Low-pass filter

A low-pass filter is a filter that passes low-frequency signals but attenuates (reduces the amplitude of) signals with 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 low-pass filter is the opposite of a high-pass filter, and a band-pass filter is a combination of a low-pass and a high-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

Figure 1 shows a low-pass RC filter for voltage signals, discussed in more detail below. Signal V_out contains frequencies from the input signal, with high frequencies attenuated, but with little attenuation below the cutoff frequency of the filter determined by its RC time constant. For current signals, a similar circuit using a resistor and capacitor in parallel works the same way. See current divider.

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 subwoofers and other types of loudspeakers, to block high pitches that they can't efficiently broadcast.

Radio transmitters use low-pass filters to block harmonic emissions which might cause interference with other communications.

The tone knob found on many electric guitars is a low-pass filter used to reduce the amount of treble in the sound.

An integrator is another example of a low-pass filter.

DSL splitters 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 synthesisers. See subtractive synthesis.

Ideal and real filters

An ideal low-pass filter completely eliminates all frequencies above the cutoff frequency while passing those below unchanged: its frequency response is a rectangular function, and is a brick-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 its impulse response, 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.

An ideal low-pass filter results in ringing artifacts via the Gibbs phenomenon. These can be reduced or worsened by choice of windowing function, and the design and choice of real filters involves understanding and minimizing these artifacts. For example, "simple truncation [of sinc] causes severe ringing artifacts," in signal reconstruction, and to reduce these artifacts one uses window functions "which drop off more smoothly at the edges."^[1]

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 converters use real filter approximations.

Continuous-time 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, and the filter is characterized by its cutoff frequency and rate of frequency rolloff. In all cases, at the cutoff frequency, the filter attenuates the input power by half or -3 dB. So the order of the filter determines the amount of additional attenuation for frequencies higher than the cutoff frequency.

A first-order filter, for example, will reduce the signal amplitude by half (so power reduces by –6 dB) every time the frequency doubles (goes up one octave); more precisely, the power rolloff approaches 20 dB per decade in the limit of high frequency. The magnitude Bode plot for a first-order filter looks like a horizontal line below the cutoff frequency, and a diagonal line above the cutoff frequency. There is also a "knee curve" at the boundary between the two, which smoothly transitions between the two straight line regions. If the transfer function of a first-order low-pass filter has a zero as well as a pole, the Bode plot will flatten 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. See Pole–zero plot and RC circuit.

A second-order filter attenuates higher 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-order Butterworth filter will reduce the signal amplitude to one fourth 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 their Q 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. See RLC 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 (i.e., $20n$ dB per decade).

On any Butterworth filter, if one extends the horizontal line to the right and the diagonal line to the upper-left (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' – 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.

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).

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

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

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

Electronic low-pass filters

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 $\tau =RC$ (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:

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

or equivalently (in radians per second):

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

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:

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 with the idea of reactance at a particular frequency:

Since DC cannot flow through the capacitor, DC input must "flow out" the path marked $V_{\mathrm {out} }$ (analogous to removing the capacitor).
Since 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).

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.

Active electronic realization

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:

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 ${\frac {-R_{2}}{R_{1}}}$ , and the stopband drops off at −6 dB per octave as it is a first-order filter.

Sometimes, a simple gain amplifier (as opposed to the very-high-gain operation amplifier) is turned into a low-pass filter by simply adding a feedback capacitor C. This feedback decreases the frequency response at high frequencies via the Miller effect, and helps to avoid oscillation in the amplifier. For example, an audio amplifier can be made into a low-pass 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.

Discrete-time realization

The effect of a low-pass filter can be simulated on a computer by analyzing its behavior in the time domain, and then discretizing the model.

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

{\begin{cases}v_{\text{in}}(t)-v_{\text{out}}(t)=R\,i(t)&{\text{(V)}}\\Q_{c}(t)=C\,v_{\text{out}}(t)&{\text{(Q)}}\\i(t)={\frac {\operatorname {d} Q_{c}}{\operatorname {d} t}}&{\text{(I)}}\end{cases}}

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

And rearranging terms gives the recurrence 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 simple RC low-pass filter is the exponentially-weighted moving average

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

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

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

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

Algorithmic implementation

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 will simulate 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[0..n] x, real dt, real RC)
   var real[0..n] y
   var real α := dt / (RC + dt)
   y[0] := x[0]
   for i from 1 to n
       y[i] := α * x[i] + (1-α) * y[i-1]    
   return y

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

   for i from 1 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 $\alpha$ 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 will have more inertia.

References

^ Mastering Windows: Improving Reconstruction

External links

Low-pass filter
Low-pass visual simulation
RC Filter Calculator
Cutoff frequency calculator
Low Pass Filter java simulator
ECE 209: Review of Circuits as LTI Systems — Short primer on the mathematical analysis of (electrical) LTI systems.
ECE 209: Sources of Phase Shift — Gives an intuitive explanation of the source of phase shift in a low-pass filter. Also verifies simple passive LPF transfer function by means of trigonometric identity.

[1] Mastering Windows: Improving Reconstruction

[1]