Filter_(signal_processing)

# Filter (signal processing) In signal processing, a **filter** is a device or process that suppresses, passes, or otherwise reshapes chosen frequencies or other features of a signal while leaving the rest largely intact. Most filtering targets frequency content, such as removing hum from an audio recording, or picking one radio channel out of the whole band arriving at an antenna, but the same idea applies wherever a wanted feature can be separated mathematically from an unwanted one, including outside the frequency domain entirely. Filters are sorted along several independent axes at once: linear or nonlinear, time-invariant or not, continuous-time or discrete-time, and analog or digital, with the last distinction driving most of the practical choice between a circuit built from real components and an algorithm running on sampled numbers. The primary microsim on this page puts one composite test signal through all four classic filter shapes at once, letting the reader drag the cutoff frequency and the order and watch the same edit happen twice, once to a trace in time and once to the spectrum that explains it. The oldest and still most transparent filters are built from resistors, capacitors, and inductors, or from mechanical and acoustic resonators that play the same role at frequencies a circuit's own parasitic reactance cannot reach cleanly. Digital arithmetic on sampled numbers has since taken over most filtering that does not need to happen at radio frequency, because a difference equation is exact and reprogrammable in a way no physical resonator is. Both routes converge on the same description, a transfer function relating a filter's output to its input frequency by frequency, and both are judged by the same trade-off between how sharply a filter separates what it keeps from what it discards and how much it costs to build. ## Linear continuous-time filters A continuous-time filter acts on a signal defined at every instant rather than at discrete samples, and its input–output relationship is a linear differential equation or, equivalently, a convolution with the filter's [[Impulse_response|impulse response]]. When the equation's coefficients do not change with time, the system is [[Linear_time-invariant_system|linear and time-invariant]], the assumption behind nearly all classical filter theory: a sinusoid presented at the input produces a sinusoid at the output, scaled and phase-shifted but never changed in frequency. The simplest physical example is a single resistor and capacitor wired as a [[Low-pass_filter|low-pass filter]], with response `H(f) = 1/(1 + jf/f_c)` and corner frequency `f_c = 1/(2*pi*R*C)`. Because `H(0) = 1`, such a filter cannot touch a signal's average value: feeding it a periodic pulse train and sweeping the corner from 100 Hz through 1 kHz to 10 kHz rounds the pulse edges by very different amounts, yet the output's mean stays locked to the input's mean at every one of the three settings.[^john119][^john146] ### Terminology Filters are described with a common vocabulary regardless of how they are built. The passband is the range of frequencies a filter is meant to leave alone, the stopband is the range it is meant to remove, and the transition band is the unavoidable stretch between them where the response is neither. A filter's order sets how quickly it moves from one band to the other: an idealized "brick-wall" response would fall in zero bandwidth, but every real filter trades a wider transition for a lower order, and the primary microsim's own Butterworth-style prototype, `H(f) = 1/(1 + (f/f_c)^(2n))`, shows the mechanism directly — raising the order slider n steepens the shoulder around the cutoff without moving the cutoff itself. Passband ripple and stopband attenuation describe how flat, or how deep, those two regions actually are, and insertion loss is the fraction of signal power a real, passive filter removes even inside its own passband. ## Technologies Before an algorithm can filter a sampled sequence of numbers, most twentieth-century filtering was done by a physical resonator chosen for the frequency range and the quality factor it could reach. Lumped inductors and capacitors serve from audio up to a few hundred megahertz; above that, the parasitic reactance built into the components themselves competes with the response they are supposed to produce, and other technologies take over. ### Digital filters A [[Digital_filter|digital filter]] replaces the resonator with arithmetic: an [[Analog-to-digital_converter|analog-to-digital converter]] samples the incoming waveform, a processor evaluates a difference equation on the resulting numbers, and, if the result must return to the analog world, a [[Digital-to-analog_converter|digital-to-analog converter]] reconstructs a voltage from them. Because the same hardware can be reprogrammed to a different response in software, digital filters have displaced analog ones wherever [[Real-time_computing|real-time]] computation is cheap enough to keep pace with the sample rate. ### Quartz filters and piezoelectrics A quartz crystal exploits [[Piezoelectricity|piezoelectricity]] — the same mechanical deformation that runs a crystal oscillator — to resonate at a sharply defined mechanical frequency with very little energy loss. Several such crystals coupled as [[Transducer|transducers]] give a bandpass filter far narrower and more stable than an equivalent arrangement of coils and capacitors, at the cost of being fixed to the one frequency the crystal is cut for. ### SAW filters A surface acoustic wave (SAW) filter converts an electrical signal into a mechanical wave that travels across the surface of a piezoelectric substrate between patterned electrodes, and back again; the spacing of the electrodes sets the passband directly. SAW filters cover the range from hundreds of megahertz to a few gigahertz used by television tuners and mobile handsets, in a package a few millimetres across. ### BAW filters A bulk acoustic wave (BAW) filter drives the same piezoelectric effect through the thickness of a thin suspended film rather than along its surface, which pushes the usable resonant frequency higher still. BAW devices are common in the radio-frequency front end of a modern smartphone, where several must sit close together without interfering. ### Garnet filters A garnet filter tunes a sphere of yttrium iron garnet, a ferrimagnetic crystal of [[Yttrium|yttrium]], [[Iron|iron]] and oxygen, by varying the strength of a surrounding electromagnet; the resonant frequency follows the applied field over an unusually wide range at a very high quality factor. That tunability makes garnet filters common in microwave spectrum analysers and signal generators, at a size and cost that rule out cheaper mass-market equipment. ### Atomic filters The narrowest filter available is not built from a circuit at all: a vapour of [[Caesium|caesium]] atoms absorbs and re-emits light, or a beam of the same atoms shifts phase, only within the extremely narrow linewidth of an atomic transition. Locking an oscillator to that transition is how atomic clocks define a frequency standard far more stable than any resonator made of bulk material.[^citeneeded1] ## The transfer function Whatever the technology, a linear filter is completely described by its [[Transfer_function|transfer function]] `H`, the ratio of output to input at each frequency once every transient has died away. Its magnitude sets how much a component at a given frequency is scaled, and its phase sets how much that component is delayed relative to the others — a distinction that matters whenever a signal's shape, and not only its energy, needs to survive the filter. The poles and zeros of `H`, found from the [[Z-transform|z-transform]] for a discrete-time filter or the [[Laplace_transform|Laplace transform]] for a continuous one, summarise the entire response in a handful of numbers, since the response at any frequency is fixed once the location of every pole and zero is fixed.[^stib072] ### Classification Filters are sorted along several independent axes. By passband shape they are low-pass, high-pass, band-pass or band-stop, the same four choices the primary microsim's dropdown offers. By how the output is built from the input they are [[Finite_impulse_response|finite impulse response]], with no feedback and a response that dies out after a fixed number of samples, or [[Infinite_impulse_response|infinite impulse response]], which reuses past outputs and can ring indefinitely.[^stib076] Beside these frequency-selective filters sits a second family, also called filters because each extracts a wanted signal from measurements corrupted by noise using statistics rather than a fixed passband: the [[Kalman_filter|Kalman filter]], the [[Wiener_filter|Wiener filter]] and the [[Particle_filter|particle filter]] among them. A filter is finally either time-invariant, responding the same way regardless of when a signal arrives, or time-variant, deliberately or otherwise. ## Impedance matching A passive filter built from inductors and capacitors is designed for a specific source and load impedance, commonly 50 ohms in radio-frequency work, and its measured response departs from the design the moment either end presents a different impedance than assumed. Reflections at a mismatched boundary can be read directly off a [[Smith_chart|Smith chart]], the standard graphical tool for translating an impedance into a reflection coefficient and back, and the same reflections turn a supposedly matched [[Transmission_line|transmission line]] feeding the filter into a source of passband ripple that no adjustment of the filter's own components will remove. Impedance-matching networks — simple L, T or pi arrangements of reactive parts — are therefore treated as part of the filter design problem rather than as an afterthought, particularly in narrowband radio-frequency filters where the matching network and the filter's own reactances are folded into a single design. ## Some filters for specific purposes Many filters exist to serve one named job rather than to realise a textbook response shape. An anti-aliasing filter ahead of an [[Analog-to-digital_converter|analog-to-digital converter]] removes energy above half the sampling rate so that it cannot fold back as a false low frequency; a reconstruction filter after a [[Digital-to-analog_converter|digital-to-analog converter]] smooths the staircase the converter produces back into a continuous waveform. A [[Matched_filter|matched filter]], shaped to the exact form of a known pulse, gives the best possible detection of that pulse in noise and sits at the centre of [[Radar|radar]] and [[Sonar|sonar]] receivers. ### Filters for removing noise from data Filtering a recorded dataset for noise does not require a single cutoff frequency at all. A moving-average filter — an equally weighted finite impulse response filter — smooths a noisy sequence by trading resolution for a reduction in random fluctuation, and the same idea generalises to the [[Wiener_filter|Wiener filter]], which sets each frequency's gain from the known statistics of the signal and the noise rather than from a fixed passband. Where the process being measured is understood well enough to write as a state that evolves in time, a [[Kalman_filter|Kalman filter]] can outperform any fixed frequency response by updating its estimate from a physical model as well as from the newest reading — the toolkit used to track anything from a spacecraft's position to a noisy sensor. A bank of parallel bandpass filters splitting one signal into adjacent sub-bands is a [[Filter_bank|filter bank]], the structure behind a graphic equaliser and much of audio and image compression. ## Microsims The primary microsim, *Filter (signal processing)*, puts one composite test signal through all four classic filter shapes at once. A dropdown chooses low-pass, high-pass, band-pass or band-stop; three sliders set the cutoff frequency f_c (1–200 Hz), the bandwidth used by the two band-shaped modes (5–100 Hz), and the order n (1–8), which controls how steep the roll-off is, following the sketch's own prototype `H(f) = 1/(1 + (f/f_c)^(2n))`. Four further sliders build the test signal from four component sinusoids f1–f4 (1–120 Hz each). The upper panel overlays the unfiltered (cyan) and filtered (orange) waveform in the time domain; the lower panel plots the same two signals' spectra together with the filter's own |H(f)| traced in yellow, so that a component's height in the spectrum and its position relative to the shaded passband explain, frequency by frequency, why it survives or disappears from the trace above. A three.js companion, built around the same transfer-function idea, renders the four filter classes as one system viewed three ways, in more depth than is shown here. *Try:* Switch the dropdown to band-pass, pull the bandwidth slider down toward 5 Hz, and watch every component but the one sitting inside the shaded passband disappear from the orange trace. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Filter_(signal_processing)) : [Wikitube](https://en.wikitube.io/wiki/Filter_(signal_processing)) Skeleton mirrored at revision 1366575707. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Digital_filter]] - [[Low-pass_filter]] - [[Filter_bank]] - [[Linear_time-invariant_system]] - [[Finite_impulse_response]] - [[Infinite_impulse_response]] - [[Transfer_function]] ## References The general algebra of linear systems used above — transfer functions, poles and zeros, and the classification of filters by order and by feedback — is standard textbook material and is not separately footnoted beyond the figures below, per the Wikitube style guide §6.1. [^john119]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, p. 119 (PDF page). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1. CC BY. [^john146]: Johnson, D. *Fundamentals of Electrical Engineering I*. 2014, p. 146 (PDF page). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/fundamentals-of-electrical-engineering-1. CC BY. [^stib072]: Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain*. 2020, p. 72 (PDF page). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain. CC BY-SA. [^stib076]: Stiber, M.; Stiber, B.; Larson, E. *Signal Computing: Digital Signals in the Software Domain*. 2020, p. 76 (PDF page). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/signal-computing-digital-signals-in-the-software-domain. CC BY-SA. [^citeneeded1]: Citation needed: manufacturer datasheets or the relevant IEC/EIA standards for the frequency ranges and package sizes quoted for quartz, SAW, BAW, garnet and atomic-reference filters in "Technologies"; no single manufacturer or standard is asserted here.