, the terms are often conflated, with "ringing" referring to both the initial overshoot and the subsequent rings. If one has a linear time invariant (LTI) filter, then one can understand the filter and ringing in terms of the impulse response (the time domain view), or in terms of its Fourier transform, the frequency response (the frequency domain view). Ringing is a time domain artifact, and in filter design is traded off with desired frequency domain characteristics: the desired frequency response may cause ringing, while reducing or eliminating ringing may worsen the frequency response.
sinc filter
The central example, and often what is meant by "ringing artifacts", is the ideal (brick-wall) low-pass filter, the sinc filter. This has an oscillatory impulse response function, as illustrated above, and the step response – its integral, the sine integral – thus also features oscillations, as illustrated at right. These ringing artifacts are not results of imperfect implementation or windowing: the ideal low-pass filter, while possessing the desired frequency response, necessarily causes ringing artifacts in the time domain.
Time domain In terms of impulse response, the correspondence between these artifacts and the behavior of the function is as follows:
impulse undershoot is equivalent to the impulse response having negative values, impulse ringing (ringing near a point) is precisely equivalent to the impulse response having oscillations, which is equivalent to the derivative of the impulse response alternating between negative and positive values, and there is no notion of impulse overshoot, as the unit impulse is assumed to have infinite height (and integral 1 – a Dirac delta function), and thus cannot be overshot. Turning to step response, the step response is the integral of the impulse response; formally, the value of the step response at time a is the integral ∫ − ∞ a {\displaystyle \int _{-\infty }^{a}} of the impulse response. Thus values of the step response can be understood in terms of tail integrals of the impulse response. Assume that the overall integral of the impulse response is 1, so it sends constant input to the same constant as output – otherwise the filter has gain, and scaling by gain gives an integral of 1.
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