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A low-pass filter is a filter that passes low frequencies well, but attenuates (or reduces) frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter when used in audio applications. A high-pass filter is the opposite, and a bandpass filter is a combination of a high-pass and a low-pass. The concept of a low-pass filter exists in many different forms, including electronic circuits (like a hiss filter used in audio), digital algorithms for smoothing sets of data, acoustic barriers, blurring of images, and so on. Low-pass filters play the same role in signal processing that moving averages do in some other fields, such as finance; both tools provide a smoother form of a signal which removes the short-term oscillations, leaving only the long-term trend.
Examples of low-pass filters
Ideal and real filters An ideal low-pass filter completely eliminates all frequencies above the cut-off frequency while passing those below unchanged. The transition region present in practical filters does not exist. An ideal low pass filter can be realized mathematically (theoretically) by multiplying a signal by the rectangular function in the frequency domain or, equivalently, convolution with a sinc function in the time domain. However, this filter is not realizable for practical, real signals because the sinc function extends to infinity. The filter would therefore need to predict the future and have infinite knowledge of the past in order to perform the convolution. It is effectively realizable for pre-recorded digital signals (by padding the ends of the signal with zeros to the point that the error after filtering is less than the quantization error), or perfectly cyclic signals that repeat for infinity. Real filters for real-time applications approximate the ideal filter by delaying the signal for a small period of time, allowing them to "see" a little bit into the future. This is manifested as phase shift. Greater accuracy in approximation requires a longer delay. The Nyquist–Shannon sampling theorem describes how to use a perfect low-pass filter (equivalent to the Whittaker–Shannon interpolation formula) to reconstruct a continuous signal from a sampled digital signal. Real digital-to-analog converters use real filter approximations. Electronic low-pass filters There are a great many different types of filter circuits, with different responses to changing frequency. The frequency response of a filter is generally represented using a Bode plot. On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upper-left (the asymptotes of the function), they will intersect at exactly the "cutoff frequency". The frequency response at the cutoff frequency in a first-order filter is −3 dB below the horizontal line. The various types of filters — Butterworth filter, Chebyshev filter, Bessel filter, etc. — all have different-looking "knee curves". Many second-order filters are designed to have "peaking" or resonance, causing their frequency response at the cutoff frequency to be above the horizontal line. See electronic filter for other types. The meanings of 'low' and 'high' — i.e. the cutoff frequency — depend on the characteristics of the filter. (The term "low-pass filter" merely refers to the shape of the filter's response. A high-pass filter could be built that cuts off at a lower frequency than any low-pass filter. It is their responses that set them apart.) Electronic circuits can be devised for any desired frequency range, right up through microwave frequencies (above 1000 MHz) and higher. Passive electronic realization
Passive digital realization A model of a simple digital implementation of a low-pass RC filter is: where: is the current output value is the previous output value input value is the time interval between samples is the time constant Active electronic realization Another type of electrical circuit is an active low-pass filter. In this example, the cutoff frequency (in hertz) is defined as: or equivalently (in radians per second): The gain in the passband is , and the stopband drops off at −6 dB per octave, as it is a first-order filter. Many times, a simple gain or attenuation amplifier (See operational amplifier) is turned into a lowpass filter by adding the capacitor C. This decreases the frequency response at high frequencies and helps to avoid oscillation in the amplifier. For example, an audio amplifier can be made into a lowpass filter with cutoff frequency 100 kHz to reduce gain at frequencies which would otherwise oscillate. Since the audio band (what we can hear) only goes up to 20 kHz or so, the frequencies of interest fall entirely in the passband, and the amplifier behaves the same way as far as audio is concerned. See also | ||||||||||||
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