Information AboutLow Pass |
| CATEGORIES ABOUT LOW-PASS FILTER | |
| digital signal processing | |
| linear filters | |
| synthesiser modules | |
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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 Average s 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 realized by an RC Circuit ]] Acoustic A stiff physical barrier tends to reflect higher sound frequencies, and so acts as a low-pass filter for transmitting sound. When music is playing in another room, the low notes are easily heard, while the high notes are attenuated. Electronic Electronic low-pass filters are used to drive Subwoofer s and other types of Loudspeaker s, to block high pitches that they can't efficiently broadcast. Radio transmitters use lowpass filters to block Harmonic emissions which might cause interference with other communications. An Integrator is another example of a low-pass filter. DSL Splitter s use low-pass and High-pass filters to separate DSL and POTS signals sharing the same Pair of wires. Low-pass filters also play a significant role in the sculpting of sound for Electronic Music as created by analogue Synthesiser s. ''See Subtractive Synthesis .'' 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 in an ideal filter. 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, the ideal filter is impossible to realize without also having signals of infinite extent, and so generally needs to be approximated for real ongoing signals, because the sinc function's support region extends to all past and future times. The filter would therefore need to have infinite delay, or knowledge of the infinite future and past, in order to perform the convolution. It is effectively realizable for pre-recorded digital signals by assuming extensions of zero into the past and future, but even that is not typically practical. Real filters for Real-time applications approximate the ideal filter by truncating and Windowing the infinite impulse response to make a Finite Impulse Response ; applying that filter requires delaying the signal for a moderate period of time, allowing the computation to "see" a little bit into the future. This delay is manifested as Phase Shift . Greater accuracy in approximation requires a longer delay. The Whittaker–Shannon Interpolation Formula describes how to use a perfect low-pass filter to reconstruct a Continuous Signal from a sampled Digital Signal . Real Digital-to-analog Converter s 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 Asymptote s 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' — that is, 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 One simple Electrical Circuit that will serve as a low-pass filter consists of a Resistor in series with a Load , and a Capacitor in parallel with the load. The capacitor exhibits Reactance , and blocks low-frequency signals, causing them to go through the load instead. At higher frequencies the reactance drops, and the capacitor effectively functions as a short circuit. The combination of resistance and capacitance gives you the Time Constant of the filter (represented by the Greek letter Tau ). The break frequency, also called the turnover frequency or cutoff frequency (in hertz), is determined by the time constant: or equivalently (in Radians per second): One way to understand this circuit is to focus on the time the capacitor takes to charge. It takes time to charge or discharge the capacitor through that resistor:
Another way to understand this circuit is with the idea of Reactance at a particular frequency:
It should be noted that the capacitor is not an "on/off" object (like the block or pass fluidic explanation above). The capacitor will variably act between these two extremes. It is the Bode Plot and Frequency Response that show this variability. 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. Laplace notation Continuous-time filters can also be described in terms of the Laplace Transform of their impulse response in a way that allows all of the characteristics of the filter to be easily analyzed by considering the pattern of poles and zeros of the Laplace transform in the complex plane (in discrete time, one can similarly consider the Z-transform of the impulse response). A first-order low-pass filter can be described in Laplace notation as : where ''s'' is the Laplace transform variable and ''τ'' is the filter Time Constant . SEE ALSO
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