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Low-pass filter

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Type of signal filter

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Alow-pass filter is afilter that passessignals with afrequency lower than a selectedcutoff frequency andattenuates signals with frequencies higher than the cutoff frequency. The exactfrequency response of the filter depends on thefilter design. The filter is sometimes called ahigh-cut filter, ortreble-cut filter in audio applications. A low-pass filter is the complement of ahigh-pass filter.

In optics,high-pass andlow-pass may have different meanings, depending on whether referring to the frequency or wavelength of light, since these variables are inversely related. High-pass frequency filters would act as low-pass wavelength filters, and vice versa. For this reason, it is a good practice to refer to wavelength filters asshort-pass andlong-pass to avoid confusion, which would correspond tohigh-pass andlow-pass frequencies.^[1]

Low-pass filters exist in many different forms, including electronic circuits such as ahiss filter used inaudio,anti-aliasing filters for conditioning signals beforeanalog-to-digital conversion,digital filters for smoothing sets of data, acoustic barriers,blurring of images, and so on. Themoving average operation used in fields such as finance is a particular kind of low-pass filter and can be analyzed with the samesignal processing techniques as are used for other low-pass filters. Low-pass filters provide a smoother form of a signal, removing the short-term fluctuations and leaving the longer-term trend.

Filter designers will often use the low-pass form as aprototype filter. That is a filter with unity bandwidth and impedance. The desired filter is obtained from the prototype by scaling for the desired bandwidth and impedance and transforming into the desired bandform (that is, low-pass, high-pass,band-pass orband-stop).

Examples

Examples of low-pass filters occur inacoustics,optics andelectronics.

A stiff physical barrier tends to reflect higher sound frequencies, acting as an acoustic 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.

Anoptical filter with the same function can correctly be called a low-pass filter, but conventionally is called alongpass filter (low frequency is long wavelength), to avoid confusion.^[1]

In an electronic low-passRC filter for voltage signals, high frequencies in the input signal are attenuated, but the filter has little attenuation below thecutoff frequency determined by itsRC time constant. For current signals, a similar circuit, using a resistor and capacitor inparallel, works in a similar manner. (Seecurrent divider discussed in more detailbelow.)

Electronic low-pass filters are used on inputs tosubwoofers and other types ofloudspeakers, to block high pitches that they cannot efficiently reproduce. Radio transmitters use low-pass filters to blockharmonic emissions that might interfere with other communications. The tone knob on manyelectric guitars is a low-pass filter used to reduce the amount of treble in the sound. Anintegrator is anothertime constant low-pass filter.^[2]

Telephone lines fitted withDSL splitters use low-pass filters to separateDSL fromPOTS signals (andhigh-pass vice versa), which share the samepair of wires (transmission channel).^[3]^[4]

Low-pass filters also play a significant role in the sculpting of sound created by analogue and virtual analoguesynthesisers.Seesubtractive synthesis.

A low-pass filter is used as ananti-aliasing filter beforesampling and forreconstruction indigital-to-analog conversion.

Ideal and real filters

Thesinc function, the time-domainimpulse response of an ideal low-pass filter. The ripples of a true sinc extend infinitely to the left and right while getting smaller and smaller, but this particular graph is truncated.

The gain-magnitude frequency response of a first-order (one-pole) low-pass filter.Power gain is shown in decibels (i.e., a 3dB decline reflects an additional ${\textstyle 1/{\sqrt {2}}}$ attenuation).Angular frequency is shown on a logarithmic scale in units of radians per second.

{\textstyle 1/{\sqrt {2}}} — The gain-magnitude frequency response of a first-order (one-pole) low-pass filter.Power gain is shown in decibels (i.e., a 3dB decline reflects an additional ${\textstyle 1/{\sqrt {2}}}$ attenuation).Angular frequency is shown on a logarithmic scale in units of radians per second.

Anideal low-pass filter completely eliminates all frequencies above thecutoff frequency while passing those below unchanged; itsfrequency response is arectangular function and is abrick-wall filter. 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 itsimpulse response, asinc function, in the time domain.

However, the ideal filter is impossible to realize without also having signals of infinite extent in time, 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, to perform the convolution. It is effectively realizable for pre-recorded digital signals by assuming extensions of zero into the past and future, or, more typically, by making the signal repetitive and using Fourier analysis.

Real filters forreal-time applications approximate the ideal filter by truncating andwindowing the infinite impulse response to make afinite 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 asphase shift. Greater accuracy in approximation requires a longer delay.

Truncating an ideal low-pass filter result inringing artifacts via theGibbs phenomenon, which can be reduced or worsened by the choice of windowing function.Design and choice of real filters involves understanding and minimizing these artifacts. For example, simple truncation of the sinc function will create severe ringing artifacts, which can be reduced using window functions that drop off more smoothly at the edges.^[5]

Time response

The time response of a low-pass filter is found by solving the response to the simple low-pass RC filter.

A simple low-passRC filter

UsingKirchhoff's Laws we arrive at the differential equation^[6]

v_{\text{out}}(t)=v_{\text{in}}(t)-RC{\frac {\operatorname {d} v_{\text{out}}}{\operatorname {d} t}}

Step input response example

If we let $v_{\text{in}}(t)$ be a step function of magnitude $V_{i}$ then the differential equation has the solution^[7]

v_{\text{out}}(t)=V_{i}(1-e^{-\omega _{0}t}),

where $\omega _{0}={1 \over RC}$ is the cutoff frequency of the filter.

Frequency response

The most common way to characterize the frequency response of a circuit is to find its Laplace transform^[6] transfer function, $H(s)={V_{\rm {out}}(s) \over V_{\rm {in}}(s)}$ . Taking the Laplace transform of our differential equation and solving for $H(s)$ we get

H(s)={V_{\rm {out}}(s) \over V_{\rm {in}}(s)}={\omega _{0} \over s+\omega _{0}}

Difference equation through discrete time sampling

A discretedifference equation is easily obtained by sampling the step input response above at regular intervals of $n T {\displaystyle nT}$ where $n=0,1,...$ and $T {\displaystyle T}$ is the time between samples. Taking the difference between two consecutive samples we have

v_{\rm {out}}(nT)-v_{\rm {out}}((n-1)T)=V_{i}(1-e^{-\omega _{0}nT})-V_{i}(1-e^{-\omega _{0}((n-1)T)})

Solving for $v_{\rm {out}}(nT)$ we get

v_{\rm {out}}(nT)=\beta v_{\rm {out}}((n-1)T)+(1-\beta )V_{i}

Where $\beta =e^{-\omega _{0}T}$

Using the notation $V_{n}=v_{\rm {out}}(nT)$ and $v_{n}=v_{\rm {in}}(nT)$ , and substituting our sampled value, $v_{n}=V_{i}$ , we get the difference equation

V_{n}=\beta V_{n-1}+(1-\beta )v_{n}

Error analysis

Comparing the reconstructed output signal from the difference equation, $V_{n}=\beta V_{n-1}+(1-\beta )v_{n}$ , to the step input response, $v_{\text{out}}(t)=V_{i}(1-e^{-\omega _{0}t})$ , we find that there is an exact reconstruction (0% error). This is the reconstructed output for a time-invariant input. However, if the input istime variant, such as $v_{\text{in}}(t)=V_{i}\sin(\omega t)$ , this model approximates the input signal as a series of step functions with duration $T {\displaystyle T}$ producing an error in the reconstructed output signal. The error produced fromtime variant inputs is difficult to quantify^{[citation needed]} but decreases as $T\rightarrow 0$ .

Discrete-time realization

For another method of conversion from continuous- to discrete-time, seeBilinear transform.

Manydigital filters are designed to give low-pass characteristics. Bothinfinite impulse response andfinite impulse response low pass filters, as well as filters usingFourier transforms, are widely used.

Simple infinite impulse response filter

The effect of an infinite impulse response low-pass filter can be simulated on a computer by analyzing an RC filter's behavior in the time domain, and thendiscretizing the model.

From the circuit diagram to the right, according toKirchhoff's Laws and the definition ofcapacitance:

v_{\text{in}}(t)-v_{\text{out}}(t)=R\;i(t)

V

Q_{c}(t)=C\,v_{\text{out}}(t)

Q

i(t)={\frac {\operatorname {d} Q_{c}}{\operatorname {d} t}}

I

where $Q_{c}(t)$ is the charge stored in the capacitor at time t. Substituting equationQ into equationI gives $i(t)\;=\;C{\frac {\operatorname {d} v_{\text{out}}}{\operatorname {d} t}}$ , which can be substituted into equationV so that

v_{\text{in}}(t)-v_{\text{out}}(t)=RC{\frac {\operatorname {d} v_{\text{out}}}{\operatorname {d} t}}.

This equation can be discretized. For simplicity, assume that samples of the input and output are taken at evenly spaced points in time separated by $\Delta _{T}$ time. Let the samples of $v_{\text{in}}$ be represented by the sequence $(x_{1},\,x_{2},\,\ldots ,\,x_{n})$ , and let $v_{\text{out}}$ be represented by the sequence $(y_{1},\,y_{2},\,\ldots ,\,y_{n})$ , which correspond to the same points in time. Making these substitutions,

x_{i}-y_{i}=RC\,{\frac {y_{i}-y_{i-1}}{\Delta _{T}}}.

Rearranging terms gives therecurrence relation

y_{i}=\overbrace {x_{i}\left({\frac {\Delta _{T}}{RC+\Delta _{T}}}\right)} ^{\text{Input contribution}}+\overbrace {y_{i-1}\left({\frac {RC}{RC+\Delta _{T}}}\right)} ^{\text{Inertia from previous output}}.

That is, this discrete-time implementation of a simpleRC low-pass filter is theexponentially weighted moving average

y_{i}=\alpha x_{i}+(1-\alpha )y_{i-1}\qquad {\text{where}}\qquad \alpha :={\frac {\Delta _{T}}{RC+\Delta _{T}}}.

By definition, thesmoothing factor is within the range $0\;\leq \;\alpha \;\leq \;1$ . The expression for α yields the equivalenttime constantRC in terms of the sampling period $\Delta _{T}$ and smoothing factor α,

RC=\Delta _{T}\left({\frac {1-\alpha }{\alpha }}\right).

Recalling that

f_{c}={\frac {1}{2\pi RC}}

so

RC={\frac {1}{2\pi f_{c}}},

note α and $f_{c}$ are related by,

\alpha ={\frac {2\pi \Delta _{T}f_{c}}{2\pi \Delta _{T}f_{c}+1}}

and

f_{c}={\frac {\alpha }{(1-\alpha )2\pi \Delta _{T}}}.

If α=0.5, then theRC time constant equals the sampling period. If $\alpha \;\ll \;0.5$ , thenRC is significantly larger than the sampling interval, and $\Delta _{T}\;\approx \;\alpha RC$ .

The filter recurrence relation provides a way to determine the output samples in terms of the input samples and the preceding output. The followingpseudocode algorithm simulates the effect of a low-pass filter on a series of digital samples:

// Return RC low-pass filter output samples, given input samples,// time intervaldt, and time constantRCfunction lowpass(real[1..n] x,real dt,real RC)varreal[1..n] yvarreal α := dt / (RC + dt)    y[1] := α * x[1]for ifrom 2to n        y[i] := α * x[i] + (1-α) * y[i-1]return y

Theloop that calculates each of then outputs can berefactored into the equivalent:

for ifrom 2to n        y[i] := y[i-1] + α * (x[i] - y[i-1])

That is, the change from one filter output to the next isproportional to the difference between the previous output and the next input. Thisexponential smoothing property matches theexponential decay seen in the continuous-time system. As expected, as thetime constantRC increases, the discrete-time smoothing parameter $\alpha$ decreases, and the output samples $(y_{1},\,y_{2},\,\ldots ,\,y_{n})$ respond more slowly to a change in the input samples $(x_{1},\,x_{2},\,\ldots ,\,x_{n})$ ; the system has moreinertia. This filter is aninfinite-impulse-response (IIR) single-pole low-pass filter.

Finite impulse response

Finite-impulse-response filters can be built that approximate thesinc function time-domain response of an ideal sharp-cutoff low-pass filter. For minimum distortion, the finite impulse response filter has an unbounded number of coefficients operating on an unbounded signal. In practice, the time-domain response must be time truncated and is often of a simplified shape; in the simplest case, arunning average can be used, giving a square time response.^[8]

Fourier transform

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For non-realtime filtering, to achieve a low pass filter, the entire signal is usually taken as a looped signal, the Fourier transform is taken, filtered in the frequency domain, followed by an inverse Fourier transform. Only O(n log(n)) operations are required compared to O(n²) for the time domain filtering algorithm.

This can also sometimes be done in real time, where the signal is delayed long enough to perform the Fourier transformation on shorter, overlapping blocks.

Continuous-time realization

Plot of the gain of Butterworth low-pass filters of orders 1 through 5, withcutoff frequency $\omega _{0}=1$ . Note that the slope is 20n dB/decade wheren is the filter order.

{\displaystyle \omega _{0}=1} — Plot of the gain of Butterworth low-pass filters of orders 1 through 5, withcutoff frequency $\omega _{0}=1$ . Note that the slope is 20n dB/decade wheren is the filter order.

There are many different types of filter circuits, with different responses to changing frequency. The frequency response of a filter is generally represented using aBode plot, and the filter is characterized by itscutoff frequency and rate of frequencyrolloff. In all cases, at thecutoff frequency, the filterattenuates the input power by half or 3 dB. So theorder of the filter determines the amount of additional attenuation for frequencies higher than the cutoff frequency.

Afirst-order filter, for example, reduces the signal amplitude by half (so power reduces by a factor of 4, or6 dB), every time the frequency doubles (goes up oneoctave); more precisely, the power rolloff approaches 20 dB perdecade in the limit of high frequency. The magnitude Bode plot for a first-order filter looks like a horizontal line below thecutoff frequency, and a diagonal line above the cutoff frequency. There is also a "knee curve" at the boundary between the two, smoothly transitioning between the two straight-line regions. If thetransfer function of a first-order low-pass filter has azero as well as apole, the Bode plot flattens out again, at some maximum attenuation of high frequencies; such an effect is caused for example by a little bit of the input leaking around the one-pole filter; this one-pole–one-zero filter is still a first-order low-pass.SeePole–zero plot andRC circuit.
Asecond-order filter attenuates high frequencies more steeply. The Bode plot for this type of filter resembles that of a first-order filter, except that it falls off more quickly. For example, a second-orderButterworth filter reduces the signal amplitude to one-fourth of its original level every time the frequency doubles (so power decreases by 12 dB per octave, or 40 dB per decade). Other all-pole second-order filters may roll off at different rates initially depending on theirQ factor, but approach the same final rate of 12 dB per octave; as with the first-order filters, zeroes in the transfer function can change the high-frequency asymptote. SeeRLC circuit.
Third- and higher-order filters are defined similarly. In general, the final rate of power rolloff for an order- n all-pole filter is 6n dB per octave (20n dB per decade).

On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upper-left (theasymptotes of the function), they intersect at exactly thecutoff frequency, 3 dB below the horizontal line. The various types of filters (Butterworth filter,Chebyshev filter,Bessel filter, etc.) all have different-lookingknee curves. Many second-order filters have "peaking" orresonance that puts their frequency responseabove the horizontal line at this peak.

The meanings of 'low' and 'high'—that is, thecutoff 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 1 GHz) and higher.

Laplace notation

Continuous-time filters can also be described in terms of theLaplace transform of theirimpulse response, in a way that makes it easy to analyze all characteristics of the filter by considering the pattern of poles and zeros of the Laplace transform in the complex plane. (In discrete time, one can similarly consider theZ-transform of the impulse response.)

For example, a first-order low-pass filter can be described by thecontinuous time transfer function, in theLaplace domain, as:

H(s)={\frac {\text{Output}}{\text{Input}}}=K{\frac {1}{\tau s+1}}=K{\frac {\alpha }{s+\alpha }}

whereH is the transfer function,s is the Laplace transform variable (complex angular frequency),τ is the filtertime constant, $\alpha$ is the cutoff frequency, andK is thegain of the filter in thepassband. The cutoff frequency is related to the time constant by:

\alpha ={1 \over \tau }

Electronic low-pass filters

Further information:Electronic filter

First-order passive

RC filter

Main article:RC circuit § Series circuit

Passive, first order low-pass RC filter

One simple low-pass filtercircuit consists of aresistor in series with aload, and acapacitor in parallel with the load. The capacitor exhibitsreactance, and blocks low-frequency signals, forcing them 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 thetime constant of the filter $\tau \;=\;RC$ (represented by the Greek lettertau). The break frequency, also called the turnover frequency, corner frequency, orcutoff frequency (in hertz), is determined by the time constant:

f_{\mathrm {c} }={1 \over 2\pi \tau }={1 \over 2\pi RC}

or equivalently (inradians per second):

\omega _{\mathrm {c} }={1 \over \tau }={1 \over RC}

This circuit may be understood by considering the time the capacitor needs to charge or discharge through the resistor:

At low frequencies, there is plenty of time for the capacitor to charge up to practically the same voltage as the input voltage.
At high frequencies, the capacitor only has time to charge up a small amount before the input switches direction. The output goes up and down only a small fraction of the amount the input goes up and down. At double the frequency, there's only time for it to charge up half the amount.

Another way to understand this circuit is through the concept ofreactance at a particular frequency:

Sincedirect current (DC) cannot flow through the capacitor, DC input must flow out the path marked $V_{\mathrm {out} }$ (analogous to removing the capacitor).
Sincealternating current (AC) flows very well through the capacitor, almost as well as it flows through a solid wire, AC input flows out through the capacitor, effectivelyshort circuiting to the ground (analogous to replacing the capacitor with just a wire).

The capacitor is not an "on/off" object (like the block or pass fluidic explanation above). The capacitor variably acts between these two extremes. It is theBode plot andfrequency response that show this variability.

RL filter

Main article:RL circuit § Series circuit

A resistor–inductor circuit orRL filter is anelectric circuit composed ofresistors andinductors driven by avoltage orcurrent source. A first-order RL circuit is composed of one resistor and one inductor and is the simplest type of RL circuit.

A first-order RL circuit is one of the simplestanalogue infinite impulse response electronic filters. It consists of aresistor and aninductor, either inseries driven by avoltage source or inparallel driven by a current source.

Second-order passive

RLC filter

RLC circuit as a low-pass filter

AnRLC circuit (the letters R, L, and C can be in a different sequence) is anelectrical circuit consisting of aresistor, aninductor, and acapacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols forresistance,inductance, andcapacitance, respectively. The circuit forms aharmonic oscillator for current and willresonate in a similar way as anLC circuit will. The main difference that the presence of the resistor makes is that any oscillation induced in the circuit will die away over time if it is not kept going by a source. This effect of the resistor is calleddamping. The presence of the resistance also reduces the peak resonant frequency somewhat. Some resistance is unavoidable in real circuits, even if a resistor is not specifically included as a component. An ideal, pure LC circuit is an abstraction for the purpose of theory.

There are many applications for this circuit. They are used in many different types ofoscillator circuits. Another important application is fortuning, such as inradio receivers ortelevision sets, where they are used to select a narrow range of frequencies from the ambient radio waves. In this role, the circuit is often called a tuned circuit. An RLC circuit can be used as aband-pass filter,band-stop filter, low-pass filter, orhigh-pass filter. The RLC filter is described as asecond-order circuit, meaning that any voltage or current in the circuit can be described by a second-orderdifferential equation in circuit analysis.

Second-order low-pass filter in standard form

The transfer function $H_{LP}(f)$ of a second-order low-pass filter can be expressed as a function of frequency $f {\displaystyle f}$ as shown in Equation 1, the Second-Order Low-Pass Filter Standard Form.

H_{LP}(f)=-{\frac {K}{f_{FSF}\cdot f_{c}^{2}+{\frac {1}{Q}}\cdot jf_{FSF}\cdot f_{c}+1}}\quad (1)

In this equation, $f {\displaystyle f}$ is the frequency variable, $f_{c}$ is the cutoff frequency, $f_{FSF}$ is the frequency scaling factor, and $Q {\displaystyle Q}$ is the quality factor. Equation 1 describes three regions of operation: below cutoff, in the area of cutoff, and above cutoff. For each area, Equation 1 reduces to:

$f\ll f_{c}$ : $H_{LP}(f)\approx K$ - The circuit passes signals multiplied by the gain factor $K {\displaystyle K}$ .
${\frac {f}{f_{c}}}=f_{FSF}$ : $H_{LP}(f)=jKQ$ - Signals are phase-shifted 90° and modified by the quality factor $Q {\displaystyle Q}$ .
$f\gg f_{c}$ : $H_{LP}(f)\approx -{\frac {K}{f_{FSF}\cdot f^{2}}}$ - Signals are phase-shifted 180° and attenuated by the square of the frequency ratio. This behavior is detailed by Jim Karki in "Active Low-Pass Filter Design" (Texas Instruments, 2023).^[9]

With attenuation at frequencies above $f_{c}$ increasing by a power of two, the last formula describes a second-order low-pass filter. The frequency scaling factor $f_{FSF}$ is used to scale the cutoff frequency of the filter so that it follows the definitions given before.

Higher order passive filters

Higher-order passive filters can also be constructed (see diagram for a third-order example).

A third-order low-pass filter (Cauer topology). The filter becomes a Butterworth filter withcutoff frequency ω_c=1 when (for example) C₂=4/3 farad, R₄=1 ohm, L₁=3/2 henry and L₃=1/2 henry.

First order active

An active low-pass filter

See also:operational amplifier applications § Inverting integrator, andOp amp integrator

Anactive low-pass filter adds anactive device to create anactive filter that allows forgain in the passband.

In theoperational amplifier circuit shown in the figure, the cutoff frequency (inhertz) is defined as:

f_{\text{c}}={\frac {1}{2\pi R_{2}C}}

or equivalently (in radians per second):

\omega _{\text{c}}={\frac {1}{R_{2}C}}

The gain in the passband is −R₂/R₁, and thestopband drops off at −6 dB per octave (that is −20 dB per decade) as it is a first-order filter.

See also

References

^^a ^bLong Pass Filters and Short Pass Filters Information, retrieved2017-10-04
^Sedra, Adel; Smith, Kenneth C. (1991).Microelectronic Circuits, 3 ed. Saunders College Publishing. p. 60.ISBN 0-03-051648-X.
^"ADSL filters explained". Epanorama.net. Retrieved2013-09-24.
^"Home Networking – Local Area Network". Pcweenie.com. 2009-04-12. Archived fromthe original on 2013-09-27. Retrieved2013-09-24.
^Mastering Windows: Improving Reconstruction
^^a ^bHayt, William H. Jr. and Kemmerly, Jack E. (1978).Engineering Circuit Analysis. New York: McGRAW-HILL BOOK COMPANY. pp. 211–224,684–729.{{cite book}}: CS1 maint: multiple names: authors list (link)
^Boyce, William and DiPrima, Richard (1965).Elementary Differential Equations and Boundary Value Problems. New York: JOHN WILEY & SONS. pp. 11–24.{{cite book}}: CS1 maint: multiple names: authors list (link)
^Whilmshurst, T H (1990)Signal recovery from noise in electronic instrumentation.ISBN 9780750300582
^Active Low-Pass Filter Design" (Texas Instruments, 2023)

External links

Wikimedia Commons has media related toLowpass filters.

Low Pass Filter java simulator
ECE 209: Review of Circuits as LTI Systems, a short primer on the mathematical analysis of (electrical) LTI systems.
ECE 209: Sources of Phase Shift, an intuitive explanation of the source of phase shift in a low-pass filter. Also verifies simple passive LPFtransfer function by means of trigonometric identity.
C code generator for digital implementation of Butterworth, Bessel, and Chebyshev filters created by the late Dr. Tony Fisher of the University of York (York, England).

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