Low-pass filter

If we let ${\displaystyle v_{\text{in}}(t)}$ be a step function of magnitude ${\displaystyle V_{i}}$ then the differential equation has the solution[8]

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

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

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

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 then discretizing the model.

A simple low-pass RC filter

From the circuit diagram to the right, according to Kirchhoff's Laws and the definition of capacitance:

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

(V)

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

(Q)

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

(I)

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

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

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 ${\displaystyle \Delta _{T}}$ time. Let the samples of ${\displaystyle v_{\text{in}}}$ be represented by the sequence ${\displaystyle (x_{1},\,x_{2},\,\ldots ,\,x_{n})}$, and let ${\displaystyle v_{\text{out}}}$ be represented by the sequence ${\displaystyle (y_{1},\,y_{2},\,\ldots ,\,y_{n})}$, which correspond to the same points in time. Making these substitutions,

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

Rearranging terms gives the recurrence relation

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

That is, this discrete-time implementation of a simple RC low-pass filter is the exponentially weighted moving average

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

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

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

Recalling that

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

note α and ${\displaystyle f_{c}}$ are related by,

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

and

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

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

The filter recurrence relation provides a way to determine the output samples in terms of the input samples and the preceding output. The following pseudocode 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 interval dt, and time constant RC
function lowpass(real[1..n] x, real dt, real RC)
    var real[1..n] y
    var real α := dt / (RC + dt)
    y[1] := α * x[1]
    for i from 2 to n
        y[i] := α * x[i] + (1-α) * y[i-1]
    return y

The loop that calculates each of the n outputs can be refactored into the equivalent:

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

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

Finite-impulse-response filters can be built that approximate to the sinc 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, a running average can be used, giving a square time response.[9]

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 filters can also be described in terms of the Laplace transform of their impulse response, in a way that lets all characteristics of the filter 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.)

For example, a first-order low-pass filter can be described in Laplace notation as:

${\displaystyle {\frac {\text{Output}}{\text{Input}}}=K{\frac {1}{\tau s+1}}}$

where s is the Laplace transform variable, τ is the filter time constant, and K is the gain of the filter in the passband.

Passive, first order low-pass RC filter

One simple low-pass filter circuit 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, 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 the time constant of the filter ${\displaystyle \tau \;=\;RC}$ (represented by the Greek letter tau). The break frequency, also called the turnover frequency, corner frequency, or cutoff frequency (in hertz), is determined by the time constant:

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

or equivalently (in radians per second):

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

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 of reactance at a particular frequency:

• Since direct current (DC) cannot flow through the capacitor, DC input must flow out the path marked ${\displaystyle V_{\mathrm {out} }}$ (analogous to removing the capacitor).
• Since alternating current (AC) flows very well through the capacitor, almost as well as it flows through solid wire, AC input flows out through the capacitor, effectively short circuiting to 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 the Bode plot and frequency response that show this variability.

A resistor–inductor circuit or RL filter is an electric circuit composed of resistors and inductors driven by a voltage or current 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 simplest analogue infinite impulse response electronic filters. It consists of a resistor and an inductor, either in series driven by a voltage source or in parallel driven by a current source.

RLC circuit as a low-pass filter

An RLC circuit (the letters R, L and C can be in a different sequence) is an electrical circuit consisting of a resistor, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively. The circuit forms a harmonic oscillator for current and will resonate in a similar way as an LC 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 called damping. 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 of oscillator circuits. Another important application is for tuning, such as in radio receivers or television sets, where they are used to select a narrow range of frequencies from the ambient radio waves. In this role the circuit is often referred to as a tuned circuit. An RLC circuit can be used as a band-pass filter, band-stop filter, low-pass filter or high-pass filter. The RLC filter is described as a second-order circuit, meaning that any voltage or current in the circuit can be described by a second-order differential equation in circuit analysis.

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 with cutoff frequency ω_c=1 when (for example) C₂=4/3 farad, R₄=1 ohm, L₁=3/2 henry and L₃=1/2 henry.

An active low-pass filter

Another type of electrical circuit is an active low-pass filter.

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

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

or equivalently (in radians per second):

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

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